Parabolic Trough Collector Defocusing Analysis: Two control stages vs four control stages

This is a repository copy of Parabolic Trough Collector Defocusing Analysis: Two control stages vs four control stages in the Depósito de Investigación de la Universidad de Sevilla Version: Author Accepted Version Citation: A.J. Sánchez, A.J. Gallego, J.M. Escaño, E.F. Camacho, Parabolic Trough Collector Defocusing Analysis: Two control stages vs four control stages, Solar Energy, Volume 209, October 2020, Pages 30-41. 10.1016/j.solener.2020.09.001 To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright: Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy: Please contact us ([email protected]) and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim Parabolic Trough Collector Defocusing Analysis: two control stages vs four control stages A. J. S´ancheza,∗, A. J. Gallegoa, J. M. Esca˜noa, E. F. Camachoa aDepartamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract In solar thermal plants, as in any industrial process, it is important to maintain good control of the system and, more importantly, to have a good security system to avoid exceeding the safety limits of the components and avoid their degradation. In the case of solar thermal plants, one of the main components is the Heat Transfer Fluid (HTF), which must be kept below a maximum temperature. Although the temperature of the fluid, in general, will be controlled by modifying the flow-rate, when the plant is saturated HTF temperature is kept under limits by defocusing of the collectors. In this paper, an analysis of the control of the defocus control applied to the different collectors is presented. A Model Predictive Control technique will be applied to control the temperature by defocusing two and four collectors in different situations. It is shown how controlling the temperature by defocusing only two collectors is not sufficient in all situations and that controlling by defocusing the four collectors solves this problem in addition to maintaining the defocus actions in areas with high control authority. Keywords: Solar Energy, Collector defocus analysis, Model Predictive Control, Electric power limitation 1. Introduction1 The sun floods the earth with huge amounts of energy2 every day. An energy that will only run out on the day3 of the star’s death. Only a small part of this energy is4 absorbed by the earth and is the base of all living beings5 on this planet (in one way or another), the rest of the en-6 ergy returns to space. Due to the current great interest in7 the reduction of CO2levels caused, among others, by the8 emissions generated by conventional power plants (fossil)9 (Romero and Gonz´alez-Aguilar, 2014; Blanco and Miller,10 2017), research and development in relation to renewable11 energy sources is being promoted more and more, being12 solar the most abundant and promising of all.13 This paper focuses in Concentrating Solar Power (CSP)14 plants with Parabolic Trough Collectors (PTC). Currently,15 electricity generation by thermal solar plants is a fact with16 almost 100 commercial solar plants producing in 2017 (Pitz-17 Paal, 2018). It is important to highlight one of the best18 characteristics that solar thermal technology plants have:19 thermal energy storage (Liu et al., 2016; Alva et al., 2017;20 Pelay et al., 2017; Sarbu and Sebarchievici, 2018). Al-21 though molten salts are generally used in storage tanks22 (Roca et al., 2016; Peir´o et al., 2018), energy storage us-23 ing steam is also possible (Prieto et al., 2018). How-24 ∗Corresponding author Email addresses: [email protected] (A. J. S´anchez), [email protected] (A. J. Gallego), [email protected] (J. M. Esca˜no), [email protected] (E. F. Camacho) ever, the use of molten salts seems to be the most recom- 25 mended when a large storage capacity is needed (Gonz´alez- 26 Roubaud et al., 2017), as occurs in commercial solar plants. 27 Currently, there are parabolic-trough solar plants operat- 28 ing in Spain (Majadas I (50 MW) (NREL Majadas, 2020)), 29 USA (Solana 280 MW (NREL Solana, 2020)), and South 30 Africa (KAXU 100 MW (NREL KAXU, 2020) among oth- 31 ers. There is currently a 600 MW project (3 CSP plants, 32 200 MW each), with a total solar field of 28 square kilo- 33 meters and 15 hours of thermal storage, under construc- 34 tion by Abengoa Solar in Dubai, United Arab Emirates, 35 launched by Dubai Electricity & Water Authority (DEWA) 36 (NREL DEWA, 2020; Helioscsp.com News, 2020). 