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Version: Author Accepted Version Citation: Antonio J Gallego, Adolfo J. Sánchez, M, Berenguel & Eduardo F. Camacho.

Adaptive UKF-based model predictive control of a Fresnel collectorfield.

Journal of Process Control – ISSN 0959-1524. Vol. 85, pp. 76-90. Enero 2020. 10.1016/j.jprocont.2019.09.003

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Adaptive UKF-based Model Predictive Control of a Fresnel Collector Field

Antonio J. Gallego1, Adolfo J. Sanchez1, M. Berenguelb, Eduardo F. Camacho1

aDepartamento de Ingenierıa de Sistemas y Automatica, Universidad de Sevilla, Camino de los Descubrimientos s/n., 41092 Sevilla, SpainbCentro Mixto CIESOL, ceiA3, Departamento de Informatica, Universidad de Almerıa. Ctra. Sacramento s/n, 04120, Almerıa, Spain

Abstract

One of the ways to improve the efficiency of solar energy plants is by using advanced control and optimizationalgorithms. In particular, model predictive control strategies have been applied successfully in their control.

The control objective of this kind of plant is to regulate the solar field outlet temperature around a desired set-point.Due to the highly nonlinear dynamics of these plants, a simple linear controller with fixed parameters is not able tocope with the changing dynamics and the multiple disturbance sources affecting the field.

In this paper, an adaptative model predictive control strategy is designed for a Fresnel collector field belonging tothe solar cooling plant installed at the Escuela Superior de Ingenieros in Sevilla. The controller changes the linearmodel used to predict the future evolution of the system with respect to the operating point.

Since only the inlet and outlet temperatures of the heat transfer fluid are measurable, the intermediate temperatureshave to be estimated. An unscented Kalman filter is used as a state estimator. It estimates metal-fluid temperatureprofiles and effective solar radiation.

Simulation results are provided comparing the proposed strategy with a PID+feedforward series controller showingbetter performance. The controller is also compared to a gain scheduling generalized predictive controller (GS-GPC)which has previously been tested at the actual plant with a very good performance. The proposed strategy outperformsthese two strategies.

Furthermore, two real tests are presented. These tests show that the proposed controller achieves adequate set-pointtracking in spite of strong disturbances.

Keywords: Solar Energy, Fresnel, Model Predictive Controller, Kalman Filter

1. Introduction

The great impulse experienced by renewable energysystems is driven by the need of reducing the negativeenvironmental impact that fossil fuels produce. Particu-larly, solar energy is the most abundant renewable en-ergy source. The only limitation of the use of solarthermal energy is harnessing it in an efficient and cost-effective way [1, 2]. Other solar technologies such asphotovoltaics (PV) are a very low-cost alternative, butthe difficulty is storing electrical energy which is moredifficult and expensive than storing thermal energy [3].

Solar energy has multiple applications such as elec-tricity generation or solar cooling systems. The use of

Email addresses: gallegolen@hotmail.com (Antonio J.Gallego), beren@ual.es (M. Berenguel), efcamacho@us.es(Eduardo F. Camacho)

URL: adolfo.spf@gmail.com (Adolfo J. Sanchez)1+34 954487347

solar energy for electricity production has been widelyextended in the last 15 years. A solar thermal plant con-sists of a solar field which heats up a fluid and thenpowers a steam generator to produce electricity. Ex-amples of solar thermal energy plants are the SOLANAand Mojave Solar parabolic trough plants constructed inArizona and California, each with a power production of280 MW [4, 5]. The 50 MW CSP plant of Yumen can becited as a more recent solar trough plant [6]. For furtherinformation concerning the solar energy plants commis-sioned or currently under construction, the reader is re-ferred to [7].

The use of solar energy for cooling systems has beenincreasing for several decades, spurred by the fact thatthe need for air conditioning is usually well correlatedto high levels of solar radiation [8, 9]. The plant usedin this paper as a test-bench for control purposes isthe solar cooling plant located on the roof of the Es-cuela Superior de Ingenieros (ESI) in Seville [10]. The

Preprint submitted to Journal of Process Control September 25, 2019

plant consists of a Fresnel collector field, a double ef-fect LiBr+ water absorption chiller and a storage tank.The Fresnel collector delivers pressurized water to theabsorption machine for producing air conditioning. Ifsolar radiation is not high enough for heating the waterup to the required temperature, the storage tank can beused. If neither the solar field nor the storage tank areable to heat the water up to the operation temperature,the absorption machine uses natural gas.

Usually, the control goal in solar energy plants isto regulate the solar field outlet temperature around aset-point [11]. Many works related to the control ofsolar plants have been developed and published since1980. Most of them were tested in the parabolic troughACUREX solar collector field. In [12], a review of dif-ferent control strategies applied to the ACUREX plantis presented. Moreover, the books by Camacho and co-workers [13] and Lemos and co-workers [14] providean overview of different approaches related to predic-tive and adaptive control applied to this kind of system.

The regulation of the outlet temperature of a solarfield around a set-point is hindered by the effect of mul-tiple disturbance sources and its dynamics are greatlyaffected by the operating conditions [15]. This factmeans that conventional linear control strategies do notperform well throughout the entire range of operations.In general, adaptative, robust or nonlinear schemes areneeded to cope with the changing dynamics of a so-lar field. In [16] a practical nonlinear MPC is devel-oped. More recently, in [17] a nonlinear continuoustime generalized predictive control (GPC) is developedand tested by simulation. In [18], an improvement ofa Gain Scheduling Model Predictive controller is pro-posed and tested on a model of the ACUREX solar field.

The development of control strategies for Fresnel col-lector fields has not been as popular as for parabolictrough fields. As examples, in [19], control approachesfor this kind of plant are discussed. The application ofan explicit model predictive control is also tested bysimulation. In [20] a sliding model predictive controlbased on a feedforward compensation is developed fora Fresnel collector field and tested on a nonlinear modelof a Fresnel collector field. In [21], control systems fordirect steam generation in linear concentrating powerplants are investigated. In [22], control laws for lin-ear Fresnel plants have been developed. In [23] a GainScheduling Generalized Predictive Controller was de-veloped and tested for a Fresnel collector field.

