parabolic collector heat storage tank oil tube

Transcription

1 Distributed Model Based Control and Passivity of a Solar Collector Field Tor A. Johansen and Camilla Storaa Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Tor.Arne.Johansen@itk.ntnu.no January 14, 2 Abstract Control of the outlet temperature of a distributed solar collector æeld is considered. A distributed model based controller is derived using internal energy as a storage function and controlled variable. Moreover, it is shown that the plant can be made passive using a feedforward from the solar irradiation. Stability of the closed loop is proved using Lyapunov-like arguments, and the practical usefulness of the method is illustrated by a simulation example using an experimentally veriæed model of a pilot-scale solar power plant. 1 Introduction Energy control of a pilot-scale solar collector æeld, Plataforma Solar de Almeria èpsaè, is studied. A distributed æeld of parabolic collectors focus the solar radiation onto a tube where a æuid èoilè is circulated and heated in order to collect the solar power. An important control problem is to keep the temperature of the oil æow at the outlet of the æeld at its setpoint, despite variations in solar radiation and oil inlet temperature, in order to avoid disturbances downstream to the energy conversion system and to avoid damage due to overheating of the oil. The manipulated variable is the oil volumetric æow rate. 1

2 The distributed solar collector æeld may be described by a distributed parameter model of the temperature èklein et al. 1974, Rorres et al. 198, Orbach et al. 1981, Carotenuto et al. 1985, Carotenuto et al. 1985, Camacho et al. 1997è. Here we suggest a control design based on such a distributed model, using ideas from passivity theory èdesoer and Vidyasagar 1975, Ydstie and Alonso 1997è. The general idea in our work is to use internal energy as a storage function, and then use energy considerations and Lyapunov-like arguments to derive stable and robust control laws relying on feedback from the distributed collector æeld's internal energy. It is shown that if the internal energy is controlled, the outlet temperature is under control as well. In order to achieve passivity and high disturbance rejection performance, the design may also incorporate feedforward from the measured disturbances. Computation of the internal energy relies on knowledge of the distributed temperature parameter of the solar collector æeld, which can be reconstructed using a state estimator based on the distributed parameter model. Rorres et al. è198è and Orbach et al. è1981è suggests an optimal control formulation where the objective is to maximize net produced power when the pumping power is taken into consideration. An alternative approach is taken by ècarotenuto et al. 1985, Carotenuto et al. 1986è, where a quadratic control Lyapunov function is formulated for the distributed parameter model, and a stabilizing control law is derived. The approach presented in this paper is similar, but relies on using a storage function with a physical interpretation leading to a conceptually simpler stabilizing control law with more transparent tuning parameters and a less involved analysis. Other control strategies for this solar power plant based on ænite-dimensional models with experimentally evaluated performance can be found in e.g. ècamacho et al. 1997, Silva et al. 1997, Rato et al. 1997, Meaburn and Hughes 1993, Meaburn and Hughes 1995è and the references therein. This paper is organized as follows: First we give an overview of the plant in section 2 and a mathematical model is introduced in section 3. Passivity is discussed in section 4, before some model-based control strategies are suggested and analysed in section 5. Some aspects of controller implementation, such as state estimation and setpoint proæle computation, are discussed in section 6. Simulation results are shown in section 7 before the conclusions. 2

3 2 Plant Description The ACUREX-æeld of Plataforma Solar de Almeria èpsaè is located in the southern part of Spain, see Figures 1-2. The æeld is composed of 48 distributed solar collectors, arranged in 1 parallel loops. Figure 1: ACUREX, the distributed collector æeld at PSA, Almeria, Spain. A collector uses the parabolic surface to focus the solar radiation onto a receiver tube, which is placed in the focal line of the parabola, cf. Figure 2. The heat-absorbing æuid èoilè is pumped through the receiver tube, causing the æuid to collect heat which is transferred through the receiver tube walls. The thermal energy developed by the æeld is pumped to the top of the thermal storage tank, see Figure 1, whereupon the oil from the top of the storage tank can be fed to a power generating system, a desalination plant or to an oil-cooling system, if needed. The oil outlet from the storage tank to the æeld is at the bottom of the storage tank. To ensure that the collectors give optimum solar absorption, every collector row has a 1 d.o.f. sun tracking system ætted to it. 3