37 PTC CSPs generally operate in a nominal high tem- 38 perature zone. One of the main objectives in this type 39 of plants is to maintain the outlet temperature around a 40 designated value or nominal set-point. However, this is 41 not the only objective pursued in research. Significant ef- 42 forts are also being made in research related to production 43 optimization, cost reduction and improvements of ther- 44 mal storage methods to name just a few. In Camacho 45 and Gallego (2013) an optimization of a solar plant is pre- 46 sented by applying a hierarchical structure of 3 layers to 47 calculate the optimal solar field temperature to increase 48 the plant performance according to environmental condi- 49 tions. In Khoukhi et al. (2015), authors presented nonlin- 50 ear continuous-time Generalized Predictive Control (GPC) 51 of solar plants. A control for solar field temperature based 52 on a Dynamic Matrix Control (DMC) is proposed in Lima 53 Preprint submitted to Solar Energy Journal June 12, 2020 et al. (2016) where a filter is included for the prediction54 of the error, improving the properties of disturbance rejec-55 tion and robustness of the DMC. A model-based predictive56 control (MPC) strategy is presented in Vasallo and Bravo57 (2016) which is used in conjunction with short-term direct58 normal irradiance forecast to perform optimal scheduling59 in CSP plants. In Gallego et al. (2016) a mathematical60 model of the new TCP-100 solar field of the Plataforma61 Solar de Almer´ıa is developed. In Cojocaru et al. (2019),62 the authors propose to include a term that penalizes the63 generation variation (cycling) to reduce it without losing64 benefit in the power cycle increasing the useful life of the65 cycle. In S´anchez et al. (2019b), a nonlinear optimization66 analysis and strategy is presented along with clustering to67 calculate the necessary control actions on the loops inlet68 valves to obtain a thermal balance of the solar field reduc-69 ing the need for unnecessary defocus actions and maintain-70 ing loops at similar temperatures. Aguilar et al. (2019)71 discusses the use of super-critical carbon dioxide (sCO2)72 to replace current HTFs in PT CSP plants in order to73 increase the solar-to-electric efficiency of the plant.74 A very important aspect is the safety of the plant com-75 ponents, including the heat transfer fluid. It is important76 never to exceed the temperature limit of the HTF provided77 by the manufacturer. In the case of diphenyl oxide (DPO)78 and biphenyl mixture fluids such as Therminol VP1 or79 similar, this temperature limit is around 400 ◦C. Defo-80 cus control is primarily focused on controlling the fourth81 (last) collector outlet temperature. However, the defocus82 control should only have to be applied in flow saturation83 situations.84 However, applying defocus over the fourth collector is85 often not enough to keep the temperature of the loops86 within the established safety limits. This will generally87 occur during the summer season on days of high radiation88 and in cases of power limitation. Defocus control of the89 other collectors is needed to keep the temperature below90 the safety limit.91 In this paper the defocus control on the different col-92 lectors is analyzed. Applying the defocus on the last two93 collectors can, in certain situations, meet the objective and94 keep the outlet temperature of the loops below the safety95 limit. However, this may be accomplished at the cost of96 making the defocus controller work with high defocusing97 angles in a zone of low control authority. The behavior of98 defocus controllers on different collectors and with differ-99 ent temperature set-points will be simulated and analyzed.100 It will be shown that applying the defocus control to four101 collectors provides a better control, solving problems of ex-102 cess temperature, although at the cost of a greater number103 of actions when using more control levels in the collectors.104 The paper is organized as follows: Section 2 section105 briefly describes the work prior to this work. In section 3106 the model of the 50 MW plant and mathematical models107 are presented. Section 4 describes the GS-GPCs controller108 for defocus and power control. Section 5 presents the pa-109 rameters and simulations of the controllers when applied110 to collectors 3 and 4. The simulation results and param- 111 eters when applying the GS-GPC on the four collectors 112 are shown in section 6. In Section 7 the numerical re- 113 sults of the simulations are presented and a discussion is 114 made regarding the advantages and disadvantages of both 115 strategies. Finally, the papers draws to and end in Section 116 8 with some conclusions and future work. 117 2. Related work 118 Model-based predictive control strategies were presented 119 to control the outlet temperature of the third