In this paper an adaptative model predictive control(MPC) strategy is applied to the Fresnel solar collec-tor field. The MPC strategy uses a state-space formula-tion whose linear matrices are obtained by linearizing a

nonlinear distributed parameter model of the field. Thenon-measurable states are estimated using an unscentedKalman filter (UKF) [24]. Many examples of the useof the UKF as a state estimator when controlling solarenergy systems can be found in literature. In [25], adiscrete-time adaptative scheme for temperature controlin molten-salt solar collector-fields is proposed. In [26],a neural model-based MPC strategy using a UKF forestimating the weights and biases was tested on the ac-tual ACUREX solar field. In [27], an iterative extendedKalman filter is used to estimate the states and solar ra-diation for control purposes. It was tested by simulationfor a model of the Shiraz solar plant. Regarding solarcooling plants, the UKF has also been used for estimat-ing parameters [28].

In this paper, the UKF estimates not only the temper-ature states but also the effective solar radiation reach-ing the metal tube. As stated in [29], pyrheliometersmeasure direct solar radiation locally. Current commer-cial plants cover great extensions of land [30]. Extrapo-lating the local measure from pyrheliometers to the restof the field can lead to important errors, especially ontransient days when passing clouds affect part of thefield while the rest is not affected. Moreover, obtain-ing an estimation of the optical efficiency composed bymultiple parameters such as mirror reflectivity, metalabsorptance, shade factors etc, is a difficult task. Theeffectiveness of the proposed strategy is validated usinga nonlinear distributed parameter model. Furthermore,two tests carried out at the real Fresnel solar field arealso shown.

The control strategy proposed here is based on previ-ous works such as [31]. However there are significantdifferences which are enumerated below:

1. The control strategy proposed in this paper, usesan incremental formulation of the MPC problem.This formulation avoids the need for using distur-bance estimators to correct tracking reference er-rors in steady state. The formulation used hereachieves offset-free tracking when step referencesare considered, and the system has no zero at z=1,as shown in [32].

2. The solar plant is based on a Fresnel solar collec-tor field which uses water as HTF, while the plantused in [31] is a solar trough using oil as HTF. Themodel of the plant and the controller itself are dif-ferent.

3. In most previous works, the UKF+MPC controlstrategy has been tested using simulation models.In this paper, tests performed at the actual plant areprovided. One of them is on a day when strong

2

GS-GPC Gain Scheduling Generalized PredictiveController

HTF Heat Transfer FluidISE Integral of square errorsMPC Model Predictive ControlPCM Phase Change MaterialPDE Partial Differential EquationPID Proportional+Integral+Derivative ControllerPV PhotovoltaicsUKF Unscented Kalman Filter

Table 1: List of Abbreviations

radiation disturbances affect the solar field. Thisis one of the most extreme operating conditions insolar energy plants.

4. Although the tests were carried out using Matlab,the final control strategy will be programmed in aPLC installed at the plant for control purposes. Forthis reason, the computational burden and com-plexity of the proposed control strategy has to beas low as possible without deteriorating the perfor-mance substantially.

The paper is organized as follows: section 2 brieflydescribes the solar cooling plant. Section 3 presents themathematical model of the Fresnel collector field usedin this paper. Section 4 describes the MPC control strat-egy and the UKF nonlinear estimator. Section 5 showsthe simulation results. Section 6 presents the tests per-formed at the actual plant. Finally, some concludingremarks are given.

2. Brief plant description

The solar cooling plant at the Escuela Supe-rior de Ingenieros in Sevilla (Latitude=37.4108972°,Longitude=-6.0006621°), was constructed in 2008. Fig-ure 1 shows the overall scheme of the solar coolingplant. The plant consists of three subsystems which arebriefly described below.

Water absorption chiller: this is a double-effect cy-cle LiBr+ absorption machine with 174 kW and a the-oretical COP of 1.34, which transforms the thermal en-ergy coming from the Fresnel solar field or the PCMstorage tank, into cold water to be used by the ESI ofSeville [10].

Solar Field: the solar field consists of a set of Fresnelsolar collectors (see Figure 2) which concentrate solarradiation onto a 64 m metal tube through which pres-surized water circulates.

PCM storage tank: the PCM storage is a 18 metrelong shell-tube heat exchanger with a 1.31 metre diam-eter. It consists of a series of tubes containing a heattransfer fluid and PCM fills in the space between thetubes and the shell. In [33], further information aboutthe storage tank can be found.

3. Mathematical modeling of the Fresnel collectorfield

In this section, the mathematical model of the solarcollector field is presented. The equations governing thesystem dynamics are similar to those used in the case ofthe parabolic trough fields [34]. The difference dependson the computation of the geometrical efficiency and theshade factor.

There are two ways of modeling this kind of system:the concentrated parameter model and the distributedparameter model [35]. In this paper, the distributed pa-rameter model is used for simulation purposes and fortesting the performance of the proposed control strategy.The concentrated parameter model is used for obtaininga feedforward controller.

3.1. Distributed parameter modelAs described in section 2, the Fresnel solar collector

field consists of a set of Fresnel solar collectors whichconcentrate solar radiation onto a line where a 64 mlong absorption tube is placed. The model dynamics isgoverned by the following system of partial differentialequations (PDE) describing the energy balance [36]:

ρmCmAm∂Tm(l, t)

∂ t= IKoptnoGa −HlG(Tm −Ta)

−LHt(Tm −Tf )

ρ fC f A f∂Tf (l, t)

∂ t+ρ fC f q

∂Tf

∂ l= LHt(Tm −Tf ) (1)

where m subindex refers to metal and f subindexrefers to a fluid. In Table 2, parameters and their unitsare shown.

The PDE system is solved by dividing the metal andfluid into 64 1 m long segments. The integration step ischosen to be 0.5 seconds and the integration techniqueis a Euler forward method. This method has been cho-sen for its simplicity, although there are more precisemethods such as the runge-kutta integration algorithm.