4 Figure 2: Parabolic collector. A control system for this plant has the objective of maintaining the outlet temperature èin this case the average outlet temperature of all the parallel loopsè at a desired value in spite of disturbances like solar irradiation èclouds and atmospheric phenomenaè, irregularities in the sun tracking control system, mirror reæectivity and inlet oil temperature. The oil æow rate is manipulated by the control system through commands to the pump. It should be noticed that the primary energy source, solar radiation, cannot be manipulated. The performance measures of the control system are to keep the oil outlet temperature close to its setpoint. 3 Mathematical Model The dynamics of the distributed solar collector æeld, cf. Figure 3, are described by the following energy balance èt; xè = G c Iètè è1è with boundary condition T èt; è = T in ètè è2è The model variables are the following T èt; xè, oil temperature at position x along the tube è3è qètè, oil pump volumetric æow rate è4è 4

5 x l T inètè T èt; xè T outètè parabolic collector heat storage tank oil tube qètè Figure 3: Sketch of heat collector loop. Iètè, solar radition è5è T in ètè, oil inlet temperature è6è T out ètè, oil outlet temperature è7è and the model parameters are A, tube inner cross-sectional area èm 2 è è8è, mirror optical eæciency è9è G, mirror aperture èmè è1è c, speciæc oil heat capacity èj=k æ kgè è11è, oil mass density èkg=m 3 è è12è l, oil tube length èmè è13è The objective is to control the variable T out ètè = T èt; lè è14è to its speciæed setpoint. The oil volumetric æow rate éq min qètè q max is the control input. The upper constraint q max is due to pump capacity limitations, and the lower constraint q min is a safety limit in order to reduce the possibility of overheating of the oil. Iètè and T in ètè can be viewed as measured disturbances. 5

6 4 Energy Considerations and Passivity Deæne the internal energy Z l Uètè = ct èt; xèadx è15è Assuming time-invariant model parameters, the power equation is Z l dt ètè = èt; è16è and from è1è dt ètè = èt; xè+ GIètè dx è17è =,cqètèèt èt; lè, T èt; èè + GlIètè è18è The interpretation of è18è is that the change in internal energy is balancing the net power transported out of the tube èærst termè and the solar power èsecond termè. Since T in ètè =T èt; è and Iètè are both measured, one may design a feedforward control q ff ètè that cancels the supplied solar power èlast termè from this equation: =,cq ff ètèèt èt; lè, T èt; èè + GlIètè è19è which may be solved for q ff ètè: q ff ètè = Gl cèt èt; lè, T èt; èè Iètè è2è Decomposing qètè into a feedforward q ff ètè and a feedback q fb ètè qètè = q ff ètè+q fb ètè è21è we get dt ètè = è,q fbètèè æ ècèt èt; lè, T èt; èèè è22è = Mètè æ æhètè è23è where the mass æow perturbation is Mètè =,q fb ètè and the speciæc entalphy diæerence is æhètè = cèt èt; lè, T èt; èè. Eq. è23è proves the following passivity result èdesoer and Vidyasagar 1975è: Theorem 1 The system è1è with feedforward è2è and è21è, input M ètè and output æhètè is passive with storage function U ètè. 6

