1 Traffic equilibrium and charging facility locations for electric vehicles Hong Zheng 1 , Xiaozheng He 1 , Yongfu Li 1,2 , Srinivas Peeta 3* 1. NEXTRANS Center, Purdue University, 3000 Kent Ave, West Lafayette, IN 47906, USA 2. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3. School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA * Corresponding author; Email: [email protected]; Tel: +1-765-496-9729; Fax: +1-765-807-3123 Abstract This study investigates the electric vehicle (EV) traffic equilibrium and optimal deployment of charging locations subject to range limitation. The problem is similar to a network design problem with traffic equilibrium, which is characterized by a bi-level model structure. The upper level objective is to optimally locate charging stations such that the total generalized cost of all users is minimized, where the user’s generalized cost includes two parts, travel time and energy consumption. The total generalized cost is a measure of the total societal cost. The lower level model seeks traffic equilibrium, in which travelers minimize their individual generalized cost. All the utilized paths have identical generalized cost while satisfying the range limitation constraint. In particular, we use origin-based flows to maintain the range limitation constraint at the path level without path enumeration. To obtain the global solution, the optimality condition of the lower level model is added to the upper level problem resulting in a single level model. The nonlinear travel time function is approximated by piecewise linear functions, enabling the problem to be formulated as a mixed integer linear program. We use a modest-sized network to analyze the model and illustrate that it can determine the optimal charging station locations in a planning context while factoring the EV users’ individual path choice behaviours. Keywords: Electric vehicle; traffic equilibrium; network design; charging location; single level model 1. Introduction 1.1. Background Electric vehicles (EVs) have received much attention in the past few years with the promise of achieving reduced fossil fuel dependency, enhanced energy efficiency, and improved environmental sustainability. While EVs can achieve lower operating costs and are more energy efficient (US Department of Energy 2014a; Weaver 2014), they have not yet been widely accepted by the traveling public. A primary reason is range anxiety which denotes the driver concerns that the vehicle will run out of battery power before reaching the destination. This is a serious issue, particularly for long or intercity trips (Mock et al. 2010; 2011). Given the current state of battery technologies, an EV typically has a range of around 100 miles with a full charge, depending on the motor type, vehicle size and battery pack style (2014b). For example, the 2015 Nissan Leaf and Chevrolet Spark EVs have a driving range of about 80 miles. The Tesla (model X and S) EV with its advanced battery technology has a higher range of around 250-350 miles which is expected to improve further
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Traffic equilibrium and charging facility locations for electric vehicles
Hong Zheng1, Xiaozheng He1, Yongfu Li1,2, Srinivas Peeta3*
1. NEXTRANS Center, Purdue University, 3000 Kent Ave, West Lafayette, IN 47906, USA
2. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3. School of Civil Engineering, Purdue University, 550 Stadium Mall Drive, West Lafayette, IN 47907, USA
Figure 4. Equilibrium flows and range limitation verification for paths of Origin 1
The equilibrium flows for origins 1 to 3 are plotted in Figures 4 to 6, respectively. Bolded links indicate that
they are used, i.e., 0 or 1. The distance of links is also plotted in the figures. It can be verified that in the
equilibrium flow every utilized path satisfies the range limitation condition, i.e., an EV must stop to recharge before
it exceeds the range of 10.
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Note that, in Figure 5, nodes 21, 23 and 24 are not reached by EVs from origin 2; this is because the demands
from origin 2 to nodes 21, 23 and 24 are zero. This is also the reason for nodes 18, 19 and 20 not being reached by
EVs from origin 3 in Figure 6.
1 2
3 4 5 6
8 79
101112 16 18
17
191514
23 22
2124 2013
Legend: Road
Node Charging station
6
4 5
4 2 4
4 6
5 2
10
3 5
3
2
6 5 4 3
3
4
4
4
2
6
8
2
2
5
4
3
3
2
3
4
5
6
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distance Road utilized in equilibrium flow
Figure 5. Equilibrium flows and range limitation verification for paths of Origin 2
18
1 2
3 4 5 6
8 79
101112 16 18
17
191514
23 22
2124 2013
Legend: Road
Node Charging station
6
4 5
4 2 4
4 6
5 2
10
3 5
3
2
6 5 4 3
3
4
4
4
2
6
8
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5
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distance Road utilized in equilibrium flow
Figure 6. Equilibrium flows and range limitation verification for paths of Origin 3
4. Conclusions
This paper addresses the EV traffic equilibrium and optimal deployment of charging locations subject to the range
limitation. The structure of the problem is similar to the bi-level model of network design problem with traffic
equilibrium. The upper level model is a network design problem to optimally locate charging stations such that the
total generalized cost is minimized. The lower level problem models travelers’ path choice decisions, in which each
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traveler minimizes his/her individual generalized cost. The generalized cost is composed of travel time and energy
consumption; the latter is proportional to the travel distance. All utilized paths have identical generalized costs, and
satisfy the range limitation constraint. Note that the objectives of the problems in the two levels do not align, and
often conflict, with each other. To obtain the global optimal solution, a single level model is formulated by adding
the optimality condition of the lower level model to the upper level problem. The nonlinear link travel time function
is approximated by piecewise linear functions, enabling the problem to be formulated as a mixed integer linear
program that can be solved using off-the-shelf commercial software.
In contrast to the shortest path problem and the network design problem with relays studied by the operations
research community, our problem is more complicated as it models flow-dependent cost in traffic equilibrium. The
traffic equilibrium is enabled on top of range feasibility specified by the upper level network design problem, i.e., all
utilized paths are range feasible and have equal and minimum generalized cost. In previous studies (He et al. 2014),
range-feasible paths are equilibrated using path enumeration. In this regard, we use origin-based flows in the multi-
commodity flow model formulation to equilibrate range-feasible paths in a link-node representation of the network
without explicit path enumeration. This is because the origin-based flows contain some path information in the
traffic equilibrium but do not necessarily require path enumeration. This property enables us to formulate the EV
travel range constraint on a set of subnetworks containing feasible paths, and circumvent the time-consuming task of
verifying the travel range constraint on each individual path.
For the Sioux Falls network example, it has been verified that the model can solve for the equilibrium flow
combined with the optimal deployment of charging locations. In particular, it has been numerically verified that all
utilized paths are equilibrated while satisfying the range limitation condition.
In this study, using CPLEX, it takes about 95 minutes to reach optimality for the Sioux Falls network with three
origins. Though the global solution is obtained, the computational time is significant for a network of modest size
(with about 400 binary variables). With increased problem size, the computational time may increase substantially.
Hence, there is a need to explore efficient solution algorithms to apply the proposed model to large-scale networks.
Acknowledgements
This research is based on the funding provided by the U.S. Department of Transportation through the NEXTRANS
Center, the USDOT Region 5 University Transportation Center, and partly supported by the National Natural
Science Foundation of China (Grant No.61304197), and the Scientific and Technological Talents Project of
Chongqing (Grant No. cstc2014kjrc-qnrc30002). The authors are solely responsible for the contents of this paper.
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