TIAGO DE ARA ´
UJO ELIAS
HYBRID MODEL PREDICTIVE
CONTROL FOR SOLAR FIELDS
FLORIAN ´
OPOLIS
FEDERAL UNIVERSITY OF SANTA CATARINA
GRADUATE PROGRAM IN AUTOMATION AND SYSTEMS
ENGINEERING
HYBRID MODEL PREDICTIVE CONTROL FOR SOLAR FIELDS
Master Thesis submitted to the Graduate Program in Automation and Systems Engineering as a requirement for the degree of Master in
Automation and Systems Engineering.
TIAGO DE ARA ´UJO ELIAS
HYBRID MODEL PREDICTIVE CONTROL FOR SOLAR FIELDS
Tiago de Ara´ujo Elias
Esta disserta¸c˜ao de mestrado foi julgada aprovada para a obten¸c˜ao do t´ıtulo de Mestre em Engenharia de Automa¸c˜ao e Sistemas, ´area de Concentra¸c˜ao em Controle, Automa¸c˜ao e Sistemas, e aprovada em sua forma
final pelo Programa de P´os-Gradua¸c˜ao em Engenharia de Automa¸c˜ao e Sistemas da Universidade Federal de Santa Catarina.
Julio Elias Normey-Rico, Dr. Orientador
Paulo Renato da Costa Mendes, Dr. Coorientador
Werner Kraus Junior, Dr.
Coordenador do Programa de P´os-Gradua¸c˜ao em Engenharia de Automa¸c˜ao e Sistemas
Banca Examinadora:
Daniel Martins Lima, Dr.
Eduardo Camponogara, Dr.
Gustavo Artur de Andrade, Dr.
Resumo estendido da disserta¸c˜ao de mestrado submetida `a Universidade Federal de Santa Catarina como parte dos requisitos para a obten¸c˜ao
do grau de Mestre em Engenharia de Automa¸c˜ao e Sistemas.
Controle Preditivo H´ıbrido Baseado em Modelo para Campos Solares
Tiago de Ara´
ujo Elias
´
Area de Concentra¸c˜ao: Controle, Automa¸c˜ao e In-form´atica Industrial
Palavras-chave: Campo de Coletores Solares, Cont-role Preditivo Baseado em Modelo Pr´atico N˜ao Lin-ear, Sistemas Mistos L´ogico Dinˆamico.
N´umero de P´aginas: xxv + 150
O objetivo deste trabalho ´e desenvolver algo-ritmos de controle para campos de coletores solares. O calor gerado nos campos solares ´e utilizado como fonte de energia para sistemas t´ermicos e para sis-temas de gera¸c˜ao de energia el´etrica. Considerando que os combust´ıveis f´osseis possuem uma natureza finita e os impactos ambientais causados pelo seu uso, existe hoje um interesse em utilizar fontes de energia que sejam renov´aveis. Nesse cen´ario o Sol aparece como a fonte renov´avel mais abundante que pode ser aproveitada em plantas fotovoltaicas (que transformam diretamente a irradia¸c˜ao em energia el´etrica) ou termosolares (que usam a irradia¸c˜ao do Sol para gerar energia t´ermica, que posteriormente ´
e convertida em eletricidade em uma unidade ter-moel´etrica). Plantas termosolares s˜ao complexas e o uso de controle autom´atico pode melhorar a sua eficiˆencia, reduzindo assim o custo por kilowatt hora produzido. Em sistemas solares a fonte de ener-gia n˜ao pode ser controlada, dessa forma controlar e limitar a temperatura m´axima do fluido de tra-balho ´e uma preocupa¸c˜ao para manter a seguran¸ca dos equipamentos. O controle de temperatura do campo normalmente ´e realizado alterando a vaz˜ao
do fluido que circula pelos coletores. Por´em, muitas vezes esse controle n˜ao tem a capacidade de limi-tar a temperatura em situa¸c˜oes extremas. Existem duas maneiras de limitar a temperatura do fluido em situa¸c˜oes extremas: reduzir a capta¸c˜ao da ener-gia solar no coletor ou remover o excesso de calor. Nesse trabalho um algoritmo de controle avan¸cado ´e desenvolvido para reduzir a capta¸c˜ao de energia solar atrav´es da desfocagem dos coletores. Outro problema importante nos sistemas heliot´ermicos ´e o desbalanceamento energ´etico causado pela irra-dia¸c˜ao n˜ao uniforme. Em grandes campos solares podem ocorrer situa¸c˜oes em que parte do campo tenha cobertura por nuvens. Na presen¸ca de nu-vens o ganho energ´etico ´e baixo e a parte coberta por nuvens come¸ca a funcionar como um dissipador de energia. Para evitar essas situa¸c˜oes, nesse tra-balho ´e desenvolvido um controlador para desativar as partes do campo solar que s˜ao afetadas pela pas-sagem de nuvens. A formula¸c˜ao dos algoritmos de controle propostos neste trabalho para os dois ca-sos citados ´e baseada em representa¸c˜ao mista l´ og-ica dinˆamica dos campos solares com a utiliza¸c˜ao de controle preditivo baseado em modelo pr´atico n˜ao linear para calcular a a¸c˜ao de controle ´otima. Os principais objetivos dos controladores s˜ao: (i) de-sativar campos com temperatura de entrada maior do que a temperatura de sa´ıda e manipular o fluxo volum´etrico dos campos ativos para seguir referˆ en-cia de temperatura; (ii) desfocar coletores solares quando a temperatura atinge o valor m´aximo e ma-nipular o fluxo do fluido para manter a temperatura de sa´ıda do campo em uma referˆencia desejada. Os resultados dos algoritmos propostos s˜ao
utiliza¸c˜ao do algoritmo de desfocagem e cen´arios com passagem de nuvens sobre alguns setores do campo solar, para apresentar as vantagens da uti-liza¸c˜ao do algoritmo de desativa¸c˜ao de campos.
Abstract of the master thesis presented to the Federal University of Santa Catarina as a partial fulfillment of the
requirements for the degree of Master in Automation and Systems Engineering.
HYBRID MODEL PREDICTIVE CONTROL FOR SOLAR FIELDS
Tiago de Ara´
ujo Elias
Area of Concentration: Control, Automation and Industrial Informatics
Keywords: Solar Collectors Field, Practical Nonlin-ear Model Predictive Control, Mixed Logical Dy-namical Systems.
Number of Pages: xxv + 150
This master thesis presents an advanced con-trol algorithm for reducing heat losses caused by clouds in large-scale solar fields and an algorithm for defocusing the collectors in order to avoid oil de-composing. Large-scale solar fields can have partial cloud cover, and the covered part of the field works as an energy dissipator. If the volumetric flow is increased the output temperature rises, and the sys-tem loses energy. Limiting maximum fluid sys- temper-ature is also a concern in solar systems, considering that the source of energy cannot be manipulated. There are two ways to prevent over-temperature: re-duce solar energy input into the collector, or remove excess heat from the collector. The formulation of the algorithms is based on a Mixed Logical Dynam-ical (MLD) representation of the solar field plus the application of a Practical Nonlinear Model Predic-tive Controller (PNMPC) for calculating the optimal control action. The main purposes of the controllers
are: (i) to deactivate fields with the inlet temper-ature greater than the outlet tempertemper-ature and to manipulate the oil flow rate of the activated fields for tracking the reference of the field outlet temper-ature; (ii) to defocus the solar collectors when the output temperature reaches the maximum value and to manipulate the fluid rate to maintain the field output temperature in the desired reference. Sim-ulation results using irradiation profiles with cloud variations are presented for illustrating the advan-tages of the proposed fields deactivation approach. The results of the defocusing approach are presented supposing extreme scenarios of overheating.