7 If model uncertainty or measurement errors need to be taken into consideration, one might preserve passivity by replacing the q ff ètè given by è2è by a larger æow rate that satisæes the inequality q max q ff ètè Gl cèt èt; lè, T èt; èè Iètè è24è Intuitively, increasing q ff ètè beyond è2è means that more net power is transported out of the tube. Thus, eq. è24è leads to a dissipation inequality: ètè Mètè æ æhètè è25è dt Note that introducing heat losses in the model will not change the passivity property, since for qètè é the losses will contribute with a negative term in the power equation. The feedforward è2è is implementable since all variables in the overall energy balance are known. The input Mètè is related to the oil volumetric æowrate qètè through è21è and the known constant. The output æhètè can be computed since T èt; lè and T èt; è are measured and c is known. Note that feedforward is widely used in the control of distributed solar collector æelds, e.g. èrorres et al. 198, Carotenuto et al. 1986, Camacho et al. 1997è. Since the plant is passive with input Mètè and output æhètè it is clear that any strictly passive feedback from æhètè to M ètè will make the closed loop asymptotically stable èdesoer and Vidyasagar 1975è: Corollary 1 The system è1è with feedforward è2è is BIBO stable with any limited PID feedback from æhètè to Mètè. The stabilizing properties of PID type controllers for this plant are well known from experiments and simulations. It is also widely recognized that the performance with such controllers will be inferior to model based approaches ècamacho et al. 1992, Meaburn and Hughes 1995, Camacho et al. 1997è. However, the design of a model based controller is not straightforward. The two primary reasons for this is that the plant is highly nonlinear èbilinearè as well as of inænite dimension. Even when the plant is linearized about some operating point and approximated by a ænite dimensional model, the frequency response contains anti-resonant modes near the bandwidth that must be taken into consideration in the controller èmeaburn and Hughes 1993è. Thus, to achieve high performance, the controller must be high-order and nonlinear. 7

8 5 Model Based Control Next, we study some control strategies with guaranteed stability that explicitly utilizes the power equation è22è and the distributed parameter model è1è. Note that the passivity property is not explicitly used in the control design or analysis below. Assume we deæne a linear setpoint proæle derived from T æ outètè: T æ èt; xè = T in ètè+ x l èt æ outètè, T in ètèè è26è and deæne the internal energy associated with the setpoint proæle: Z l U æ ètè = ct æ èt; xèadx è27è Eq. è26è corresponds to a linear temperature increase through the tube, and it is easy to prove that è26è is a steady-state solution to è1è for some constant q æ é, if I é, T in and T æ out ét in are time-invariant. Theorem 2 Let qètè be deæned by qètè = K p Z de eètè+t d cèt èt; lè, T èt; èè dt ètèè + 1 t T i eètè = Uètè, U æ ètè = Z l cèt èt; xè, T æ èt; xèèadx eè èd è28è è29è where K p ;T i é, T d, and assume T èt; lè étèt; è for all t. If T in ètè;t æ outètè and Iètè I min é are time-invariant then Uètè! U æ, qètè! q æ, and T èt; xè! T æ èxè for all x 2 ë;lë as t!1. Proof. The power equation è22è becomes æ dt ètè = K pèu æ ètè, Uètèè + K p T d ètè, dt dt ètè + K p T i Z t èu æ èè, Uèèèd + GlIètè Laplace transformation of this linear 2nd order ordinary diæerential equation leads to, è1 + Kp T d ès 2 + K p s + K p =T i æ Uèsè =, Kp T d s 2 + K p s + K p =T i æ U æ èsè+ Gl sièsè è31è è3è Since U æ ètè and Iètè are time-invariant, it follows from è31è that Uètè! U æ and dt ètè! as t! 1. Note that stability of è31è follows from e.g. Hurwitz' criterion since all coeæcients of the left-hand-side polynomial are positive. Since eètè! and de dt ètè! it follows from è28è that dq dt ètè! ast!1, i.e. qètè! qæ é. Next, consider the steady-state solution T æ è1;xè: q è1;xè = Gl c I è32è 8

9 and deæne æètè =q æ, qètè. Introducing the new variable èt; xè =T èt; xè, T æ è1;xè, combining è1è and è32è we get the èt; xè = 1 A T èt; è33è with boundary condition èt; è = and æow velocity v = q æ =Aé. From the results above, we know æètè! ast!1. Since juètèj and æ dt ètè æ are uniformly bounded and I, Tin and T out are bounded, it follows that xèj, jt èt; xèj and jèt; xèj are uniformly bounded as well, and it is clear that the right hand side of è33è is uniformly bounded and asymptotically vanishing, i.e. sup x2ë;lëæ 1 T èt; xè æ! as t!1 è34è The result follows using Lemma 1, see below, since è33è satisæes its assumptions. 2 The following lemma is used in the proof of Theorem 2. It establishes that the partial diæerential equation under consideration with constant æow velocity is ègloballyè asymptotically stable with respect to asymptotically vanishing perturbations èsee ècrawford and Kastenberg 197è for a more general treatment of this topicè. Lemma 1 Consider the inhomogeneous hyperbolic partial èt; xè+v èt; xè = "èt; with boundary condition èt; è=. Suppose vé, j"èt; xèj is uniformly bounded for all x 2 ë;lë and sup x2ë;lë j"èt; xèj! as t!1. Then èt; xè! for all x 2 ë;lë as t!1. Proof. Deæne the Lyapunov-like functional Z d V d è; tè = èt; xèdx è36è where d 2 è; lë is arbitrary, but æxed. Uniform boundedness of jèt; xèj, i.e. boundedness of k = sup jèt; xèj t;x2ë;lë è37è follows from the uniform boundedness of "ètè = sup j"èt; xèj x2ë;lë è38è and it follows that V d ètè is uniformly bounded. Its time-derivative along any solution to è35è is Z d dv d dt ètè = 2 èt; xèdx è39è 9