One of the problems of the Euler forward method isthat if the integration step is not small enough, numer-ical instabilities may appear [37]. In the case of solar

3

Solar System

ESI

Cooling System

PCM

Storage Tank

Natural Gas

Burner

Absorption

Machine

ESI

Guadalquivir

River

Cool water 7 C◦

Figure 1: Plant general scheme

Symbol description Unitst Time sl Space mρ Density kgm−3

C Specific heat capacity JK−1kg−1

A Cross sectional area m2

T (l, t) Temperature K,°Cq(t) Water flow rate m3s−1

I(t) Direct Solar Radiation Wm−2

no geometric efficiency UnitlessKopt Optical efficiency UnitlessGa Collector aperture m

Ta(t) Ambient temperature K,°CHl Global coefficient of

thermal lossWm−2°C−1

Ht Coefficient of heat trans-mission metal-fluid

Wm−2°C−1

L wetted perimeter mS Total reflective surface m2

Cth Thermal capacity of thesolar field

J/K

Pcp Parameter of solar field J m−3K−1

T average Temperature ofthe solar field

°C,K

Table 2: Parameter description

Figure 2: Fresnel collector field

energy systems, the most abrupt changes are producedwhen transients in solar radiation appear and produceabrupt changes in the temperatures. It has been found bytest and error that if the integration step is chosen higherthan 1.5 seconds, these numerical instabilities appear.

The computation of geometric efficiency, which takesinto account the effect of the cosine of the incidenceangle of the solar beam and the shade factor, involvescomplex trigonometric formulas. These formulas areexplained in [38].

The density, specific heat and heat transmission coef-ficients have been obtained as polynomial functions ofthe segment temperature and water flow using thermo-dynamical data of pressurized water. The thermal losscoefficient has been obtained using real data from the

4

Fresnel solar field. These expressions are valid from0 °C to 250 °C. This is sufficient for the model usedhere, because the solar cooling plant operating range isbetween 0 °C and 190 °C. At 190 °C, the solar fieldis defocused and the plant is shut down to prevent anypossible damage to the equipment. The expressions areas follows:

ρ f =−0.0025 T 2f −0.203 Tf +1003.91 (2)

C f = 5.16e−7 T 4f −1.56e−4 T 3

f +0.0277 T 2f

−1.627 Tf +4207.403 (3)

Hv = 1.34e−4 T 4f −7.78e−2 T 3

f +18.73 T 2f

−2.57e3 Tf +4.11e5

Ht = Hv q0.8 (4)

The thermal loss coefficient can be calculated usingequation (5):

Hl = (0.00122)(Tm −Ta)+0.001763 (5)

Figure 3 shows a comparison between the model andthe real outlet temperature from the solar field. As canbe seen, the model dynamics is very similar to the realone.

Time (Local hour)13 13.5 14 14.5 15

Te

mp

era

ture

(ºC

)

60

80

100

120Temperatures: Comparison between model and real data

Model Real ToutTin

Time (Local hour)13 13.5 14 14.5 15S

ola

r R

ad

iatio

n (

W/m

2)

200

400

600

800

Direct Solar Radiation

Time (Local hour)13 13.5 14 14.5 15

Wa

ter

Flo

w (

m3/h

)

4

6

8

Water Flow

Figure 3: Solar field evolution Model vs Real: 31/07/2017

Number of segments Error (°C)64 1.4532 1.5716 1.78 1.844 1.962 2.461 2.83

Table 3: Maximum Error between the model output and real temper-ature

3.2. Concentrated parameter modelThe concentrated parameter model provides a lumped

parameter description of the whole field. The variationin the internal energy of the fluid can be described bythe equation [38]:

CthdTout

dt= KoptnoSI −qPcp(Tout −Tin)−Hl(T −Ta)

(6)The parameters of the equation are shown in Table 2.

The total value of the reflective surface is 352 m2.

4. Model Predictive Control Strategy

In this section, the proposed model predictive con-trol strategy is presented. The linear model used in theMPC control strategy is a linearization of the nonlineardistributed parameter model (equations (1)). The tube isdivided into 4 segments for the metal and fluid respec-tively, instead of the 64 segments used in the distributedparameter model. This simplification reduces the com-putational burden of the control strategy and the size ofthe tuning matrices of the unscented Kalman filter es-timator [24]. This fact means that, since the numberof the tuning parameters is smaller, the control strategycan be tuned more easily. The main drawback is theprecision loss. Since the final control strategy is aimedat being installed on a PLC, reducing the computationalburden becomes an important issue.

In order to choose the above-mentioned number ofsegments, a study has been carried out comparingthe maximum error between the distributed parametermodel and the actual plant temperature. The numericalresults are shown in table 3. As can be seen, the lowerthe number of segments, the higher the error expected.Moreover, when the number of segments is smaller than4, the error grows faster.

Since only the inlet and outlet temperature of thefluid are available, the intermediate segment tempera-tures and the metal segment temperature are estimated

5

by means of a UKF. One of the advantages of using aUKF as a state estimator instead of other simpler ap-proaches such as the Luenberger observer [39] is thatthe UKF uses the full nonlinear model without simplifi-cations to obtain the estimation of the state and param-eters. The problem is that, although results concerningthe stability of nonlinear estimators have been obtained,as shown in [40], to mathematically prove the stabilityof the MPC+UKF set is still a challenge and an openproblem, especially for PDE systems of the Fresnel col-lector field. In this paper, the UKF estimates the ef-fective solar radiation reaching the metal tube, namely,product IKoptno.

Firstly, the MPC algorithm is introduced. Sec-ondly, the procedure for obtaining the linear matricesis shown. Then, the UKF estimator is described andthe incremental formulation of the state-space MPC ispresented. This formulation avoids the necessity of adisturbance estimator to achieve offset-free referencetracking. Lastly, the final control scheme is shown andexplained.

4.1. Control Objectives in solar energy plantsIn solar thermal plants, the solar field heats up a heat

transfer fluid. This fluid can be used for producing elec-tricity in a turbine, or in an absorption machine in a so-lar cooling plant to deliver air conditioning as is the casehere.

The control objective in a solar energy plant is to reg-ulate the outlet temperature around a desired set-point.This set-point can be computed in order to satisfy opti-mal production [41], or any other criteria [17]. To ac-complish this task the manipulated variable is the HTFflow. The fulfillment of this objective is not an easytask due to the complex dynamics and multiple distur-bances which affect the solar field. The transport-delaydepends strongly on the control signal (HTF flow) [42].

4.2. MPC control problemAn MPC control strategy consists of the following

steps [43, 44]: firstly, it uses a model to predict the pro-cess evolution depending on a given control sequence,then it computes the control sequence by minimizing anobjective function. Only the first element of the con-trol sequence is applied to the system (receding horizonstrategy).

