Elsevier

Applied Energy

Volume 103, March 2013, Pages 135-144
Applied Energy

A two-stage stochastic programming model for the optimal design of distributed energy systems

https://doi.org/10.1016/j.apenergy.2012.09.019Get rights and content

Abstract

A distributed energy system is a multi-input and multi-output energy system with substantial energy, economic and environmental benefits. The optimal design of such a complex system under energy demand and supply uncertainty poses significant challenges in terms of both modelling and corresponding solution strategies. This paper proposes a two-stage stochastic programming model for the optimal design of distributed energy systems. A two-stage decomposition based solution strategy is used to solve the optimization problem with genetic algorithm performing the search on the first stage variables and a Monte Carlo method dealing with uncertainty in the second stage. The model is applied to the planning of a distributed energy system in a hotel. Detailed computational results are presented and compared with those generated by a deterministic model. The impacts of demand and supply uncertainty on the optimal design of distributed energy systems are systematically investigated using proposed modelling framework and solution approach.

Highlights

► The optimal design of distributed energy systems under uncertainty is studied. ► A stochastic model is developed using genetic algorithm and Monte Carlo method. ► The proposed system possesses inherent robustness under uncertainty. ► The inherent robustness is due to energy storage facilities and grid connection.

Introduction

A distributed energy system is typically based on the concept of “local production of energy for local consumption”. It refers to an advanced energy supply system which consists of many small scale energy generation technologies and locates at or near end-users [1]. Recently, interest has been intensifying in the development of distributed energy systems owing to their high overall efficiency, excellent environmental performance and other benefits [2], [3], [4]. However, balancing demand and supply is a challenging task for a distributed energy system, because it always faces instantaneously varying loads and small number of equipments within the system provide limited operation flexibility to cope with the fluctuation in energy demands [4]. Therefore, distributed energy systems may not be able to produce the potential benefits due to lack of appropriate system configurations and suitable operation strategies [1], [5].

To address issues related to the optimal design and operation of distributed energy systems, several mathematical models have been proposed using different mathematical programming techniques such as mixed-integer programming (MIP) and multi-objective programming (MOP) [6], [7]. Most models focus on the combined heating cooling and power (CCHP) system, which is a representative type of distributed energy system [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], whereas a few models investigate more complex systems where renewable energy resources are introduced [1], [5], [20], [21], [22], [23]. A common feature of most of these approaches is that they are formulated as deterministic mathematical programming models, ignoring the uncertainties of parameters. If model uncertainties are not adequately identified and handled, the actual economic and feasible operation of the designed distributed energy system may deviate from the optimal one. Thus, the assessment of uncertainty in the modelling of distributed energy systems has been recently received a lot of attention. Gamou et al. proposed an optimal unit sizing method for cogeneration systems taking into consideration uncertainty in energy demands [24]. Yoshida et al. performed a sensitivity analysis on uncertainties caused by volatile equipment performance, energy prices and possible decline in equipment costs [25]. Li et al. conducted a sensitivity analysis on fluctuating energy demands to evaluate their influence on the performance of CCHP systems [26]. Smith et al. analyzed a CCHP system model under different operating strategies with uncertainty in inputs and parameters, such as energy demand and process efficiency [27]. Li et al. proposed a mathematical programming model to optimize a CCHP system under uncertainty in energy demands [28]; Mavrotas et al. developed an energy planning framework that can be used in buildings taking into account uncertainties in economic parameters [29].

There are different types of uncertainty in the optimal design of distributed energy systems, for example, uncertainty in energy demand and energy supply, uncertainty in economic parameters such as unit investment cost and energy price and uncertainty in technological parameters such as efficiency. In most of the aforementioned studies considering uncertainty in the planning of distributed energy systems, energy demands uncertainty catches the most significant attention among different sources of uncertainties. Energy demands uncertainty arises from the simplified description of energy demands patterns using 24-h profiles of representative days thus reducing the dimension of the underlying optimization problem and improving the computational efficiency. However, there is always a trade-off between the computational efficiency and the accuracy of modelling results. Much information about energy demands is lost due to the adoption of representative days, leading to less accurate model predictions. Hawkes and Leach and Mavrotas et al. concluded in their studies independently that grouping of demand data will jeopardize the accuracy of the optimization results [29], [30], which implies that considering the volatility of energy demands is of great importance towards the optimal design of CCHP system.

