Event-Based MPC for defocusing and power production of a parabolic trough plant under power limitation

This is a repository copy of “Event-based MPC for defocusing and power production of a parabolic trough plant under power limitation” 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. “Event-based MPC for defocusing and power production of a parabolic trough plant under power limitation”. Solar Energy – ISSN 0038-092X. Vol.: 174, pp. 570-581. Noviembre 2018. 10.1016/j.solener.2018.09.044 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 Event-Based MPC for defocusing and power production of a parabolic trough plant under power limitation A.J. S´ancheza,∗ , A.J. Gallegoa, J.M. Esca˜nob, E.F. Camachoa aDepartamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain bDepartamento de Ingenier´ıa, Universidad Loyola Andaluc´ıa, Energ´ıa Solar 1. Edificio E, Campus Palmas Altas, 41014 Sevilla, Spain Abstract Optimal operation of a solar plant is generally understood as a tracking of the optimal working temperatures which maximize the net electric power. However, a commercial solar plant may receive a limitation from the Transmission System Operator due to saturation of the electrical grid. In these situations the plant moves to an operation mode in which the objective is not maximum production but compliance with the orders of the Transmission System Operator. The paper proposes an Event-Based Gain Scheduling Generalized Predictive Control strategy for electric power production reference tracking when power limitations are imposed by the Transmission System Operator. Gain Scheduling Generalized Predictive Controllers are proposed to control fourth and third collector defocus in order to prevent heating fluid temperature from exceeding the limits of the manufacturer and therefore, avoid oil degradation. A 50 MW parabolic solar trough plant model has been used to design and validate the strategy. Simulation results are presented showing the advantages of using the proposed strategy. Keywords: Solar parabolic, Model Predictive Control, Collector defocus, Electric power limitation 1. Introduction During the second half of the 70s interest in renewable energies experienced a boost. This happened after the first oil crisis driven by economical factors when oil prices soared. After oil prices decreased, interest in renewable energies also decreased. Due to global warming and with the objective of reducing harmful emissions from conventional fossil power plants, interest in renewable energies has, once again, resurged (Goswami et al., 2000; Blanco and Santigosa, 2017). Currently, the renewable energies with the greatest impact on society are solar, wind and hydraulic, solar energy being the most abundant renewable energy by far. This paper deals with the operation of parabolic solar plants. A parabolic solar plant consists of a field of parabolic-cylinder collectors arranged in loops with a cold oil inlet and hot oil outlet. The collectors focus direct solar radiation on a receiver tube through which oil circulates to heat it and send it to a heat exchanger. The next step is the steam cycle or power cycle where a steam turbine will produce electric energy. Several Concentrating Solar Power (CSP) plants with Parabolic Trough Collectors (PTC) have been built around the world in the last decade. Examples of commercial ∗Corresponding author Tel.: +34 600064314 Email addresses: [email protected] (A.J. S´anchez), [email protected] (A.J. Gallego), [email protected] (J.M. Esca˜no), [email protected] (E.F. Camacho) CSPs currently producing are: 50 MW Andasol solar plants (Solar Millennium AG, 2018), Helios 1/2, 50 MW CSPs (220 hectares, 90 loops) (Helios 1, 2018), Khi Solar One in South Africa (operational since 2016) (Khi Solar One, 2018) and Solana CSP (Arizona, USA) with a gross turbine capacity of 280 MW (777 hectares, 808 loops) (Solana Generating Station, 2018). One of the advantages of CSP plants is the possibility of using thermal energy storage (Camacho et al., 2011; Alva et al., 2017; Sarbu and Sebarchievici, 2018). Generally, this is done by using molten salt tanks (hot and cold) (Solana Generating Station, 2018; Kaxu Solar One, 2018). Generally, the main objective of the control systems in solar trough plants is to maintain the outlet temperature of the field around a desired set-point. Unlike