Contents
1 Introduction 1 1.1 Objectives . . . 10 1.1.1 General objectives . . . 10 1.1.2 Specific objectives . . . 10 1.2 Contributions . . . 11 1.3 Thesis Organization . . . 122 Thermosolar Energy Systems 13 2.1 Photovoltaic Plants . . . 17
2.2 Thermosolar Plants/Concentrating So-lar Thermal Systems . . . 19
2.2.1 Fresnel Collectors . . . 21
2.2.3 Parabolic Dish Collectors . . . 23 2.2.4 Enclosed Trough Collector . . 24 2.2.5 Parabolic Trough Collector . . 25 2.3 Final Remarks . . . 28
3 Solar Fields Modeling 29
3.1 Distributed Collector System . . . 29 3.1.1 Distributed Parameters Model 33 3.1.2 Linear Model of the Distributed
Collector Field . . . 38 3.1.3 Bilinear Models . . . 39 3.1.4 Lumped Parameters Model . 40 3.2 Large Scale Solar Collector Fields . . 43 3.3 Final Remarks . . . 45
4 Model Predictive Control 47
4.1 Model Predictive Control . . . 49 4.1.1 Nonlinear Model Predicitive
Con-trol Schemes . . . 58
4.1.2 Practical Nonlinear Model
Pre-dictive Control . . . 61
4.2 Mixed Logical Dynamical Systems . . 69
4.3 Final Remarks . . . 74
5 Optimal Solar Collector Fields Deacti-vation 75 5.1 MLD modeling of the solar subfields 76 5.2 Proposed Control Strategy . . . 79
5.2.1 Process constraints . . . 82
5.2.2 Operational constraints . . . . 83
5.3 Results . . . 91
5.4 Final Remarks . . . 98
6 Solar Collectors Defocusing Based on Maximum Temperature 99 6.1 Solar Fields Modeling . . . 100
6.2 Proposed Control . . . 103
6.2.1 Constraints . . . 109
6.3 Results . . . 117
6.3.1 High Irradiation Scenario . . . 119
6.4 Final Remarks . . . 133
7 Conclusions and Future Works 135 7.1 Contributions . . . 137 7.2 Future Works . . . 138
Nomenclature
Acronyms
AC Alternating current
CIEMAT Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas CNF Conjunctive Normal Form
CSP Concentrated Solar Power
DC Direct current
DMC Dynamic Matrix Control DNI Direct Normal Irradiance DTC Dead-Time Compensator DTC-GPC Dead-Time Compensator
Generalized Predictive Controller EHAC Extended Horizon Adaptive Control EPSAC Extended Prediction Self Adaptive
Control
FL Feedback linearization FLC Fresnel lens collector
GPC Generalized Predictive Control
IEA International Energy Agency MAC Model Algorithmic Control
MIQP Mixed Integer Quadratic Programming MLD Mixed Logical Dynamical
MPC Model Predictive Control
MUSMAR Multipredictor Adaptive Regulators NEPSAC Nonlinear Extended Prediction
Self-Adaptive Control
NMIQP Nonlinear Mixed Integer Quadratic Programming
NMPC Nonlinear Model Predictive Control PNMPC Practical Nonlinear
Model Predictive Control PSA Plataforma Solar de Almeria PTC Parabolic trough collectors
PV Photovoltaic
RHC Receding Horizon Control
SEGS Solar Energy Generating Systems SSPS Small Solar Power Systems UKF Unscented Kalman filter
List of Symbols
Indexes m Metal f Fluid in Input out outputf ield Solar field
loop Loop of solar collectors
P Prediction C Control max Maximum min Minimum SP Set-point Variables
R Radius of the Earth dx length control volume
ρ Density
A Absorver section
c Specific thermal capacity T Temperature
I Irradiation
l Tube circumference ˙
m Mass flow
v Volumetric flow rate hl Coefficient of global losses
ht Coefficient of thermal transfer
between metal and fluid c Specific thermal capacity b Collector aperture
Ta Ambient temperature
η0 Mirror optical efficiency
A,B State-space system matrices x State vector
u Model input y Model output r Reference
Hl Function of thermal losses
c1 Coefficient of thermal losses
c2 Coefficient of thermal losses
L Length
td Flow-output temperature HTF
related dead time
tr Input-output temperature transport delay TS Sampling time N Horizon λ Weighting factor δ Binary variable α Binary variable ϕ Auxiliary variable m Lower bound of f (x) M Upper bound of f (x) ζ Machine precision Slack variable
z Continuous auxiliary variable w Continuous auxiliary variable vM IN Minimum volumetric flow
vM AX Maximum volumetric flow
R Weighting matrix Q Weighting matrix
List of Figures
1.1 Direct normal irradiation in Brazil [7] 7 1.2 Average cloud cover [1] . . . 9
2.1 Thermosolar plants around the world [8] . . . 15 2.2 Solar energy conversion system [5] . . 16 2.3 Diagram Thermosolar Plant [22] . . . 20 2.4 Fresnel Lens collector [44] . . . 21 2.5 Linear Fresnel reflector [44] . . . 21 2.6 Central receiver system [44] . . . 22 2.7 Parabolic dish collector [44] . . . 24 2.8 Diagram of a GlassPoint solar steam
generator system [3] . . . 25
3.1 Solar Collectors Field Scheme . . . . 30 3.2 Solar Collectors Field and Tank Scheme
[56]. . . 31 3.3 ACUREX distributed solar collector
field ([12], [26], [27], [34]). . . 32 3.4 Control volume of the distributed
pa-rameters model [21] . . . 34 3.5 Large Scale Field Serial Configuration 43 3.6 Large Scale Field in Parallel
Config-uration . . . 44
4.1 MPC strategy . . . 52 4.2 MPC structure . . . 53
5.1 Hysteresis band . . . 86 5.2 Hysteresis graph . . . 86 5.3 Ambient temperature Ta and input
temperature Tin . . . 92
5.4 Solar irradiation . . . 93 5.5 Output Temperatures and
Volumet-ric Flow . . . 94
5.6 Magnified DeltaTemperature of Sub-field 2 . . . 95 5.7 Magnified DeltaTemperature of
Sub-field 3 . . . 96 5.8 Field Output Temperature . . . 97
6.1 Solar Field Configuration . . . 101 6.2 Ambient Temperature . . . 118 6.3 Input Temperature . . . 119 6.4 Irradiation . . . 120 6.5 Results partial defocusing approach
high irradiation scenario . . . 121 6.6 Results ON/OFF defocusing approach
high irradiation scenario . . . 124 6.7 Irradiation profile pump failure scenario126 6.8 Results partial defocusing approach
pump failure scenario . . . 127 6.9 Results ON/OFF defocusing approach
pump failure scenario . . . 130
List of Tables
3.1 Model parameters and variables . . . 37 3.3 Model parameters . . . 41
4.1 Conversion of logic into integer inequal-ities [15] . . . 72
5.1 Truth table . . . 87 5.2 Energy Comparison Subfields
Deacti-vation . . . 98
6.1 Energy Comparison . . . 131 6.3 Problem Solving Time Comparison . 132
Chapter 1
Introduction
After the crisis of 1973, during the rise of the oil prices, the use of renewable energy had a great impulse, caused by economic issues. At that time, initiatives appeared for constructing solar power plants at Coolidge, Arizona in 1979 and at Plata-forma Solar de Almer´ıa (PSA), Spain in 1981 [24]. Nowadays, there is an interest in the use of renew-able energies for reducing the environmental impact produced by fossil energy systems [22]. This is very important, mainly because the fossil fuel reserves are being reduced every year while, and at the same time, energy demand increases in every country ([49], [13]). Also, the world total final energy consumption and CO2 emissions have doubled in the last years
de-mand is expected to be about 32 TW. This expected energy demand cannot be covered by increasing the energy production through improvements in the effi-ciency of the traditional methods [22]. On the other hand, only a fraction of the solar energy reaching the Earth’s surface would be enough to cover it. More-over, renewable energy generation can constitute an important part of the energy scenario with the uti-lization of small distributed energy resources [49].
