Progressive hedging for stochastic energy management systems The mixed-integer linear case

Energy systems have increased in complexity in the past years due to the ever-increasing integration of intermittent renewable energy sources such as solar thermal or wind power. Modern energy systems comprise different energy domains such as electrical power, heating and cooling which renders their control even more challenging. Employing supervisory controllers, so-called energy management systems (EMSs), can help to handle this complexity and to ensure the energy-efficient and cost-efficient operation of the energy system. One promising approach are optimization-based EMS, which can for example be modelled as stochastic mixed-integer linear programmes (SMILP). Depending on the problem size and control horizon, obtaining solutions for these in real-time is a difficult task. The progressive hedging (PH) algorithm is a practical way for splitting a large problem into smaller sub problems and solving them iteratively, thus possibly reducing the solving time considerably. The idea of the PH algorithm is to aggregate the solutions of subproblems, where artificial costs have been added. These added costs enforce that the aggregated solutions become non-anticipative and are updated in every iteration of the algorithm. The algorithm is relatively simple to implement in practice, re-using almost all of a possibly existing deterministic implementations and can be easily parallelized. Although it has no convergence guarantees in the mixed-integer linear case, it can nevertheless be used as a good heuristic for SMILPs. Recent theoretical results shown that for applying augmented Lagrangian functions in the context of mixed-integer programmes, any norm proofs to be a valid penalty function. This is not true for squared norms, like the squared L-2-norm that is used in the classical progressive hedging algorithm. Building on these theoretical results, the use of the L-1 and L-infinity-norm in the PH algorithm is investigated in this paper. In order to incorporate these into the algorithm an adapted multiplier update step is proposed. Additionally a heuristic extension of the aggregation step and an adaptive penalty parameter update scheme from the literature is investigated. The advantages of the proposed modifications are demonstrated by means of illustrative examples, with the application to SMILP-based EMS in mind.