conventional fossil fuel plants where the main source of energy (fuel) can be manipulated, in solar plants the main source of energy is considered a disturbance since the plant controller will have to deal with radiation transients due to clouds. In addition to clouds the daily beam radiation profile cycle is another source concern, as it affects the available solar energy. Research to improve performance in order to optimize solar power plants from the control point of view has been addressed in many ways. In most of the research works, the design of controllers is based on the ACUREX model, a parabolic trough field plant for research and experimental purposes, located in Almer´ıa, Spain (Camacho et al., 2012; Beschi et al., 2014; Lemos et al., 2014; Khoukhi et al., 2015; Alsharkawi and Rossiter, 2017). As stated above, most of the works focus on new opti- 1 mized control methods for temperature set-point tracking, constraint compliance, state estimation, just to name out a few. One of the most widespread applied techniques is Model Predictive Control (MPC). MPC techniques applied to solar plants have shown to be effective, as in Lim´on et al. (2008) where a Robust control of ACUREX is proposed using MPC for tracking. A Neural Network based MPC is presented in Gil et al. (2014). In Alsharkawi and Rossiter (2016), authors developed a linear time-variant state space estimation to design a Dual mode MPC. A different MPC approach is developed in Lima et al. (2016) in which the authors designed a Filtered Dynamic Matrix Control (FDMC). Authors used a filter for the prediction error so that the robustness of the control strategy is ensured. However, these works are focused on the tracking of a temperature set-point, rejection of disturbances, robustness, estimation and stability, just to name out a few. Some of the most important topics in solar plants are costs reduction and the optimization of the plant operation (N.A Engineering, 2008; Blanco and Miller, 2017). Some examples can be found in Montes et al. (2009) where a standard methodology for the economic optimization of the solar multiple in parabolic trough plants is presented. Wittmann et al. (2011) presented a methodology on how to set up an economically optimized bidding strategy at the energy exchange. The objective was to achieve maximum benefit from the production and selling price point of view. In Camacho and Gallego (2013) an optimization of the temperature operating point for solar plants is presented. The optimal temperature set-point is obtained throughout the day depending on the environmental conditions. An optimal turbine inlet pressure of a CSP plant is calculated in Desai and Bandyopadhyay (2015). This works states that the optimal turbine inlet pressure is a weak function of design radiation. However, the optimum value increases with plant size and various modifications of Rankine cycle. In S´anchez et al. (2018), the authors proposed an online non-linear model based optimization to control the inlet valves. The objective is to homogenize the solar field to avoid the loss of electric production due to dirt (different loop efficiencies). However, there are situations in which the plant has to move into an operation mode in which the optimization of the produced power does not make sense. A commercial solar plant may receive commands of power limitation from the Transmission System Operator (TSO). Typically, when the electrical grid is saturated. In these cases, the plant is forced to decrease its electric production and maintain the power set-point determined by the TSO. Therefore, maximum power production no longer makes sense. In this situations the objective is double: fulfilling the TSO power set-point and temperature tracking. The nominal operating point of a commercial plant is generally around the 393 ◦C field outlet temperature (Ibersol 1, 2018; Solaben 2, 2018). The plant will have a time period to reduce its generated electric power to the setpoint determined by the TSO. If the plant does not comply with the determined set-point it would face economic sanctions. To decrease power it is necessary to decrease the oil flow-rate that reaches the heat exchanger where the steam phase begins. Plants with Thermal Energy Storage (TES) are able to deal with this situation, at least for a while, by diverting part of the flow-rate to the TES