Parallel methods for solving stochastic optimal control problems: control of drinking water networks
Creators: name: Sampathirao, Ajay Kumar
Publisher: name: IMT Alti Studi Lucca
Year: 2016
Subjects: QA75 Electronic computers. Computer science

description: This thesis is concerned with the development of optimisation methods to solve stochastic Model Predictive Control (MPC) problem and employ them in the management of DrinkingWater Networks (DWNs). DWNs are large-scale, complex both in topology and dynamics, energy-intensive systems subjected to irregular
 demands. Managing these networks play a crucial role in the economic sustainability of urban cities. The main challenge associated with such infrastructures is to minimise the energy required for pumping water while simultaneously maintaining
 uninterrupted water supply. State-of-the-art control 
 methodologies as well as the current engineering practices use predictive models to forecast upcoming water demands but do not take into consideration the inevitable forecasting error. This way, the water network is operated in a deterministic fashion
 disregarding its inherent stochastic behaviour which accrues from the volatility of water demand and, often, electricity prices. In this thesis, we address two challenges namely: optimisation methods for solving stochastic MPC problems and closedloop feedback control for the management of drinking
 water networks. MPC is an advanced control technology that copes with complex control problem by repeatedly solving a finite horizon constrained optimal control problem; uses only the first decision as input and discards the rest of the sequence. This methodology decides the control action based on present state of
 the system and thus provides an implicit feedback to the system. Instead of historical demand profile, time-series models were developed to forecast the future water demand. The economic and the social aspects involved in operation of the DWN were 
 captured in a cost function. Now the MPC controller combined with online forecaster minimise the cost function across a prediction horizon of 1 day with sampling time equal to 1 hour and thus the closed-loop strategy for DWN management is devised.
 The forecasts are just nominal demands and differ from the actual demands. There exist several approaches when it comes to working with uncertain forecasts: (i) to assume that forecast errors are negligible and disregard them, (ii) to assume knowledge of their worst-case values (maximum errors), (iii)
 to assume knowledge of probabilistic information. These three approaches lead to the three principal flavours of MPC: the certainty-equivalent (CE), the worst-case robust and the stochastic MPC. CE-MPC is simple but not realistic (because the errors are not negligible), worst-case MPC is more meaningful but
 it is too conservative (because it is highly improbable that the errors admit their worst-case values) and then we have stochastic MPC which is the approach pursued in this thesis. A stochastic MPC allows a systematic framework as trade-off performance against constraint violation by modelling the uncertainty as stochastic process and quantifying its influence. However, this
 formulation is an infinite dimensional optimisation problem and its corresponding discrete approximation is deemed to be a large-scale problem with millions of decision variables. Therefore, the applicability of stochastic MPC in control applications is
 limited due to the unavailability of algorithms that can solve them efficiently and within the sampling time of the controlled system. Here we developed optimisation algorithms that solve
 stochastic MPC problem by exploiting their structure and using parallelisation. These algorithms are (i) accelerated proximal gradient algorithm also known as forward-backward splitting and (ii) LBFGS method for forward-backward envelope (FBE) function.
 Both these algorithms employ decomposition to solve the Fenchel dual and make them suitable for parallel implementation. Graphics processing units (GPUs) are capable of perform parallel  computation and are therefore perfect hardware to solve
 the stochastic MPC problem with the accelerated proximal gradient method. The water network of the city Barcelona is considered to study the validity of the proposed algorithm.
 The GPU implementation is found to be 10 times faster than commercial solvers like Gurobi running in multi-core environment and made the problem computationally tractable in the sampling time. The efficiency of the stochastic MPC to manage theDWN
 is quantified in terms of key performance indicators
 like economic utility, network utility and quality of
 service. The forward-backward splitting is a first-order method
 and has slow convergence for ill-conditioned problems. We constructed a continuously differentiable real-valued forward-backward envelope function that has the same set of minimisers as the actual problem. Then we use quasi-Newton method, in particular LBFGS method, that utilises secondorder information to solve the FBE. The computations with this algorithm are also parallelisable and it demonstrated fast convergence compared to accelerated dual proximal gradient algorithm.