Online prediction and control in nonlinear stochastic systems

Online prediction and control in nonlinear stochastic systems The present thesis consists of a summary report and ten research papers. The subject of the thesis is on-line prediction and control of non-linear and non-stationary systems based on stochastic modelling. The thesis consists of three parts where the rst part deals with on-line estimation in linear as well as non-linear models and advances a class of non-linear models which are particularly useful in the context of on-line estimation. The second part considers various aspects of using predictive controllers in connection with control of supply temperature in district heating systems { a class of systems which are inherently non-stationary. The third part concerns the issue of predicting the power production from wind turbines in the presence of Numerical Weather Predictions (NWP) of selected climatical variables. Here the transformation through the wind turbines from (primarily) wind speed to power production give rise to non-linearities. Also the nonstationary characteristics of the weather systems are considered. The summary report commences by considering some aspects of on-line estimation of linear as well as non-linear models for time-varying and/or non-stationary system. In the following chapters the presented papers are brought into their corresponding context with respect to optimal control of supply temperature in district heating systems and prediction of power production from wind turbines located in a given geographical area. The papers A to C focus primarily on issues regarding modelling and estimation. Paper A considers on-line estimation of conditional parametric models and a new model class Conditional Parametric Auto-RegressiveeXtraneous (CPARX) is suggested. The models are estimated using local polynomial regression an estimation method with close resemblance to that of ordinary linear Least Squares (LS) regression. Furthermore it is shown how the relationship between supply temperature and network temperature in a district heating system can be described using CPARX models. In paper B a method for on-line or adaptive estimation of time-varying CPARX models is proposed. Essentially the method is a combination of Recursive Least Squares (RLS) with exponential forgetting and local polynomial regression. The paper also suggests a method for varying the forgetting factor in order to avoid ushing information from the model in seldomly visited regions of data. These methods are a prerequisite for employing the various conditional parametric models considered in the papers in on-line applications. Paper C considers on-line estimation of linear time-varying models with a partly known seasonal variation of the model parameters. An estimation method based on local polynomial regression in the dimension of time is suggested and it is indicated that the new method is superior to ordinary RLS, if the parameter variations are smooth. Paper D presents two predictive controllers eXtended Generalized Predictive Controller (XGPC) and a predictive controller derived using a physical relation and considers the various issues arising when the two controllers are applied in district heating systems with the purpose of controlling the supply temperature. The proposed controllers are mplemented in a software system PRESS and installed at the district heating system of Høje Tåstrup in the Copenhagen area, where it is demonstrated that the system can indeed lower the supply temperature without sacri cing the safe operation of the system or consumer satisfaction. The PRESS control system is also the subject of paper E. Here the results obtained for a PRESS installation at the district heating utility of Roskilde is evaluated with respect to energy and monetary savings as well as security of supply. The papers F to J consider prediction of wind power. Paper F proposes a new reference predictor as a supplement or replacement for the often used persistence predictor. It is shown in the paper, that it is not reasonable to use the persistence predictor for prediction horizons exceeding a few hours. Instead a new statistical reference for predicting wind power, which basically is a weighting between the persistence and the mean of the power, is proposed. The papers G , H and J investigate models and methods for predicting wind power from a wind farm on basis of observations and numerical weather predictions. All three papers consider multistep prediction models, but uses di erent estimation methods as well as dierent models for the diurnal variation of wind speed and the relationship between (primarily) wind speed and wind power (the power curve). In paper G the model parameters are estimated using a RLS algorithm and any systematic time-variation of the model parameters is disregarded. Two di erent parameterizations of the power curve is considered { by a double exponential Gompertz model or by a Hammerstein model { and the diurnal variation of wind speed is explained directly in the prediction models using a rst order Fourier expansion. In paper H the model parameters are assumed to exhibit a systematic time-variation and the model parameters are estimated using the algorithm proposed in paper C. The power curve and the diurnal variation of wind speed is estimated separately using the local polynomial regression procedure described in paper A . In paper J the parameters of the prediction model is assumed to be smooth functions of wind direction (and prediction horizon) and the functions are estimated recursively and adaptively using the algorithm proposed in paper B. As in paper G the diurnal variation of wind speed is taken into account directly in the prediction model using a first order Fourier expansion, whereas the power curve is estimated separately. One of the prediction models considered in paper G { the model based on a Hammerstein parametrization of the power curve { is implemented in a software system { WPPT { and installed at the control centres of Elsam and Eltra, the power production and transmission utilities in the Jutland/Funen area, respectively. Predictions of wind power for the Jutland/Funen area are calculated by upscaling predictions from 14 wind farms in the area to cover the total production. Paper I describes WPPT as used by Eltra and Elsam and evaluates the predictions of wind power for the total area. Three cases are analyzed in order to illustrate, how the operators use the predictions and with which consequences. It is concluded that WPPT generally produces reliable predictions, which are used directly in the economic load dispatch and the day to day power trade.