Temperature homogenization of a solar trough field for performance improvement

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Solar Energy, Vol. 165 on May 2018, available at: https://doi.org/10.1016/j.solener.2018.03.001 © 2018 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Temperature homogenization of a solar trough field for performance improvement 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 solar plant is a challenge due to the multiple disturbances affecting the plant. Research in optimal temperature set-point tracking is extensive whereas research concerning optimal power production under disturbances is not. A solar plant has to deal not only with temperature and radiation disturbances but also with dirt accumulated on the collectors which lead to a great disparity in collectors reflectivity. Cleaner collectors produce higher output temperature because of the higher reflectivity. In fact the temperature in this collector may be so high that they have to be defocused with the corresponding energy losses. The paradox is that the most efficient loops may be the one collecting less energy because of defocusing. This paper proposes a new solution based in the manipulation of the loops input valves in order to homogenize the temperatures of the different loops and avoid defocusing. The paper presents an original non linear model based optimization to homogenize the loop temperatures by manipulating the inlet valves of the loops. The nonlinear model is based on a distributed parameters model. In order to perform the optimization, temperature profiles of the loops and its reflectivities are needed. These are obtained by means of a Classification and Regression Trees (CARTs) trained with the full distributed parameters model. Simulations are carried out using a model of the Plataforma Solar de Almer´ıa (PSA) solar trough plant to show the results of the proposed control scheme, temperature homogenization and production benefit. Keywords: Online nonlinear optimization, Model Predictive Control, Solar trough, Regression Trees 1. Introduction The use of renewable energies had a boost in the second half of the 70s after the first oil crisis although the main reason for the interest in renewable energies was economic. Currently, and driven by global warming, there is a global awareness to search for inexhaustible sources of energy. Solar is the most abundant source of renewable energy in the world. In the last decade, several Concentrated Solar Plant (CSP) have been built around the world. Examples of commercial CSPs currently producing are: Solaben 50 MW CSPs (220 hectares, 90 loops) (Solaben 2, 2017), 50MW Andasol solar plants (Solar Millennium AG, 2017), Solana CSP (Arizona, USA) with a gross turbine capacity of 280 MW and molten salts thermal energy storage (777 hectares, 808 loops) (Solana Generating Station, 2017) and Khi Solar One in South Africa (operational since 2016) (Khi Solar One, 2017). The possibility of using thermal energy storage is an added advantage of CSP over other renewable technologies (Wittmann et al., 2008; Gil et al., 2010; Camacho et al., 2011). Thermal energy storage is ∗Corresponding author Tel.: +34 954487347 Email addresses: [email protected] (A.J. S´anchez), [email protected] (A.J. Gallego), [email protected] (J.M. Esca˜no), [email protected] (E.F. Camacho) usually done using molten salts which are currently used in Solana and in Kaxu 100MW CSP located in Pofadder, South Africa (Kaxu Solar One, 2017). 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. While in conventional fossil fuel power generating processes the main source of energy (the fuel) can be manipulated as the main control variable. Furthermore, the main source of energy itself is considered as a disturbance since the plant controller will have to deal with radiation transients due to clouds. Research to improve the performance of solar power plants from the control and optimization point of view has been addressed in many ways. Most of them have been implemented and tested at the ACUREX plant, a parabolic trough field plant, for research and experimentation purposes, located in Almer´ıa, Spain (Camacho et al., 1992; Silva et al., 1997; Igreja et al., 2003; Camacho et al., 2007a,b). Research in solar fields control is extensive and covers both linear and nonlinear control methods for tracking the outlet temperature of the field. Within the Model Predictive Control (MPC) it can be found, inter alia, an adaptive MPC to control a parabolic