and fourth 120 collectors of 50 MW solar plants by defocusing in S´anchez 121 et al. (2018, 2019a). It was shown that defocusing the 122 fourth collector was not sufficient, in all situations, to keep 123 the oil temperature below the safety limit. A controller 124 for defocusing the third collector was added to help the 125 fourth and prevent the defocus angle from reaching the 126 control limit. However, in these works, the main purpose 127 was to design and test predictive controllers for defocus- 128 ing. In fact, the proposed controllers provided good results 129 in tracking the temperature reference for both collectors, 130 keeping the oil temperature below the limits during tran- 131 sients, and under given plant circumstances. However, it 132 is important to emphasize that the level at which the col- 133 lectors have to be defocused will largely depend on the 134 radiation level, plant flow rate, collector efficiency, plant 135 operating point and possible power limitations. There- 136 fore, it is highly likely that the third and fourth defocus 137 controllers will not be sufficient in all circumstances in 138 order to properly track the loop outlet temperature and 139 avoid the safety limit. In the next sections, an analysis 140 will be presented in various circumstances, presenting the 141 need and convenience of applying the defocus control in 142 the four collectors of the loops of a solar plant. 143 Since the main objective of the paper is to perform a 144 defocus analysis, a GS-GPC, (S´anchez et al., 2018), will 145 be applied to each of the 4 collectors. 146 3. 50 MW solar plant model 147 This section describes the 50 MW plant used, (S´anchez 148 et al., 2018, 2019b). Two mathematical models, a dis- 149 tributed parameter model and a concentrated parameter 150 model, are used for simulation purposes and controller de- 151 sign. 152 3.1. Parabolic trough field 153 The solar field of the plant to be simulated occupies 154 around 110 hectares, with 90 loops of 4 collectors each. 155 Each loop is 600 meters long (NREL Guzm´an, 2020; 156 NREL Helios, 2020; NREL Solaben, 2020). The main com- 157 ponents of a parabolic trough solar plant are the collector, 158 the number of loops, the receiver tube, the HTF and the 159 power cycle. 160 2 The HTF used is Therminol VP1, widely used for this161 type of solar application. This fluid begins to degrade from162 400 ◦C. HTF parameters, such as specific heat capacity163 (Cf) and fluid density (ρf), are temperature dependent164 and can be obtained through equations (1) and (2). All165 the parameters approximations can be found in Thermi-166 nol VP1 HTF (2020). The collector used in this plant167 model is the EuroTrough ET150, with similar character-168 istics to those used in 50 MW commercial plants. The169 parameters of this collector are shown in Table 1 (Geyer170 et al., 2002; Kearney, 2007; System Advisor Model (SAM).171 NREL, 2018). The reflectivity and collector form factor172 values are assumed to be 0.92 and 0.96 respectively. For173 the receiver tube, the Schott PTR70, very common in com-174 mercial plants, is used. The tube is made of steel-type DIN175 1.4541 or similar, external diameter of 70mmm and inter-176 nal diameter of 66mm (Burkholder et al., 2007; SCHOTT177 Solar CSP GmbH, 2020). The tube efficiency has been178 assumed to be 0.9.179 Table 1 EuroTrough ET150 parameters. Description Value Unit Focal length 1.71 m Aperture width 5.77 m Aperture area 817.5m2 Number of Modules per Drive 12 Unitless Length per Solar Collector Assembly (SCA) 148.5m SCAs per loop 4 Unitless Heat Collection Element (HCE) Type Evacuated tube Unitless ρf=−0.90797 ·T+ 0.00078116 ·T2−2.367 ×10−6·T3 +1083.25 (1) Cf= 4.5904 ×10−8·T4−3.1536 ×10−5·T3+ 0.006498 ·T2 +2.3458 ·T+ 1500.8 (2) In general, 50 MW commercial plants operate at nom-180 inal with 393 ◦C at the output of the solar field and a re-181 turn temperature of 293 ◦C, therefore, the thermal jump182 is 90-100 ◦C. Assuming only the losses produced by the183 parasitic effects (0.9) and the rankine cycle (0.381) (Anda-184 sol 1, 2018; NREL Extresol, 2020; System Advisor Model185 (SAM). NREL, 2018), an approximation of the flow nec-186 essary to produce 50 MW (3000 m3/h approx.) can be187 obtained using the equation 3, (S´anchez et al., 2019b).188 Q=P·106 ∆T·Cf·µrankine ·µparasitic (3) 3.2. Distributed parameter model189 The distributed solar field dynamics can be described190 by a partial differential equations (PDE) system shown in191 equation 4. The system energy balance is described in this192 set of PDEs (Carmona, 1985; Camacho et al., 1997):193 ρmCmAm ∂Tm ∂t =IKoptnoG−HlG(Tm−Ta)−LHt(Tm−Tf) (4a) ρfCfAf ∂Tf ∂t +ρfCfq∂Tf ∂x =LHt(Tm−Tf) (4b) Subindexes fand mare used referring to the fluid and 194 metal. Geometric efficiency depends on declination, day 195 of the year, local latitude, collector parameters, solar hour 196 and hourly angle. Coefficients and parameters Hl, specific 197 heat Cand density ρdepends on the temperature of the 198 fluid. Coefficient Htdepends on fluid temperature and 199 HTF flow-rate (Camacho et al., 1997). An approximation 200 for Hlcan be obtained from Burkholder et al. (2007), 201 L¨upfert et al. (2008). To obtain Htvalue, equations (5) 202 are used, where the dependency of the flow-rate can be 203 observed. 