until these are saturated. Plants that do not have TES cannot cope with this so easily. This work focuses on plants that do not have TES such as Ibersol 1 (2018), Solaben 2 (2018), Guzm´an (2018), Helios 1 (2018). Decreasing the flow-rate increases the outlet temperature. However, if the flow-rate is used to control the power, a new mechanism is necessary to control the field outlet temperature at the nominal operating point. This mechanism is the defocusing of collectors. This is done by augmenting the incidence angle between the solar beam and the normal to the mirror plane and thus the efficiency decreases. Typically, this control is only used in commercial plants to prevent the oil temperature from exceeding a maximum from which the oil begins to degrade. The main motivation of this paper is dealing with the operation of a parabolic trough plant under power limitation. In this work, a novel Event based Gain Scheduling Generalized Predictive Control (EGS-GPC) for electric power reference tracking when a power limitation appears is proposed. A new control strategy for defocusing the fourth and third collectors of each loop is developed in order to control the outlet temperature and to avoid loop temperature from exceeding the limit temperature, since the flow-rate control will be devoted to electric power reference tracking. To design the control strategy a 50 MW parabolic trough plant model will be used instead of the ACUREX model. This paper is organized as follows: In section 2 the 50 MW solar plant model and the mathematical models that have been used are described. In section 3, a flow-rate Gain Scheduling Generalized Predictive Control scheme (GS-GPC) is explained as well as a series Feed-Forward (FF). In section 4, the fourth collector defocus EGS-GPC is developed. In section 5, power EGS-GPC is detailed as well as a GS-GPC for the third collector. Simulation results are presented in section 6. Finally, in section 7, the paper draws to a close with some concluding remarks and future work. 2. Parabolic trough field model Research works have used the ACUREX solar plant model for design and testing. The ACUREX field, located at the Plataforma Solar de Almer´ıa, consists of 480 parabolic trough collectors. The collectors are arranged in 10 loops, each one composed of two rows of 12 modules. The total length of each loop is 172 m, which comprises active parts (142m) and passive parts, i.e. joints and other parts not reached by concentrated radiation (30 m). However, to design, simulate and present the results of the proposed strategy in this paper, it seems reasonable 2 Nomenclature ACross-sectional area of the pipe (m2)tTime (s) CSpecific heat capacity J/(kg◦K)TTemperature (◦C, K) DHydraulic diameter of the pipe (m)TaAmbient temperature GCollector aperture (m)Ti C3Third collector temperature, loop i Hl Global coefficient of thermal loss (W/(m2◦K)) Tin Inlet temperature Ht Coefficient of heat transmission metal-fluid (W/(m2◦K)) Tout Outlet temperature ISolar radiation (W/m2)Tmean Mean temperature between inlet and outlet temperature kThermal conductivity (W/(m·K)) Tlow Field outlet temperature to consider the plant is not saturated Kopt Optical efficiency Thigh Field outlet temperature to consider the plant is saturated LLength of pipeline (m)Tref Temperature reference provided by the GS-GPC controller to the FeedForward noGeometric efficiency Tref−C3 Temperature set-point applied to the 3rdcollector Nu Nusselt number Tref−C4 Temperature set-point applied to the 4thcollector PPower (MW)Tref−sat Temperature set-point for the 4th collector in saturation Pcp Fixed factor (loop geometrical and thermal properties) Tref−nosat Temperature set-point for the 4th collector not in saturation phi Fixed factor TSOL Boolean variable indicating a power limitation arrived Pr Prandtl number xSpace (m) PWref Reference to the Power GS-GPC ∆Tthe thermal difference (◦C) PWset−point Power set-point by TSO βi k Defocus angle, 4th collector, loop i, instant k(deg) PWT SO Boolean variable indicating the plant is on limitation mode βi k−1 Defocus angle, 4th collector, loop i, instant k−1 QOil flow rate (m3/s,m3/h,kg/s)γi k Defocus angle, 3th collector, loop i, instant k qff Computed flow-rate for one loop γi k−1 Defocus angle, 3th collector, loop i, instant k−1 Qff Computed flow-rate for the complete field µDynamic viscosity