As pointed out in [11], in 2015, 5534 TW h was generated by renewable sources in the world. It rep-resents 7.1% of the total electricity generated in the world, this contribution is ten times greater com-pared to 1973. In Brazil it was generated 430 TW h, in United States 568 TW h, and in China 1398 TW h. The renewable spending has increased 30% in 2010, the solar energy attracted 53% more investment than in 2009 and the wind energy 34%. Clean energy gen-erating technology has doubled in the period 2008–2010 [22]. About 85% of the electricity gen-erated in Brazil comes from renewable sources. It is estimated the Sun and the wind can provide four times the Brazilian current energy demand [4].
In this scenario, the Sun is the most abun-dant sustainable source of energy. It provides over 15 × 104TW to the Earth, from nuclear fusion
reac-tions at its core - where hydrogen atoms are fused into helium forced by intense pressure created by the
3
gravity. The expectation of Sun’s life is about five bilion years; half of the hydrogen has already been converted into helium. The irradiance in the Sun is about 63 MW m−2 and on the Earth’s surface it is around 1 kW m−2 [33], the Earth’s disk intercepts solar radiation with an area πR2; where R is the
radius of the Earth [22].
Solar energy is already being used for gener-ation of electricity by means of photovoltaic (PV) plants, and indirectly, by solar-thermal plants to produce steam used to drive a turbine to generate electrical power. Also it is used for many other pur-poses such as refrigeration [19], detoxification [17], desalination ([65], [45]), and for generating hydrogen gas, which is expected to be another form of clean energy [61]. It is worth mentioning that renewable sources of energy such as wind energy, wave energy and bio-fuels depend on the Sun’s energy, and nonre-newable energy sources as fossil fuels were originated by solar energy.
This way, solar energy is becoming the most promising source of energy, however the costs are not yet competitive and it is not always available. For overcoming these issues scientific research and tech-nology have been developed during the last years to improve the efficiency of solar power plants from the control and optimization viewpoints [22]. If com-pared to photovoltaic devices, which is a direct
con-version process, a solar thermal plant has better conversion efficiency and the possibility of efficient storage of the thermal energy to be subsequently converted into electricity [13]. Research efforts have been devoted to techniques which may help to col-lect, convert, store and utilize the solar energy ([22], [42], [24]). As pointed out in [13], the major in-vestors are in Spain (Abengoa Solar, Acciona, Iber-drola Renovables, Torresol) and USA (BrightSource Energy, Enerworks, eSolar, FPL Energy). In solar power systems, the main source of energy cannot be manipulated as in other power generating pro-cesses. Moreover, a solar plant needs to deal with problems of seasonal and daily variations that are not encountered in other thermal power plants: the solar radiation depends on atmospheric conditions such as cloud cover, humidity and air transparency. Thermosolar plants are represented as a com-plex grid where each element has stationary and transient losses with changing dynamics, nonlinear-ities and uncertainties. The use of control can im-prove the efficiency and increase the operational hours, reducing the cost per kilowatt hour produced. There are numerous examples in the literature, about advanced control techniques in solar fields using Model Predictive Control (MPC), adaptive control, gain scheduling, time-delay compensation, optimal control, robust control, fuzzy logic control, and neu-ral network controllers.
5
In [21] some control strategies for thermal so-lar fields are studied: a basic control scheme using feedforward control action for improving disturbance rejection; an adaptive controller for compensating changes in the dynamic characteristics of the solar field; a frequency domain controller design; a robust optimal controller using frequency-based interpola-tion models to take into account systems with res-onance and antiresres-onance modes; an heuristic fuzzy logic controller for incorporating the experience in operating the solar plant in terms of qualitative rules and a model predictive controller (MPC) for taking the process constraints into account. Model Predic-tive Control is a well-known technique for control-ling solar systems and has been used with success in several plants [23]. In [51] a PI controller with dead-time compensation based on adaptive filtered Smith predictor structure is proposed as a robust compensator to solar collector fields. In [57] feed-back linearization control is used for transforming the nonlinear system into a linear one to use a linear controller for the field outlet temperature. In [63] a cascade control structure is proposed for automat-ically compensating the losses in the pipe. On the inner loop the field outlet temperature is controlled employing the Multi-predictor Adaptive Regulators (MUSMAR) presented in [50] and, on the outer loop, the tank outlet temperature is controlled using clas-sical control. The main control objectives are to
force the outlet temperature to follow a desired ref-erence and to reject disturbances caused by clouds and ambient temperature, using the fluid flow as the manipulated variable.
Considering the main source of energy cannot be manipulated, the appropriate site locations for solar plants based on thermodynamic cycles are in arid to semi-arid regions for achieving high efficiency [13]. Regions with these conditions are located in; the North African Desert, the Arabian Peninsula, major portions of India, central and western Aus-tralia, the high plateaus of the Andean states, north-eastern Brazil, northern Mexico, the United States Southwest, southern Spain, and several Mediterranean islands [33]. Brazil has high potential with semi-arid regions close to the equator with the consequential optical advantage receiving a direct normal irradiation on the order of 2200 kW h m−2 as shown in figure 1.1.
In Brazil immense land areas with good clima-tological conditions are available for solar thermal applications, showing competitive costs [49]. Arid areas are commonly vacant, since they are unpro-ductive for agriculture and livestock. The installa-tion of a solar thermal plant can drive regional de-velopment, creating jobs in the construction, main-tenance, and supply of this new market. Besides the good climatological condition, Brazil has a
es-7
Figure 1.1: Direct normal irradiation in Brazil [7]
tablished productive chain of thermoelectric plants. Some parts used in conventional thermal power plants can be used in thermosolar, and the metallic structures and mirrors can be produced in Brazil. Brazilian researchers have already started survey in this area for adapting the technology to the local conditions, currently two projects are being imple-mented. The main challenges to implement this technology are; the high initial investment, the reg-ulatory standards are still being created, the high Brazilian taxes, and the lack of professionals to plan, construct and operate the plants [2].
The daily solar cycle, cloud cover, and weather conditions of the site location must be considered in the control algorithms for solar plants. Once the source of energy cannot be manipulated in solar sys-tems, limiting maximum fluid temperature has also become a concern [37]. In PV systems, 27.6% of incoming solar radiation is converted into electric-ity while the remaining is reflected or converted into heat elevating the temperatures. Considering the encapsulating and bonding materials have temper-ature limits, a strategy to deal with elevated tem-peratures is studied in [38]. In thermosolar systems, the manufacturer defines a maximum temperature for reducing the risk of fluid decomposition and to avoid excessive pressures in the collector loop [22]. There are two ways to prevent over-temperature: re-duce solar energy input into the collector, or remove excess heat from the collector [37]. If the mass flow reaches an upper limit the heat cannot be removed from the collector and the collector needs to be de-focused for reducing the energy absorbing. In this work it is proposed an algorithm for defocusing the solar collectors when the maximum temperature is reached.
