Modelling Free Response of a Solar Plant for Predictive Control

Copyright © IFAC System Identification, Kitakyushu, Fukuoka,Japan,1997 MODELLING FREE RESPONSE OF A SOLAR PLANT FOR PREDICTIVE CONTROL* Manuel Berenguel, Manuel R. Arahal and Ed uardo F. Camacho Departamento de Ingenieria de Sistemas y Automatica, Universidad de Se villa, Escuela Superior de Ingenieros, Avda. Reina Mercedes sin, 41012 Se villa, Spain Abstract: This paper deals with the identification of a nonlinear plant by means of a neural network (NN) modelling approximation. The problem of neural identification is tackled using a static NN in a NARX configuration. A method is proposed to obtain the number of past values needed to feed the network. The on-line adaptation of the model and other issues are discussed. In order to show the benefits that can be achieved with the proposed methods, the NN model is used within a Model Predictive Control (MPC) framework. The MPC scheme uses the prediction of the output of the system calculated as the sum of the free response (obtained using the nonlinear NN model) and the forced response (obtained linearizing around the current operating point) to optimize a performance index. The control scheme has been applied and tested in a solar power plant. Keywords: non-linear identification, neural networks, sotar 'Power plant. 1. INTRODUCTION A key feature of the MPC schema is the necessity of a reliable model of the plant being controlled. The methodology of MPC for linear plants has to be changed when coping with nonlinear behaviour, because the superposition principle does not hold. Two aspects have to be revised: the model used for the output prediction and the optimization procedure to obtain the desired control signal. Model for nonlinear output prediction- The nonlinear behaviour of the plant can be taken into account by using a nonlinear model. Neural network approaches constitute an adequate way to 'Work supported in part by CICYT Contract #TAP- 95-0370 and IBERDROLA (Program" Ayudas a la Investigaci6n Cientifica y al Desarrollo Tecnol6gico" - Proyecto 036/96). The experiments described in this paper were performed within the project "Enhancement and Development of Industrial Applications of Solar Energy Technologies ", supported by EEC Program " Human Capital and Mobility - Large Installations Program " and promoted by CIEMAT - Plataforma Solar de Almeria, Spain. 1241 obtain such kind of models. They have been applied to the identification of dynamical systems (Narendra and Parthasarathy, 1990) and used in a variety of model based controllers. When using static neural networks to predict future outputs of a system based on input-output information a problem is to determine the number of past values that the network needs to construct the prediction. A new selection method is proposed in this paper. Off-line identification allows to develop and test a model before using it . It has the disadvantage that a large amount of data has to be stored. Selection of training examples among the available data has the benefits of reducing the training time and improving generalization. A simple procedure to select examples is commented in the paper. After an off-line training phase, the network needs further on-line training. Fast on-line adaptation is achieved adjusting only a sub-network that faces small discrepancies and temporal changes. Copyright © IFAC System Identification, Kitakyushu, Fukuoka,Japan,1997 MODELLING FREE RESPONSE OF A SOLAR PLANT FOR PREDICTIVE CONTROL* Manuel Berenguel, Manuel R. Arahal and Ed uardo F. Camacho Departamento de Ingenieria de Sistemas y Automatica, Universidad de Se villa, Escuela Superior de Ingenieros, Avda. Reina Mercedes sin, 41012 Se villa, Spain Abstract: This paper deals with the identification of a nonlinear plant by means of a neural network (NN) modelling approximation. The problem of neural identification is tackled using a static NN in a NARX configuration. A method is proposed to obtain the number of past values needed to feed the network. The on-line adaptation of the model and other issues are discussed. In order to show the benefits that can be achieved with the proposed methods, the NN model is used within a Model Predictive Control (MPC) framework. The MPC scheme uses the prediction of the output of the system calculated as the sum of the free response (obtained using the nonlinear NN model) and the forced response (obtained linearizing around the current operating point) to optimize a performance index. The control scheme has been applied and tested in a solar power plant. Keywords: non-linear identification, neural networks, sotar 'Power plant. 1. INTRODUCTION A key feature of the MPC schema is the necessity of a reliable model of the plant being controlled. The methodology of MPC for linear plants has to be changed when coping with nonlinear behaviour, because the superposition principle does not hold. Two aspects have to be revised: the model used for the output prediction and the optimization procedure to obtain the desired control signal. Model for nonlinear output prediction- The nonlinear behaviour of the plant can be taken into account by using a nonlinear model. Neural network approaches constitute an adequate way to 'Work supported in part by CICYT Contract #TAP- 95-0370 and IBERDROLA (Program" Ayudas a la Investigaci6n Cientifica y al Desarrollo Tecnol6gico" - Proyecto 036/96). The experiments described in this paper were performed within the project "Enhancement and Development of Industrial Applications of Solar Energy Technologies ", supported by EEC Program " Human Capital and Mobility - Large Installations Program " and promoted by CIEMAT - Plataforma Solar de Almeria, Spain. 