Stochastic systems with embedded parameter variations. Applications to district heating

Henning T. Søgaard

This thesis is about statistical models and methods for stochastic systems with embedded parameter variations. This includes a discussion of dynamic state-space models in continuous time with a view to description, forecasting and control. The embedded parameters are the physical parameters being embedded in the system, and embedded parameter variations refer to variation in time of the embedded parameters.

Most of the treated statistical models and methods are to some extent motivated from a need for automatic tools for forecasting and control in connection with district heating systems. However, approaches which do not assume very specific technical or physical description of the considered system are emphasized. Thus the methods are applicable to a wide class of dynamic systems. Furthermore, models and algorithms which are operatiional and easy to implement are emphasized.

In Chapter 2 some methods for adaptive estimation of time-varying time-delays and dynamics are discussed. First three algorithms based upon recursive estimation with exponential forgetting and next two methods based upon explicit models of the embedded parameter variations are proposed. All the methods assume that the basic model belongs to an ARIMAX model structure. The methods have been tested with simulated data, data from a district heating system in Ish›j and Swedish business cycle data. The simulated data represents a slowly varying system and the variations of the time-delay are tracked with close accuracy. Concerning the district heating data, the results from two of the methods indicate that the time-delay is exposed to large diurnal variations. However, the presence of these large variations results in bad performance of a third methods. The results of the experiments with Swedich business cycle data show certain variations of the time-delays between various economic indicators. Furthermore, the experiments illustrate that the proposed methods can be applied in technically very different fields.

Chapter 3 is about modelling and forecasting of ambient air temperature. In the first part forecast procedures based on exponential smoothing are described and applied. The procedures are applied for prediction of the air temperature up to 24 hours ahead. Various aspects of Winters' seasonal forecast procedure are reviewed, and four alternative forecast procedures which can be considered as modified versions of Winters' forecast procedure are proposed Two of them are non-linear procedures which turn out to give very good prediction results. However, as expected a traditional ARIMA model provides closer prediction accuracy for short prediction horizons (here up to 9 hours ahead). Investigations of a method which combines predictions from two simple exponential smoothing procedures through an adaptively estimated regression model have also been done. This method gives even better results than all the previously mentioned methods for prediction horizons longer than 5 hours.

In the second part of Chapter 3 a local climatic system is modelled by means of linear state-space models in continuous time. Discrete time data is used and the imbedded parameters of the continuous time model are estimated by a maximum likelihood method. The state-space models are obtained by approximating the distributed system by heat capacities and thermal resistances. It is assumed that the primary input of the system is the net radiation. The deviations from the real system and the measurement errors are described through stochastic terms in the model. The data consists of hourly measurements of climatic variables from a period of almost eight years. The estimates of the physically interpretable parameters are discussed, and it is concluded that some of the estimates are in accordance with the expected values while others deviate significantly from the expected values.

Chapter 4 is about multi-step predictive control with special emphasis laid upon controllers for systems with embedded parameter variations. First the use of multi-step predictive control in connection with district heating systems is motivated. Next a weighted predictive control strategy is described. This strategy can be used in case the dynamic relationship between input and output is not sufficiently described for application of generalized predictive control (GPC). The strategy is implemented and applied in the district heating system in Esbjerg/Varde, and results from this implementation are discussed. After this the traditional GPC strategy which does not allow embedded parameter variations in the model is reviewed. Then an extended strategy which permits multi-step predictive control of systems with embedded parameter variations is proposed. This extended strategy implies a generali- zation of the loss function which defines the optimality of the control. Furthermore, it is possible to expose the control signal to general equality constraints. It is shown that the traditional GPC is a special case of the extended control strategy. Finally, simulation experiments with multi-step generalized predictive control are performed. The results show that the proposed strategy leads to significantly better control than minimal variance control.

IMSOR Ph.D Thesis 66, 1993

Last modified August 18, 1996
Finn Kuno Christensen

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