solar trough plant in Gallego and Camacho (2012), a Gain- Scheduled Control of solar power plant is design in Johansen et al. (2000), Alsharkawi and Rossiter (2017), and Preprint submitted to Elsevier February 27, 2018 in Gallego et al. (2013) an Observer-based Model Predictive Control is developed. Lima et al. (2016) presents a filtered Dynamic Matrix Control (FDMC) where a filter is used for the prediction error so that the robustness of the control strategy is ensured applied to a solar collector field and Alsharkawi and Rossiter (2016) proposes a Dual mode MPC for a concentrated solar thermal power plant based on an estimated linear time-invariant state space model around a nominal operating point. Within the nonlinear MPC, in Andrade et al. (2013) a non linear MPC with robustness of stability ensured by using a Lyapunov criterion is presented and in Gil et al. (2014) a Neural Network based MPC and Kalman Filter weighting computation is proposed for a distributed collector field. ACUREX solar field can be considered as one single collector loop, though is formed by 10 loops, to obtain an overall dynamic model of the complete system in order to design a controller for tracking a set-point on the outlet temperature of the field (which will be the weighted average temperature of the 10 loops). In Cirre et al. (2009) a multilayer control strategy is proposed for solar trough plants which uses a lumped description of the solar field to solve the temperature tracking problem. When developing a control strategy based on an overall dynamic model, either lumped or distributed parameters model, certain parameters can be considered constant for all loops, such as reflectivity, efficiency of the receiver tubes or the shape factor of the collector. It is possible to approximate the full dynamics of the field by a single loop when considering constant these parameters. Even if these parameters are not the same in all loops, i.e. reflectivity, a welldesigned controller can continue performing a good field outlet temperature tracking at the cost of having some loops at higher temperatures than others. For example in Camacho and Gallego (2013), a procedure to obtain the best working temperature is presented. In this work, the ACUREX solar field is considered a lumped model for the calculation of the optimal set-point. If the plant is required to work at a high average temperature, which usually happens with high levels of solar radiation, a situation where some loops work at a much higher temperature than the rest of the field may appear. This problem is produced mainly by two facts: a) these loops are more efficient than the rest of the field and b) the input valves of the solar field are not well balanced so that the flow is not equally distributed (loops with low flow are hotter than loops with high flow levels). If the temperature of hotter loops exceeds the safe limits of the thermal fluid, the collectors have to be defocused to reduce the fluid temperature leading to energy losses. The main motivation of this work is the research of new control strategies for performance improvement, in terms of power production, in solar trough plants under disturbances. In this work, an online nonlinear model based optimization to control the loop’s inlet valves in parallel with a linear global dynamic model based Gain-Scheduling Generalized Predictive Control (GS-GPC) for the tracking of the outlet temperature of the field is proposed. The global GS-GPC controller, manipulating the flowrate, is responsible for the control of the outlet temperature of the field while the nonlinear optimization will include a nonlinear distributed parameter simplified model of each of the loops to control the aperture of the inlet valves in order to obtain a field temperature homogenization to achieve a power production improvement when the solar field is not well balanced. This paper is organized as follows: In section 2 the ACUREX field and mathematical models that have been used are described. In section 3, a GS-GPC control scheme is explained as well as a Feed-Forward in series. Temperature profiles estimations using CARTs and the proposed nonlinear optimization to obtain a temperature homogenization of the field is described in section 4. Simulation results are presented in section 5. Finally, in section 6, the paper draws to a close with some concluding remarks and future work. 2. ACUREX solar trough field 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 172m, which comprises active parts (142m) and passive parts, i.e. joints and other parts not reached by concentrated radiation (30m). Figure 1 Acurex Field schematic. In this paper two types of model are proposed for the plant: a concentrated parameter model will be used in the feed-forward module of the control loop, 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. A schematic of the plant is shown in Fig. 1. For a complete description of the plant and the modeling procedure, refer to Carmona (1985), Camacho et al. (1997). 