When solar radiation is affected by clouds the gain can vary over 100%, making the control prob-lem more difficult to solve [65]. Large scale ther-mosolar fields can have partial cloud cover, in the covered part of the field the output temperature
9
drops because of low radiation. In this case the input temperature is greater than the output temperature, and the covered part works as an energy dissipator. Over this low radiation condition if the flow rate is increased the output temperature rises, it is the op-posite of the normal case; the gain is reversed. For avoiding these problems of gain inversion and energy loss, a control algorithm for large scale solar fields with the possibility of turning off covered parts of the field is proposed in this work. Using this algo-rithm, the efficiency of thermosolar plants in regions with cloud variations can be increased.
Figure 1.2: Average cloud cover [1]
To quantify this fact, figure 1.2 shows the world average cloud cover, period 2002-2015. As can be seen in Brazil there is cloud cover in sites with high direct normal irradiation; in the extreme south and in the northeast coast, for example, this way the
proposed algorithm can contribute for the national thermosolar technology.
1.1
Objectives
1.1.1 General objectives
The general objective of this master thesis is to develop two algorithms, one for controlling and turning off solar fields in the presence of clouds and another one for defocusing solar collectors to avoid over-temperature. The algorithms are generic and can be used in systems involving dynamical and log-ical operations.
1.1.2 Specific objectives
The specific objectives for performing the study defined in the general objectives are presented be-low:
• To review the literature about thermosolar sys-tems and its modeling.
• To study methodologies for representing dynamic systems with binary variables.
• To formalize the control problems and propose a solution.
1.2. Contributions 11
• To analyze the simulation results according to the expected behavior.
1.2
Contributions
The main contributions of this work are: • The proposal of a methodology for controlling
large scale solar fields avoiding heat losses us-ing a parallel structure and a methodology for defocusing solar collectors avoiding fluid over-heating;
• Modeling of the solar plant considering the en-ergy loss problem and the overheating fluid prob-lem using a Mixed Logical Dynamical (MLD) system;
• The use of a Hybrid PNMPC strategy to com-pute the optimal control action avoiding the use of nonlinear mixed integer quadratic optimiza-tion, simplifying the problem to a mixed integer quadratic one.
1.3
Thesis Organization
This thesis is organized as follows. Chapter 2 describes thermosolar techonologies current avail-able. Chapter 3 presents a review of solar collector
fields modeling. Chapter 4 describes the MPC algo-rithms and the MLD systems. The MPC strategy, linear and nonlinear schemes are introduced in this chapter. Chapters 5 and 6 present the main contri-butions of this thesis. The proposal of modeling so-lar systems with logical and dynamical variables are introduced, and the proposed control structures are formulated followed by simulation results. Finally, chapter 7, presents the conclusions, contributions of this work and suggestions for future works.
Chapter 2
Thermosolar Energy
Systems
The oldest use of solar energy is credited to Archimedes, though not proved, which has burned the Roman fleet of ships in the Syracuse Bay by con-centrating solar rays. In 1973 a scientist, Dr. Ioan-nis Sakkas, curious about the Archimedes story, fired a ship using 60 oblong mirrors tipped for catching the sun’s rays [10]. In the 18th century, in Europe and in Middle East, solar furnaces were developed for metal funding, mainly iron and copper. During 19th century came up the first trials of generating steam from solar radiation; in 1866, Auguste Mou-chout used parabolic troughs for producing steam for the first solar steam engine, and in 1886 the first
patent for a solar collector was obtained by Alessan-dro Battaglia in Genoa, Italy. Near Genoa, in 1968 professor Giovanni Francia designed and built the first Concentrated Solar Power (CSP) plant. A. G. Eneas installed a solar collector for water pumping in a farm located in California, and Frank Shuman and Charles Vernon Boys built a water pumping plant in Egypt in the 20th century. The oil crises in 1973 and 1978 stimulated the investments in alternative sources of energy and the development of the cur-rent solar collectors in the United States of Amer-ica (USA). The first commercial solar plant was in-stalled by the Sandia National Laboratory in 1979 for generating heat for industrial processes in New Mexico. In the same year a 200 kW plant was con-structed at Coolidge, Arizona, and in 1981 a 500 kW plant was built at the Plataforma Solar de Almeria (PSA), Spain, as a part of the International Energy Agency (IEA) project entitled Small Solar Power Systems (SSPS) and the 10 MW Solar One power tower was developed in Southern California [24]. In 1982 the company LUZ International Limited devel-oped parabolic-trough solar collectors and was re-sponsible for the first commercial plant of electricity of the world. The parabolic-trough technology of the 354 MW Solar Energy Generating Systems (SEGS), begun in 1984, in California and was the largest solar power plant in the world, until the 390 MW Ivan-pah power tower project reached full power in 2014.
15
The abundant cheap oil in 1986 led to a reduction in the investments in renewable energy, that returned just in the 2000’s [48]. The acquired knowledge in re-search and development and the economic incentives along the years have contributed USA and Spain to be the reference in solar thermal technology, as can be seen in figure 2.1.
Figure 2.1: Thermosolar plants around the world [8]
From figure 2.1, it can be seen that 5 GW of CSP is installed globally. Spain is the world leader with a total capacity of 2.3 GW, followed by United States with 1.7 GW. Interest is also notable in Chile, North Africa and China. In Brazil the development in CSP is not significant in the international sce-nario. In figure 2.2, it is shown a diagram with three basic solar energy conversion system, able to con-vert the solar resource into a desired form of energy; thermal energy, shown in the first diagram or
elec-tricity. The solar energy is converted into electricity by using photovoltaic (PV) cells and by collecting solar energy as heat and converting it into electric-ity using a power plant or engine. The solar resource is converted into heat for supplying the demand of thermal load [5].
Figure 2.2: Solar energy conversion system [5]
If the solar energy conversion system is the only source of electricity, storage and auxiliary en-ergy supply are needed. In thermosolar plants the storage of heat is used instead of electricity storage for extending the operating time of the system and the auxiliary energy may either be supplied as heat or as electricity after the power conversion system. In the photovoltaic system extra electricity may be
2.1. Photovoltaic Plants 17
stored in batteries and the external electricity source is the only choice for auxiliary power [5]. Concen-trated Solar Power plants were originally built with-out energy storage and treated as the competitor to PV [31]. In 2015 the CSP technology presented thermal energy storage from 3 to 12 hours. The amount of storage is a key design parameter for PV and CSP, and it has a significant impact on which is the cheapest technology; if three hours of stor-age is the desired, PV tends to have lower cost; if nine hours of storage is the desired, CSP tends to have lower cost. As pointed out in [32], others fac-tors influence the value of energy. For example, CSP plants require time to start up and ramp and have minimum run levels, in contrast , PV with battery systems respond quickly to changing system needs. In ([52], [31]) a hybrid PV-CSP power plant is stud-ied. The idea is that PV operates during the day at low cost, and CSP operates supplementing the PV and also operates at night.
2.1
Photovoltaic Plants
The electricity generation in photovoltaic plants is based on the photovoltaic effect; photons of light hitting certain materials, commonly crystalline silicon, will knock electrons into a higher energy level producing electrical current. The first photovoltaic
systems were used to generate electrical power for spacecrafts. PV power generation systems rooftop-mounted, building-integrated and in large fields con-nected to the grid are promising to provide clean, safe, and strategically alternative to current meth-ods of electricity generation. Operating silently and without any moving parts or environmental emis-sions, PV systems have a range of capacity from a few to several tens of kilowatts, to large scale power stations of hundreds of megawatts [5]. Top installers countries in terms of capacity are currently China, Japan and the United States.
A conversion of direct current (DC) to alter-nating current (AC) is required when the PV sys-tems is connected to the grid. There are many others applications of photovoltaic systems such as houses isolated from the grid, pumps for water extraction, electric cars, roadside emergency telephones, remote sensing and cathodic protection of pipelines [22]. The photovoltaic devices have low conversion effi-ciency, limitations of electricity storage and high costs to compete with others nonrenewable fuels [13]. The advances in technology and the increases in manufacturing scale and sophistication are declin-ing the cost of photovoltaics. Crystal silicon solar cells have been replaced by less expensive multicrys-talline silicon solar cells, and thin film silicon solar cells have been developed at lower costs of produc-tion ([36], [39]).