10 = = Z d Z d èt; èt; è4è èt; =,v, 2 èt; dè, 2 èt; è æ + Z èt; xè+"èt; xè dx è41è èt; xè"èt; xèdx è42è,v 2 èt; dè+kd"ètè è43è where the inequality follows from èt; è = for all t. Let æé be arbitrary. Since "ètè! as t!1there exists a t 1 such that for t t 1 "ètè væ2 kd è44è and consequently for t t 1 dv d dt ètè,v2 èt; dè+væ 2 è45è Since vé and 2 èt; dè is uniformly bounded there exists a t 2 t 1 such that for t t 2 jèt; dèj æ è46è Since æémay be arbitrarily small, èt; dè! ast!1. The result follows since d 2 è;lëis arbitrary, and èt; è = for all t. 2 The feedback è28è in Theorem 2 is a PID feedback with nonlinear ètime-varyingè gain. The diæerence between this PID feedback and the PID feedback considered in Corollary 1 must be emphasized. While Theorem 2 considers feedback from internal energy èa macroscopic variable containing information about the whole distributed æeldè, Corollary 1 considers feedback from the outlet temperature èa microscopic variable containing only information about a single point in the distributed æeldè. The assumptions T èt; lè é T èt; è and Iètè I min é are non-restrictive since this will always hold during normal operation of the plant. The reason for this is that the purpose of the plant is to produce energy in terms of increased temperature of the oil. The above assumption will not necessarily hold at startup and when that solar radition is very low, but to handle such cases it is common practice shut down the plant when the solar power is very low, and in other abnormal situations to rely on a supervisory system that overrides the controller that is used during normal operation. 1

11 Adding a feedforward to this control strategy will be beneæcial from the disturbance rejection performance point of view: Corollary 2 Let qètè be deæned by either Gl qètè = cèt èt; lè, T in ètèè Iètè+ K p Z de eètè+t d cèt èt; lè, T èt; èè dt ètèè + 1 t T i or qètè = Gl cètoutètè æ, T in ètèè Iètè+ K p de eètè+t d cèt èt; lè, T èt; èè dt ètèè + 1 T i Z t eè èd è47è eè èd è48è where K p ;T i é, T d, and assume T èt; lè étèt; è for all t. If T in ètè;t æ outètè and Iètè I min é are time-invariant, then Uètè! U æ, qètè! q æ, and T èt; xè! T æ èxè for all x as t!1. Proof. Consider ærst è47è. The additional feedforward term is time-invariant under the stated assumptions and the power equation reduces to ècf. è3èè: Z dt ètè = K pèu æ æ ètè, Uètèè + K p T d ètè, dt dt ètè + K t p èu æ èè, Uèèèd T i è49è since the feedforward cancels the solar power. The rest of the proof is similar to the proof of Theorem 2. Next, consider è48è. In this case it can be seen that the power equation can be written where q fb ètè = dt ètè = cq fbètèèt èt; lè, T èt; èè + GlI, cq æ èt èt; lè, T èt; èè è5è K p æ èu æ ètè, Uètèè + T d ètè, cèt èt; lè, T èt; èè dt dt ètè Z + 1 t èu æ èè, Uèèèd è51è T i It is clear that è5è is equivalent to the power equation è3è derived in the proof of Theorem 2 since the last term in è5è can be taken into the inital value q fb èè èi.e. the integral term in the PID controllerè. Hence, the result follows from Theorem 2. 2 Note that unlike the time-varying feedforward è47è, the steady-state feedforward term in è48è will not render the system passive. 6 Controller implementation with state estimator A block diagram of the control structure is shown in Figure 4. The controller è48è is represented in the block diagram by a feedforward corresponding to the ærst term, and a PID feedback corre- 11

