LQG/LTR Control of the Distributed Collector Field of a Solar Power Plant

6975 6976 fect when the state is accf'ssible. In the literature this procedure is called Loop Transfer Recovery (LTR). The use of this methodology leads to more robust controllers (Doyle and Stein, 1979), (Rubio and Aracil, 1990). The application of this approach is covered and results obtained by simulation using a non-linear model of the plant alld results from the plant itself (lre presented. The paper is organized as follows: in section 2 it brief description of the distributed collector field is given. Section 3 is dedicated to presenting the control structure used. In section 4 the controller design procedure and simulation studies are given. Section 5 presents results obtained when applying the LQG/LTR controller to the distributed solar collector field. Conclusions arc given in section 6. 2. SYSTEM DESCRIPTION The system considered corresponds to the ACUREX distributed collector field of the sSPS solar plant, located in Almcrfa (Southern Spain). The main object.ive of the distributed field is to collect solar energy by heating oil passing through the field. The field consists of 480 distribut.ed solar collectors. These collectors are (lrranged in 20 rows which form 10 parallel loops, as indicated schematically in Fig. 1, and lie along an east west axis. !o.(()KAGf.: TAN'" TO I STEA~1 • GENERATOR • ()R :IWSAIJ:-.lATION I PI.ANT Fig. 1. Schematic Diagram of Collector Field The collector uses a parabolic surface to concentrate a direct normal beam onto the receiver tube which is located at t.he focal point of the parahola. The heat transfer fluid is pumped through the receiver tube and picks up the heat transferred though the receiver t.ube walls. The field is also provided with a tracking system which causes the mirrors to revolve around an axis parallel to that of the pipe enabling the varying inclination of the sun to be followed. The cold inlet oil is extracted from the bottom of the storage tank and is passed through the field using a pump located at the field inlet. This fluid is heated and then introduced into a storage tank to be llsed for electrical energy generation. Tlw system is provided with a three way valve located at the field outlet that allows the oil to be recycled whithin the field until its outlet temperature is adequate for entering into the top of the storage tank. Each of the loops mentioned above is formed by four twelve module collect.ors, suitably connected in series. The loop is 172 metres long, the active part of the loop measuring 142 metres and the passive part 30 metres. 3. CONTROL STRUCTURE High order linear models of the field have been obtained taking into account the frequency response of the plant, showing antiresonance characteristics (Camacho et al., 1994b), but in the development of the robust LQG/LTR controller a low order model is used to show that even when the model uncertainty is high, the robust control scheme proposed provides very good results and a low order controller. In this way, a sirnple linear model for control purposes relating changes in fluid flow, the (ldjustable input variable, to changes in outlet temperature is used (Camacho et al., 1992). Observations of step responses obtained from the plant indicate that in the continuolls time domain behaviour can be closely approximated by a first order transfer function with a time delay: g(8) = esr" _K __ (1 + T8) The time delay Td, time constant T and gain K of the system vary with the oil flow-rate and at the lower operational flowrate the delay is approximately twice that of at the maximum flow rate. It has been shown (Camacho et al., 1992), that one way of accommodating this variation in time delay is to use a model of the form: 3.1 Series Compensation In order to define input and output signals of the LQG /- LTR, some aspects have to be taken into account. The control signal used is the oil flow. The outlet temperature of the plant, however, is also influenced by changes in system variables such as solar radiation and fluid inlet temperature. Hence, dynamically, the outlet temperature To can be expressed as a nonlinear function f of oil flow Uj, solar radiation I and inlet temperature Tin: To = f(uf, I, Tin). The mathematical model which accounts for these additional influences dynamically, even when linearized, is complex, especially when fect when the state is accf'ssible. In the literature this procedure is called Loop Transfer Recovery (LTR). The use of this methodology leads to more robust controllers (Doyle and Stein, 1979), (Rubio and Aracil, 1990). The application of this approach is covered and results obtained by simulation using a non-linear model of the plant alld results from the plant itself (lre presented. The paper is organized as follows: in section 2 it brief description of the distributed collector field is given. Section 3 is dedicated to presenting the control structure used. In section 4 the controller design procedure and simulation studies are given. Section 5 presents results obtained when applying the LQG/LTR controller to the distributed solar collector field. Conclusions arc given in section 6. 