204 Re =Q·D/(ν·A) (5a) Pr =Cf·µ/k (5b) Nu = 0.025 ·(Re0.79)·(P r0.42)·phi (5c) Ht=Nu ·k/D (5d) 3.3. Concentrated parameter model 205 The concentrated parameter model (CPM) is a simpli- 206 fication of the spatially distributed solar field (Camacho 207 et al., 2007, Gallego et al., 2019). This simplification pro- 208 vides an overall description of the solar field in terms of 209 the fluid internal energy variation by equation 6. 210 Cloop dTout dt =KoptnoSI −qCfρf(Tout −Tin) −HlS(Tmean −Ta) (6) where qis the HTF flow-rate, Tout and Tin are the 211 outlet and inlet oil temperatures of the model, Tmean is 212 the average value between outlet and inlet temperatures 213 and Tais the ambient temperature. Cloop is the thermal 214 capacity, approximated by 3.8×106J/◦C, Kopt is the op- 215 tical efficiency (mirror reflectivity, tube absorptance, and 216 interception factor), Iis the direct solar irradiance and S217 is the reflective surface of the loop, 3427 m2.218 4. Generalized predictive control 219 The GPC algorithm is based on the following single- 220 input single-output model (Camacho and Bordons, 2007): 221 A(z−1)yk=z−dB(z−1)uk−1+C(z−1) ∆ek(7) where ukand ykare the control and output sequences 222 of the plant, ekis a zero mean white noise term and ∆ is 223 the integrator operator. A, B and Care polynomials in 224 the backward shift operator z−1:225 3 A(z−1) = 1 + a1z−1+... +anaz−na 226 B(z−1) = b0+b1z−1+... +bnbz−nb 227 C(z−1) = 1 + c1z−1+... +cncz−nc 228 where dis the dead time of the system and ∆ is the op-229 erator 1−z−1. This model is known as a Controller Auto-230 Regressive Integrated Moving-Average (CARIMA) model.231 Consider a multistage cost function of the form:232 J(N1, N2, Nu) = N2 X j=N1 δ(j)[ˆy(k+j|k)−w(k+j)]2 + Nu X j=1 λ(j)[∆u(k+j−1)]2 (8) where ˆy(k+j|k) is an optimum j step ahead predic-233 tion of the system output, N1and N2are the minimum234 and maximum costing horizons, Nuis the control hori-235 zon, δ(j) and λ(j) are weighting sequences and w(k+j)236 is the future reference trajectory. The aim of GPC is to237 minimise J(N1, N2, Nu) in order to compute a future se-238 quence of control actions u(k), u(k+ 1), ... that drives the239 future plant output y(k+j) close towards w(k+j).240 241 Lets consider the following Diophantine equation:242 1 = Ej(z−1)˜ A(z−1) + z−jFj(z−1) (9) where ˜ A(z−1) = ∆A(z−1). The polynomials Ejand Fj 243 are uniquely defined with degrees j−1 and narespectively.244 By operating with the Diophantine equation (9) and245 the CARIMA plant model (7) the following expression for246 the system output can be obtained:247 y=Gu +F(z−1)y(t) + G0(z−1)∆u(t−1) (10) The Eq. (10) can be compacted into two parts, (11),248 since the last two terms depend only on the past:249 y=Gu +f(11) where fis known as the free response of the system.250 Finally the expression (8) can be written as:251 J= (Gu +f−w)T(Gu +f−w) + λuTu(12) Hence given a CARIMA plant model and suitable cost252 function, the minimum of the cost function can be ob-253 tained by setting the gradient of Jequal to zero and solving254 the control sequence ∆uby the following equation (Cama-255 cho and Bordons, 2007):256 ∆u= (GGT+λI)−1GT(w−f) (13) where matrix Gcontains the step response coefficients 257 of the forced response model (Camacho et al., 2012), Iis 258 the eye matrix, fis the free response of the plant, wis 259 the future reference trajectory vector and λis the control 260 weighting vector (Camacho and Bordons, 2007). 261 4.1. Defocus GS-GPC Control 262 For the control of the temperature by defocus collec- 263 tors, a GPC control with a Gain Scheduling (GS) will be 264 used. In order to design this controller it is necessary to 265 consider the defocus curve. This curve is presented in Fig. 266 1 (Goswami et al., 2000) and it can be seen how it is not 267 only nonlinear but also has 3 clearly identifiable zones. 268 Two areas where it is necessary to send important actions 269 to the collector in order to decrease or increase the col- 270 lector’s efficiency level (0-1 and 4-5 degrees). Notice that 271 above 3 degrees of defocus the efficiency goes below 20 %. 