2.1. Distributed Parameters Model The dynamics of the distributed solar collector field are described by the following system of partial differential 2 equations (PDE) describing the energy balance (Carmona, 1985; Camacho et al., 1997): ρmCmAm ∂Tm ∂t =IKoptnoG−HlG(Tm−Ta)−LHt(Tm−Tf) (1a) ρfCfAf ∂Tf ∂t +ρfCf˙q∂Tf ∂x =LHt(Tm−Tf) (1b) where the subindex mrefers to the metal and frefers to the fluid. The model parameters and units are shown in Table 1. Table 1 Parameters description. Symbol Description Unit tTime s xSpace m ρDensity kg/m3 CSpecific heat capacity J/(kg◦C) ACross sectional area m2 T(x, y) Temperature ◦C Ta(t) Ambient Temperature ◦C q(t) Oil flow rate m3/s I(t) Solar radiation W/m2 noGeometric efficiency Unitless Kopt Optical efficiency Unitless GCollector aperture m HlGlobal coefficient of thermal loss W/(m2◦C) HtCoefficient of heat transmission metal-fluid W/(m2◦C) LLength of pipeline m 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 coefficient of heat transmission Htdepend on fluid temperature. The coefficient of heat transmission depends on temperature and oil flow (Camacho et al., 1997). The model is then 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 meter has been chosen for the implementation of the simulation model (172 segments per loop). 2.2. Concentrated Parameters 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(Tm−Ta) (2) where Cis the thermal capacity of the loop, Kopt is the optical efficiency which takes into account elements such as reflectivity and absorptance. The geometric efficiency nois determined by the position of the mirrors respect the radiation beam vector. The factor Pcp takes into account some geometrical and thermal properties of the loop. Tin is the inlet temperature, Hlis the global thermal losses coefficient, Tmis the mean temperature between inlet and outlet temperature, and Tais the ambient temperature. The coefficient Hlcan be approximated by 1.05kW/◦C, Pcp by 1.924 ×106kJ/◦C,Cby 2267kW/◦C, and total reflective surface Stakes the value of 2674m2. These values have been obtained based on data relative to measures on the plant operating at different set-points by multiple regression analysis (Camacho et al., 1997). 3. Model Predictive Control Scheme This section describes the control scheme used to perform the outlet temperature tracking. A GS-GPC is used because it is one of the MPC strategies which performed best on the ACUREX solar trough plant. A Feed-Forward is used for disturbance rejection. The Feed-Forward series 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(3) where u(t) and y(t) are the control and output sequences of the plant and e(t) is a zero mean white noise term. A, B and Care polynomials in the backward shift operator z−1: A(z−1) = 1 + a1q−1+... +anaq−na B(z−1) = b0+b1q−1+... +bnbq−nb C(z−1) = 1 + c1q−1+... +cncq−nc where dis the dead time of the system and ∆ is the operator 1−q−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(t+j|t)−w(t+j)]2 + Nu X j=1 λ(j)[∆u(t+j−1)]2 (4) where ˆy(t+j|t) 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(t+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(t), u(t+ 1), ... that drives the future plant output y(t+j) close towards w(t+j). 3 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 for the control sequence ∆uthrough the following equation (Camacho and Bordons, 2007): ∆u= (GGT+λI)−1GT(w−f) (5) 3.2. Gain Scheduling The design of the GS-GPC is described in Camacho et al. (1994), and thus far, is one of the best performing control schemes on the ACUREX field. Depending on the point at which the system operates, the GS-GPC feedback gain is adjusted in order to compensate the variation in the plant response at 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 (2.8,5.6,7.9 and 9.8 l/s). 