A linear diversity constraint – Application to scheduling in microgrids
Highlights
► A linear diversity constraint that can be used in MILP models. ► A detailed/rigorous mathematical model for scheduling in microgrids connected to the national grid by incorporating various realistic features. ► Time constraints for the purchase/sale of power from/to the national grid. ► Round trip efficiency of batteries and hydrogen generation. ► Limits on storage and retrieval rates from batteries/hydrogen tanks/natural gas tanks.
Introduction
In today’s world, advances in electricity generation and transmission are being rewarded by economic, environmental and technological incentives. This is leading to the development of many small distributed facilities to supplement the traditional centralized power generation. This rising popularity of distributed power generation is due to its ability to penetrate distant and isolated areas. The distributed energy generation encompasses a wide range of emerging and environmentally friendlier (than traditional power plants) technologies like fuel cells, photovoltaic energy, wind farms, biomass etc. [1], [2]. Such an infrastructure offers improved reliability by permitting the use of excess energy stored in the form of electricity or hydrogen within itself. In addition, the ability to buy and sell from the national grid enhances the reliability and profitability of a microgrid.
The sources of electricity for the microgrids are largely non-conventional such as natural gas, biomass, photo-voltaics, wind, etc. Natural gas and biomass are combusted to produce electricity. A part of the biomass is gasified to produce syngas and this is used by the fuel cells to produce electricity. The fuel cells also make use of the hydrogen produced through electrolysis. This cycle of operation can be visualized in Fig. 1. The excess electricity is stored in the battery, the excess hydrogen is stored in a hydrogen storage facility. The stored electricity can be utilized whenever required. Depending on the supply and demand, electricity is exchanged between the microgrid and the national grid.
Several researchers have considered the design of microgrids from the strategic perspective and have considered investment costs, carbon tax etc. Ren & Gao [3] have presented a MILP model for minimizing costs while installing different power generation units. Kavvadias & Maroulis [4] used multiobjective optimization methodology for capacity sizing in this sector. Söderman & Pettersson [5] presented an MILP considering transmission of electricity, transportation of liquid fuels, transport of water in district heating with an objective of minimizing costs and designing an optimal infrastructure, while Mavrotas et al. and Howing et al. [6], [7] also present a model to systematically analyze uncertainities facing this sector. Others have also presented studies involving heating and cooling networks in distributed energy systems [8], [9], [10]. Li et al. [11] presented a mixed integer nonlinear model for designing a plant considering gas turbine, internal combustion engine, absorption chiller and gas boiler and solved it using a genetic algorithm. Hanschin et al. [12] presented a stochastic model to handle uncertainty in a distributed energy system considering gas turbines, fuel cells and other combined heat and power units. Other researchers [13], [14] have also presented models for analyzing investments in the distribution energy system. Ruan et al. [15] analyzed four commercial building and presented options for optimal operation of the microgrids. Maribu et al [16]. presented a market diffusion model for the spread of microgrid technology in the US.
Chinese & Meneghetti [17] presented two models for profit maximation and emissions minimization for a district heating system in the Italian context. Hawkes & Leach [18] presented a linear programming model for minimizing costs for design and operation of a microgrid system containing CHP generators and storage in the UK context. Mohamed & Koiva [19] presented a nonlinear model considering photo-voltaics, wind turbine, generators among others and solved it using Mesh Adaptive Direct Search algorithm. Considerable research has been done in the area of scheduling area by various research organizations [20], [21], [22], [23]. A simple MILP model [24] exists in the literature for scheduling in microgrids and its features are very limited. Kalantar & Mousavi [25] presented a simulation model for studying the dynamic performance of a microgrid, while Medrano et al. [26] presented a simulation model and studied the integration of microgrids in commercial buildings in California. Other researchers have presented an agent based methodology for scheduling in microgrids [27]. It can be seen that most of the works presented in literature are focusing on strategic issues and there is limited literature in the area of mixed integer linear programming models for scheduling.
In view of the increasing focus on microgrids employing various technologies, a good mathematical model is required to optimize its operation to achieve a certain objective (e.g. profit maximization). Hence, we develop a detailed/rigorous mathematical model for scheduling in microgrids connected to the national grid by incorporating various realistic features. Further, in view of their relative isolation, the various generation capacities in microgrids must be sufficiently diverse in order to meet the demand as best as possible in the event that one source of generation becomes unavailable, thus improving reliability. It is common to use nonlinear diversity indices such as Shannon-Weiner Index and Herfindahl–Hirschman index [28] which makes the implementation difficult particularly when the rest of the model is an MILP. Hence, in this work we present a linear diversity constraint that can be used in MILP models.
Section snippets
Problem statement
In this work, we consider the scheduling of operations in a microgrid based on distributed and diverse energy sources. Among the alternatives, we consider the production of electricity via solar and wind farms, combustion of biomass and natural gas, fuel cells that use syngas produced from biomass, and hydrogen fuel cells that use hydrogen generated from excess electricity. Among the storage options for excess energy, we consider direct storage of electrical energy in the batteries and indirect
A linear diversity constraint
We explain diversity through a small example. Consider three electricity generating units (x1, x2 and x3) that always operate at full capacity but run on different fuels. The decision maker must decide the capacities of the three production units such that the diversity of energy sourcing is maintained while meeting the demand (say demand is 1 unit). One option is to select the same capacity for the three generating units, i.e., 1/3 unit, so that in the event that one unit does not work due to
A mathematical model for scheduling in microgrids
We have developed a discrete time, MILP (mixed integer linear programming) model in this work for a planning horizon of one day. The details are presented next. First, we divide one day into several periods of a fixed length, and let t denote the time at the end of period t.
Results and discussion
The use of the model to study various policies was previously presented [29]. The MILP developed was used to study various scenarios to help identify the best strategy considering economic, reliability and operational issues. The various scenarios studied were: a.) Connected to the National Grid with penalty for not meeting local demand, b.) Not connected to National Grid with penalty for not meeting demand, c.) Not connected to National Grid with a constraint that demand must always be met,
Conclusion
In this work, we presented a mathematical model for scheduling in microgrids. In addition, we have introduced a linear diversity constraint that can be easily implemented. The use of the mathematical model in analyzing various policies such as conditions that demand must always be met, national grid not being available, etc were presented earlier. The effectiveness of the diversity constraint in maintaining diversity in electricity generation as well as in providing solutions that are easier to
Nomenclature
- t
- time period
- r
- range
Binary variables
- 1, if biomass is purchased at time t, 0 otherwise
- 1, if combustion occurs in range r at time t, 0 otherwise
- 1, if electricity is purchased from national grid at time t, 0 otherwise
- 1, if syngas is produced in range r at time t, 0 otherwise
- 1, if electricity is sold to the national grid at time t, 0 otherwise
- 1, if fuel cell is operating at time t, 0 otherwise
- 1, if hydrogen is used in fuel cell in range r at time t, 0 otherwise
- 1, if combustion unit is
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