The difference in various MPC strategies is mainlyrelated to the model used to predict the system evolu-tion. The use of linear models and quadratic cost func-tions gives rise to a quadratic programming problem. Ifthe model used in the MPC strategy is nonlinear, the re-sulting optimization problem is computationally harder

to solve and attaining the global optimum is not, in gen-eral, ensured [45].

In this paper, the model used to predict the future evo-lution of the system is a linear state-space model and thecost function is considered to be quadratic.

In general, the mathematical expression of the MPCproblem can be posed as follows:

min∆u

J =Np

∑t=1

(yk+t|k − yre f

k+t

)ᵀ(yk+t|k − yre f

k+t

)+λ

Nc−1

∑t=0

∆uᵀk+t|k∆uk+t|k

s.t.

yk+t|k = f (∆u,yk+t−1,yk+t−2, ...)

uk+t|k = uk+t−1|k +∆uk+t|k

umin ≤ uk+t|k ≤ umax

t = 0, . . . ,Np −1

where Np and Nc stand for the prediction and the con-trol horizons respectively. The parameter λ penalizesthe control effort. Then uk ≡ uk|k is applied to the sys-tem. In this case, only constraints in the amplitude ofthe water flow have been considered.

4.3. Obtaining the linear Matrices

In this subsection, the procedure for obtaining thelinear matrices is presented. The linear model of thePDE system (1) consists of a set of matrices depend-ing on inputs and system states. Let x be the state vec-tor formed by the temperatures of the 4 metal and fluidsegments, Tin the inlet temperature, q the water flow ,Ie f f = IKoptno the effective solar radiation, and Ta theambient temperature. ∆x is the length of the segmentsinto which the tube is divided (in this case 4 16 m seg-ments).

The linear model in continuous time is computed us-ing (7):

xt = Axt +But +Bd dt

yt =Cxt (7)

ut = q dt =[

Tin Ie f f Ta]ᵀ (8)

6

The linear matrices are computed as follows:

P0 =−HlGa −LHt

ρmCmAmP1 =

LHt

ρmCmAm(9)

P2 =LHt

ρ fC f A fP3 =

qA f ∆l

(10)

P4 =−P2 −P3 (11)

A =

P0 0 0 0 P1 0 0 00 P0 0 0 0 P1 0 0... ... ... ... ... ... ... ...0 0 0 P0 0 0 0 P1P2 0 0 0 P3 0 0 00 P2 0 0 P4 P3 0 0... ... ... ... ... ... ... ...0 0 0 P2 0 0 P4 P3

(12)

where the first 4 states correspond to the metal tem-peratures and the last 4 correspond to the fluid tempera-tures. The value of the control signal q in matrix A is thevalue around which the nonlinear model is linearized.

B =[

01x41

A f ∆x . . . 1A f ∆l

]ᵀ(13)

BTin =[

01x4q

A f ∆x 01x3

]ᵀ(14)

BIe f f =[

11x4 · GaρmCmAm

01x4

]ᵀ(15)

BTa =[

HlGρmCmAm

. . . HlGaρmCmAm

04x1

]ᵀ(16)

Bd =[

BTin BIe f f BTa]ᵀ (17)

C =[

0 0 0 0 0 0 0 1]

(18)

Notice that the A and BTa matrices depend on the sys-tem states, the water flow and the system parameters;BTin is a function of the water flow u.

It is worth pointing out that only ρm, Cm, A f , Am andG are constants, whereas each parameter Hl , p f , C f andHt depends on the temperature. The linear matrices arediscretized using a sampling time of 20 s. The sam-pling time has been chosen taking into account the typ-ical time constants of the plant in closed loop. Fromnow on we will refer to the discrete-time matrices keep-ing the notation A, B and Bd . The discretization methodchosen is a zero order hold method which assumes thecontrol inputs are piecewise constant over the samplingperiod.

4.4. Nonlinear state estimator: the unscented Kalmanfilter

In this subsection the unscented Kalman filter (UKF)is described. For the model predictive control strategy,the temperature of metal and fluid segments and theeffective solar radiation are needed. The temperatureprofiles and the estimation of effective solar radiationare used in the model predictive controller to obtain theprediction of the outlet temperature along the predictionhorizon (see section 4.2).

In this plant, only the inlet and outlet temperaturesare measurable, so the intermediate fluid segment tem-perature, the metal segment temperatures and the effec-tive solar radiation have to be estimated. The nonlinearmodel used for the UKF is a simplified version of theone described by equations (1). The tube is divided into4 segments instead of the 64 used in the full nonlinearmodel. This simplification is carried out in order to re-duce the computational burden and the complexity ofthe control strategy [31].

The Kalman filter is a widely used tool for state es-timation in linear systems [46]. The Kalman filter hasbeen extended to nonlinear state estimation through al-gorithms such as the extended Kalman filter (EFK) orthe unscented Kalman filter (UKF) [24].

The UKF is based on the unscented transformation,which represents a method for calculating the mean andcovariance of a random variable that undergoes a non-linear transformation [47, 48]. Since the appearanceof the UKF algorithm, several improvements and vari-ations have been developed [49, 50]. In [51], an im-proved unscented transformation by incorporating therandom parameters into the state vector in order to en-large the number of sigma points is proposed. For moredetails about the UKF implementation, the reader is re-ferred to [24].

Another important point to be addressed is that thechanges in solar radiation can be abrupt and very fast.Since adequate solar field sampling time for control pur-poses is around 20 or 30 seconds, this may be too longfor the UKF to capture the dynamics of radiation fastenough. In this paper the sampling time for control pur-poses is chosen to be 20 s and for the UKF, 10 seconds.

As far as the evolution of the effective solar radiationwithin the sampling time is concerned, several modelsfor predicting solar radiation evolution have been pro-posed in literature [52, 53]. The models provide only anestimation (nowcasting) of the future evolution. Thesemodels depend on multiple parameters such as the sun-shine stability number and cloud transmittance whichare not easy to know a priori. An estimation of these

7

parameters would be necessary at the cost of increas-ing the model complexity. The approach used here isto consider that the effective solar radiation is constantalong the sampling time. This is reasonable because thesampling time is small.