12 T in I feedforward T æ out - Integrator ~T æ out compute setpoint T æ èxè desired internal U æ e profile energy - ^U PID feedback q plant T out æt out compute internal energy model ^T èxè - ^T out compute outlet temperature Figure 4: Block diagram of control structure. sponding to the second term. This diagram also includes a model-based state-estimator, a reference model used to compute the setpoint proæle, and an outer feedback loop with integral action in order to compensate for unmodelled dynamics or unmeasured disturbances. The outer feedback loop will eæectively reduce steady-state error, which is reasonable since the main unmodelled dynamics are due to neglecting heat losses, which are fairly independent of temperature variations in the plant since the oil temperature is much higher than the ambient temperature during normal operation. The model block contains a real-time numerical integration of the distributed plant model, and its state ^T èt; xè is the estimated temperature in the tube. Spatial discretization intervals at 1 m and temporal discretization intervals at 1 sec is utilized in the integration. For a constant ètime-invariantè setpoint Tout, æ the setpoint proæle T æ èt; xè is a linear function of the spatial variable x, see è26è. This corresponds to a steady-state solution of the plant model. After a setpoint change this setpoint proæle is modiæed such that if there is perfect match between the plant and the model, the PID feedback component is zero, in order to avoid interactions between the two loops. The sampling interval is 3 s, i.e. the pump æow rate is allowed to change every 3 seconds. The controller parameters are K p = 1 l=s=m 3, T d = 15 sec and T i = 6 sec. The integrator in the outer feedback loop has gain.3. The saturation limits of the pump are q min =2l=s and q max =1l=s. 12

13 Nominal stability is guaranteed since the 2nd order diæerential equation for the internal energy is completely speciæed to have desired risetime and damping through the choice of T i and T d. The gain K p is tuned to maximize the disturbance rejection performance. 7 Simulation results This section provides simulation results èboth disturbance rejection and setpoint trackingè using the controller è48è. Two realistic scenaria with experimental disturbances and some model mismatch are included to illustrate the robustness and practical usefulness of the approach. The model è1è with parameter values similar to ècamacho et al. 1997è are used internally in the controller, while the plant is simulated using the model èt; xè = G c Iètè, hèt èt; xè, T è è52è where T is the ambient temperature and héaccounts for the heat transfer through the pipe wall. The last term models heat losses ècorresponding to about 7 è of the solar powerè, the value of is reduced by 5 è compared to the model, and temperature-dependent values of c and ècamacho et al. 1997è are utilized in the plant simulator but not in the internal model in the controller. This leads to some realistic simulation of modelèplant mismatch. In the ærst scenario the experimental solar irradiation and inlet temperature disturbances from 18 May 1998, together with a preprogrammed setpoint proæle, are used as inputs to the simulations, cf. Figure 5. We observe that there are only minor atmospheric disturbances in the solar radiation Iètè, except a light cloud at around 14:15 that causes a deviation in the outlet temperature of less than 1.5 æ C. The main disturbance is the periodic variation in solar power due to the sun's motion on the sky. Figure 6 shows a diæerent scenario èfrom 13 May 1998è, with constant setpoint at 2 æ C but very large disturbances in both solar radiation èseveral large cloudsè and inlet temperature. Note that during the periods with signiæcant disturbances it is evident that the constraints on the oil pump æow rates severly limits the achievable performance, and no control could have signiæcantly better performance. Both examples shows that the model-based controller is able to accurately control the outlet temperature and internal energy. Comparing these results with other simulated and experimental results from this pilot plant, e.g. ècamacho et al. 1997è, we emphasize that the performance with the suggested controller seems to be highly accurate and the controller seems to behave remarkably calm. We believe the use of 13