2.2. Thermosolar Plants/Concentrating Solar
Thermal Systems 19
2.2
Thermosolar
Plants/Concentrating Solar
Thermal Systems
In figure 2.3, it is shown a diagram of a ther-mosolar plant. In the collector fields, the solar col-lectors concentrate sunlight to heat a heat transfer fluid (HTF) to a high temperature. The hot work-ing fluid can be stored, or used directly for different thermal applications and for generating steam that drives the power conversion system for producing electricity.
Solar collectors are heat exchangers which con-vert solar radiation into heat. There are two types of solar collectors: concentrating and non-concentrating. In the non-concentrating solar collector the interception and the absorption area are the same, this type of collector is used in low temperature systems. The concentrating solar col-lector has a reflecting surface which directs the ra-diation to a focus point, where the receptor with a HTF is located. The concentrating collectors are used in electricity generation systems and others high temperature systems. The thermal storage is cru-cial to provide heat for operation during periods without enough radiation [55]. The HTF may be air, water, oil or some organic solvent [42]. There is a wide range of solar collectors for different
ap-Figure 2.3: Diagram Thermosolar Plant [22]
plications; flat-plate, compound parabolic, evacu-ated tube, parabolic trough collectors (PTC), Fres-nel lens, parabolic dish and heliostat field collectors [22]. In PTC the temperature range is 350◦C to 400◦C and in flat-plate is 120◦C to 140◦C. There are stationary and tracking solar collectors; the tracking collectors can track the Sun in one or two axes.
2.2. Thermosolar Plants/Concentrating Solar
Thermal Systems 21
2.2.1 Fresnel Collectors
The Fresnel collector has two variations: the Fresnel lens collector (FLC) shown in figure 2.4 and the linear Fresnel reflector (LFR) shown in figure 2.5. The FLC consists of a transparent plastic ma-terial to concentrate the solar rays to a receptor. LFR is composed by various thin mirror strips to concentrate the irradiation onto tubes with circulat-ing fluid located at the focal point of the mirrors [44].
Figure 2.4: Fresnel Lens collector [44]
Multiple absorbers can be used to improve the system efficiency [22]. The Fresnel technology is not mature and it is used in low power plants operating in USA and Spain [48].
2.2.2 Heliostat Field Collector (HFC)
Heliostat field collector has an array of flat mir-rors, called heliostats, that tracks the Sun’s move-ment with two axes and concentrates solar radiation onto a central receiver at the top of a tower. The heat energy absorbed by the receiver is transferred to a circulating HTF that can be stored and used for power generation ([22], [44]). In figure 2.6, it is shown a schema of a HFC.
Figure 2.6: Central receiver system [44]
in-2.2. Thermosolar Plants/Concentrating Solar
Thermal Systems 23
stalled on the same pillar, with a total reflecting area of 50 m2 to 150 m2. The development of power
tower systems is less advanced than trough systems but they offer several advantages [44]: single receiver which minimizes thermal energy transport require-ments, high efficiency in collecting energy and in converting it into electricity, convenient storage of thermal energy, quite large and thus benefit from economies of scales.
2.2.3 Parabolic Dish Collectors
A dish engine system is composed by a parabolic collector dish that concentrates the Sun’s light onto a receiver positioned at the focal point of the dish reflector. The receiver absorbs the radiant solar energy, converting it into thermal energy in a circulating HTF. The thermal energy can be either converted into electricity using a Stirling engine po-sitioned at the focal point of the parabola and an electric generator, or transported through pipes to a central power conversion system [22]. The reflec-tor tracks the Sun along two axes pointing directly from sunrise to sunset, enhancing the efficiency of the system [48]. A schematic diagram of a parabolic dish collector is shown in figure 2.7.
Figure 2.7: Parabolic dish collector [44]
Parabolic dish systems are modular and have the following advantages: highly efficient at thermal energy absorption and power conversion systems, the most efficient of all collector systems, and possi-bility to function either independently or as part of a large system of dishes [44].
2.2.4 Enclosed Trough Collector
The GlassPoint Solar company created the en-closed trough design, that encapsulates the solar thermal energy within a greenhouse, as shown in fig-ure 2.8. Lightweight curved mirrors are suspended by wires from the ceiling of the greenhouse for con-centrating the sunlight and focus it on a tube re-ceiver, which is also suspended in the glasshouse structure. The glasshouse shelters the mirrors from
2.2. Thermosolar Plants/Concentrating Solar
Thermal Systems 25
the wind, allowing them to achieve higher tempera-ture [9].
Figure 2.8: Diagram of a GlassPoint solar steam generator sys-tem [3]
2.2.5 Parabolic Trough Collector
Parabolic trough systems are the most used and advanced solar thermal technology because of the considerable experience and the development of commercial industry to produce and market these systems. Parabolic trough collector is a high-performance collector that can produce heat at temperatures between 50◦C to 400◦C, compound-ing a solar system with light structures and low-cost technology [43]. A parabolic trough is made by bending a sheet of reflective material into a parabolic shape. In the focal line of the receiver is placed a black metal tube, covered with glass tube to reduce heat losses and plated with a selective coating that
has a high absorptance for solar radiation but a low emittance for thermal radiation loss [42]. The lin-ear parabolic mirror reflects and concentrates the received solar radiation onto the receiver. The size of the receiver, and consequently the concentration ratio, is determined by the size of the reflected sun image and the manufacturer tolerances. The concen-trated radiation heats the pumped HTF by transfer-ring the heat through the receiver tube walls. The HTF is routed either to a heat exchanger when the fluid is oil, to produce steam, to an ash tank when the fluid is pressurized water, or to a turbine when superheated and pressurized steam is produced di-rectly in the solar field [22]. In figure 2.9, it is shown the parabolic trough schema.
Figure 2.9: Parabolic trough collector [44]
Parabolic trough collectors can only use Di-rect Normal Irradiance (DNI); the fraction of solar radiation which is not deviated by clouds, fumes or dust [33]. The parabolic mirror follows the Sun
us-2.2. Thermosolar Plants/Concentrating Solar
Thermal Systems 27
ing just a single-axes tracking, therefore long col-lector modules are oriented in an east–west direc-tion, tracking the sun from north to south, or in a north–south direction, tracking the sun from east to west [44]. Using Fresnel collectors higher concen-tration can be obtained when compared to parabolic trough, but a more complex tracking mechanism and a increased spacing between reflectors for avoiding shading between adjacent reflectors are needed ([44], [22]). Parabolic trough applications are divided into two groups; Concentrated Solar Power (CSP) plants with temperatures from 300◦C to 400◦C connected to steam power cycles, and low-temperature appli-cations, mainly industrial processes [33].
The biggest application of parabolic trough col-lectors is the Southern California power plants; Solar Electric Generating Systems (SEGS) I, which has a total capacity of 14 MW, SEGS II, III, IV, V, VI, and VII with 30 MW each, and SEGS VIII and IX with 80 MW each [22]. A number of companies en-tered the field of parabolic trough collectors after a period of research and commercial development in the 1980s, producing collectors with a temperature range 50◦C to 300◦C and one-axis tracking. The de-velopments aim cost reduction and improvements in technology, like automatic washing of the collector mirrors which reduces the maintenance cost [43].