of the fluid (Pa ·s=N·s/m2 = kg/(m·s)) Qhigh Flow limit to consider the plant is saturated νKinematic viscosity (m2/s) Qlow Flow limit to consider the plant is not saturated ρDensity (kg/m3) QP W Flow-rate computed by the Power GS-GPC µrankine Rankine cycle efficiency Re Reynolds number µexchanger Heat exchanger efficiency STotal reflective surface (m2)µparasitics Parasitic effects efficiency 3 to use a plant model of approximate dimensions and electric production as existing commercial plants. The chosen plant model is a 50 MW of electric production without TES (Helios 1, 2018; Guzm´an, 2018; Ibersol 1, 2018). Plants that have thermal storage may, for a time, deal with power limitations by diverting part of the flow-rate to the TES. Plants that do not have thermal storage cannot divert flow-rate at any time. In this paper, two types of mathematical models are proposed for the plant: a concentrated parameter model will be used in the FF module of the control strategy, and a distributed parameter model will be used for simulation purposes. Both models have been obtained through tests and validations conducted at the plant and have been used by many authors. For a complete description of the plant and the modeling procedure, refer to Carmona (1985), Camacho et al. (1997). 2.1. Parabolic trough field The solar field of a 50 MW plant is, unlike the ACUREX field, extensive. The total length of each loop in ACUREX, as previously mentioned, is 172 m, whereas in a 50 MW commercial plant the length is around 600 m with 4 collectors each loop (Extresol 1, 2018; Solaben 2, 2018; Guzm´an, 2018). The number of loops is another important factor. In commercial plants, the number of loops is 80-100 loops (Ibersol 1, 2018; Helios 1, 2018; Solaben 2, 2018; Guzm´an, 2018; Majadas I, 2018; Palma del R´ıo I, 2018). Power cycle b b b b b b b b b b b b 1st Collector 2nd Collector 3rd Collector 4th Collector 2nd Collector 1st Collector 3rd Collector4th Collector . Figure 1 Parabolic trough plant general schematic The position of the field of a commercial solar plant, for efficiency reasons, is north-south, unlike ACUREX, which is east-west. The plant model that will be used for the rest of the paper consists of 90 600 meter loops where each collector is 150 m long. In Fig. 1 a schematic parabolic trough plant is shown. 2.2. Collectors, receiver tube and heat transfer fluid For the simulation model of this paper the collector EuroTrough ET150 (Rohani et al., 2017; Andasol 1, 2018; System Advisor Model (SAM). NREL, 2018) is selected. In order to simulate the field, it is necessary to describe the collector in terms of parameters. In Table 1 the main parameters of the EuroTrough ET150 collector are shown (Geyer et al., 2002; Kearney, 2007; System Advisor Model (SAM). NREL, 2018). 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 The receiver tube used in the model is the Schott PTR70 since it is one of the most used in commercial plants (Extresol 1, 2018; Ibersol 1, 2018; Andasol 1, 2018). The reader should refer to SCHOTT Solar CSP GmbH (2018) for a complete description of Schott PTR70. The Heat Transfer Fluid (HTF) is used to generate the necessary steam for the steam power cycle. Therminol Vp1 and DOWTHERM  HTFs are the most used fluids in commercial solar plants. In this paper, Therminol Vp1 is used. It is a synthetic thermal oil with a Biphenyl oxide/Diphenyl composition. This HTF can operate at temperatures between 12 ◦C and 400 ◦C (Therminol VP1 HTF, 2018). Above 400 ◦C the fluid degrades. Fluid density (ρf) and specific heat capacity (Cf) are temperature dependent and can be obtained through equations (1) and (2). The reader should refer to the manufacturer, Therminol VP1 HTF (2018), for more parameters approximations. 