Multiscale modeling in mathematical programming: Application of Clustering
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Integration across decision levels of a supply chain is a key point in improving returns on investment. For example, planning and scheduling are usually carried out separately although they are interdependent of each other. Integration of planning and scheduling results in better coordination between decision levels and a reduction in operating costs. Integration of different time scales leads to large scale problems which are usually computationally intractable. Different approaches have been proposed to tackle the problem in terms of modeling and solution methods. However, most of them are problem specific or applicable only to short time horizons. Clustering has the potential to handle such a problem by grouping similar input parameters (like demand or price) together. This will considerably shrink the model size and make it more computationally tractable while at the same time not compromising solution accuracy. Therefore, the aim of this thesis is to develop a new class of clustering algorithms that are based on mathematical programming techniques in order to support the integration of planning applications of different time scales (strategic, tactical, and operational) in process systems engineering. The clustering algorithms were formulated using integer programming with IAE (integral absolute error) as a similarity measure. The initial formulation was a Mixed Integer Nonlinear Program (MINLP) and then reduced to a Mixed Integer Linear Program (MILP) using exact linearization techniques. The model resulted in two different clustering algorithms: normal and sequence clustering. Two case studies were presented to assess outputs and computational performance of the algorithms. Electricity demand and solar radiation data were clustered in these case studies. Both clustering algorithms captured the trend in the data. However, the computational burden of the model was prohibitive to tackle large planning horizons. In order to deal with computational complexity, a heuristic algorithm was developed utilizing an iterative scheme. The heuristic was first applied to clustering the electricity demand in the original cases studied for validation purposes. The quality of the solutions from the heuristic algorithm were checked against the MILP optimal solutions and it was found that the heuristic algorithm is able to provide good quality solutions and even succeeded in finding the optimal solution for simulation runs carried out. The heuristic algorithm was applied to clustering the electricity demand for a whole year with a small computational effort and providing clusters with high intra-cluster similarity and low inter-cluster similarity. In order to illustrate the use of the clustering procedure in solving large scale planning model, the clustered electricity demand was used as input to a Unit Commitment (UC) model with the objective to evaluate the solution quality when clustered demand is applied. The UC problem is a classical problem in electrical power production where the production of a set of electrical generators is coordinated in order to meet the energy demand at minimum cost or maximize revenues from energy production. The results showed a great advantage in term of solution time for the clustering technique compared to the regular solution when no clustering of demand was applied. Moreover, the error of objective function was within 0.5 % of the non-clustered demand for all cases. In addition, a sensitivity analysis study suggested that high quality solutions could still be achieved with smaller number of clusters. The clustering algorithm was extended in order to take into account multiple attributes at the same time such as clustering simultaneously demand for electricity and heat. In this respect, the objective function had different scales due to the different units of measurements of the attributes, and the problem was dealt with as a multi-objective optimization problem. The weighting method was chosen as the optimization approach and to be able to appropriately scale the different attributes. The clustering algorithm was successfully applied to simultaneously cluster hourly electricity and heat demands for the whole year. The Pareto front was captured for all runs with the weight factor combinations considered in this study. The results show that a better objective function is achieved when the number of clusters increases for both normal and sequence clustering. Normal clustering and as expected leads to a better objective function, error average and standard deviation than sequential clustering due to the additional restrictions of sequencing requirements imposed on the model. Clusters that take into account the time of occurrence of events and abide to certain minimum sequencing restrictions are also needed in planning operations in order to minimize the number of set-ups and inconvenience to operators. The statistical analysis of the heat demand was challenging as suggested by the results, due to the huge fluctuation in the heat demand. Moreover, calculations of relative error were problematic for the demand that was close to zero. The results indicated that in the case when operations are flexible or in the case of just classifying demand patterns, normal clustering should be used since it has a major advantage in terms of solution quality over sequence clustering. For the case of simultaneously clustering heat and electricity, it was required to employ many clusters of electricity that sometimes overlap with each other. These clusters could not be merged since they correspond to different days and the clusters of heat demand for these days are different. Nevertheless, the proposed algorithm was able to obtain groups that simultaneously cluster the two attributes and hence can provide computational advantages when solving integrated planning models that deal with more than one demand attribute. The clustered electricity and heat demands were used as inputs to an energy hub model, with the objective of evaluating the solution quality when multiple clustered demand attributes are applied to planning models. The average error of objective function was -1.7 % for normal clustering while for sequence clustering it was -4.2 %. Increasing the number of clusters was found to enhance the solution quality for both normal and sequence clustering. For this particular example, varying the weight factors did not have a drastic effect on the values of the objective function. This is due mainly to a symmetry or inverse similarity in the heat and electricity demands.
Cite this work
Falah Alhameli (2017). Multiscale modeling in mathematical programming: Application of Clustering. UWSpace. http://hdl.handle.net/10012/12165