0 200 400 600 800 1000 1200 1400 1600 1800 Time(s) 0 0.2 0.4 0.6 0.8 1 1.2 Amplitude Step Response 2.8 l/s 5.6 l/s 7.9 l/s 9.8 l/s Figure 2 Step response of linear models. A suitable identification of the model parameters at four different set-points for the oil flow (2.8, 5.6, 7.9 and 9.8 l/s) 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 across its operation range. The Feed-Forward input signal is a temperature setpoint from the GS-GPC control while control output is the oil flow qff which is computed by the lumped parameter description (Camacho et al., 1992): qff =KoptnoSI −SHl(Tm−Ta) qPcp(Tref −Tin)(6) where Tref is the temperature reference provided by the GPC controller. 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 Feed-Forward. The Feed-Forward, computes an oil flow taking into account the virtual reference and the measured disturbances to track the desired set-point. Figure 3 GS-GPC + FeedForward control scheme. If the model were perfect, the virtual reference would be equal to the desired set-point in steady-state. A description of the other parameters can be found in Section II. In Fig. 4 a simulation of the control scheme with a sampling time of Ts = 39s(Camacho et al., 1997), when all the loops have the same parameters values, is shown. To compute the estimation of the flow to be applied to the whole solar field, the temperature to be controlled is the mean of all the loops. The geometric efficiency nois considered to be the same for all the loops and is computed by the equations described in (Camacho et al., 1997). The optical efficiency Kopt used in the Series Feed-Forward controller is the mean of the estimation obtained for each loop. Time(local hour) 11 12 13 14 15 16 17 18 Temperature (ºC) 220 240 260 280 300 Tref Tout Time(local hour) 11 12 13 14 15 16 17 18 Flowrate (l/s) 2 4 6 8 10 12 14 q Time(local hour) 11 12 13 14 15 16 17 18 I (W/m2) 0 500 1000 I Figure 4 GS-GPC+FeedForward outlet temperature tracking results. 4 4. Nonlinear Optimization for Temperature Homogenization As presented in Fig. 4 the GS-GPC in series with a Feed-Forward tracks properly the field outlet temperature set-point. Though each loop will have different parameters values, leading to slightly different dynamic behaviour. Some loops will be dirtier than others or the efficiency of the tubes will be different due to manufacturing faults or degradation. In Fig. 5 a simulation of what would happen when applying the GS-GPC controller when the loops have different reflectivities is shown. It can be observed that even if the parameters of the loops differ, as the reflectivity, the GS-GPC is able to perform a good temperature tracking but at the cost of having some loops at higher temperatures than others. This is not critical when working at medium temperatures, but when the setpoint of the outlet temperature of the field is within the limit allowed before oil degradation, due to rankine cycle (Goswami et al., 2000) as in commercial plants, some of the loops could be at temperatures beyond the limit. 11 12 13 14 15 16 17 Time (local hour) 260 270 280 290 300 Temperature (ºC) Tout L1 Tout L2 Tout L3 Tout L4 Tout L5 Tout L6 Tout L7 Tout L8 Tout L9 Tout L10 Set-Point Tout Field 11 12 13 14 15 16 17 Time (local hour) 0 500 1000 Irradiance (W/m2) 0 5 10 Oil Flow (l/s) Figure 5 GS-GPC+Feedforward outlet temperature tracking and Loops outlet temperatures. Loops Reflectivities = [0.65, 0.67, 0.68, 0.7, 0.715, 0.73, 0.75, 0.77, 0.79, 0.8] Generally, parabolic trough plants have safety mechanisms to prevent the outlet temperature of a loop exceeds the allowed maximum. This security procedure is carried out by defocusing the collectors causing the loop to cool due to the decreasing of the amount of effective radiation received in the tube. Defocusing will lead to production losses. To avoid defocusing a homogenization of the field is needed. To perform the field homogenization an online nonlinear model based optimization, in which the manipulated variables are the apertures of the valves that perform the flow distribution in each of the loops see Fig. 1, is developed. The nonlinear model used for the optimization is a simplification of the distributed parameters model due to computational costs. The use of the nonlinear model gives the possibility to adjust all the time variant parameters according to the operating zone unlike the linear models used in the GS-GPC more inaccurate. For the optimization, to calculate the apertures of the valves, an estimation of the reflectivity of each of the loops is needed and for it, an estimation of the temperatures along the length of the loops to feed the nonlinear model is also needed. The following subsections detail each of the elements that form the proposed nonlinear optimization. 