272 The curve presents a third zone, around 2.5 degrees of de- 273 focus, where a high slope can be observed, that is, small 274 control actions will cause large changes in the efficiency of 275 the collector. 276 Using the GS makes it possible to approximately in- 277 clude the nonlinear characteristic of this curve in the GPC 278 controller. For this, several linear models are obtained at 279 different defocus and flow-rate operating points. Similar to 280 (S´anchez et al., 2018), the GS is designed at nine different 281 points of defocus angle (0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 and 4.5 282 degrees). In addition, in order to cover the entire possible 283 operating range of the plant, the nine linear models will 284 be obtained at different flow operation points (1494, 1908, 285 2322 and 2736 m3/h). 286 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Collector Defocus Angle (deg) 0 0.2 0.4 0.6 0.8 1 Collector Defocus Efficiency Defocus Curve ( col) Figure 1 Collector efficiency-defocus angle curve. 4.2. Power limitation GPC control 287 Solar plants may receive orders commanded by the 288 Transmission System Operator (TSO) to limit its gener- 289 ated power, mainly due to the saturation of the grid. In 290 order to reduce the power, the flow must be reduced and 291 the HTF temperature may rise to undesirable values if not 292 appropriate measures are taken. 293 In order to control the power generated a GS-GPC will 294 be used which uses the HTF flow as manipulated variable. 295 4 As done with the defocus GPC, different linear models296 can be obtained to capture the dynamics of the plant that297 strongly depends on the flow-rate (Schenk et al., 2015;298 Monta˜n´es et al., 2018). Linear models at 3 different flow299 points are used to capture the non-linearity of the plant,300 approximately, and include it in the GPC: 167.06, 334.1301 and 501.16 kg/s (855, 1710 and 2565 m3/h). Upon receiv-302 ing a power limitation, the plant will have a known time,303 specified by the TSO, to reach the power set-point. One304 of the advantages of using MPC strategies is the sliding305 horizon. Since the time at which the power set-point has306 to be reached is known, set-point ramps can be created307 for the controller at each control instant and thus achieve308 better tracking of the trajectory during the power drop309 until reaching the power set-point, (S´anchez et al., 2018).310 5. Defocus control: 3rd and 4th collectors311 In this section, various results will be presented when312 applying the control strategy in the third and fourth col-313 lectors of the loops. The temperature set-points used in314 previous works will be used for collector 3 and 4 (S´anchez315 et al., 2018). The temperature of the fourth collector will316 be 393 ◦C if the plant is in power limitation and close to or317 in flow saturation, otherwise the temperature will be some-318 what higher, 395 ◦C, to avoid coupling between the defocus319 controllers and the global temperature flow-rate controller.320 The temperature of the third collector was chosen taking321 into account the defocus curve (S´anchez et al., 2018). The322 defocus curve is non-linear with a steep slope around 2.5323 degrees. Above 3 degrees, the efficiency of the collector324 is observed to be very close to zero. Working above this325 angle implies very little control authority, i.e., big incre-326 ments in the control actions implies small changes in the327 efficiency. Therefore, a temperature of 385 ◦C was chosen328 by simulation for the third collector. This temperature329 caused the third collector to defocus out at high temper-330 ature, keeping the action of the fourth collector below 3331 degrees of defocusing.332 5.1. Control sample time and parameters333 In commercial plants, the defocusing strategy is carried334 out using partial and total defocusing, which can cause335 significant fluctuations in the loops outlet temperatures.336 The fact of proposing and applying MPC strategies such337 as the GS-GPC does not imply that this safety strategy338 is eliminated, although modifying the operating temper-339 ature for it. In previous works, (S´anchez et al., 2018,340 2019a), a sampling time of 30 s was selected for the defocus341 GS-GPC controller. This time was found to be sufficient,342 in general, to properly tract the temperature set-point.343 It was observed that the fluid temperature did not, un-344 der any circumstances, exceed the limit temperature. In345 very extreme cases, where the temperature could exceed346 this limit, the partial and/or total defocus safety strat-347 egy could perfectly coexist with the proposed GS-GPC348 strategy. However, to try to avoid activating this security 349 strategy, the sampling time of the GPC controllers that 350 are applied to each of the loops has been modified. 