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To attain the highest performance of energy supply systems, it is necessary to
rationally determine types, capacities, and numbers of equipment in consideration of
their operational strategies corresponding to seasonal and hourly variations in energy
demands. In the combinatorial optimization method based on the mixed-integer linear
programming (MILP), integer variables are used to express the selection, numbers, and
on/off status of operation of equipment, and the number of these variables increases
with those of equipment and periods for variations in energy demands, and affects the computation efficiency significantly. In this paper, a MILP method utilizing the
hierarchical relationship between design and operation 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 independently at each period 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.

Energy field is one of the practical areas to which optimization can contribute significantly. In this chapter, the application of mixed-integer linear programming (MILP) approaches to optimal design and operation of distributed energy systems is described. First, the optimal design and operation problems are defined, and relevant previous work is reviewed. Then, an MILP method utilizing the hierarchical relationship between design and operation variables is presented. In the optimal design problem, integer variables are used to express the types, capacities, numbers, operation modes, and on/off states of operation of equipment, and the number of these variables increases with those of equipment and periods for variations in energy demands, and affects the computation efficiency significantly. The presented method can change the enumeration tree for the branching and bounding procedures, and can search the optimal solution very efficiently. Finally, future work in relation to this method is described.

To attain the highest performance of energy supply systems, it is necessary to rationally determine types, capacities, and numbers of equipment in consideration of their operational strategies corresponding to seasonal and hourly variations in energy demands. In the combinatorial optimization method based on the mixed-integer linear programming (MILP), integer variables are used to express the selection, numbers, and on/off status of operation of equipment, and the number of these variables increases with those of equipment and periods for variations in energy demands, and affects the computation efficiency significantly. In this paper, a MILP method utilizing the hierarchical relationship between design and operation 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 independently at each period 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 to be minimized 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.

Generation of multiple best solutions in multiobjective optimal design of energy supply systems
(2015)

Optimization approaches based on the mixed-integer linear programming (MILP) have been utilized to design energy supply systems. In this paper, an MILP method utilizing the hierarchical relationship between design and operation is extended to search not only the optimal solution but also suboptimal ones which follow the optimal one without any omissions, what are called K-best solutions, efficiently in a multiobjective optimal design problem. At the upper level, the values of design variables for the K-best solutions are searched by the branch and bound method. At the lower level, the values of operation variables are optimized independently at each period by the branch and bound method under the values of design variables given tentatively. Incumbents for the K-best solutions and an upper bound for all the values of the objective function for the K-best solutions are renewed if necessary between both the levels. This method is implemented into a commercial MILP solver. A practical case study on the multiobjective optimal design of a cogeneration system is conducted, and the validity and effectiveness of the method are clarified.

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 effi- ciently. An original problem has been solved by dividing it into a relaxed optimal design problem at the upper level and optimal operation problems which are independent of one another at the lower level. In addition, some strategies have been proposed to enhance the computation efficiency furthermore. In this paper, a method of reducing model by time aggregation is proposed as a novel strategy to search design candidates efficiently in the relaxed optimal design problem at the upper level. In addition, the previous strategies are modified in accordance with the novel strategy. This method is realized only by clustering periods and averaging energy demands for clustered periods, while it guarantees to derive the optimal solu- tion. The method can decrease the number of design variables and constraints at the upper level, and thus may decrease the computation time at the upper level. Through a case study on the optimal design of a gas turbine cogeneration system, it is clarified how the model reduction is effective to enhance the computation efficiency in comparison and combination with the modified previous strategies.

To attain the highest performance of energy supply systems, it is necessary to rationally determine types, capacities, and numbers of equipment in consideration of their operational strategies corresponding to seasonal and hourly variations in energy demands. Mixed-integer linear programming (MILP) approaches have been applied widely to such optimal design problems. The authors have proposed a MILP method utilizing the hierarchical relationship between design and operation variables to solve the optimal design problems of energy supply systems efficiently. In addition, some strategies to enhance the computation efficiency have been adopted: bounding procedures at both the levels and ordering of the optimal operation problems at the lower level. In this paper, as an additional strategy to enhance the computation efficiency, parallel computing is adopted to solve multiple optimal operation problems in parallel at the lower level. In addition, the effectiveness of each and combinations of the strategies adopted previously and newly is investigated. This hierarchical optimization method is applied to an optimal design of a gas turbine cogeneration plant, and its validity and effectiveness are clarified through some case studies.