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) The last approximation is the maximum amount of HTF needed to generate 50 MW. This calculation is performed under nominal operating conditions, at a field outlet temperature of 393 ◦C. To calculate the maximum flowrate to produce 50 MW, equation (3) is used. The power generation is proportional to the oil flow-rate and the temperature difference that occurs in the heat exchanger. This difference is around 90-100 ◦C approximately in current plants, so the field inlet temperature is 293 ◦C approximately at nominal operation. Q=P·106 ∆T·Cf·µrankine ·µexchanger ·µparasitics (3) 4 The efficiency of the Rankine cycle in current 50 MW solar plants goes up to 38.1 % (Andasol 1, 2018; Extresol 1, 2018). Parasitic effects and heat exchanger efficiencies reduce the generated power and therefore there is a gross and a net power (System Advisor Model (SAM). NREL, 2018). Table 2 shows the selected values for these efficiencies. To generate approximately 50 MW, the maximum flow-rate calculated for the simulation model is 655 kg/s (3350 m3/h) approximately at 393 ◦C. Table 2 Flow-rate parameter values. Parameter ∆T µrankine µexchanger µparasitics Value 95◦C 38.1% 90% 90% 2.3. Distributed parameter model The dynamics of the distributed solar collector field are described by the following system of partial differential equations (PDE) describing the energy balance (Carmona, 1985; Camacho et al., 1997): ρmCmAm ∂Tm ∂t =IKoptnoG−HlG(Tm−Ta)−LHt(Tm−Tf) (4a) ρfCfAf ∂Tf ∂t +ρfCfQ∂Tf ∂x =LHt(Tm−Tf) (4b) where the subindex mrefers to the metal and frefers to the fluid. The geometric efficiency depends on hourly angle, solar hour, declination, day of the year, local latitude and collector dimensions. The density ρ, specific heat Cand coefficients Htand Hldepend on fluid temperature. The coefficient of heat transmission depends on temperature and oil flow (Camacho et al., 1997). Hlcan be obtained from Burkholder et al. (2007), L¨upfert et al. (2008) while Htcan be calculated with the equations (5) for a turbulent flow-rate inside a pipeline where phi is a fixed factor and Qis the flow-rate in m3/s. Re =Q·D/(ν·A) (5a) Pr =Cf·µ/k (5b) phi = 1.023 (5c) Nu = 0.025 ·(Re0.79)·(Pr0.42)·phi (5d) Ht=Nu ·k/D (5e) The model is discretized in the longitudinal dimension of the tube so the dynamics of each loop can be simulated as a chain of sub-models. A segment length of 1.98 meter has been chosen for the implementation of the simulation model (300 segments per loop). For the ACUREX plant, the chosen number of segments is typically 172 with a 1 meter length segment. This amount of segments provides good performance for the simulation of the field by using the distributed parameters model. However, ACUREX loop length is 172 meter while in this case a 600 meter loop length is considered for the 50 MW plant simulation. A simulation with 300 segments of approximately 2 meters length each also provides good simulation performance and reduces the computation time. Number of segments can always be increased but with a higher computational cost. Reducing too much the number of segments will entail a loss in the nonlinear dynamic resolution of the field simulation. 2.4. Concentrated parameter model The concentrated parameter model provides a general description of the whole field. The variation in the internal energy of the fluid can be described by the equation: CdTout dt =KoptnoSI −QPcp(Tout −Tin)−Hl(Tmean −Ta) (6) This model will be used to implement a series Feed- Forward controller. 3. Flow-Rate Model Predictive Control Scheme This section describes the control scheme used to perform the outlet temperature tracking. A GS-GPC is used for the flow-rate control scheme and a series FF is used for disturbance rejection. The series FF has proved to be very effective at rejecting measurable disturbances affecting the solar field. (Camacho et al., 1997). 3.1. Generalized Predictive Control The GPC algorithm is based on the following singleinput single-output model (Camacho and Bordons, 2007): A(z−1)yk=z−dB(z−1)uk−1+C(z−1) ∆ek(7) where ukand ykare the control and output sequences of the plant, ekis a zero mean white noise term and ∆ is the integrator operator. A, B and Care polynomials in the backward shift operator z−1: A(z−1) = 1 + a1z−1+... +anaz−na B(z−1) = b0+b1z−1+... +bnbz−nb C(z−1) = 1 + c1z−1+... +cncz−nc where dis the dead time of the system and ∆ is the operator 1−z−1. This model is known as a Controller Auto- Regressive Integrated Moving-Average (CARIMA) model. Consider a multistage cost function of the form: 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) 5 where ˆy(k+j|k) is an optimum j step ahead prediction of the system output, N1and N2are the minimum and maximum costing horizons, Nuis the control horizon, δ(j) and λ(j) are weighting sequences and w(k+j) is the future reference trajectory. The aim of GPC is to minimise J(N1, N2, Nu) in order to compute a future sequence of control actions u(k), u(k+ 1), ... that drives the future plant output y(k+j) close towards w(k+j). Hence given a CARIMA plant model and suitable cost function, the minimum of the cost function can be obtained by setting the gradient of Jequal to zero and solving the control sequence ∆uby the following equation (Camacho and Bordons, 2007): ∆u= (GGT+λI)−1GT(w−f) (9) where matrix Gcontains the step response coefficients of the forced response model (Camacho et al., 2012), Iis the eye matrix, fis the free response of the plant, wis the future reference trajectory vector and λis the control weighting vector (Camacho and Bordons, 2007). 