The Fresnel solar collector field has slow dynamicswith a time constant of minutes. It was found that reduc-ing the sampling time of the UKF did not produce anynoticeable improvement in the estimation, because theevolution of the temperature is not significant for such asmall sampling time. The minimum sampling time forthe UKF would be 5 seconds because it is the minimumtime to ensure correct communication between the PCwhere the controller is running and the data acquisitionsystem. Figure 4 shows the evolution of the error in theeffective solar radiation for different sampling times. Ascan be seen the error decreases with respect to the sam-pling time but the difference between 5 s and 10 secondsis quite small.

Time (local hour)14.4 14.5 14.6 14.7 14.8 14.9 15 15.1 15.2 15.3 15.4

Estim

atio

n e

rro

r (W

/m2)

-40

-30

-20

-10

0

10

20

30

40

Error between estimated effective radiation (UKF) and the real value for different UKF sampling times

5 seconds10 seconds20 seconds30 seconds

Figure 4: Effective solar radiation error for different UKF samplingtimes

4.5. Incremental formulation of the state-space MPCIn this subsection, the incremental formulation of the

state-space formulation is presented.The main problem when using state-space models in

model predictive control strategy is that steady-state er-rors may appear due to model-plant mismatches. Thisproblem is usually addressed by using a disturbance es-timator [54, 55]. This approach has the drawback thatadditional dynamics is added to the control system andmore parameters have to be tuned properly. Thus, thecomplexity of the overall control scheme increases.

In this paper, an incremental formulation is used[32]. This formulation has the advantage of avoiding theneed of a disturbance estimator for offset-free referencetracking when step references are given. The procedureis fully described in [32].

Considering a linear state-space model with n states,m outputs, r inputs and l disturbance sources:

xk+1 = Axk +Buk +Bddk

yk =C xk (19)

The key, as shown in [32], is to use an incrementalform of the state space formulation and augment thestates to include the system output. The incremental for-mulation is described as follows:

∆xk+1 = A∆xk +B∆uk +Bd∆dk (20)

yk = yk−1 +C∆xk (21)

where ∆xk = xk − xk−1. Augmenting equations (20)and (21), the complete state-space model is posed asfollows:

xk+1︷ ︸︸ ︷[∆xk+1

yk

]=

A︷ ︸︸ ︷[A 0n×mC Im×m

] xk︷ ︸︸ ︷[∆xkyk−1

]+

B︷ ︸︸ ︷[B

0m×r

]∆uk+

(22)

Bd︷ ︸︸ ︷[Bd

0l×r

]∆dk (23)

yk =

C︷ ︸︸ ︷[C Im×m

][ ∆xkyk−1

](24)

The new system can be posed as follows:

xk+1 = Axk + B∆uk + Bd∆dk (25)

yk = Cxk (26)

Since the model used is linear, the predicted responseof the system can be described by the sum of the forcedresponse, a term which depends on the future controlactions, and the free response which does not dependon the future control actions [43], as follows:

y = G∆u+ f (27)

8

Matrix G can be obtained from the model matrices(equations (25) and (26)) and the free response can becomputed as a matrix depending on the past values ofthe states, outputs and measured disturbances. Since itis difficult to know the future evolution of the distur-bances, the approach used in this paper is to assume thatthey remain constant along the prediction horizon.

Although estimating the future evolution of the maindisturbance sources affecting the solar field (in thiscase, the evolution of solar radiation and inlet temper-ature) would produce better disturbance rejection, pre-cise models to estimate this evolution would be requiredto do this.

In the case of solar radiation, several models havebeen proposed to predict its evolution [52, 53]. Thesemodels depend on multiple parameters which are diffi-cult to estimate. Furthermore, although an estimationof the future evolution on a clear day is possible at thecost of a more complex control strategy, obtaining thisestimation on a cloudy day is not an easy matter. Anestimation of the position of the clouds and how theyaffect the solar field would be necessary, and this is avery difficult and open problem.

With regard to the inlet temperature of the solar field,to be able to estimate its evolution, a nonlinear modelof the plant would be required in order to estimate thetemperature evolution of all the subsystems and the pipeconnecting these subsystems with the solar field alongthe prediction horizon. The resulting control strategywould be a nonlinear model predictive control strategywhich is more complex to solve. Since the proposedcontrol strategy is aimed at being installed on a PLCsystem, this approach has to be ruled out.

The free response can be calculated as follows [43]:

f = F[

∆xkyk−1

]+GIe f f ∆Ie f f +GTin ∆Tin +GTa∆Ta

(28)where matrix F can be obtained from the model, and

matrices GIe f f , GTa and GTin are the contribution of eachvariable to the future evolution of the linear system.

4.6. Final control Scheme

This subsection explains the final control schemeshown in Figure 5. It works as follows:

Each estimation sampling time (10 seconds), the un-scented Kalman filter receives the variables from the so-lar field and environmental conditions: inlet and out-let temperatures, ambient temperature, solar time andthe water flow applied in the previous instants. Tak-ing into account this information, it estimates a fluid-

metal temperature profile and the effective solar radia-tion. This estimation is transmitted to the control blockwhich is computed every 20 seconds. The linear con-trol block computes the linear matrices associated to thecurrent operation point of the solar field and solves thelinear MPC control problem obtaining a control signal,qpred. This signal is added to a feedforward compen-sation, q f f . This sum is the final control signal sent tothe Fresnel solar field. The feedforward compensationis computed as follows:

q f f =Ie f f S−Hl(T −Ta)

Pcp(Tre f −Tin)(29)

Although the MPC control strategy adapts its modelto the current operating conditions, the control prob-lem is linear, that is, the control signal computed by theMPC is a variation around the operating point. In or-der to help the linear MPC algorithm, the feedforwardprovides a steady-state water flow for the desired tem-perature reference and the MPC controller is added tocorrect it. If the model were perfect the feedforwardcompensation would be sufficient to track the desiredreference without errors.

Tin Tamb Solar Time q(k − 1) Tref

UKF

TEMPERATURE

EFFECTIVE SOLAR

RADIATION

METAL-FLUID

PROFILES

LINEAR CONTROL

MPC ALGORITHM

MATRICES COMPUTATION

FEEDFORWARD

BLOCK

COMPENSATION

FRESNEL

COLLECTOR FIELD

Ts = 10s

Ts = 20s

IDN

+

+

qpred

qff

q Tout

Tin Tamb Solar Time

Ts = 20s

Figure 5: Final control Scheme

5. Simulation results

In this section, simulations testing the performance ofthe proposed control strategy are presented. The plat-form used to perform the simulations was an i3 with 4GB of RAM. The software used is Matlab 2014 ®. Themaximum computational time obtained for the overallcontrol strategy was 0.7 seconds.