In designing energy supply systems, designers should heighten the robustness in performance criteria against the uncertainty in energy demands. In this paper, a robust optimal design method using a hierarchi- cal mixed-integer linear programming (MILP) method is proposed to maximize the robustness of energy sup- ply systems under uncertain energy demands based on a mixed-integer linear model. A robust optimal design problem is formulated as a three-level min-max-min MILP one by expressing uncertain energy demands by intervals, evaluating the robustness in a performance criterion based on the minimax regret cri- terion, and considering relationships among integer design variables, uncertain energy demands, and inte- ger and continuous operation variables. This problem is solved by evaluating upper and lower bounds for the minimum of the maximum regret of the performance criterion repeatedly outside, and evaluating lower and upper bounds for the maximum regret repeatedly inside. Since these different types of optimization problems are difficult to solve even using commercial MILP solvers, they are solved by applying a hierarchi- cal MILP method developed for ordinary optimal design problems with its modifications. In a case study, the proposed approach is applied to the robust optimal design of a cogeneration system. Through the study, its validity and effectiveness are ascertained, and some features of the obtained robust designs are clarified.

The mixed-integer linear programming (MILP) method has been applied widely to optimal design of energy supply systems. A hierarchical MILP method has been proposed to solve such optimal design problems effi- ciently. As one of the strategies to enhance the computation efficiency furthermore, a method of reducing model by time aggregation has been proposed to search design candidates accurately and efficiently in the relaxed optimal design problem at the upper level. In this paper, the hierarchical MILP method and model reduction by time aggregation are applied to the multiobjective optimal design. In applying the model reduc- tion, the methods of clustering periods by the order of time series, based on an operational strategy, and by the k-medoids method are applied. As a case study, the multiobjective optimal design of a gas turbine cogeneration system with a practical configuration is investigated by adopting the annual total cost and pri- mary energy consumption as the objective functions to be minimized simultaneously, and the clustering methods are compared with one another in terms of the computation efficiency. It turns out that the model reduction by any clustering method is effective to enhance the computation efficiency when importance is given to minimizing the first objective function. It also turns out that the model reduction only by the k- medoids method is effective very limitedly when importance is given to minimizing the second objective function.

To attain the highest performance of energy supply systems, it is necessary to rationally determine design specifications in consideration of operational strategies corresponding to energy demands. Mixed-integer linear programming (MILP) approaches have been applied widely to such optimal design problems. A MILP method utilizing the hierarchical relationship between design and operation variables have been proposed to solve them efficiently. However, it cannot necessarily be effective to multi-objective optimal design problems because of the existence of a large number of competing design candidates. In this paper, the hierarchical MILP method is revised from the viewpoint of computation efficiency so that it can be applied practically to multi-objective optimal design problems. At the lower level, the order of the optimal operation problems to be solved is changed based on incumbents obtained previously to increase a lower bound for the optimal value of the combined objective function and reduce the number of the optimal operation problems to be solved. At the upper level, a lower bound for the optimal value of the combined objective function is incorporated into the solution method to reduce the number of the design candidates to be generated. This revised hierarchical MILP method is applied to a multiobjective optimal design of a gas turbine cogeneration plant, and its validity and effectiveness are clarified.

In designing energy supply systems, designers are requested to rationally determine equipment types, capacities, and numbers in consideration of equipment operational strategies corresponding to seasonal and hourly variations in energy demands. However, energy demands have some uncertainty at the design stage, and the energy demands which become certain at the operation stage may differ from those estimated at the design stage. Therefore, designers should consider that energy demands have some uncertainty, evaluate the performance robustness against the uncertainty, and design the systems to heighten the robustness. Especially, this issue is important for cogeneration plants, because their performances depend significantly on both heat and power demands. Although robust optimal design methods of energy supply systems under uncertain energy demands were developed, all of them are based on linear models for energy supply systems. However, it is still a hard challenge to develop a robust optimal design method even based on a mixed-integer linear model. At the first step for this challenge, in this paper, a method of evaluating the performance robustness of energy supply systems under uncertain energy demands is proposed based on a mixed-integer linear model. This problem is formulated as a bilevel mixed-integer linear programming one, and a sequential solution method is applied to solve it approximately by discretizing uncertain energy demands within their intervals. In addition, a hierarchical optimization method in consideration of the hierarchical relationship between design and operation variables is applied to solve large scale problems efficiently. Through a case study on a gas turbine cogeneration plant for district energy supply, the validity and effectiveness of the proposed method and features of the performance robustness of the plant are clarified.