14 22 setpoint (dashed), outlet temperature (solid), inlet temper ature (dashed dotted) temperatures (C) time (h) oil flow rate (l/s) 8 7 oil flow rate (l/s) time (h) 9 solar radiation (W/m 2 ) 85 8 solar radiation (W/m 2 ) time (h) 6 x 17 setpoint (dashed), computed (solid) internal energy (J) time (h) Figure 5: Simulation scenario 18 May 1998 with experimental disturbances. 14

15 22 setpoint (dashed), outlet temperature (solid), inlet temper ature (dashed dotted) temperatures (C) time (h) oil flow rate (l/s) oil flow rate (l/s) time (h) 1 solar radiation (W/m 2 ) solar radiation (W/m 2 ) time (h) 6.5 x setpoint (dashed), computed (solid) 17 6 internal energy (J) time (h) Figure 6: Simulation scenario 13 May 1998 with experimental disturbances. 15

16 a macroscopic variable such as internal energy as a feedback variable, rather than a microscopic variable such as the temperature at the outlet, contributes strongly to the high performance and calm behaviour since the eæect of the changing the æowrate on the whole distributed collector æeld is taken into consideration when adjusting the æow rate. 8 Conclusions We have presented a nonlinear controller for a solar power plant based on an distributed parameter model of the collector æeld. A conceptually simple control design based on controlling the internal energy of the plant is suggested. The advantage of this approach is that it allows simple and transparent tuning of the nonlinear controller through some PID parameters, and a stability proof is provided. Furthermore, it is shown how the solar collector æeld can be made passive by a feedforward from the solar irradiation. Realistic simulation results are included to show the practical usefulness of the suggested control structure. The similaritybetween the solar collector æeld and e.g. industrial heat exchangers suggests that the control strategy might be useful for other process control applications, see also èydstie and Alonso 1997è where the general idea of exploiting macroscopic thermodynamic variables and passivity in the design and analysis of process control systems is discussed in a much more general framework than this case study. References Camacho, E. F., M. Berenguel and F. R. Rubio è1997è. Advanced Control of Solar Plants. Springer- Verlag, London. Camacho, E. F., R. F. Rubio and F. M. Hughes è1992è. Self-tuning PI control of a solar power plant with a distributed collector æeld. IEEE Control Systems Magazine 12è2è, 72í78. Carotenuto, K., M. La Cava, P. Muraca and G. Raiconi è1986è. Feedforward control for the distributed parameter model of a solar power plant. Large Scale Systems 11, 233í241. Carotenuto, L., M. La Cava and G. Raiconi è1985è. Regulator design for the bilinear distributed parameter of a solar power plant. Int. J. Systems Science 16, 885í9. 16

17 Crawford, R. M. and W. E. Kastenberg è197è. Stability analysis of distributed parameter systems in a Banach space. Int. J. Control 12, 929í943. Desoer, C. A. and M. Vidyasagar è1975è. Feedback Systems: Input-Output Properies. Academic Press, New York. Klein, S. A., J. A. Duæe and W. A. Beckman è1974è. Transient considerations of æat-plate solar collectors. Trans. ASME J. Engng. Power 96A, 19í. Meaburn, A. and F. M. Hughes è1993è. Resonance characteristics of distributed solar collector æelds. Solar Energy 51, 215í221. Meaburn, A. and F. M. Hughes è1995è. Pre-scheduled PID control of a solar thermal power plant. Transactions of the Institute of Measurement and Control 17, 132í142. Orbach, A., C. Rorres and R. Fischl è1981è. Optimal control of a solar collector loop using a distributed-lumped model. Automatica 17, 535í539. Rato, L., D. Borrelli, E. Mosca, J. M. Lemos and P. Balsa è1997è. MUSMAR based switching control of a solar collector æeld. In: Proceedings of the European Control Conference, Brussels. Rorres, C., A. Orbach and R. Fischl è198è. Optimal and suboptimal control policies for a solar collector system. IEEE Trans. Automatic Control 25, 185í191. Silva, R. N., L. M. Rato, J. M. Lemos and F. Coito è1997è. Cascade control of a distributed collector solar æeld. J. Process Control 7, 111í117. Ydstie, B. E. and A. A. Alonso è1997è. Process systems and passivity via the Clausius-Planck inequality. Systems and Control Letters 3, 253í