2.3
Final Remarks
This chapter briefly reviewed the basic con-cepts of the thermosolar energy systems. The reader is referred to ([44], [5]) for a detailed review of the
technologies. The parabolic
trough collector will be studied throughout this the-sis. In chapter 3 the modeling of solar fields is pre-sented and, in chapter 4, algorithms to control solar collector fields are proposed.
Chapter 3
Solar Fields
Modeling
This chapter introduces a review of the solar fields modeling to be used throughout this thesis.
3.1
Distributed Collector System
A solar field consists of loops of solar collectors in parallel with a solar tracking system. Each loop is a basic subsystem formed by collectors in series. The solar field to be studied in this work is composed by parabolic trough collectors with single axis tracking described in chapter 2. In figure 3.1, it is shown a scheme of a solar collectors field.
Figure 3.1: Solar Collectors Field Scheme
The HTF is pumped from the bottom of stor-age tank through the solar field where it gets the heat transferred through the receiver to the top of the tank 3.2.
The considered solar collectors field is the ACUREX field prototype of the Plataforma Solar de Almer´ıa (PSA) located in the Tabernas Desert (Southern Spain) (see figure 3.3). The PSA be-longs to the Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT), the biggest European center of research and develop-ment in CSP technologies [6]. The ACUREX Dis-tributed Collector Field consists of 480 east-west aligned single axis tracking collectors organized in 10 parallel loops with a mirror area of 2672 m2. The
3.1. Distributed Collector System 31
Figure 3.2: Solar Collectors Field and Tank Scheme [56].
loop is 172 m long formed by 4 modules of 12 col-lectors connected in series. The operation limits for
HTF volumetric rate are
2.0 × 10−3m3s−1 to 12.0 × 10−3m3s−1. The
maxi-mum value is for avoiding the HTF decomposing, which happens when the temperature is higher than 305◦C. For avoiding stress in the absorber pipe ma-terial, due to high HTF pressure in the pipe system, the difference between the inlet and outlet HTF tem-peratures must be less than 80◦C. A good overview of the field can be found in [22].
Figure 3.3: ACUREX distributed solar collector field ([12], [26], [27], [34]).
A collectors loop is a basic subsystem that rep-resents the characteristics of a collectors field. This way if the loop can be modeled in steady or transi-tory state, so is the solar field, by adding the loops and delays [25]. The objectives of modeling the solar collector field are:
• To simulate the field dynamic for an optimal control system.
3.1. Distributed Collector System 33
• For studying the system characteristics to pro-pose modifications and improvements.
3.1.1 Distributed Parameters Model
This subsection is a literature review based in [25], where the following considerations are made:
• The properties of the HTF are functions of the temperature, which depends on the time and space.
• The incident heat flow in each section is con-sidered radially uniform and equal to the mean flow.
• Variations in the radial temperature of the ab-sorber walls are negligible; this is a reasonable assumption considering a thin wall with good thermal conductivity.
• The fluid is considered incompressible, the HTF flow and the irradiation are time functions and are the same in each element.
• The axial conduction in the tube walls and in the fluid are considered negligible; high thermal resistance in the tube and low thermal conduc-tivity of the HTF.
• The heat capacity at constant pressure and vol-ume are equivalents. This is a reasonable as-sumption considering the range of pressures and temperatures of the process.
Considering the above assumptions and apply-ing the Energy Conservation Law in the metal tube of a length control volume dx (figure 3.4) over a time interval dt:
Figure 3.4: Control volume of the distributed parameters model [21]
• The variation of the internal energy U can be expressed as:
dU
dt = ρmAmdxcm dTm
dt dx (3.1) • This variation of energy is equal to the inci-dent energy minus the energy lost to the envi-ronment (radiation, convection) and the energy
3.1. Distributed Collector System 35 transferred to the HTF: dU dt =Iη0bdx − hlb(Tm− Ta)dx − htl(Tm− Tf)dx (3.2)
The global coefficient of losses hl is the power
lost per square meter and per degree Celsius. The coefficient of transfer htcorresponds to the
thermal power that cross through the contact surface between the metal and fluid.
The energy balance in the metal can be ex-pressed as:
ρmcm
dTm
dt = Iη0b−hlb(Tm−Ta)−htl(Tm−Tf) (3.3)
In the same way, the first principle of thermo-dynamic can be applied to the control volume in the fluid:
dU
dt = htl(Tm− Tf)dx − ˙m(Hx+dx− Hx) (3.4) where Hx+dx− Hx is the difference of enthalpy in
variation can be expressed as: ∂H
∂xdx = cf ∂Tf
∂x dx (3.5)
where the specific thermal capacity to constant pres-sure is supposed to be invariant. Therefore the vari-ation of the internal energy in the fluid element can be expressed as:
dU
dt = htl(Tm− Tf)dx − vρfcf ∂Tf
∂x dx (3.6) The variation of the internal energy of the fluid element can be defined as a function of the variation of the fluid temperature:
dU
dt = ρfcfAfdx ∂Tf
∂t (3.7)
where Afdx is the volume of the fluid element. This
way the energy balance in the fluid element can be expressed as (3.8): ρfcfAf ∂Tf ∂t + vρfcf ∂Tf ∂x = htl(Tm− Tf) (3.8) The model parameters and variables are shown in table 3.1 , the subindex m is referred to the metal tube and the subindex f to the fluid ∗.
Plata-3.1. Distributed Collector System 37
Table 3.1: Model parameters and variables
Symbol Description Units
A Absorver section m
c Specific thermal capacity kJ kg−1 ◦C−1
b Collector aperture m
hl Coefficient of global losses W m−2 ◦C−1 ht Coefficient of thermal transfer
between metal and fluid
W m−2 ◦C−1
T Temperature ◦C
Ta Ambient temperature ◦C
η0 Mirror optical efficiency
v Volumetric flow rate m3s−1
˙
m Mass flow kg s−1
v Volumetric flow rate m3s−1
ρ Density kg m−3
l Tube circumference m
I Normal irradiation over the tube
W m−2
The density and specific thermal capacity of the fluid can be expressed as polynomial functions of the temperature; ρf = 903 − 0.672Tf and cf =
1820 + 3.478Tf. The coefficient of global losses has
been evaluated by tests [22] and it is given by:
hl = 0.00249∆T − 0.06133 (3.9)
where ∆T is the difference between the average
put Tin = Tf(t, 0) and output Tout = Tf(t, x)
tem-perature Tout+Tin
2 and the ambient temperature Ta.
The convective heat transfer coefficient can be ex-pressed in function of the volumetric flow rate and a coefficient hv:
ht= hvv0.8 (3.10)
where hv = 2.17 × 106−5.01 × 104Tf+4.53 × 102Tf2−
1.64T3
f + 2.1 × 10−3Tf4.
3.1.2 Linear Model of the Distributed Collector Field
A linear model can be obtained from the dis-tributed parameters model equations (3.8), (3.3) con-sidering the assumptions: (i) The loop is divided into active and passive (isolated parts) segments; (ii) The temperature of the fluid and the metal are sidered the same (it is a reasonable assumption con-sidering that the fluid and the metal are designed to have high coefficient of heat transmission). This way the energy balance can be expressed as (3.11).
ρfcfAf ∂Tf ∂t +vρfcf ∂Tf ∂x = Iη0b−hlb(Tf−Ta) (3.11) Discretizing in space,
3.1. Distributed Collector System 39 ρfcfAf dTf(n, t) dt = − vρfcf (Tf(n, t) − Tf(n − 1, t)) ∆x Iη0b − hlb(Tf(n, t) − Ta) (3.12) where ∆x is the length of the segments and n is the nth segment in which the pipe is divided. This model is studied in [35] neglecting the heat losses and in [34] using a linearized state space representa-tion.