Firstly, a simulation showing the estimation of thesolar radiation provided by the UKF estimator using

9

data taken from the actual Fresnel solar collector fieldis performed. Then, two simulations are carried outcomparing the proposed MPC strategy to a PID+seriesfeedforward. The effective solar radiation used in thePID+series feedforward is the one measured, not theone estimated by the UKF. The simulations are per-formed using the nonlinear distributed parameter model.The tuning parameters for the MPC are Nc = 10, Ny =14 and λ = 2e8. The water flow is constrained to 3.6and 10 m3/h. These values were obtained in order toprovide fast responses without producing great oscilla-tions and brisk control actions.

The PID controller has the following expression:

upid = 0.1 e(k)+0.8∆e(k))

Ts+6.29e−3 Ierror(k)

(30)

where Ierror(k) is the value of the error integral at in-stant k and ∆e(k) is the increment of the error computedas e(k)− e(k−1). Ts is the sampling time. The param-eters of the PID were adjusted using data from differentenvironmental conditions (clear and cloudy days). Theinitial design of the PID was for the medium water flowconditions. However, since the PID is a fixed parametercontroller, when the plant evolves far from these condi-tions, the performance deteriorates [56]. Taking this ini-tial design, the parameters were tuned to work properly(without great oscillations) throughout the entire rangeof operations by a test and error method.

The unscented Kalman filter has been tuned using aset of data taken from different operation days of theactual Fresnel collector field. The covariance matricesQ and R for the UKF [24], were tuned by a test anderror method. Obtaining the Q and R matrices is one ofthe problems of using a Kalman filter, since they are notgenerally known and have to be estimated.

The covariance matrix of sensor R was set to 0.01.The actual value is not known, but the temperature sen-sor measurement is considered to have a small variance.Matrix Q is the covariance of the states. In this paper,the size of the Q matrix is 9× 9 since the states to beestimated are the eight segment temperatures (4 metaland 4 fluid segments) and the effective solar radiation.The covariance value of the temperature states is con-sidered to be low, that is, the estimation uncertainty issmall and the parameter with a higher uncertainty is theeffective solar radiation. The matrix has been chosen asa diagonal for simplicity.

The final Q matrix obtained is as follows:

Q =

1e−2 0 0 0 0 0 ...

0 1e−2 0 0 0 0 ...0 0 1e−2 0 0 0 ...... ... ... ... ... ... ...... 0 0 0 0 0 0.5

(31)

5.1. UKF tuning

Figure 6 shows a day when the Fresnel collector fieldwas connected to the absorption chiller. The top partof the figure shows the evolution of the inlet and outlettemperatures. As shown, they present oscillatory be-havior due to the return temperature of the absorptionchiller. The bottom part of the figure depicts the waterflow, which was about 12 m3/h and the solar radiationmeasured by the pyrheliometer and that estimated bythe UKF. To obtain the measured effective solar radia-tion, the overall Kopt parameter is chosen from severalstudies to be 0.35. As can be seen, the effective solarradiation estimated by the UKF approximately followsthe measured effective solar radiation.

Time (local hour)12 13 14 15 16 17 18 19

Te

mp

era

ture

(ºC

)

60

80

100

120

140

160

Operation day: 29/06/2009ToutTin

Time (local hour)12 13 14 15 16 17 18 19

Ieff

(W

/m2)

an

d q

*10

(m

3/h

)

50

100

150

200

250

300

350

400Effective solar radiation (W/m 2) and water flow*10 (m 3/h)

q*10Ieff (UKF)Ieff( Measured)

Figure 6: UKF tuning with real data

5.2. Comparison with a fine-tuned PID

For the simulations shown below, parameter Kopt ischosen to be 0.25. This parameter was estimated by a

10

set of data collected on several operational days in 2017.There are 4 inoperative rows out of 22 and some mis-calibrated mirrors. For the sake of clarity, the effectivesolar radiation is scaled by multiplying by a factor of 4,and the water flow is multiplied by 100, in order to plotthem in the same figure.

5.2.1. Clear day at high temperatureFigure 7 shows the first simulation which consists of

a series of increasing temperature references on a clearday. This kind of situation is typical when the plantis stable at a nominal point, the absorption machine isoperating and connected to the solar field. The objectiveis to change the set-point of the solar field according tothe temperature needed by the absorption chiller.

As can be seen, the proposed control strategy perfor-mance maintains good tracking behavior throughout thesimulation whereas the PID performance deteriorates atlow water flow levels. As seen at the bottom of figure7, the estimated effective solar radiation is very close tothe real one. This steady-state error may be produced bythe fact that the model used to estimate it is a simplifiedversion of the full distributed nonlinear model. Anotherpoint is that the MPC control strategy better rejects theinlet temperature disturbances due to its prediction ca-pability.

Time (local hour)12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5

Te

mp

era

ture

(ºC

)

120

125

130

135

Simulation 1: Temperature evolutionTout (MPC)RefTout (PID)Tin

Time (local hour)12 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5

Ieff

(W

/m2),

q (

m3/h

)

300

400

500

600

700

800

900

1000Ieff evolution (W/m 2) and water flow (m 3/h)

Measured*4Estimated*4 (UKF)q*100 (MPC)q*100 (PID)

Figure 7: Simulation 1: clear day

Figure 8 shows the contribution of the feedforwardand the MPC to the final control signal. As can be seen,

the feedforward acts as a high gain signal and the MPCcontributes as a small gain signal to track the desiredset-point and remove the steady-state error. Since themodel is not perfect, the feedforward compensation isnot sufficient to remove the steady-state error. This isdone by the MPC signal.

Local hour11 12 13 14 15 16 17 18 19

Wa

ter

Flo

w (

m3/h

)

-2

0

2

4

6

8Control signals of Feedforward and MPC:Simulation 1

FeedforwardMPC

Local hour11 12 13 14 15 16 17 18 19

Wa

ter

Flo

w (

m3/h

)

3

4

5

6

7

8

9

10

Total control signal

Figure 8: Simulation 1: Feedforward and MPC control signals

5.2.2. Start-up stage: recirculation modeFigure 9 shows the simulation performed using real

data of the Fresnel solar plant. On this day, the plant wasworking in recirculation mode, where the outlet temper-ature is recirculated through a long pipe and returns tothe solar field. This situation is typical when the plantis at the start-up stage, when the objective is to raise thetemperature of the solar field in order to reach a valuehigh enough to feed the water chiller.