351 9 10 11 12 13 14 15 16 17 18 19 Time (h) 350 360 370 380 390 400 410 (ºC) 0 200 400 600 800 1000 1200 (W/m2) Treffield Toutfield Tin+70 I 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 100 200 300 (m3/h) 0 20 40 60 80 100 (MW) Q/10 PWnet Figure 2 High radiation day. 4th and 3rd collectors GS-GPCs. Field and inlet temperatures, flow and power results. 9 10 11 12 13 14 15 16 17 18 19 Time (h) 300 320 340 360 380 400 (ºC) MaxTemp TrefC4 TrefC3 ToutC4 ToutC3 ToutC2 ToutC1 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 1 2 3 4 5 Defocus Angle (deg) Def C4 Def C3 Def C2 Def C1 Figure 3 High radiation day. 4th and 3rd collectors GS-GPCs. Collectors temperatures and defocus actions. It must be taken into account that since a control for 352 reference tracking is being applied, the sampling times 353 must be in line with the dynamics of the process. That 354 is, very long sampling times should not be chosen since 355 then the GS-GPC would not work properly for tracking 356 the temperature reference applied to each of the collec- 357 tors. Therefore, the following sampling times have been 358 applied for each of the collector controllers: 359 1. Sample time GS-GPC Collector 3: 30s 360 5 2. Sample time GS-GPC Collector 4: 5s361 The sampling time of 30 seconds in collector 3 has been362 maintained since it has already given a good result in pre-363 vious works. However, the fourth collector sampling time364 has been reduced to 5 seconds since it is the one that will365 mainly prevent the temperature from exceeding the safety366 limit in very extreme cases, thus avoiding the possible use367 of the final security strategy.368 Regarding the weights of the control actions, although369 correct tracking of the different temperatures set-points370 is intended, it is also important to avoid activating high-371 frequency dynamics and causing oscillations in the actions372 and on the temperatures. A weight λC4= 2000 has been373 chosen for the fourth collector. Since the controller now374 sends control actions every 5 seconds, it can also be more375 sensitive to activating the commented modes. For the376 third collector, a λC3= 1000 has been chosen, providing377 temperature tracking and smooth control actions. These378 values have also been chosen in relation to the following379 subsection which talks about the precision in the perfor-380 mance of the defocus actuators. As for the control and381 prediction horizons, for the GS-GPC of the third collector,382 which is controlled every 30 seconds, the same horizons are383 used as in (S´anchez et al., 2018), that is, N2 = 12 and Nu384 = 6. Regarding the GS-GPC of the fourth collector, since385 it has been sampled at 5 seconds, the horizons have been386 increased and the chosen values for this work are N2 = 90387 and Nu = 50.388 Figs. 2 and 3 show the results of applying GS-GPC389 controllers to the third and fourth collector on a day of390 high Direct Normal Irradiance (DNI). It can be seen how391 the control signals are smooth, producing a good tracking392 of the reference temperatures for collectors 3 and 4. The393 control actions of the third collector are centered on ap-394 proximately 2 degrees while the fourth is a little below 3395 degrees, as intended when selecting 385 ◦C as the reference396 temperature for the third collector.397 5.2. Minimum defocus angle constraint398 Although a good behavior of the GS-GPC controller399 for defocusing is observed, it should be noted that in these400 simulations, the control signal that is applied to the col-401 lector does not undergo any modification. In reality, due402 to the actuators physical limitations, there will be a min-403 imum angle increment on the collector. For this work, a404 precision of the collector increments of 0.1 degrees is as-405 sumed based on the actuators manufactured by HELAC406 Corp. (model L30-380) (Heney, 2020).407 Obviously, having a more unprecise control signal will408 cause loss of precision in the set-point tracking. From now409 on, all the simulations will be done using the 0.1 degrees in410 the precision of the increments and it will shown that no411 different control signals will be sent to the actuator in each412 iteration of the controllers. Moreover, despite the precision413 in the control, it will be checked that the temperature414 tracking will still have a good performance.415 9 10 11 12 13 14 15 16 17 18 19 Time (h) 350 360 370 380 390 400 410 (ºC) 0 200 400 600 800 1000 1200 (W/m2) Treffield Toutfield Tin+70 I 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 100 200 300 (m3/h) 0 20 40 60 80 100 (MW) Q/10 PWnet Figure 4 High radiation day with transients. 