3.1.3 Bilinear Models
A simplified partial differential equation model of the PTC described by equation (3.13) is used by many authors. Diffusion and heat losses are ne-glected and the fluid is assumed incompressible.
Af ∂Tf ∂t (t, x) + v ∂Tf ∂x (t, x) = η0b ρfcfAf I(t) (3.13)
Assuming a smooth variation of the HTF tem-perature along the pipe, the temtem-perature distribu-tion can be approximated by ∂Tf
∂x ≈
Tfi−Tfi−1
l , i =
1, . . . , n. Where l is the length of each segment, n is the number of segments, nl is the pipe length and Tfi = Tf(t, il). Defining xi = Tf(t, il), the process
can be described by a system of bilinear ordinary differential equations: ˙x = h(x) + g(x)v (3.14) where x = x1 .. . xn , h(x) = η0b ρfcfAfI 1 1 .. . 1 , and g(x) = −1 l x1 x2− x1 . . . xn− xn−1
. This bilinear model reasonably
describes the transport and heating phenomena and can be used for feedback linearization control pur-poses, considering a number of states n high enough [22].
3.1.4 Lumped Parameters Model
A lumped-parameter physical model of the plant can be used for control purposes. The model is an approximation original partial differential equation model:
3.1. Distributed Collector System 41 ρfcfAf dTout dt =η0bI − Hl Lf ield − ρfcfv(t − td) Tout− Tin(t − tr) ηopLloop (3.15) where the thermal loss coefficient is given by Hl =
c1∆T −c2. The model parameters are shown in table
3.3.
Table 3.3: Model parameters
Symbol Description Units
Hl Function of thermal losses J s−1K−1 c1 Coefficient of thermal losses J s−1K−2 c2 Coefficient of thermal losses J s−1K−1
Lf ield Total solar field length m
Lloop Loop length m
ηop Number of operative loops
td Flow-output temperature HTF re-lated dead time
s tr Input-output temperature
trans-port delay
s
The dead time between a change in the HTF flow rate and the output temperature td is due to
delays in pump response and sensors. The variable delay between the measured input and output tem-peratures tr is given by equation (3.16), this
expres-sion was obtained experimentally in the real plant [22].
tr = A1e −q(t−td)
q1 + y0 (3.16)
This approximation for tr can be used for
con-trol purposes if the flow rate does not change much. If the flow rate changes significantly, the variable de-lay can be estimated by numerical integration using the method proposed in [51]. The delay is depen-dent on the flow rate and the distance Lloopthrough
which the oil flows in the pipe given by equation (3.17). Lloop= TS A i=n−1 X i=0 v(k − i) (3.17)
The transport delay can be estimated as an in-teger multiple n of the sampling time; n is calculated as a function of the past flows until the volume of the pipe is filled, resulting in tr = nTS. It is worth
noting the method accuracy depends on the sample time. As it increases, the transport delay calculation becomes less accurate [22].
The lumped model presented in this subsec-tion is suitable for describing the process and for studying control techniques as can be seen in [26]
3.2. Large Scale Solar Collector Fields 43
and in [13], where the lumped model is compared to the distributed model presenting suitable results. Because of that, this model will be used in the rest of this master thesis.
3.2
Large Scale Solar Collector
Fields
Large scale fields supply the thermal energy used to produce steam supplied to a Rankine steam turbine-generator cycle for producing electricity. In a large scale field the solar collectors are organized in groups configured in parallel and in series [29]. The serial configuration is represented in figure 3.5.
Figure 3.5: Large Scale Field Serial Configuration
As can be seen the output temperature of the field i is the input temperature of the field i + 1 and
the volumetric flow rate is the same in all fields. In cloudy regions it may happen nonuniform solar irra-diation distribution and in this configuration if the irradiation over a field is low, thermal energy can be lost to the environment and the large field out-put temperature Tf ield is decreased. For attenuating
energy losses in the presence of clouds, a parallel configuration shown in figure 3.6 is proposed.
Figure 3.6: Large Scale Field in Parallel Configuration
In this case, the volumetric flows of the fields are controlled individually and the large field output temperature Tf ield is a mixture of each field output
temperature Touti and volumetric flow rate vi,
mod-eled as a weighted average described by equation (3.18).
3.3. Final Remarks 45 Tf ield = Pn i=1Toutivi Pn i=1vi † (3.18)
As can be seen, in the parallel configuration, the output temperature can be controlled in the presence of clouds, by manipulating the volumet-ric flow rate of the individual fields for reducing the energy losses.
3.3
Final Remarks
This chapter briefly reviewed the complete dis-tributed model of the solar collectors field, which is essentially a very large heat exchanger. Approaches used for control purposes and for modeling large scale fields were also presented. The lumped model will be studied in chapters 5 and 6 to develop control algorithms.
†This is a reasonable hypothesis because the dynamic of the mix is
Chapter 4
Model Predictive
Control
As pointed out in this work, in a solar plant the primary energy source cannot be manipulated and it depends on the atmospheric conditions; cloud cover, humidity and air transparency. The objective is to maintain the HTF output temperature in a desired level for providing viable power production, consid-ering the atmospheric conditions and changes in the mirror reflectivity and in the HTF input tempera-ture. During operation, the fluid flow is adjusted to achieve this objective leading to significant vari-ations in the response rate and in the dead time, which complicates obtaining adequate performance with a fixed parameter controller. Therefore the
control algorithm is designed robust enough to cope with the following characteristics [22]:
• the solar radiation acts as a fast disturbance compared to the dominant time constant of the process;
• the variable delay depends on the HTF flow rate;
• there are strong unmodeled dynamics and the linearized dynamics vary with the operating point;
Because of these characteristics, advanced con-trol strategies have to be used to concon-trol these plants, and MPC appears as an interesting approach. Dif-ferent MPC strategies have been applied with suc-cess to control solar collector fields. In [45] a Filtered DMC (Dynamic Matrix Control) algorithm is stud-ied for improving the robustness and disturbance re-jection considering multiple delays, in [30] a bilin-ear filtered smith predictor subspace predictive con-troller with numerical robustness is proposed consid-ering dead-time caused by multiple physical factors. In [14] a multi-model MPC is applied to control the temperature in a solar field updating the predictions and tuning accordingly to the model variation, and in [13] and [60] nonlinear and linear MPC controllers with robust stability and dead-time compensation
4.1. Model Predictive Control 49
are proposed, respectively, to the same problem. In [34], it is proposed an adaptive MPC which uses an unscented Kalman filter (UKF) to estimate the metal-fluid temperature and the effective solar radi-ation. In [57] a combined feedback linearization (FL) with dead-time compensator generalized predictive controller (DTC-GPC) is used as a robust solution to handle dead-time errors. In [65] is presented the nonlinear extended prediction self-adaptive control (NEPSAC) algorithm combined with a dead-time compensator (DTC) to control the output tempera-ture in a solar desalination plant.
Thus, in this work, an MPC is also used to control the solar field, considering a hybrid MPC algorithm which utilizes a MLD model. To under-stand the proposed controller, a brief MPC review is presented in section 4.1, and the MLD framework is introduced in section 4.2.