It can be observed that the MPC is generally fasterand tracks the temperature references better than thePID controller. At 14.1 h, the solar field was com-pletely defocused, causing the effective solar radiationto fall to 0 W/m2. The estimated effective solar radi-ation follows this drop in a smoother way. This is adrawback when using the estimated effective solar ra-diation: the UKF estimation evolves when the effect ofthe real effective solar radiation is observed in the out-let temperature. This fact produces a loss of anticipativeaction, partially compensated by the prediction capabil-ities of the MPC strategy. To make the UKF faster, ahigher gain can be chosen but it may produce a more

11

oscillatory behavior. One possible solution could be touse a fault tolerant algorithm that selects which mea-sure to use: that provided by the pyrheliometer or theestimation provided by the UKF. In any case, as seen insimulation results, the MPC algorithm rejects the distur-bances properly, achieving good reference tracking.

Time (local hour)14 14.5 15 15.5 16 16.5

Te

mp

era

ture

(ºC

)

50

60

70

80

90

100

110

Simulation 2: Temperature evolutionTout (MPC)Tout (PID)TinRef

Time (local hour)13.5 14 14.5 15 15.5 16 16.5 17E

ffe

ctive

so

lar

rad

iatio

n (

W/m

2)

0

50

100

150

200

250

300Effective solar radiation evolution

MeasuredEstimated (UKF)

Figure 9: Simulation 2

Figure 10 shows the water flow signals of the MPCand PID.

Time (local hour)13.5 14 14.5 15 15.5 16 16.5 17

Wat

er F

low

(m

3/h

)

3

4

5

6

7

8

9

10Water flow evolution

MPCPID

Figure 10: Simulation 2: water flow evolution

As in the first simulation, figure 11 shows the con-tribution of the feedforward controller and the MPC to

the overall control signal. The conclusions obtained aresimilar to those obtained in the first simulation.

Local hour13.5 14 14.5 15 15.5 16 16.5 17

Wa

ter

Flo

w (

m3/h

)

-4

-2

0

2

4

6

8

10Control signals of Feedforward and MPC

FeedforwardMPC

Local hour13.5 14 14.5 15 15.5 16 16.5 17

Wa

ter

Flo

w (

m3/h

)

3

4

5

6

7

8

9

10

Total control signal

Figure 11: Simulation 2: Feedforward and MPC control signals

Figure 12 shows the metal-fluid temperature profilesestimated by the UKF algorithm. As can be seen, theestimation is very close to that obtained by the nonlineardistributed parameter model, with a maximum error of0.5 .

5.2.3. Performance comparison between using a paral-lel feedforward or not

In this subsubsection, a performance comparison be-tween using a parallel feedforward or not is carried out.

The MPC uses the measured disturbances and vari-ables from the plant to predict the evolution of the out-let temperature. The model used to do this is linear. Inconsequence, the control signal computed by the MPCcan be interpreted by a small gain signal to be addedto a high gain signal. This high gain signal is the flowcomputed by the feedforward.

If the feedforward is not used, the MPC has worseperformance, as shown in figure 13. The use of thefeedforward helps by rejecting the disturbances. Bet-ter performance and a higher degree of robustness whencontrolling the actual plant are achieved [56].

An ISE (Integral of square error) performance indexhas been calculated for the two strategies. The result is:ISEno f f = 1420.6 and ISE f f = 920.6.

12

Time (local hour)14 14.5 15 15.5 16 16.5

Te

mp

era

ture

(ºC

)

50

60

70

80

90

100

110

Metal temperature: estimated (red solid) and distributed parameter model (blue dashed)

Time (local hour)14 14.5 15 15.5 16 16.5

Te

mp

era

ture

(ºC

)

50

60

70

80

90

100

110

Fluid temperature: estimated (red solid) and distributed parameter model (blue dashed)

Figure 12: Simulation 2: Metal-fluid temperature profiles estimation

Time (local hour)11 12 13 14 15 16 17 18

Te

mp

era

ture

(ºC

)

110

115

120

125

130

135Comparison between MPC with ff vs MPC without ff

Tout (MPC with ff)RefTout (MPC no ff)Tin

Time (local hour)11 12 13 14 15 16 17 18

Effe

ctive

so

lar

rad

iatio

n (

W/m

2)

200

250

300

350Effective solar radiation evolution

Measured

Figure 13: Simulation 3: Performance comparison between using ffvs no ff

5.3. Comparison with Gain Scheduling GPC

In this subsection, a performance comparison witha Gain scheduling GPC (GS-GPC) is carried out [57].The GS-GPC control strategy has been tested at the ac-tual plant with very good performance.

Basically the control strategy consists of a general-ized model predictive control strategy which adapts thelinear model (discrete transfer function) with respect tothe flow level. The control strategy uses a series feedfor-ward to reject the disturbances affecting the solar field.For a complete description of the control strategy see[23].

5.3.1. Clear day at stable temperatureOn this day, shown in figure 14, the field is operating

with a stable inlet temperature and the set-points for theoutlet temperature are changed. As can be observed,both control strategies correctly track the temperaturereference, performing well at all flow levels. Slightlyfaster responses are obtained with the MPC controller.

Some transients affect the plant and both controlstrategies reject the disturbance and track the temper-ature reference correctly. The effective solar radiationestimated by the UKF is very close to the measured one.

Time (local hour)12 13 14 15 16 17

Te

mp

era

ture

(ºC

)

110

115

120

125

130

135Comparison between GS-GPC and UKF+MPC

Tout (UKF+MPC)RefTout (GS-GPC)Tin

Time (local hour)12 13 14 15 16 17

Ieff (

W/m

2),

q (

m3/h

)

200

400

600

800

1000

1200

Ieff evolution (W/m2) and water flow (m3/h)Measured*4Estimated*4 (UKF)q*100 (MPC)q*100 (GS-GPC)

Figure 14: Simulation 4: Performance comparison between MPC andGS-GPC: tracking references

13

MPC GS-GPCSim 1 941.1503 1.2619e+03Sim 2 3.6022e+03 3.9478e+03

Table 4: ISE criterion for both simulations: MPC vs GS-GPC

5.3.2. Recirculation modeThis simulation is very similar to the second simula-

tion of the PID case. The plant is working in recircu-lation mode where the outlet temperature returns to theinlet temperature after circulating through a pipe. In thismode, the plant is constantly affected by inlet tempera-ture disturbances which hinder the set-point tracking.