4.1. CART Loops Temperatures Estimation This methodology is a non-parametric statistical classification method which uses historical data to construct so-called decision trees to solve classification and regression problems. The method was developed by Breiman et al. (Breiman et al., 1984) in order to build a decision tree which describes one response variable (univariate CART). It can select from a large number of explanatory variables x, those that are most important in determining the response variable yto be explained. Given an input X= [x1, ..., xN] it will produce an output ¯ Ythat estimates some random variables of interest Y, stochastically related to X. Generally, CARTs are used for classification of binary or categorical (discrete) variables. In those cases, the result is a classification tree and the output of the tree is a class (discrete value). When the variable is continuous the result obtained is a regression tree. Instead of classes there are response vectors Y, which represents the response values for each observation in variable matrix X. Examples of classification trees research can be found in (Malinowska, 2014), where CART methodology is used for the prognosis of the concrete building damage, a Multivariate data analysis and modeling using CART is proposed in (Siciliano and Mola, 2000) and an MPC scheme for freeway traffic systems based CART theory is develop in (Oleari et al., 2015). Classification trees are built in accordance with splitting rule, the rule that performs the splitting of the learning sample (data-set) into smaller parts. Each time, dataset has to be divided into two parts with maximum homogeneity, see Fig. 6. Each split generates a node in the tree. Each split is defined by a simple rule based on a single explanatory variable. In Fig. 6 NPdenotes the parent node. NCL and NCR are respectively the left and right child nodes. xjis the variable j; and xR jis best splitting value of variable xjbeing PLand PRthe probabilities of the left and right nodes. Trees are grown by selecting the splits in such a way that the so-called homogeneity and the impurity of the response variable within each node is maximized and minimized respectively. The maximum homogeneity of child nodes is defined by the impurity function i(N). 5 Figure 6 Classification and Regression Tree node splitting. Since the impurity of the parent node NPis constant for any of the possible splits xj≤xR j,j= 1, ..., M, the maximum homogeneity of left and right child nodes will be equivalent to the maximization of the change of the impurity function. Should the reader refer to (Breiman et al., 1984) for a complete explanation on how these are obtained by splitting, maximization and minimization of the homogeneity and impurity functions. The CART steps can be summarized as follows: 1. Assign all objects to root node and split each explanatory variable at all its possible split points. 2. For each split point, split the parent node into two child nodes and select the variable and split point with the highest reduction of impurity. 3. Perform the split of the parent node into the two child nodes according to the selected split point. 4. Repeat 2-4, using child nodes as a new parent. To obtain the regression tree a data-set needs to be created. This data-set is composed of irradiance, geometric efficiency, inlet temperature, ambient temperature, flow and outlet temperature. While irradiance, flow, geometric efficiency ambient and inlet temperatures can be created, the outlet temperature has to be computed. The distributed parameters model is simulated with different values for those parameters in order to obtain the outlet temperature. Since attempting to obtain a tree containing all the dynamics of the plant can be very costly and complex due to CART being inherently static, the data to build the tree will be obtained with the distributed parameters model once it reaches the steady-state. During the operation the temperature set-point will be modified by operators. GS-GPC will modify significantly the flow rate to achieve the target temperature so it will be necessary to filter the obtained estimation to prevent undesirable values of temperature when the set-point is modified. The estimation is obtained through the following steps: Every 13 seconds, instant k: 1. Measure Tin(k), Tout(k), and q(k). 