4.1
Model Predictive Control
As of 1970, articles appeared showing an terest in Model Predictive Control (MPC) in the in-dustry, being the most important the publications of Richalet presenting Model Algorithmic Control (MAC) [58], and Cutler and Ramaker with Dynamic Matrix Control (DMC) [28]. These formulations were heuristic and took advantage of the increasing
potential of digital computers at the time. Different from other control schemes, MPC originated from the industrial practitioners with the academia stepping in later to provide the backbone of the theory [53]. In 1963 within the frame of “open-loop optimal feedback“, Propoi proposed the reced-ing horizon control (RHC) principle, one of the cen-tral ideas of MPC. MPC became popular due to the simplicity of the algorithm and to the use of the impulse response model (MAC algorithm) or step response model (DMC algorithm), which is intuitive and requires less information for identification. Gen-eralized Predictive Control (GPC) was developed by Clarke in 1987 for monovariable process formulated with input-output model. MPC has also been for-mulated in the state space context allowing the use of well known theorems of the state space theory, and facilitating the generalization to more complex cases. MPC is an ample range of control methods which have practically the same structure [18]:
• Explicit use of a model to predict the process output at a future horizon;
• Optimization procedure to obtain the control action by optimizing the system’s predicted evo-lution;
• Application of the first control signal of the cal-culated sequence.
4.1. Model Predictive Control 51
The MPC algorithms only differ amongst them-selves in the model used to represent the process and the noises and cost function. This type of con-trol has been applied in the process industry, robots [40], clinical anaesthesia [41], solar plants [22]. In [20] is presented a series of advantages of the MPC over other methods;
• It is attractive to staff because the concepts are intuitive and the tuning is relatively easy. • It can be used to control process with simple
and complex dynamics.
• Multivariable problems can easily be dealt with. • It has intrinsic compensation for dead times. • It has natural feed forward control for
measur-able disturbances.
• The control law is easy to implement and re-quires little computation, but its derivation is more complex than classical controllers.
• The treatment of constraints is conceptually simple.
• It is very useful when future references are known.
• It is an open methodology which allows future extensions.
MPC and others different Receding Horizon Control variants are very powerful control techniques to address complex control problems in practice. At present there is no others techniques to design con-trollers for general large linear multivariable sys-tems with input and output constraints with sta-bility guarantee [18]. The main idea of the MPC strategy is illustrated in figure 4.1.
Figure 4.1: MPC strategy
At each sampling time an open-loop optimal control problem is solved over a finite horizon [18]. The future outputs are determined along the predic-tion horizon NP using the prediction model. These
predicted outputs ˆy(t + k|t)∗, k = 1 . . . NP depend
on the past inputs and outputs and on the future
∗x(t + k|t) indicates the value of the variable x at the instant t + k
4.1. Model Predictive Control 53
control signals u(t + k|t), k = 0 . . . NC− 1, to be
cal-culated in the control horizon NC. The future inputs
are calculated by optimizing a determined criterion defined in the objective function. The control sig-nal u(t|t) is sent to the process and the next con-trol signals calculated are rejected using the reced-ing horizon concept. The u(t + 1|t + 1) is different of u(t + 1|t) because new information is available. The basic structure of the MPC controllers is represented in figure 4.2 .
Figure 4.2: MPC structure
The MPC algorithms have the following ele-ments in common; (i) prediction model, (ii) objec-tive function, (iv) constraints and (v) control law calculation.
Prediction model The need to calculate the pre-dicted outputs ˆy(t + k|t) leads to the use of the process model. The model should be complete enough to fully capture the process dynamics for the predictions calculation and permit theo-retic analysis. The different MPC strategies use various models to represent the process and the disturbances. The disturbances model is used to describe the effect of nonmeasurable inputs, noise and process modeling errors. In the linear case, the most used process models are impulse response, step response, transfer function, and state space. A disturbance model widely used is the Controlled Auto-Regressive and Integrated Moving Average (CARIMA).
The prediction of the output ˆY is separated in two parts; the free response F, which is the evolution of the process due to the present state (measured output y(t) and past inputs u(t − i)) and the forced response G∆u, the evolution of the process due to future control increments ∆u:
ˆ
Y = G∆u + F (4.1)
The DMC algorithm uses the step response to model the process, taking into account the first N terms assuming the process is stable and without integrators. The disturbances are
as-4.1. Model Predictive Control 55
sumed to be the constants along the horizon and equal to the measured output minus the estimated output ˆy(t|t). The MAC algorithm uses a impulse response model valid only for stable processes. The disturbances can be treated as in DMC. The output predictions of the GPC are based on the CARIMA model. The Predictive Functional Control (PFC) uses a state space model allowing the application for nonlinear and unstable linear internal mod-els. The Extended Horizon Adaptive Control (EHAC) formulation considers the process mod-eled by its transfer function without the distur-bance model. In the Extended Prediction Self Adaptive Control (EPSAC) the control horizon is reduced to one reducing the calculation to a single value u(t) which can be calculated ana-lytically. A good overview of the MPC strate-gies can be found in ([47], [20]).
Objective function The objective function of the MPC algorithms is defined based on the con-trol objectives. The general objective is that the predicted outputs ˆy(t + k|t) should follow a reference signal r(t + k|t), on the considered horizon NP, and the control efforts necessary
for doing so should be penalized on the horizon NC. For a SISO case the cost function can be:
J = NP X k=1 λP(k)[ˆy(t + k|t) − r(t + k|t)]2 + NC−1 X k=0 λC(k)[∆u(t + k|t)]2 (4.2)
where λC(k) is the weighting factor to penalize
changes in the control signal at instant k and λP(k) is the weighting factor of the reference
tracking to penalize tracking errors at instant k. In processes with dead time and non-minimum phase the first tracking errors are not consid-ered in the cost function. These parameters are adjusted to give satisfactory dynamic per-formance; the prediction horizon NP is selected
based on the time constant of the system; the control horizon NC is normally chosen smaller
than the prediction horizon, and this is enough for a good performance. The reference trajec-tory also affects the closed-loop behavior; if the future evolution of the reference is known, the system can react before the change has effec-tively been made. A first order filter to smooth changes in the reference trajectory can be con-sidered to avoid peaks in the system response. Constraints Limits in the process caused by oper-ational conditions, constructive reasons, safety
4.1. Model Predictive Control 57
and actuators limitations introduce constraints to the objective function to be minimized, turn-ing the optimization problem more complex. The typical limits of the manipulated and con-trolled variables are:
umin6 u(t + k|t) 6 umax k = 0, . . . , NC− 1
∆umin6 ∆u(t + k|t) 6 ∆umax k = 0, . . . , NC− 1
ymin6 ˆy(t + k|t) 6 ymax k = 1, . . . , NP
(4.3)
Control law calculation To obtain the control sig-nal values u(t + k|t) it is necessary to minimize the objective function. If there are no con-straints an analytical solution can be obtained. Rewriting the cost function (4.2) in matricial form;
J = ( ˆY − r)0R( ˆY − r) + ∆u0Q∆u substituting (4.1),
J =∆u0(G0RG + Q)∆u + 2(F − r)0RG∆u + (F − r)0R(F − r)
where R is a diagonal matrix with the weight-ing factors λP, Q is a diagonal matrix with
the weighting factors λC and r is the vector
of future references. The future control moves are the solution of the quadratic programming problem (4.4).
minimize ∆u 1 2∆u 0 H∆u + b0∆u subject to (4.3) (4.4) where H = 2(G0RG+Q) and b0 = 2(F−r)0RG. In the unconstrained case the solution is given by ∆u = (G0RG + Q)−1G0R(F − r).
The elements of the MPC algorithms were in-troduced and illustrated using the objective function (4.2) and the constraints (4.3). As mentioned be-fore, each algorithm uses a different model. A brief overview of some MPC algorithms is presented in next section.
4.1.1 Nonlinear Model Predicitive Control Schemes
There are process in which the nonlinearities are severe and the use of a linear model is not suffi-cient for control purposes. In these cases nonlinear controllers are essential for improved performance or stable operation. The stringent specifications on product quality, tighter environmental regulation of effluent streams, and higher competition in the pro-cess industries, motivates the development of non-linear model predictive control (NMPC) techniques [64]. The extension of MPC to nonlinear process