3.2. Gain Scheduling The design of the GS-GPC is described in Camacho et al. (1994, 1997). GS-GPC controller has demonstrated to have a very good behaviour not only in respect to setpoint tracking but also in disturbance rejection capabilities (Camacho et al., 1997). Depending on the point at which the system operates, the GS-GPC feedback gain is adjusted in order to compensate variation in the plant response under different working conditions. In a solar trough plant, the dynamic is mainly dictated by the oil flow. Four oil flow levels covering the operation range of the plant are used (1494, 1908, 2322 and 2736 m3/h). Since the linear models correspond to the FF plus the plant model, the input is the temperature reference from GS-GPC (input to FF) and the output is the outlet oil temperature (output of the plant) (◦C). A suitable identification of the model parameters at four different set-points for the oil flow (1494, 1908, 2322 and 2736 m3/h) will be used. Step responses are shown in Fig. 2. The GS-GPC has been designed assuming that the parameters of all loops are the same. It is also assumed that the field is balanced, therefore the flow going into each of the loops is the same. A global dynamic model based only on one loop is used to design the GS-GPC. 3.3. Feed-Forward Control The use of a series Feed-Forward controller action has proved to be very effective at rejecting solar radiation disturbances when using linear controllers. Moreover, it significantly contributes to preserving the validity of the assumed linear description of the plant over its operation range. The FF input signal is a temperature set-point from GS-GPC control, while the control output is the oil flow-rate qff which is computed by the lumped parameter description (Camacho et al., 1992): qff =KoptnoSI −SHl(Tmean −Ta) Pcp(Tref −Tin)(10) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time(s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ∆T (ºC) Solar Field Step Response 1494 m3/h 1908 m3/h 2322 m3/h 2736 m3/h Figure 2 Solar field step response of linear models. Output increments when unitary steps are applied to the input. Pcp can be approximated by 1.868 ×106kJ/◦Cand S takes the value of 3427m2(Camacho et al., 1997, Geyer et al., 2002). The control scheme, see Fig. 3, works as follows: The GS-GPC receives the temperature set-point for the solar field and the current mean temperature and computes a virtual reference temperature, Tref , for the FF. The FF computes oil flow taking into account the virtual reference and the measured disturbances to track the desired set-point. Since the GS-GPC + FF scheme is considering a global model by one loop, the calculated flow-rate is for one loop. The GS-GPC + FF scheme is presented in Fig. 3 where Qff represents the global flowrate for the complete field and Qis the measured flow-rate. Solar Field GS-GPC Tin,Ta,Ieff Tout Tref Qff Tset−point Tout FF . Flow Q Figure 3 GS-GPC + FeedForward control scheme. Let’s assume that the model is perfect. In this case the virtual reference would be equal to the desired set-point in steady-state. Since the model is not perfect, the virtual reference will not be equal to the desired set-point. Fig. 4 shows a simulation of the control scheme with a sampling time of Ts = 30s(Camacho et al., 1997), where all the loops have the same parameter values. To compute the estimation of the flow to be applied to the whole solar 6 field, the temperature to be controlled is the mean of all the loops. Geometric efficiency nois considered to be the same for all the loops and is computed by the equations described in (Camacho et al., 1997). Optical efficiency Kopt used in the series FF controller is the mean of the estimation