4th and 3rd collectors GS-GPCs (precision 0.1 degrees). Field and inlet temperatures, flow and power results. 9 10 11 12 13 14 15 16 17 18 19 Time (h) 300 320 340 360 380 400 (ºC) MaxTemp TrefC4 TrefC3 ToutC4 ToutC3 ToutC2 ToutC1 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 1 2 3 4 5 Defocus Angle (deg) Def C4 Def C3 Def C2 Def C1 Figure 5 High radiation day with transients. 4th and 3rd collectors GS-GPCs (precision 0.1 degrees). Collectors temperatures and defocus actions. Regarding the behavior of the strategy on transient 416 days when the radiation is high, simulation results are pre- 417 sented in Figs. 4 and 5. It can be seen that both collectors 418 respond adequately to transients to avoid loss of tempera- 419 ture. Due to the high radiation and the abrupt transient, 420 the defocus controllers have to act decisively to avoid the 421 strong loss of temperature. 422 5.3. Inlet temperature disturbance rejection 423 In this section we test how the defocus controller is 424 able to cope with significant disturbance in the inlet tem- 425 perature. It is important to emphasize that the GS-GPC 426 6 controllers do not use the inlet temperature of the loop, nor427 the inlet temperature of the collectors, therefore the be-428 havior of the controller when faced with such disturbances429 should be analyzed.430 9 10 11 12 13 14 15 16 17 18 19 Time (h) 350 360 370 380 390 400 410 (ºC) 0 200 400 600 800 1000 1200 (W/m2) Treffield Toutfield Tin+70 I 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 100 200 300 (m3/h) 0 20 40 60 80 100 (MW) Q/10 PWnet Figure 6 Inlet temperature disturbance. 4th and 3rd collectors GS- GPCs (precision 0.1 degrees). Field and inlet temperatures, flow and power results. 9 10 11 12 13 14 15 16 17 18 19 Time (h) 300 320 340 360 380 400 (ºC) MaxTemp TrefC4 TrefC3 ToutC4 ToutC3 ToutC2 ToutC1 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 1 2 3 4 5 Defocus Angle (deg) Def C4 Def C3 Def C2 Def C1 Figure 7 Inlet temperature disturbance. 4th and 3rd collectors GS- GPCs (precision 0.1 degrees). Collectors temperatures and defocus actions. In the simulation presented in Figs. 6 and 7, a sine431 wave added to the inlet loop temperature of 10 degrees432 peak to peak has been applied with a period of about 30433 minutes. This type of disturbance in the input tempera-434 ture is not common in practice, since for this to happen435 in reality, the field outlet temperature would also have to436 be oscillating with a greater amplitude of the oscillation,437 which would imply a poor control of the field outlet tem- 438 perature. However, it is important to observe the behavior 439 against strong disturbances. 440 Since collectors 1 and 2 are free of control (always 441 tracking the sun), the input temperature disturbance is 442 transmitted throw both collectors. It can be seen that 443 the GS-GPCs of the third and fourth collector manage to 444 reject this disturbance. The third collector cannot com- 445 pletely reject the disturbance, having an oscillating follow- 446 up of its reference temperature, although these control ac- 447 tions decrease the amplitude of the disturbance (from 10 448 ◦C to 3 ◦C peak to peak). 449 The fourth collector is the one that, thanks to the help 450 of the third collector largely rejecting the disturbance, fi- 451 nally achieves its objective and maintains good tracking of 452 the reference temperature. Compared with the result in 453 tracking the outlet temperature of both collectors, it can 454 be seen how the actions in the third collector are more 455 important than in the fourth which can regulate its tem- 456 perature through smaller control actions due to the help 457 of the third. 458 5.4. 30 MW Power limitation 459 To observe the behavior of the GS-GPC controllers ap- 460 plied to the third and fourth collector in cases of significant 461 power limitations on days of very high radiation, a case is 462 simulated where the plant is limited to 30 MW. This case 463 is shown in Figs. 8 and 9. 464 9 10 11 12 13 14 15 Time (h) 350 360 370 380 390 400 410 (ºC) 0 200 400 600 800 1000 1200 (W/m2) Treffield Toutfield Tin+70 I 9 10 11 12 13 14 15 Time (h) 0 100 200 300 (m3/h) 0 20 40 60 80 100 (MW) Q/10 PWnet Figure 8 30 MW power limitation scenario. 4th and 3rd collectors GS-GPC (precision 0.1 degrees). Field and inlet temperatures, flow and power results. 