1241 obtain such kind of models. They have been applied to the identification of dynamical systems (Narendra and Parthasarathy, 1990) and used in a variety of model based controllers. When using static neural networks to predict future outputs of a system based on input-output information a problem is to determine the number of past values that the network needs to construct the prediction. A new selection method is proposed in this paper. Off-line identification allows to develop and test a model before using it . It has the disadvantage that a large amount of data has to be stored. Selection of training examples among the available data has the benefits of reducing the training time and improving generalization. A simple procedure to select examples is commented in the paper. After an off-line training phase, the network needs further on-line training. Fast on-line adaptation is achieved adjusting only a sub-network that faces small discrepancies and temporal changes. Optimization procedure- When nonlinear models are used within a MPC framework, computationally demanding and generally nonconvex nonlinear programs (NLP) arise (Camacho and Bord6ns, 1995). Some algorithms have appeared in the literature to obtain less computationally demanding algorithms (de Oliveira and Morari, 1994). In this paper, the approach (proposed by authors in (Camacho and Berenguel, 1994)) used is based on a generalized predictive control (GPC) strategy. When coping with linear systems, a natural division of the system response in free and forced response can be obtained. The free response is interpreted here as the output obtained if the system input is maintained at a constant value during certain control and prediction horizons. If the deviation from linearity is not too large, some approximations can be made, which acknowledge that certain systems characteristics change from operating point to operating point, but assuming linearity in the neighborhood of a specific operating point. The algorithm used in this paper relies on the allowance of an adequate nonlinear model of the plant, obtained using static neural networks. The paper is organized as follows: in section 2 a the neural identification problem is presented along with the proposed algorithm for selecting the inputs. Section 3 is devoted to describe the solar plant. In section 4 the controller structure which uses the neural model is defined. Finally, some issues regarding the application to the distributed collector field are discussed, including plant results. 2. NEURAL IDENTIFICATION The problem we are interested in is the identification of nonlinear systems of the form: YP(k+l) = h(YP(k) ,'" YP(k-L), U(k),'" U(k-M), P(k),'" P(k-N)) (1) where U(k) is the system's input vector at time k, YP(k) is the output vector, P(k) is a vector of measurable disturbances and h is an unknown nonlinear function. For the identification of plant (1), a neural network model like the one depicted in Fig. 1 can be used. Tapped Delay Lines (TDL) provide past values of the variables. This structure is called NARX model. The prediction is given by: y(k + 1) = f(YP(k),'" YP(k-P), U(k),'" U(k_Q), P(k),'" P(k-R), Wf) (2) 1242 where Wf is a vector containing all the parameters in the neural network. Two problems have to be solved. First of all, a model has to be selected among the family of NARX models. This is the model selection problem and includes selecting the number of nodes, connections, etc. of the NN. A second step is to give a value to the adjustable parameters of the model. This is called the parameter estimation problem. The model selection problem is tackled here in two steps: first the inputs that feed the network are selected, later, the neural structure is determined. 2.1 Selection of past values as network inputs. The temporal domain is considered treating past. values of variables as different inputs and feeding them to a static network. Taking an upper bound of the order (when it is unknown) can lead to inefficient models due to the large number of inputs needed. A new method is proposed, based on the descent in the gradient needed to explain the observed output when a new TDL is added. The method chooses the variable that provides the largest descent at each stage, determining P, Q and R in (2) so as to obtain a good approximation to the "supposed true" plant (1). For real processes, the larger P, Q and R , the more accurate the approximation can be made. It is clear, however, that an arbitrary large number of inputs can turn our problem insolvable. Furthermore , in order to avoid over-parametrization it is convenient to keep the number of adjustable parameters as small as possible. To expose the idea that supports the method, let us assume that the components of vector x XLMN(k) = [YP(k)"" YP(k-L), U(k),'" U(k-M), P(k),' .. P(k-N)] (3) suffice to construct the prediction as f)(k + 1) = f (XLM N(k))' If one tries to approximate function f using an input vector XPQR, with dimension