Björn Bahl

An adaptive discretization MINLP algorithm for optimal synthesis of decentralized energy supply systems

Abstract Decentralized energy supply systems (DESS) are highly integrated and complex systems designed to meet time-varying energy demands, e.g., heating, cooling, and electricity. The synthesis problem of DESS addresses combining various types of energy conversion units, choosing their sizing and operations to maximize an objective function, e.g., the net present value. In practice, investment costs and part-load performances are nonlinear. Thus, this optimization problem can be modeled as a nonconvex mixed-integer nonlinear programming (MINLP) problem. We present an adaptive discretization algorithm to solve such synthesis problems containing an iterative interaction between mixed-integer linear programs (MIPs) and nonlinear programs (NLPs). The proposed algorithm outperforms state-of-the-art MINLP solvers as well as linearization approaches with regard to solution quality and computation times on a test set obtained from real industrial data, which we made available online.

Time-series aggregation for synthesis of distributed energy supply systems by bounding error in operational expenditure

Abstract For synthesis of distributed energy supply systems, the complexity of the mathematical optimization problem is commonly reduced by time-series aggregation. Today, the accuracy of the aggregation is measured in the time-series domain, i.e., by the capability of the aggregated time-series to represent the original time-series. In this paper, we propose a method for time-series aggregation measuring the accuracy of the aggregation in the domain of the objective function: The error is evaluated between the operational expenditure resulting from calculations using the aggregated time-series and the original time-series. An adaptive procedure selects representative time-steps for the aggregated time-series. It is shown that aggregation to few time-steps is sufficient to represent the original time-series with excellent accuracy in operational expenditure.

Rigorous synthesis of energy systems by decomposition via time-series aggregation

Abstract The synthesis of complex energy systems usually involves large time series such that a direct optimization is computationally prohibitive. In this paper, we propose a decomposition method for synthesis problems using time-series aggregation. To initialize the method, the time series is aggregated to one time step. A lower bound is obtained by relaxing the energy balances and underestimating the energy demands leading to a relaxed synthesis problem, which is efficiently solvable. An upper bound is obtained by restricting the original problem with the full time series to an operation problem with a fixed structure obtained from the lower bound solution. If the bounds do not satisfy the specified optimality gap, the resolution of the time-series aggregation is iteratively increased. The decomposition method is applied to two real-world synthesis problems. The results show the fast convergence of the decomposition method outperforming commercial state-of-the-art optimization software.

Rigorous synthesis of energy supply systems by time-series aggregation

Abstract A rigorous solution method is proposed for complex synthesis problems of energy supply systems with large time series. Time-series aggregation is used to iteratively tighten feasible solutions as upper bounds and best possible solutions as lower bounds. To initialize the method, the time series is aggregated to one time step. The lower bound is obtained by relaxing and underestimating the energy demands of all time steps which makes the corresponding equations redundant allowing for an efficient solution of the relaxed synthesis problem. The upper bound results from a restriction to an operation problem for the structure obtained from the lower bound solution. If the bounds do not satisfy the specified optimality gap, the resolution of the time series aggregation is increased and the solution process is restarted. The solution method is applied to an industrial real-world synthesis problem. The results show the fast convergence of the solution method outperforming a commercial state-of-the-art solver.

Typical Periods for Two-Stage Synthesis by Time-Series Aggregation with Bounded Error in Objective Function

Two-stage synthesis problems simultaneously consider here-and-now decisions (e.g., optimal investment) and wait-and-see decisions (e.g., optimal operation). The optimal synthesis of energy systems reveals such a two-stage character. The synthesis of energy systems involves multiple large time series such as energy demands and energy prices. Since problem size increases with the size of the time series, synthesis of energy systems leads to complex optimization problems. To reduce the problem size without loosing solution quality, we propose a method for time-series aggregation to identify typical periods. Typical periods retain the chronology of time steps which enables modeling of energy systems, e.g., with storage units or start-up cost. The aim of the proposed method is to obtain few typical periods with few time steps per period, while accurately representing the objective function of the full time series, e.g. cost. Thus, we determine the error of time-series aggregation as the cost difference between operating the optimal design for the aggregated time series and for the full time series. Thereby, we rigorously bound the maximum performance loss of the optimal energy system design. In an initial step, the proposed method identifies the best length of typical periods by autocorrelation analysis. Subsequently, an adaptive procedure determines aggregated typical periods employing the clustering algorithm

Rigorous synthesis of energy systems by relaxation and time-series aggregation to typical periods

k

Integrated Synthesis of Batch Plants and Utility Systems

-medoids which groups similar periods into clusters and selects one representative period per cluster. Moreover, the number of time steps per period is aggregated by a novel clustering algorithm maintaining chronology of the time steps in the periods. The method is iteratively repeated until the error falls below a treshold value. A case study based on a real-world synthesis problem of an energy system shows that time-series aggregation from 8760 time steps to 2 typical periods with each 2 time steps results in an error smaller than the optimality gap of of the synthesis problem (2%). This corresponds to a reduction of the number time steps and thus a reduction of the size of the synthesis problem by a factor of 1000 with excellent accuracy in cost estimation. Thus, the proposed method enables an efficient and accurate synthesis of energy systems.

Time-series aggregation for synthesis problems by bounding error in the objective function

Abstract The synthesis of energy systems is a complex optimization task depending on multiple large time series. Time-coupling constraints, e.g., due to storage systems, complicate computation even further. To still efficiently solve time-coupled synthesis problems, we propose a rigorous synthesis method. In the proposed method, lower and upper bounds are calculated to obtain a feasible solution of the original synthesis problem with known quality. We compute the lower bound by binary relaxation. For the upper bound, we use time-series aggregation to obtain a feasible design for the system. Employing this feasible design, we solve an operational problem, which can be solved efficiently. To tighten the upper bound, we iteratively increase the time resolution of the aggregation. In a case study for an industrial energy system, we show that only few typical periods are required to obtain a solution of the original synthesis problem with excellent quality. The method has fast convergence outperforming a commercial state-of-the-art solver.

Optimization-based identification and quantification of demand-side management potential for distributed energy supply systems

Abstract A mathematical optimization framework is presented for the integrated synthesis of batch plants and utility systems. Usually, synthesis starts with the design of the batch plant. Subsequently, the utility system is designed to supply the batch plant with energy. In general, such a sequential synthesis approach leads to a suboptimal overall design. We therefore integrate the two synthesis problems into one optimization framework resulting in an integrated problem formulation covering structural decisions, sizing and scheduling. The presented framework is exemplified for a literature example, where we observe additional cost savings of 5.4% demonstrating the potential for integrated synthesis.