7 9 10 11 12 13 14 15 Time (h) 300 320 340 360 380 400 (ºC) MaxTemp TrefC4 TrefC3 ToutC4 ToutC3 ToutC2 ToutC1 9 10 11 12 13 14 15 Time (h) 0 1 2 3 4 5 Defocus Angle (deg) Def C4 Def C3 Def C2 Def C1 Figure 9 30 MW power limitation scenario. 4th and 3rd collectors GS-GPC (precision 0.1 degrees). Collectors temperatures and defocus actions.. As previously mentioned, it can be seen in Figs 8 and 9,465 how both controllers work correctly until the power lim-466 itation arrives. The power controller begins to decrease467 the flow to reach, in the stipulated time by the TSO, the468 30 MW set-point. As the flow rate decreases, the defocus469 angle applied by the third and fourth GS-GPCs increases470 to compensate for the lack of flow control over the loop471 outlet temperature. Reaching the point at which the third472 collector reaches its saturation (5 degrees - 0 efficiency)473 first, since it is designed to help the fourth. Secondly, the474 fourth collector ends up also reaching saturation so that,475 between the flow dedicated to power control and the lack476 of control due to defocusing, the temperature becomes un-477 controllable and it shoots up exceeding the temperature478 limit. As discussed, defocusing the third and fourth can479 be sufficient in most situations, but not all, therefore, more480 levels of control are required.481 6. Using four collectors GPC defocus control482 Although it has been observed that the control with the483 third and fourth collector can keep the outlet temperature484 within the safety limit even on very high radiation days,485 it is also observed that the fourth collector is centered ap-486 proximately around 3 degrees of defocusing. Reviewing487 Fig. 3, it can be seen that in this area the collector effi-488 ciency is 0.2 approx., while the third collector maintains489 an approximate value of about 2 degrees, which means a490 collector efficiency of 0.6. Since the controller is already491 close to the area where the control capacity is low, it could492 happen that it was not capable of maintaining good tem-493 perature control in the face of certain external events, such494 as strong transients or more power limitations. In fact, it495 has been shown in the previous section how the GS-GPC 496 applied in the third and fourth collector are not capable of 497 maintaining the temperature below the limit in a 30 MW 498 power limitation situation. 499 Power Tin,Ta,Ieff Tin Tout Tref Tref−C4 Q P PWset−point βi k Tset−point PWref Q Tout γi k FF Event Collector Reference Temperature Selection Power Event Tout Ti C3 T ime PWref PQP W Qff GS-GPC Flow Q Q βi k−1 Q Power GS-GPC Cycle Limitation Solar Field . C4 C3 C2 C1 Defocus GS-GPCs i= 1,2, ..., 90 αi kθi k Tref−C3 Tref−C2 Tref−C1 γi k−1 θi k−1 αi k−1 Figure 10 Full control strategy scheme In fact, it has been shown in the previous section, see 500 Figs. 8 and 9, how the GS-GPC applied in the third and 501 fourth collector are not capable of maintaining the tem- 502 perature below the limit in a 30 MW power limitation 503 situation. Since loops have four collectors, to prevent the 504 controller from getting too close to the low control author- 505 ity area, one option is to extend the blur controller to all 506 four collectors. In this case, it would only be necessary to 507 add a GS-GPC in the first and second collectors, which will 508 help to lower the actuation levels of the third and fourth, 509 keeping the control actions in a more appropriate control 510 area. The control scheme is illustrated in Fig. 10. 511 6.1. Temperature reference and control parameters 512 By adding the GS-GPC control on collectors 1 and 2, 513 a further stage of aid in temperature control is being in- 514 cluded by the defocus control of the fourth collector. In 515 other words, it is no longer necessary for the reference tem- 516 perature of the third collector to be around 385 ◦C as in 517 the previous simulations. Furthermore, it is now necessary 518 to select 3 new reference temperatures for the first, second 519 and third collector controllers. Once again the problem of 520 deciding which temperatures to choose arises. The case of 521 the fourth collector is simple since it is directly linked to 522 the nominal working point of the plant. However, in the 523 case of defocusing the other collectors, it may be some- 524 what complex and/or subjective, given that according to 525 the chosen operating criteria different reference tempera- 526 tures for defocusing would be obtained. 527 The criterion chosen in this article to select these tem- 528 peratures is to try to maintain, as far as possible and ac- 529 cording to environmental circumstances, the same level of 530 defocusing in the four collectors. For this, what is done is 531 8