obtained for each loop. 9 10 11 12 13 14 15 16 17 18 Time (h) 320 340 360 380 400 Temperature (ºC) Tref Tout Tin + 50 MaxTemp 400 ºC 9 10 11 12 13 14 15 16 17 18 Time (h) 0 200 400 600 800 Irradiance(W/m2) 0 200 400 600 Flow(m3/h) I Q/10 Qmax/10 Figure 4 GS-GPC + FF tracking results (medium irradiance) 4. Fourth collector defocus GPC As presented in Fig. 4, the GS-GPC in series with a FF correctly tracks the field outlet temperature set-point because this simulation corresponds to a medium irradiance day. Flow-rate in this simulation is close to the maximum. In a high irradiance day flow-rate control is not enough to control the outlet temperature and, thus, temperature will soar up. Fig. 5 shows this scenario. It can be observed that the flow-rate is at its maximum and the outlet temperature starts to increase. Generally, parabolic trough plants have safety mechanisms to prevent the outlet temperature of a loop exceeding the allowed maximum. This security procedure is carried out by defocusing the collectors, causing the loop to cool down due to the decrease in the amount of effective radiation received by the tube. The defocusing of the fourth collector is, generally, applied to simply prevent the oil temperature from exceeding a maximum temperature of 400 ◦C. This control action is carried out by modifying the angle of the collector. Moreover, fourth collector defocus of a loop is independent of the flow rate and the defocus actions of other loops since each loop is different and their parameters may vary amongst themselves so that not all loops will be at the same temperature. In this paper an Event based GS-GPC is proposed for fourth collector defocus control. An EGS-GPC per loop is needed and it is also necessary to apply a temperature set-point. 9 9.5 10 10.5 11 11.5 12 12.5 13 Time (h) 300 350 400 450 Temperature (ºC) Tref Tout Tin + 50 MaxTemp 400ºC 9 9.5 10 10.5 11 11.5 12 12.5 13 Time (h) 0 200 400 600 800 1000 Irradiance(W/m2) 0 200 400 600 Flow(m3/h) I Q/10 Qmax/10 Figure 5 GS-GPC + FF. Flow-Rate at maximum due to high irradiance leading to temperature limit violation. A function that relates the defocus angle to collector efficiency is needed. The approximation of this function to design the GS-GPC is presented in Fig. 6, (Goswami et al., 2000). Since the function is non-linear, the gainscheduling will be designed at 9 different points of defocus angle (0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 and 4.5 degrees). From a certain defocus angle, efficiency begins to decrease rapidly, since rays no longer reach the tube. Moreover, the plant responds differently depending on the flow-rate, so defocus linear models will not be the same for all flow-rates. To improve the performance of collector defocus GS-GPC, 9 linear models are obtained for each of the flow-rate points where Flow GS-GPC has been designed (1494, 1908, 2322 and 2736 m3/h) . The linear models obtained are presented in Figs. 7, 8, 9 and 10. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Collector Defocus Angle (º) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Collector Defocus Efficiency Collector Defocus Curve Figure 6 Efficiency - Defocus angle curve. 7 0 100 200 300 400 500 600 700 800 900 Time(s) -15 -10 -5 0 ∆T (ºC) Collector Step Response (Low Flow-Rate) 0.5 º 1 º 1.5 º 2 º 2.5 º 3 º 3.5 º 4 º 4.5 º Figure 7 Low flow-rate collector step response of linear models. 0 100 200 300 400 500 600 700 800 900 Time(s) -12 -10 -8 -6 -4 -2 0 ∆T (ºC) Collector Step Response (Low-Medium Flow-Rate) 0.5 º 1 º 1.5 º 2 º 2.5 º 3 º 3.5 º 4 º 4.5 º Figure 8 Low-Medium flow-rate collector step response of linear models. 0 100 200 300 400 500 600 700 800 900 Time(s) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 ∆T (ºC) Collector Step Response (Medium-High Flow-Rate) 0.5 º 1 º 1.5 º 2 º 2.5 º 3 º 3.5 º 4 º 4.5 º Figure 9 Medium-High flow-rate collector step response of linear models. 