less than dim(xL,M,N), then it is likely to find a couple of values for XL,M,N that produces the same value for XPQR. In this case, the model based on XPQR will yield the same prediction and the real plant will give different outputs. From the point of view of the model, the output of the system is multivalued. The number of past values of u, yp and p can be increased until the ambiguity in the output of the model disappears. For a couple of instancies of vector XPQR, such as e, e the gradient of function f along the line that joins both points b t· t d 1't"7 f 1- Iv 1 -v 21 'd d can e es 1ma e as v 12 - lie-ell' prov1 e that the distance lie -ell is small enough and Optimization procedure- When nonlinear models are used within a MPC framework, computationally demanding and generally nonconvex nonlinear programs (NLP) arise (Camacho and Bord6ns, 1995). Some algorithms have appeared in the literature to obtain less computationally demanding algorithms (de Oliveira and Morari, 1994). In this paper, the approach (proposed by authors in (Camacho and Berenguel, 1994)) used is based on a generalized predictive control (GPC) strategy. When coping with linear systems, a natural division of the system response in free and forced response can be obtained. The free response is interpreted here as the output obtained if the system input is maintained at a constant value during certain control and prediction horizons. If the deviation from linearity is not too large, some approximations can be made, which acknowledge that certain systems characteristics change from operating point to operating point, but assuming linearity in the neighborhood of a specific operating point. The algorithm used in this paper relies on the allowance of an adequate nonlinear model of the plant, obtained using static neural networks. The paper is organized as follows: in section 2 a the neural identification problem is presented along with the proposed algorithm for selecting the inputs. Section 3 is devoted to describe the solar plant. In section 4 the controller structure which uses the neural model is defined. Finally, some issues regarding the application to the distributed collector field are discussed, including plant results. 2. NEURAL IDENTIFICATION The problem we are interested in is the identification of nonlinear systems of the form: YP(k+l) = h(YP(k) ,'" YP(k-L), U(k),'" U(k-M), P(k),'" P(k-N)) (1) where U(k) is the system's input vector at time k, YP(k) is the output vector, P(k) is a vector of measurable disturbances and h is an unknown nonlinear function. For the identification of plant (1), a neural network model like the one depicted in Fig. 1 can be used. Tapped Delay Lines (TDL) provide past values of the variables. This structure is called NARX model. The prediction is given by: y(k + 1) = f(YP(k),'" YP(k-P), U(k),'" U(k_Q), P(k),'" P(k-R), Wf) (2) 1242 where Wf is a vector containing all the parameters in the neural network. Two problems have to be solved. First of all, a model has to be selected among the family of NARX models. This is the model selection problem and includes selecting the number of nodes, connections, etc. of the NN. A second step is to give a value to the adjustable parameters of the model. This is called the parameter estimation problem. The model selection problem is tackled here in two steps: first the inputs that feed the network are selected, later, the neural structure is determined. 2.1 Selection of past values as network inputs. The temporal domain is considered treating past. values of variables as different inputs and feeding them to a static network. Taking an upper bound of the order (when it is unknown) can lead to inefficient models due to the large number of inputs needed. A new method is proposed, based on the descent in the gradient needed to explain the observed output when a new TDL is added. The method chooses the variable that provides the largest descent at each stage, determining P, Q and R in (2) so as to obtain a good approximation to the "supposed true" plant (1). For real processes, the larger P, Q and R , the more accurate the approximation can be made. It is clear, however, that an arbitrary large number of inputs can turn our problem insolvable. Furthermore , in order to avoid over-parametrization it is convenient to keep the number of adjustable parameters as small as possible. To expose the idea that supports the method, let us assume that the components of vector x XLMN(k) = [YP(k)"" YP(k-L), U(k),'" U(k-M), P(k),' .. P(k-N)] (3) suffice to construct the prediction as f)(k + 1) = f (XLM N(k))' If one tries to approximate function f using an input vector XPQR, with dimension less than dim(xL,M,N), then it is likely to find a couple of values for XL,M,N that produces the same value for XPQR. In this case, the model based on XPQR will yield the same prediction and the real plant will give different outputs. From the point of view of the model, the output of the system is multivalued. The number of past values of u, yp and p can be increased until the ambiguity in the output of the model disappears. For a couple of instancies of vector XPQR, such as e, e the gradient of function f along the line that joins both points b t· t d 1't"7 f 1- Iv 1 -v 21 'd d can e es 1ma e as v 12 - lie-ell' prov1 e that the distance lie -ell is small enough and