Distributed solution of stochastic optimal control problems on GPUs

Stochastic optimal control problems arise in many applications and are, in principle, large-scale involving up to millions of decision variables. Their applicability in control applications is often limited by the availability of algorithms that can solve them efficiently and within the sampling time of the controlled system. In this paper we propose a dual accelerated proximal gradient algorithm which is amenable to parallelization and demonstrate that its GPU implementation affords high speed-up values (with respect to a CPU implementation) and greatly outperforms well-established commercial optimizers such as Gurobi

Water demand forecasting for the optimal operation of large-scale drinking water networks: The Barcelona case study

Trabajo presentado al 19th IFAC World Congress celebrado del 24 al 29 de agosto de 2014 en Cape Town (Sudafrica).Drinking Water Networks (DWN) are large-scale multiple-input multiple-output systems with uncertain disturbances (such as the water demand from the consumers) and involve components of linear, non-linear and switching nature. Operating, safety and quality constraints deem it important for the state and the input of such systems to be constrained into a given domain. Moreover, DWNs' operation is driven by time-varying demands and involves an considerable consumption of electric energy and the exploitation of limited water resources. Hence, the management of these networks must be carried out optimally with respect to the use of available resources and infrastructure, whilst satisfying high service levels for the drinking water supply. To accomplish this task, this paper explores various methods for demand forecasting, such as Seasonal ARIMA, BATS and Support Vector Machine, and presents a set of statistically validated time series models. These models, integrated with a Model Predictive Control (MPC) strategy addressed in this paper, allow to account for an accurate on-line forecasting and flow management of a DWN.This work was financially supported by the EU FP7 research project EFFINET “Efficient Integrated Real-time monitoring and Control of Drinking Water Networks,” grant agreement no. 318556.Peer Reviewe

Water demand forecasting for the optimal operation of large-scale drinking water networks: the Barcelona case study

Drinking Water Networks (DWN) are large-scale multiple-input multiple-output systems with uncertain disturbances (such as the water demand from the consumers) and involve components of linear, non-linear and switching nature. Operating, safety and quality constraints deem it important for the state and the input of such systems to be constrained into a given domain. Moreover, DWNs' operation is driven by time-varying demands and involves an considerable consumption of electric energy and the exploitation of limited water resources. Hence, the management of these networks must be carried out optimally with respect to the use of available resources and infrastructure, whilst satisfying high service levels for the drinking water supply. To accomplish this task, this paper explores various methods for demand forecasting, such as Seasonal ARIMA, BATS and Support Vector Machine, and presents a set of statistically validated time series models. These models, integrated with a Model Predictive Control (MPC) strategy addressed in this paper, allow to account for an accurate on-line forecasting and flow management of a DWN

Parallel methods for solving stochastic optimal control problems: control of drinking water networks

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

Uncertainty-aware demand management of water distribution networks in deregulated energy markets

We present an open-source solution for the operational control of drinking water distribution networks which accounts for the inherent uncertainty in water demand and electricity prices in the day-ahead market of a volatile deregulated economy. As increasingly more energy markets adopt this trading scheme, the operation of drinking water networks requires uncertainty-aware control approaches that mitigate the effect of volatility and result in an economic and safe operation of the network that meets the consumers’ need for uninterrupted water supply. We propose the use of scenario-based stochastic model predictive control: an advanced control methodology which comes at a considerable computation cost which is overcome by harnessing the parallelization capabilities of graphics processing units (GPUs) and using a massively parallelizable algorithm based on the accelerated proximal gradient method