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A hierarchical mixed-integer linear programming method to solve optimal design problems of energy supply systems efficiently has been proposed. The original problem is solved by dividing it into a relaxed optimal design problem at the upper level and optimal operation problems which are independent of each other at the lower level. In this paper, a method of reducing model by time aggregation is proposed to search design solution candidates efficiently in the relaxed optimal design problem at the upper level. This method is realized only by clustering periods and averaging energy demands for clustered periods, while it guarantees to derive the optimal solution. On one hand, the method may decrease the number of design variables and costraints at the upper level, and thus the computation time at the upper level. On the other hand, it may increase the numbers of generated design solution candidates and solved optimal operation problems, and thus the computation time at both the levels. Though a case study on the optimal design of a cogeneration system, it is clarified how the model reduction is effective to enhance the computation efficiency.
To attain the highest performance of energy supply systems, it is necessary to determine design specifications optimally in consideration of operational strategies corresponding to seasonal and hourly variations in energy demands. Mixed-integer linear programming (MILP) methods have been applied widely to such multi-period optimal design problems. A hierarchical MILP method has been proposed to solve the problems very efficiently. In addition, by utilizing features of the hierarchical MILP method, a method of reducing model by clustering periods has also been proposed to search design solution candidates efficiently in the relaxed optimal design problem at the upper level. In this paper, by utilizing features of the hierarchical MILP method, a method of clustering periods is proposed based on the optimal operational strategies of energy supply systems obtained by solving the relaxed optimal design problem. As a case study, the method is applied to the optimal design of a gas turbine cogeneration system, and it is clarified that the method is effective to enhance the computation efficiency in comparison with a conventional method of clustering periods regularly.
A robust optimal design method of energy supply systems under uncertain energy demands has been proposed using a mixed- integer linear model for constituent equipment. A robust optimal design problem has been formulated as a three-level min-max- min optimization one by expressing uncertain energy demands by intervals, evaluating the robustness in a performance criterion based on the minimax regret criterion, and considering hierarchical relationships among design variables, uncertain energy demands, and operation variables. However, this method takes a long computation time, and thus it can be applied only to small-scale problems. In this paper, mixed-integer linear programming method in consideration of the hierarchical relationship between design and operation variables is applied to parts of the robust optimal design method which take long computation times to solve problems efficiently. In a case study, this revised method is applied to the robust optimal design of a cogeneration system with a simple configuration, and the validity and effectiveness of the method are ascertained.
A mixed-integer linear programming (MILP) method utilizing the hierarchical relationship between design and op- eration variables is proposed to solve the optimal design problem of energy supply systems efficiently: At the upper level, the optimal values of design variables are searched by the branch and bound method; At the lower level, the values of operation variables are optimized by the branch and bound method under the values of design variables given tentatively during the search at the upper level; Lower bounds for the optimal value of the objective function are evaluated, and are utilized for the bounding operations at both the levels. This method is implemented into open and commercial MILP solvers. Illustrative and practical case studies on the optimal design of cogeneration systems are conducted, and the validity and effectiveness of the proposed method are clarified.
A mixed-integer linear programming methodutilizing the hierarchical relationship between design and operation variables proposed ot solve the optimal design problem of energy supply systems efficiently is extended ot search K-best solutions: At the upper level, the optimal values ofdesign variables are searched with operation variables relaxed to continuous ones; At the lower level, the values of operation variables are optimized with the values of design variables given tentatively; The obtained solution is used to renew K-best incumbent solutions, and the upper bound for the value of the objective function for K-best solutions is replaced correspondingly. A practical case study is conducted, and the validity and effectiveness of the proposed method are clarified.
Ahierarchical mixed-integer linear programmingmethod has been proposed to solve optimal design problems of energy supply systems efficiently. In this paper, a method of reducing model by clustering periods is proposed to search design solution candidates efficiently in the relaxed optimal design problem at the upper level. This method is realized only by clustering periods and averaging energy demands for clustered periods, while it guarantees to derive the optimal solution.
Through acase study on the optimaldesign of a cogeneration system, ti is clarified how the model reduction si effective ot enhance the computation efficiency.
A mixed-integer linear programming method utilizing the hierarchical relationship between design and operation variables proposed to solve the optimal design problem of energy supply systems efficiently is extended to search K-best solutions in multiobjective optimization: At the upper level, the optimal values of design variables are searched by the branch and bound method with operation variables relaxed to continuous ones; At the lower level, the values of operation variables are optimized independently at the respective periods set for variations in energy demands by the branch and bound method with the values of design variables given tentatively during the search at the upper level. A weighting method is employed for multiobjective optimization, and a weighted sum of the annual total cost and primary energy consumption is adopted as the objective function to be minimized. A practical case study on the optimal design of a cogeneration system is conducted. The validity and effectiveness of the proposed method are clarified, and the trade-off relationship between the annual total cost and primary energy consumption is confirmed.
Mixed-integer linear programming (MILP) methods have been applied widely to optimal design of energy supply systems. A hierarchical MILP method has been proposed to solve such optimal design problems efficiently. In addition, some strategies have been proposed to enhance the computation efficiency furthermore. As one of the strategies, a method of reducing model by time aggregation has been proposed to search design candidates efficiently in the relaxed optimal design problem at the upper level. In this paper, three clustering methods are applied to time aggregation and compared with one another in terms of the computation efficiency. Especially, the k-medoids method is applied newly in addition to the time-series and operation-based methods applied previously. A case study is conducted on the optimal design of a gas turbine cogeneration system for district energy supply. Through the study, it turns out the k-medoids method is effective to shorten the computation time as compared with the time-series method, although it is necessary to set the number of clusters artifically in both the methods. It also turns out that the operation-based method is more effective than the k-medoids method in terms of the computation efficiency even with the number of clusters set automatically.
Mixed-integer linear programming (MILP) methods have been applied widely to optimal design of energy supply systems. A hierarchical MILP method has been proposed to solve such optimal design problems efficiently. In addition, some strategies have been proposed to enhance the computation efficiency furthermore. As one of the strategies, a method of reducing model by time aggregation has been proposed to search design candidates efficiently in the relaxed optimal design problem at the upper level. In this paper, the hierarchical MILP method with the strategies has been extendedly applied to the optimal design of energy supply systems with storage units. Especially, the method of re- ducing model is extended by aggregating representative days and sampling times differently in consideration of the characteristics of storage units. A case study is conducted on the optimal design of a gas turbine cogeneration system with a thermal storage unit for district energy supply. Through the study, it turns out the hierarchical MILP method is effective to derive the optimal solution as compared with a conventional method. It also turns out that the model reduction with the special time aggregation is effective to shorten the computation time as compared with that without time aggregation in case that the number of candidates for equipment capacities is relatively small.
To attain the highest performance of energy supply systems, it is necessary to determine design specifications optimally in consideration of operational strategies corresponding to seasonal and hourly variations in energy demands. Mixed-integer linear programming (MILP) methods have been applied widely to such optimal design problems. A hierarchical MILP method has been proposed to solve the problems very efficiently. In addition, by utilizing features of the hierarchical MILP method, a method of reducing model by clustering periods based on the optimal operational strategies of equipment has been proposed to search design solution candidates efficiently in the relaxed optimal design problem at the upper level. In this paper, these methods are applied to the multiobjective optimal design of a cogeneration system by considering the annual total cost and primary energy consumption as the objective functions to be minimized. Through a case study, it turns out that the model reduction by the operation-based time-period clustering is effective in terms of the computation efficiency when importance is given to the first objective function, while it is not when importance is given to the second objective function.