0 100 200 300 400 500 600 700 800 900 Time(s) -8 -7 -6 -5 -4 -3 -2 -1 0 ∆T (ºC) Collector Step Response (High Flow-Rate) 0.5 º 1 º 1.5 º 2 º 2.5 º 3 º 3.5 º 4 º 4.5 º Figure 10 High flow-rate collector step response of linear models. For the temperature set-point, it is easier to apply a temperature set-point of 393 ◦C to each loop given that it is the nominal working temperature. However, since the loops may vary in reflectivity, tube efficiency and form factor, among others, some loops may be colder than others. In the hotter loops, the defocus would be activated for a temperature reference of 393 ◦C and the other loops would be at a slightly lower temperature. This could cause a conflict with flow-rate control at certain times. To avoid this, an event based heuristic is applied to send different temperature set-points to the GS-GPC. This heuristic is to detect an event based on flow-rate, outlet temperature, power limitation and to apply the appropriate set-point: 1. If P WT SO(k) or (Q(k)> Qhigh &Tout(k)> Thigh))  Temperature set-point = Tref−sat 2. Else  If Q<Qlow or Tout < Tlow –Temperature set-point = Tref−nosat 3. If P WT SO(k−1) & !P WT SO(k)  Temperature set-point = Tref−nosat In this work, the chosen values for previous set-points are: Qhigh = 3150 m3/h,Qlow = 3060 m3/h,Thigh = 395 ◦C, Tlow = 391 ◦C, Tref−nosat = 396 ◦C and Tref−sat = 393 ◦C. Rules work by detecting the state of the plant. Rule (1) detects if there is a power limitation or if the plant is saturated. This occurs when the flow-rate is close to the maximum by checking it with Qhigh and the outlet temperature is over Thigh. If the plant is saturated or in power limitation Tref−sat is applied to the defocus GS-GPC as temperature set-point. Rule (2) detects if the plant is not saturated. This will happen when the flow-rate is far from the maximum, Qlow, or if the outlet temperature is below Tlow. Since flow-rate is low or the field outlet temperature is far from the maximum allowed, the temperature setpoint applied to defocus the loops is higher, Tref−nosat. The last rule detects if the power limitation was removed 8 9 10 11 12 13 14 15 16 17 18 19 Time (h) 320 340 360 380 400 Temperature (ºC) TC4-ref TC3 L1 TC3 L7 TC3 L11 TC3 L23 TC3 L33 TC3 L47 TC3 L61 TC3 L74 TC3 L81 TC3 L89 MaxTemp 400ºC 9 10 11 12 13 14 15 16 17 18 19 Time (h) 0 1 2 3 4 5 Defocus Angle (deg) C3 Def L1 C3 Def L7 C3 Def L11 C3 Def L23 C3 Def L33 C3 Def L47 C3 Def L61 C3 Def L74 C3 Def L81 C3 Def L89 Figure 29 Flow GS-GPC + FF + C4/C3 EGS-GPC + PW EGSGPC, 40 M W TSO limitation at 11:40am (60 min ramp) and TSO limitation off at 15:45pm (30 min ramp). Top plot: loop fluid temperatures. Bottom plot: Third collector defocus angle control actions. 7. Conclusions It is reasonable to assume that the optimization of a solar plant concept is directly related to terms of electric power generation. However, on some occasions, a commercial solar plant may receive a power limitation from the TSO. These commands make the plant move into an operation mode in which the objective is not maximum production. In these cases, the plant is forced to decrease its electric production and maintain the power set-point commanded by the TSO. Therefore a double objective problem arises: (1) temperature tracking and (2) generated power set-point tracking. The plant will have a time period to reduce its generated electric power to the set-point determined by the TSO. In this mode, the flow-rate should be decreased until the power set-point is achieved but at the cost of increasing the outlet temperature. To control the outlet temperature, a secondary control is needed. In this paper, an Event-Based Gain Scheduling Generalized Predictive Control (EGS-GPC) for electric power reference tracking when power limitations appear, has been proposed. In order to control the loops/field outlet temperature and to avoid exceeding oil degradation limits, an event based control strategy for defocusing the fourth and third collectors of each loop is developed with set-point changes depending on events. The linear models and control strategy have been obtained and designed by means of a 50 MW parabolic trough plant model instead of the ACUREX model. Results show the proposed MPC strategy achieves power set-points determined by the TSO in the set time periods. The Power and Defocus EGS-GPCs are capable of performing good electric power tracking and nominal set-point tracking of loop outlet temperature avoiding oil degradation. Acknowledgments The authors would like to acknowledge the European Research Council for funding this work under Advanced Research Grant OCONTSOLAR (789051). References Alsharkawi, A., Rossiter, J. A., 2016. Dual mode mpc for a concentrated solar thermal power plant. 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