In this case, the MPC achieves faster responses andbetter disturbance rejection. This can be observed intable 4, which shows the ISE performance criterion. Inboth cases, the MPC achieves better performance thanthe GS-GPC strategy.

Local hour14 14.5 15 15.5 16 16.5

Te

mp

era

ture

(ºC

)

50

60

70

80

90

100

110

Temperature Evolution Comparison: MPC vs GS-GPC

MPCGS-GPCTinRef

Local hour14 14.5 15 15.5 16 16.5

Wa

ter

flo

w (

m3/h

)

300

400

500

600

700

800

900

1000q*100 (MPC)q*100 (GS-GPC)

Figure 15: Simulation 5: Performance comparison between MPC andGS-GPC: tracking references in recirculation mode

6. Real Tests

In this section, two tests carried out at the actual Fres-nel collector field are presented. In these tests the waterflow is constrained to 3.7-9 m3/h. The parameters ofthe MPC Nc = 10, Ny = 14 and λ = 4e8. Parameter λis chosen higher than that used in simulations to obtainless aggressive control actions.

Both tests were carried out in recirculation mode. Inthis mode, the heated water is recirculated and returnedto the solar field after passing through a long pipe. Thiskind of disturbance is difficult to deal with. If the con-troller is not properly tuned, oscillations are produced inthe outlet temperature. These oscillations propagate tothe input of the solar field, later producing more oscil-lations. Furthermore, the inlet temperature disturbancedelay depends on the water flow. Taking anticipated ac-tions to reject the inlet temperature disturbance is not aneasy matter.

Figure 16 shows the test carried out on 27/10/2017.It consisted of a series of steps in the temperature ref-erence. The MPC controller tracks the reference well,without being too aggressive with the control actions.At 12.5 h, communication was cut off and the temper-ature reference was the default value of 120 °C. Whencommunication was recovered shortly after, the test wasresumed. The controller tracked all the temperature ref-erences correctly.

Time (local hour)12.5 13 13.5 14 14.5 15

Te

mp

era

ture

(ºC

)

30

40

50

60

70

80

90

100

110

120Test on 27/10/2017: Temperature evolution

ToutRefTin

Time (local hour)12.5 13 13.5 14 14.5 15

Ieff

*4 (

W/m

2),

q*1

00

(m

3/h

)

200

400

600

800

1000

1200Test on 27/10/2017: Ieff estimation*4 and water flow*100

Measured*4 Estimated*4 (UKF)q*100

Figure 16: Test carried out on 27/10/2017

The bottom part of figure 16 shows the water flowand effective solar radiation. As seen, the effective so-lar radiation estimated by the UKF is slightly lower thanthe measured one, probably because some miscalibratedmirrors produced lesser optical efficiency. However,the dynamics estimation is similar to that measured, al-though some oscillations can be observed, produced by

14

the constant changes in outlet temperature, inlet temper-ature and water flow.

Figure 17 shows a test carried out on 13/10/2017. Itwas a cloudy day with strong changes in solar radiation.The environmental conditions did not allow the temper-ature to rise much, but the test showed that the controllercould track the temperature references even under suchadverse conditions. For example, from 15.2 h onwards,the controller could not reach the desired set-point be-cause strong solar radiation disturbances were affectingthe solar field. The controller went to minimum wa-ter flow but, under these conditions, the reference wasunreachable because there was not enough solar energyto heat the water up to the set-point . From 15.6 h to15.75 h, the controller reached the desired reference andrejected the disturbances correctly. From 15.75 h on-wards, the solar radiation levels were very low and theoperation finished.

Time (local hour)14.5 15 15.5 16

Te

mp

era

ture

(ºC

)

40

45

50

55

60

65

70Test on 13/10/2017: Temperature evolution

ToutRefTin

Time (local hour)14.5 15 15.5 16E

ffe

ctive

so

lar

rad

iatio

n (

W/m

2)

0

50

100

150

200

250Test on 13/10/2017: Effective solar radiation

Estimated (UKF)Measured

Figure 17: Test carried out on 13/10/2017

The effective solar radiation estimated by the UKF isa smoother version than the one measured. This can bean advantage on this kind of day, because the measuredsolar radiation is used and very abrupt changes in thewater flow to reject the disturbance may appear [56].

Finally, figure 18 shows the control signal. As can beseen, the changes in the control signal are not aggressivein spite of the strong changes in solar radiation.

Time (local hour)14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16

wat

er f

low

(m

3 /h)

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5Test on 13/10/2017: water flow

Figure 18: Test carried out on 13/10/2017: Water flow signal

7. Concluding Remarks

In this paper, an adaptative model predictive controlstrategy was designed for the Fresnel collector field lo-cated at the Escuela Superior de Ingenieros de Sevilla.The controller uses an unscented Kalman filter as stateestimator which estimates metal-fluid temperature pro-files. The UKF also estimates the effective solar radia-tion reaching the tube.

Simulation results were provided comparing the pro-posed strategy to a PID+feedforward series controller,showing better performance. A comparison to amore sophisticated control strategy, a GS-GPC strat-egy, which has been tested showing very good perfor-mance at the actual plant. The proposed strategy outper-forms these two control strategies, showing good track-ing properties throughout the entire range of operation.

Furthermore, two real tests carried out at the actualplant are presented. In both tests, a wide range of op-erating conditions are covered: temperature set-pointtracking, disturbance rejection, and operating whenstrong radiation disturbances are affecting the plant.

The proposed controller achieves adequate set-pointtracking in spite of strong disturbances. The final out-come can be considered satisfactory.

8. Acknowledgements

The authors would like to acknowledge the EuropeanResearch Council, the Spanish Ministry of Economy,Industry and Competitiveness and the UE ERDF forfunding the work under the Advanced Grant OCON-TSOLAR (Project ID: 789051), Control Predictivo

15

de Microrredes Reconfigurables con AlmacenamientoHıbrido y Movil (DPI2016-78338-R) and ENERPRO(DPI2014-56364-C2-1-R).

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