In our previous work, a superstructure based mixed-integer linear programming (MILP) model has been proposed for the optimal design of distributed energy systems under purely deterministic conditions [31]. The salient feature of this model lies in its capability to take into account a large variety of potential energy resources and alternative technologies. This feature makes it possible to utilize distributed renewable energies that are utterly fluctuating. The method of representative days is also used for the description of energy demands as well as hourly resource availabilities of solar energy and wind energy. Therefore, lessons learned from previous works on the uncertainty of CCHP systems cannot be directly applied to more complex distributed energy systems proposed by our model. The external uncertainties should be addressed by considering fluctuations in both demand side and supply side.

This work extends our previous contribution and presents a two-stage stochastic programming model for the optimal design of distributed energy systems with a stage decomposition based solution strategy. Both demand uncertainty and supply uncertainty are taken into consideration, making the uncertainty analysis in this study more comprehensive than previous studies which consider demand uncertainty only. In this model, genetic algorithm performs the search on the first stage variables while the Monte Carlo method is adopted in the second stage to deal with uncertainty in energy demand and supply. As the complex distributed energy system with multiple energy resources is a new concept proposed by us, the stochastic model, in which uncertainties in energy demand and supply are considered simultaneously, is novel in comparison to existing contributions. This paper is organized as follows: First a deterministic model for the optimal design of distributed energy systems is presented in Section 2. Then, a stage decomposition based solution strategy is briefly introduced in Section 3. The subsequent Section 4 provides a quantitative description of uncertainties of energy demands, solar energy availability and wind energy availability. Section 5 presents the results of the stochastic programming problem in comparison to those of the deterministic one. The impacts of external uncertainties on the optimal design of distributed energy systems are investigated based on the modelling results.

Section snippets

Deterministic model for the optimal design of distributed energy systems

A superstructure based mixed-integer linear programming (MILP) model for the optimal design of distributed energy systems under purely deterministic conditions is presented in this section. The superstructure representation of a distributed energy system, i.e., its potential energy flow network, is shown in Fig. 1. The structure is comprised of an energy generation section, an energy conversion section and an energy storage section. The energy conversion processes take place in the following

Quantitative description of uncertain parameters

In this study, three types of parameters, with significant volatility, are selected as uncertain parameters, i.e., energy demands, solar energy availability and wind energy availability. The probability distributions of these parameters are presented in this section.

Case study

With the optimization model and its solution strategy described in Section 2 and probability distributions of energy demands, solar energy availability and wind energy availability provided in Section 3, the deterministic and stochastic models are applied to the planning of a distributed energy system for a hotel in Beijing. The hourly demand data of this hotel in some representative days are shown in Fig. 3.

Two cases are investigated for comparison purposes. Case 1 is a deterministic

Results and discussions

The solution strategy for the two-stage stochastic optimization problem is a sample based algorithm, therefore it must employ a large number of samples in the Monte Carlo method and adequate number of generations in the genetic algorithm to guarantee the accuracy of the modelling results.

The number of samples in each Monte Carlo simulation should be large enough so that the expected value of the stochastic term fs as shown in Eq. (3) does not change significantly with the sample size. Fig. 4

Conclusion

This work presents the development and application of a two-stage stochastic model for the optimal design of distributed energy systems under energy demand and supply uncertainty. The potential impacts of uncertain parameters are accounted by reformulating a deterministic design problem into a two-stage stochastic programming problem. A solution strategy which combines genetic algorithm and the Monte Carlo method is proposed to solve the stochastic programming problem.

A case study is used to

Acknowledgements

The authors gratefully acknowledge the financial support from National Natural Science Foundation (Project No. 51106080), from the IRSES ESE Project of FP7 (Contract No: PIRSES-GA-2011-294987), from BP company in the scope of the Phase II Collaboration between BP and Tsinghua University and from Mitsubishi Heavy Industries.

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