2. Feed the CART with measured data to obtain a temperature estimation of the 16 segments for each loop. 3. Filter the estimation in kwith the obtained at instant (k−1). CART based temperature profiles estimation results are presented in Fig. 7. The set of reflectivities used is R= [0.65,0.67,0.68,0.7,0.715,0.73,0.75,0.77,0.79,0.8]. 11 12 13 14 15 16 17 Time (local hour) 200 220 240 260 280 300 Temperature (ºC) NLModel Estimation 11 12 13 14 15 16 17 Time (local hour) -5 0 5 10 Estimation Error (ºC) Collector 1 Collector 2 Collector 3 Collector 4 Collector 2 Collector 3 Collector 4 Collector 1 Figure 7 CART based temperature estimates are compare to the Non Linear Model. Temperature profiles at the output of the collectors (Top plot) and estimation errors (Bottom plot). 4.2. Loop Reflectivity Estimation The reflectivity estimation is done through a model based nonlinear optimization. It is assumed that the reflectivity only vary over long periods of time (hours). As the sampling time can be high, an estimation using an optimization problem based on a nonlinear model to estimate the reflectivity of the loop is proposed. A sampling time of 120 seconds has been chosen for the estimation of reflectivity although the sampling time may be longer if needed. The nonlinear model used for the estimation is a simplification of the distributed parameter model (16 segments instead of 172 segments) to achieve a reasonable trade-off between computational cost and accuracy. The optimization process is not global but individual as the reflectivities of the loops are independent. The formulation of the nonlinear optimization problem for the estimation tries to minimize a quadratic cost function (Eq. (7)). The cost function is computed along a prediction horizon and tries to minimize the quadratic sum of the difference between the estimate ˆynand the reference w. minJn= N2 X j=N1 δ(j)[ ˆyn(t+j|t)−w(t+j)]2 + Nu X j=1 λ(j)[∆un(t+j−1)]2 s.t : Umin < U(t+j|t)< Umax x=f(x, U), y =g(x) (7) where n= [1,2, ..., 10] is the loop being estimated, w is the field outlet temperature set-point (used in the GSGPC), ˆynis the simulated loop outlet temperature, Uis the objective variable, the reflectivity estimation. 6 Since the objective is to estimate the reflectivity of the loop, and not a sequence that minimizes that sum, the goal is to minimize the error at the end of the prediction horizon N2. Although due to the dynamics of the loop it is necessary to compute the entire horizon to solve the problem. For the same reason the terms relative to the control are not needed, as it is not necessary to weight the control signal along the prediction horizon. δ(j) is the weighting sequence for the temperature but since there is no weighting sequence for control, this is not necessary in the cost function. Eq. (7) is simplified to a simple subtraction at the end of the simulation, for each of the loops, as follows: minJn= ˆyn(N2)−w(N2) s.t : Umin < U(t+j)< Umax x=f(x, U), y =g(x) (8) 11 12 13 14 15 16 17 Time (local hour) 0.5 0.55 0.6 0.65 Global Loop Efficiency Estimated Real Figure 8 August case. Loop efficiency estimation. 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 Time (local hour) 0.5 0.55 0.6 0.65 Global Loop Efficiency Estimated Real Figure 9 December case. Loop efficiency estimation. The results obtained for the previous set of reflectivities are presented in Figs. 8 & 9. The simulations were carried out in August and December. Results show good performance estimating the efficiency of the loops with a maximum error of 0.01 (1%). 4.3. Inlet Loop Valve Optimization Once the reflectivities are estimated, the final step is to calculate the control signals (aperture of the inlet valves). The objective is to homogenize the field temperature. As with the reflectivity estimation, a nonlinear online optimization problem is proposed. Since reflectivities are considered to vary over long periods of time, actuation on the valves will have a high control sampling time. Other factors that make the sampling time high are: avoiding the coupling between the valve control and the flow rate control, as well as the valves over-actuation that would decrease its useful life. For this work a sampling time of 30 minutes for the valves control is chosen. The formulation of the nonlinear problem, manipulated variables the aperture of the valves, also tries to minimize a function of cost subject to restrictions (Eq. (9)). As commented above, weighting the control and output signal is not needed in the cost function as the valves are not controlled along a horizon. The prediction horizon is used to obtain the simulated