Robust planning of dynamic wireless charging infrastructure for battery electric buses

Highlights

• Optimal location of dynamic wireless charging facilities for battery electric buses.

• Uncertainties of energy consumption and travel time are considered.

• A robust optimization model is proposed.

• Computationally tractable reformulation of the robust optimization model is provided.

Abstract

Battery electric buses with zero tailpipe emissions have great potential in improving environmental sustainability and livability of urban areas. However, the problems of high cost and limited range associated with on-board batteries have substantially limited the popularity of battery electric buses. The technology of dynamic wireless power transfer (DWPT), which provides bus operators with the ability to charge buses while in motion, may be able to effectively alleviate the drawbacks of electric buses. In this paper, we address the problem of simultaneously selecting the optimal location of the DWPT facilities and designing the optimal battery sizes of electric buses for a DWPT electric bus system. The problem is first constructed as a deterministic model in which the uncertainty of energy consumption and travel time of electric buses is ignored. The methodology of robust optimization (RO) is then adopted to address the uncertainty of energy consumption and travel time. The affinely adjustable robust counterpart (AARC) of the deterministic model is developed, and its equivalent tractable mathematical programming is derived. Both the deterministic model and the robust model are demonstrated with a real-world bus system. The results demonstrate that the proposed deterministic model can effectively determine the allocation of DWPT facilities and the battery sizes of electric buses for a DWPT electric bus system; and the robust model can further provide optimal designs that are robust against the uncertainty of energy consumption and travel time for electric buses.

Introduction

As an integral part of public transportation, the public bus system provides people with an economical and sustainable travel mode, and it helps to reduce traffic congestion and exhaust emissions (Song, 2013). However, due to the limitations of vehicle technology, diesel-powered buses still dominate today’s bus fleet. For example, diesel buses accounted for 50.5% of all bus vehicles in the United States in 2015 (Dickens and Neff, 2016). Diesel engines are a primary source of particulate matter (PM) and nitrogen oxides (NOx) emitted by motor vehicles. Furthermore, most transit buses are operated in densely populated urban areas, and they are generally in use for large portions of the day. Battery electric buses, which produce zero tailpipe emissions, offer tremendous potential in improving the environmental sustainability and livability of urban areas. However, range limitations associated with on-board batteries as well as the problem of battery size, cost, and life, have substantially limited the popularity of electric buses.

The technology of dynamic wireless power transfer (DWPT), also called dynamic inductive charging, offers the promise of eliminating the range limitation of electric buses. DWPT provides bus operators the ability to charge buses while in motion, using wireless inductive power transfer pads embedded underneath the roadway. The technology potentially makes electric buses as capable as their diesel counterparts. DWPT technology has been implemented in a bus line in Gumi City, South Korea (Jang et al., 2015). Utah State University (USU) demonstrated the DWPT technology for an electric bus with peak power of 25 kW at its Electric Vehicle and Roadway (EVR) test track in 2016 (see Fig. 1). Additionally, the United Kingdom recently conducted a study to determine the feasibility of implementing this technology on its strategic road network (Highways England, 2015). Another benefit of DWPT technology is that it could substantially reduce on-board battery size. The battery pack on a long-range all-electric bus can account for about a quarter of the weight of the vehicle and as much as 39% of the total cost of the bus (Bi et al., 2015). Bi et al. (2015) demonstrated the potential of downsizing the battery of an electric bus to about one-third of a plug-in charged battery, assuming stationary wireless charging at bus stations is employed. The battery downsizing not only makes electric buses more affordable, but also offers additional energy savings, due to reduced vehicle weight.

Although a number of studies have investigated the problem of deploying or managing DWPT facilities for private electric vehicles in transportation networks (e.g., He et al., 2013, Riemann et al., 2015, Chen et al., 2016, Chen et al., 2017, Fuller, 2016, Deflorio and Castello, 2017), with current technologies, constructing DWPT facilities for private electric vehicles could be costly. Fuller (2016) estimated that it costs $4 million per lane mile to construct DWPT facilities for private electric vehicles. However, constructing DWPT facilities for an electric bus system is quite different from constructing such facilities for private electric vehicles. DWPT facilities consist of inverters and wireless power transfer pads. For DWPT facilities for private electric vehicles, inverters should be densely deployed to serve continuous vehicle flows. However, headways of buses can be controlled through proper scheduling. As a result, for DWPT facilities for an electric bus system, an inverter can cover a relatively long distance of roadway. Therefore, the cost for constructing DWPT facilities for an electric bus system could be significantly reduced.

To enable DWPT for an electric bus system, wireless charging infrastructure must be strategically built in the road network. Meanwhile, because DWPT provides the potential of reducing on-board battery size, battery sizes for electric buses should also be designed. The charging infrastructure planning problem is twofold. First, the combination of deployed dynamic wireless charging facilities and designed battery sizes should ensure the normal operation of electric buses. Second, one must consider the trade-off between on-board battery sizes and the number (length) of DWPT facilities.

A handful of studies have investigated the location of DWPT infrastructure for electric buses. Ko and Jang (2011) formulated a nonlinear model to simultaneously determine the optimal location of DWPT facilities and the battery sizes of electric buses for a single electric bus line. In this model, the cost of DWPT facilities is linearly related to length. Ko and Jang (2013) improved this model by separating the cost of DWPT facilities into two parts: the cost of inverters and the cost of cables. The total number of DWPT facilities determines the cost of inverters, and the cost of cables is linearly related to the total length. More recently, Jang et al. (2015) proposed a mixed-integer programming (MIP) model to optimize the location of DWPT facilities and the battery sizes of electric buses for a DWPT electric bus line in a closed environment.

The above studies only consider electric bus systems with a single bus line. However, a real-world bus system almost always contains more than one bus line. Moreover, multiple transit lines may have significant overlap, especially in areas with high transit demand, e.g., downtown or shopping malls. Overlapping transit lines could share wireless power transfer pads. The synergistic effect among different transit lines could substantially reduce the average cost of constructing DWPT infrastructure for individual bus lines and make DWPT more economically attractive for real-world implementation. Another significant drawback of previous studies lies in their strong assumption that energy consumption and travel time of electric buses are predefined. Nevertheless, in real-world traffic, energy consumption and travel time of electric buses will change along with traffic conditions and travel demands. Note that the travel time of an electric bus on a DWPT facility determines the potential dynamic charging time. Ignoring the uncertainty of energy consumption and travel time of electric buses could lead to a suboptimal or even infeasible plan for a DWPT electric bus system.

In this paper, we consider the planning problem of DWPT infrastructure in a general electric bus system with multiple lines. Moreover, the uncertainty of energy consumption and travel time of electric buses is also considered through robust optimization (RO). The primary contributions of our work are summarized as follows:

• We develop an innovative model to select the optimal location of DWPT facilities and design the optimal battery sizes of electric buses for a DWPT electric bus system with multiple lines.

• Based on the deterministic model, we formulate the corresponding robust optimization model, which can provide robust optimal solutions against the uncertainty of energy consumption and travel time of electric buses.

• We reformulate the initial robust optimization model, which is intractable, into a computationally tractable model.

The remaining portions of this paper are organized as follows. In the following section, we formulate a deterministic model to optimize the location of DWPT facilities and the battery sizes of electric buses for a DWPT electric bus system. Next, in Section 3 we propose a robust counterpart model to consider the uncertainty of energy consumption and travel time of electric buses. Section 4 presents numerical studies for both deterministic model and robust model. Finally, conclusions are discussed in Section 5.