2. SYSTEM DESCRIPTION The system considered corresponds to the ACUREX distributed collector field of the sSPS solar plant, located in Almcrfa (Southern Spain). The main object.ive of the distributed field is to collect solar energy by heating oil passing through the field. The field consists of 480 distribut.ed solar collectors. These collectors are (lrranged in 20 rows which form 10 parallel loops, as indicated schematically in Fig. 1, and lie along an east west axis. !o.(()KAGf.: TAN'" TO I STEA~1 • GENERATOR • ()R :IWSAIJ:-.lATION I PI.ANT Fig. 1. Schematic Diagram of Collector Field The collector uses a parabolic surface to concentrate a direct normal beam onto the receiver tube which is located at t.he focal point of the parahola. The heat transfer fluid is pumped through the receiver tube and picks up the heat transferred though the receiver t.ube walls. The field is also provided with a tracking system which causes the mirrors to revolve around an axis parallel to that of the pipe enabling the varying inclination of the sun to be followed. The cold inlet oil is extracted from the bottom of the storage tank and is passed through the field using a pump located at the field inlet. This fluid is heated and then introduced into a storage tank to be llsed for electrical energy generation. Tlw system is provided with a three way valve located at the field outlet that allows the oil to be recycled whithin the field until its outlet temperature is adequate for entering into the top of the storage tank. Each of the loops mentioned above is formed by four twelve module collect.ors, suitably connected in series. The loop is 172 metres long, the active part of the loop measuring 142 metres and the passive part 30 metres. 3. CONTROL STRUCTURE High order linear models of the field have been obtained taking into account the frequency response of the plant, showing antiresonance characteristics (Camacho et al., 1994b), but in the development of the robust LQG/LTR controller a low order model is used to show that even when the model uncertainty is high, the robust control scheme proposed provides very good results and a low order controller. In this way, a sirnple linear model for control purposes relating changes in fluid flow, the (ldjustable input variable, to changes in outlet temperature is used (Camacho et al., 1992). Observations of step responses obtained from the plant indicate that in the continuolls time domain behaviour can be closely approximated by a first order transfer function with a time delay: g(8) = esr" _K __ (1 + T8) The time delay Td, time constant T and gain K of the system vary with the oil flow-rate and at the lower operational flowrate the delay is approximately twice that of at the maximum flow rate. It has been shown (Camacho et al., 1992), that one way of accommodating this variation in time delay is to use a model of the form: 3.1 Series Compensation In order to define input and output signals of the LQG /- LTR, some aspects have to be taken into account. The control signal used is the oil flow. The outlet temperature of the plant, however, is also influenced by changes in system variables such as solar radiation and fluid inlet temperature. Hence, dynamically, the outlet temperature To can be expressed as a nonlinear function f of oil flow Uj, solar radiation I and inlet temperature Tin: To = f(uf, I, Tin). The mathematical model which accounts for these additional influences dynamically, even when linearized, is complex, especially when 6977 considering that the variation of the iIllet temperature (Tin) produces a change in the outlet temperature after a variable delay time (depending on the flow). One simple approach which reduces the cOlllplexity of the model is to make use of the known operating characteristics of the field. Studies have been carried out which incorporated approximate 1 ransfer fUIlctions in the feedforwa.rd term but the benefits provided were not considered to justify the added complexity or uncertainty involved. So, in order to account for the disturbances, a series feedforward controller was introduced (Camacho et al., 1992), directly calculated from steady-state relationships (steady-state energy balance), which makes an adjustment in the flow input, aimed at eliminating the change in outlet temperat1ll'c caused by the variations in solar radiation and inlet temperature. The calculation employed is: 0.7869[- 0.185(u-151.5)-80.7 uf = (u - Tin) where "I is the oil flow, If is the temperature set-point given by the LQG/LTR, Tin is the inlet oil temperature and I is the effective solar radiation. The feedforward is placed in series with the LQC/LTR. and thus, the output signal of the LQG/LTR controller is the increment in the set point temperature for the feedforward term and not the oil flow (this is calculated by the feedforward controller) . 