Optimal Control of Water Systems Under Forecast Uncertainty: Robust, Proactive, and Integrated

Abstract

Water systems consist of natural and man-made objects serving multiple essential purposes. They are affected by many types of meteorological disturbances. In order to deal with these disturbances and to serve the desired objectives, infrastructures have been built and managed by societies for specific purposes. Given a water system, and its purposes, the control of the existing infrastructures is the subject of operational water management. The system controller, either a natural person or a mathematical algorithm, takes his recursive decisions observing the state of the system and trying to bring it to the desired condition. Model Predictive Control (MPC) is an advanced method for the control of complex dynamic systems. When applied to water systems operation, MPC provides integrated and optimal management. If disturbance forecasts are available, this information can be integrated in the control policy and water management becomes proactive. Before the realization of the disturbance, the MPC controller sets the system to a state which is optimal to accommodate the expected disturbance. A typical example is lowering the water level of a reservoir before an expected storm event in order to avoid floods. In proactive control of open water systems, the main uncertainty is generally related to the difficulty of producing good forecasts. Weather and hydrological processes are difficult to predict, and meteorological or rainfall-runoff models can be wrong. Especially when using only one deterministic estimate, the control is more vulnerable to forecast uncertainty, running the risk of taking action against a predicted event that will not occur. The research question of this thesis is: How to use existing forecasting methods in optimal control schemes, thereby enhancing robustness in the face of forecasting uncertainty? In open water systems, such as rivers, canals, or reservoirs, the available forecast is generally the natural inflow, which is the output of a deterministic rainfall-runoff model. The model produces a point estimate, which is the expected value of the variable of interest. Nevertheless, the nonlinearity of the control problem requires the forecast of the entire probability distribution. When residuals are assumed independent, identically distributed, zero-mean, and Gaussian, then the variance is the only extra parameter required to build up the entire distribution, and its value can be estimated from the data. However, residuals of rainfall-runoff models are in fact heteroscedastic (i.e. the variance changes in time) and autocorrelated. In Chapter 2 it is shown how to deal with both deficiencies. Dynamic modelling of predictive uncertainty is built up by regression on absolute residuals, and applied to two test cases: the Rhone River, in Switzerland, and Lake Maggiore, at the border between Italy and Switzerland. When the information on the catchment state does not offer sufficient anticipation, for example because the catchment dynamics are fast compared to the controlled system, it is necessary to include weather forecasts. Meteorological agencies produce not only a deterministic trajectory of the future state of the weather system, but a set of them, called ensemble, to communicate the forecast uncertainty. The algorithm presented in Chapter 3, called Tree-Based Model Predictive Control (TB-MPC), exploits the information contained in the ensemble, setting up a Multistage Stochastic Programming (MSP) problem within the MPC framework. MSP is a stochastic optimization scheme that takes into account not only the present uncertainty, but its resolution in time as well. Going on in time along the control horizon, information will enter the system. Consequently, uncertainty will be reduced, and the control strategy after uncertainty reduction will change according to the occurring ensemble member. The key idea of TB-MPC is producing a tree topology from the ensemble data and using this tree in the following MSP optimization. A tree specifies in fact the moments when uncertainties are resolved. Generating a tree from ensemble data is both difficult and of critical importance. It has been considered an open problem until now, especially regarding the tree branching structure, which also strongly affects control performance. Chapter 4 shows a new methodology that produces a tree topology from ensemble data. The proposed method models the information flow to the controller. This implies the explicit definition of the available observations and their degree of uncertainty. Chapter 5 summarizes the contribution of my PhD and the research directions that, in my opinion, deserve more investigation.