prediction and perform the optimization but the objective is not a temperature set-point tracking problem minimizing the movement of the valves but to calculate only one control signal for each valve. minJ = NLoop X n=1 N2 X j=N1 δ(j)[ ˆyn(t+j|t)−w(t+j)]2 + Nu X j=1 λ(j)[∆un(t+j−1)]2 s.t : Umin < U(t+j)< Umax ∆umin <∆u(t+j)<∆umax x=f(x, U), y =g(x) (9) A control signal increment constraint is not needed as only one is being calculated. The constraint that is only taken into account for this controller is the maximum and minimum aperture, Umin < U(t+j)< Umax. Thus, the cost function is simplified (Eq. (10)). Since the objective is to control the valves in such a way in steady state a homogenization of the field temperatures is achieved, it is not necessary to include in the cost function all the steps along the prediction horizon. minJ = NLoop X n=1 N2 X j=N1 [ ˆyn(t+j|t)−w(t+j)]2 s.t : Umin < U(t+j)< Umax x=f(x, U), y =g(x) (10) Although it will continue using the complete horizon to simulate the system to obtain the optimal solution, the cost function will only contemplate the temperature at the 7 end of the horizon. A final aspect is that at least one of the loops should have the valve aperture at 100 %, though it may not be the only one. In this way the complexity of the problem is reduced by simplifying the cost function as follows: minJ = NLoop−1 X n=1 [ ˆyn(N2)−w(N2)]2 s.t : Umin < U(t+j)< Umax x=f(x, U), y =g(x) (11) The simplified nonlinear model used for the reflectivity estimation is used for the optimization problem to minimize the computation requirements. The programmed nonlinear algorithm has the following structure: 1. Measure Tinlet(k), Toutlet(k), Tloop(n, k) and q(k). 2. Inlet Valves Optimization (a) Loops simulation. (b) J computation at instant N2. (c) If J is minimum goto (3). If not go back to (a). 3. Apply control signals. The nonlinear valve control (NLVC) does not change the flow rate of the plant, but receives it as an input, so that there is no direct interaction with the GS-GPC which is controlling the field outlet temperature. In fact, it is important that it does not have direct interaction to avoid coupling or resonant modes in the plant. The final control scheme is presented in Fig. 10 and consists of 2 controllers, GS-GPC+FeedForward and NLVC. Figure 10 Proposed control scheme 5. Results The complete strategy has been simulated for three scenarios with different reflectivities for each loop. For the first scenario it has been assumed that some loops have the same reflectivity whereas for the second case all reflectivities are different, see table 2. While half the field is at a reflectivity r1 and the other half is at its maximum possible reflectivity in the third scenario. Tube efficiency and shape factor are assumed constant and equal to 0.85 and 0.9 respectively. The efficiency of each loop is obtained as Reflectivity ·TubeEfficiency ·ShapeFactor. Table 2 Simulated scenarios. Reflectivities. Reflectivity Set L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 Case 1 0.67 0.71 0.71 0.74 0.74 0.76 0.76 0.79 0.79 0.8 Case 2 0.65 0.67 0.68 0.7 0.715 0.73 0.75 0.77 0.79 0.8 Case 3 0.70 0.70 0.70 0.70 0.70 0.80 0.80 0.80 0.80 0.8 In addition to the above, simulations when medium/low and high outlet temperature set-points are presented. When the requested field outlet temperature set-point is close to the maximum allowed, higher efficiency of the rankine generation cycle is reached. In saturation periods, summer time, the loop defocus control is generally applied as CSPs are oversized by designed to obtain a good performance throughout the year. 11 12 13 14 15 16 17 Time (local hour) 260 265 270 275 280 285 290 Temperature (ºC) Tout L1 Tout L2,L3 Tout L4,L5 Tout L6,L7 Tout L8,L9 Tout L10 Set-Point Tout Field Figure 11 Case 1. GS-GPC+FeedForward results (no valve control, all at 100%) 11 12 13 14 15 16 17 Time (local hour) 260 270 280 290 Temperature (ºC) Tout L1 Tout L2,L3 Tout L4,L5 Tout L6,L7 Tout L8,L9 Tout L10 Set-Point Tout Field 11 12 13 14 15 16 17 Time (local hour) 70 80 90 100 (%) Valv L1 Valv L2,L3 Valv L4,L5 Valv L6,L7 Valv L8,L9 Valv L10 Figure 12 Case 1. GS-GPC+FeedForward results with reflectivity estimation and valve control. Loops temperatures (Top plot) and Valves apertures (Bottom plot) Fig. 11 shows the simulation of the case 1, simulated at low/medium temperatures, when only the GS-GPC control is applied. The GS-GPC behaves correctly obtaining a good performance tracking of the field outlet temperature 8