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

摘要

This thesis is concerned with the development ofudoptimisation methods to solve stochastic Model PredictiveudControl (MPC) problem and employ them inudthe management of DrinkingWater Networks (DWNs).udDWNs are large-scale, complex both in topology anduddynamics, energy-intensive systems subjected to irregularuddemands. Managing these networks play audcrucial role in the economic sustainability of urbanudcities. The main challenge associated with such infrastructuresudis to minimise the energy required forudpumping water while simultaneously maintaininguduninterrupted water supply. State-of-the-art controludmethodologies as well as the current engineeringudpractices use predictive models to forecast upcomingudwater demands but do not take into considerationudthe inevitable forecasting error. This way, theudwater network is operated in a deterministic fashionuddisregarding its inherent stochastic behaviourudwhich accrues from the volatility of water demandudand, often, electricity prices. In this thesis, we addressudtwo challenges namely: optimisation methodsudfor solving stochastic MPC problems and closedloopudfeedback control for the management of drinkingudwater networks.udMPC is an advanced control technology that copes with complex control problem by repeatedly solvinguda finite horizon constrained optimal control problem;uduses only the first decision as input and discardsudthe rest of the sequence. This methodologyuddecides the control action based on present state ofudthe system and thus provides an implicit feedbackudto the system. Instead of historical demand profile,udtime-series models were developed to forecast theudfuture water demand. The economic and the socialudaspects involved in operation of the DWN wereudcaptured in a cost function. Now the MPC controllerudcombined with online forecaster minimise theudcost function across a prediction horizon of 1 dayudwith sampling time equal to 1 hour and thus theudclosed-loop strategy for DWN management is devised.udThe forecasts are just nominal demands and differudfrom the actual demands. There exist several approachesudwhen it comes to working with uncertainudforecasts: (i) to assume that forecast errors are negligibleudand disregard them, (ii) to assume knowledgeudof their worst-case values (maximum errors), (iii)udto assume knowledge of probabilistic information.udThese three approaches lead to the three principaludflavours of MPC: the certainty-equivalent (CE), theudworst-case robust and the stochastic MPC. CE-MPCudis simple but not realistic (because the errors are notudnegligible), worst-case MPC is more meaningful butudit is too conservative (because it is highly improbableudthat the errors admit their worst-case values) and then we have stochastic MPC which is the approachudpursued in this thesis.udA stochastic MPC allows a systematic frameworkudas trade-off performance against constraint violationudby modelling the uncertainty as stochastic processudand quantifying its influence. However, thisudformulation is an infinite dimensional optimisationudproblem and its corresponding discrete approximationudis deemed to be a large-scale problem with millionsudof decision variables. Therefore, the applicabilityudof stochastic MPC in control applications isudlimited due to the unavailability of algorithms thatudcan solve them efficiently and within the samplingudtime of the controlled system.udHere we developed optimisation algorithms that solveudstochastic MPC problem by exploiting their structureudand using parallelisation. These algorithms areud(i) accelerated proximal gradient algorithm also knownudas forward-backward splitting and (ii) LBFGSudmethod for forward-backward envelope (FBE) function.udBoth these algorithms employ decompositionudto solve the Fenchel dual and make them suitableudfor parallel implementation. Graphics processingudunits (GPUs) are capable of perform parallel computationudand are therefore perfect hardware to solveudthe stochastic MPC problem with the acceleratedudproximal gradient method.udThe water network of the city Barcelona is consideredudto study the validity of the proposed algorithm.udThe GPU implementation is found to be 10 times faster than commercial solvers like Gurobi runningudin multi-core environment and made the problemudcomputationally tractable in the sampling time. Theudefficiency of the stochastic MPC to manage theDWNudis quantified in terms of key performance indicatorsudlike economic utility, network utility and quality ofudservice.udThe forward-backward splitting is a first-order methodudand has slow convergence for ill-conditionedudproblems. We constructed a continuously differentiableudreal-valued forward-backward envelope functionudthat has the same set of minimisers as the actualudproblem. Then we use quasi-Newton method,udin particular LBFGS method, that utilises secondorderudinformation to solve the FBE. The computationsudwith this algorithm are also parallelisable andudit demonstrated fast convergence compared to accelerateduddual proximal gradient algorithm.