3.2 LQG/LTR Controller It has been observed that the Linear Quadratic Gaussian Controller (LQG) method worked well when very precise mathematical models were used, but the method was extremely sensitive to imprecisions in the parameters and to structural modifications. One outstanding property is that the closed loop transfer function obtained when the state is directly accessible, so that the Kalman filter (KBF) is not necessary, is the same as if this last module were included. That is, the closed loop is the same as if the observer were inexistent; there is no influence on the global transfer function of the system. This gave the idea that the observer was irrelevant in the design of the closed loop system. However, as was discovered at the end of the seventies, although it is true that the observer does not affect the closed loop transfer function it noticeably affects the open loop transfer function. This influence can be extremely damaging to the robustness of the system (in fact it can greatly reduce the stability margins). The LQR and LQG regulators Given the plant model: x(k + 1) = Ax(k) + BU(k) y(k) = Cx(k) The purpose of the LQR regulator is to determine the control Rignal U in order to minimize the following functional: N-l J = x~HcXN + L (xT(k)Qcx(k) +uT(k)Rcu(k)) k=O x and XN being the state and the final state vectors respectively. The LQG problem is presented which is based on the separation theorem, according to which the control strategy call be divided into two parts: (1) An optimum control problem, from which the regulation by feedback of state variables is obtained. (1) Kc is found by iteration of the well known Riccati equation. (2) An optimum filtering problem which solves the observation using the Kalman filter, whose corresponding dynamic is given by the system: x(k + 1) = Ax(k) + BU(k) + Ko(k + 1) [y(k + 1) - C(Ax(k) + BU(k»)] (2) where K, is obtained from a Riccati equation dual to the previous one. The resulting structure is shown in Fig. 2. HJ-"'-~.~ G 1,~ Fig. 2. Block diagram of the LQG regulator Loop Transfer Recovery (LTR) y ) T", y, Asymptotic recovery techniques have been developed for continuous time, minimum phase systems (Doyle and Stein, 1979). There are design techniques which allow for excellent robustness properties. To dispose of a similar procedure for discrete time systems is, therefore, desirable, even though the LQR discrete controller does not considering that the variation of the iIllet temperature (Tin) produces a change in the outlet temperature after a variable delay time (depending on the flow). One simple approach which reduces the cOlllplexity of the model is to make use of the known operating characteristics of the field. Studies have been carried out which incorporated approximate 1 ransfer fUIlctions in the feedforwa.rd term but the benefits provided were not considered to justify the added complexity or uncertainty involved. So, in order to account for the disturbances, a series feedforward controller was introduced (Camacho et al., 1992), directly calculated from steady-state relationships (steady-state energy balance), which makes an adjustment in the flow input, aimed at eliminating the change in outlet temperat1ll'c caused by the variations in solar radiation and inlet temperature. The calculation employed is: 0.7869[- 0.185(u-151.5)-80.7 uf = (u - Tin) where "I is the oil flow, If is the temperature set-point given by the LQG/LTR, Tin is the inlet oil temperature and I is the effective solar radiation. The feedforward is placed in series with the LQC/LTR. and thus, the output signal of the LQG/LTR controller is the increment in the set point temperature for the feedforward term and not the oil flow (this is calculated by the feedforward controller) . 3.2 LQG/LTR Controller It has been observed that the Linear Quadratic Gaussian Controller (LQG) method worked well when very precise mathematical models were used, but the method was extremely sensitive to imprecisions in the parameters and to structural modifications. One outstanding property is that the closed loop transfer function obtained when the state is directly accessible, so that the Kalman filter (KBF) is not necessary, is the same as if this last module were included. That is, the closed loop is the same as if the observer were inexistent; there is no influence on the global transfer function of the system. This gave the idea that the observer was irrelevant in the design of the closed loop system. However, as was discovered at the end of the seventies, although it is true that the observer does not affect the closed loop transfer function it noticeably affects the open loop transfer function. This influence can be extremely damaging to the robustness of the system (in fact it can greatly reduce the stability margins). The LQR and LQG regulators Given the plant model: x(k + 1) = Ax(k) + BU(k) y(k) = Cx(k) The purpose of the LQR regulator is to determine the control Rignal U in order to minimize the following functional: N-l J = x~HcXN + L (xT(k)Qcx(k) +uT(k)Rcu(k)) k=O x and XN being the state and the final state vectors respectively. The LQG problem is presented which is based on the separation theorem, according to which the control strategy call be divided into two parts: (1) An optimum control problem, from which the regulation by feedback of state variables is obtained. (1) Kc is found by iteration of the well known Riccati equation. (2) An optimum filtering problem which solves the observation using the Kalman filter, whose corresponding dynamic is given by the system: x(k + 1) = Ax(k) + BU(k) + Ko(k + 1) [y(k + 1) - C(Ax(k) + BU(k»)] (2) where K, is obtained from a Riccati equation dual to the previous one. The resulting structure is shown in Fig. 2. HJ-"'-~.~ G 1,~ Fig. 2. Block diagram of the LQG regulator Loop Transfer Recovery (LTR) y ) T", y, Asymptotic recovery techniques have been developed for continuous time, minimum phase systems (Doyle and Stein, 1979). There are design techniques which allow for excellent robustness properties. To dispose of a similar procedure for discrete time systems is, therefore, desirable, even though the LQR discrete controller does not 6978 6979 6980