Locating Battery Swapping Stations for a Smart eBus System
Abstract
:1. Introduction
2. Model
2.1. Assumption
2.2. Notation
2.3. Mathematical Programming Formulations
2.3.1. SetCovering Formulation
2.3.2. FlowBased Formulation
2.3.3. PathBased Formulation
3. Case Study on a Smart eBus System
3.1. Experiments with SetCovering Model
3.2. Experiments with FlowBased Model
3.3. Experiments with PathBased Model
3.4. Model Comparison
3.5. Quality of Service Analysis for Electric Buses
4. Conclusions and Future Research
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Jeju Introduces Electric Buses with Swapping Battery. Available online: https://kojects.com/2016/07/25/jejuintroduceselectricbuseswithswappingbattery/ (accessed on 20 November 2019).
 China Electric Bus Market—Growth, Trends, and Forecast (2019–2024). Available online: https://www.mordorintelligence.com/industryreports/chinaautomotiveelectricbusmarket (accessed on 20 November 2019).
 Kim, S.; Pelton, R.E.; Smith, T.M.; Lee, J.; Jeon, J.; Suh, K. Environmental Implications of the National Power Roadmap with Policy Directives for Battery Electric Vehicles (BEVs). Sustainability 2019, 11, 6657. [Google Scholar] [CrossRef] [Green Version]
 The U.S. Has a Fleet of 300 Electric Buses. China Has 421,000. Available online: https://www.bloomberg.com/news/articles/20190515/inshifttoelectricbusitschinaaheadofus421000to300 (accessed on 20 November 2019).
 Seoul Introduces Electric Bus (Posted on November 15, 2018). Available online: http://koreabizwire.com/seoulintroduceselectricbus/127453 (accessed on 20 November 2019).
 Teoh, L.E.; Khoo, H.L.; Goh, S.Y.; Chong, L.M. Scenariobased electric bus operation: A case study of Putrajaya, Malaysia. Int. J. Transp. Sci. Technol. 2018, 7, 10–25. [Google Scholar] [CrossRef]
 Rogge, M.; van der Hurk, E.; Larsen, A.; Sauer, D.U. Electric bus fleet size and mix problem with optimization of charging infrastructure. Appl. Energy 2018, 211, 282–295. [Google Scholar] [CrossRef] [Green Version]
 Liu, H.; Wang, D.Z. Locating multiple types of charging facilities for battery electric vehicles. Transp. Res. Part Methodol. 2017, 103, 30–55. [Google Scholar] [CrossRef]
 Ko, J.; Shim, J.S. Locating battery exchange stations for electric taxis: A case study of Seoul, South Korea. Int. J. Sustain. Transp. 2016, 10, 139–146. [Google Scholar] [CrossRef]
 Chao, Z.; Xiaohong, C. Optimizing battery electric bus transit vehicle scheduling with battery exchanging: Model and case study. ProcediaSoc. Behav. Sci. 2013, 96, 2725–2736. [Google Scholar] [CrossRef] [Green Version]
 Kang, Q.; Wang, J.; Zhou, M.; Ammari, A.C. Centralized charging strategy and scheduling algorithm for electric vehicles under a battery swapping scenario. IEEE Trans. Intell. Transp. Syst. 2015, 17, 659–669. [Google Scholar] [CrossRef]
 Sarker, M.R.; Pandžić, H.; OrtegaVazquez, M.A. Optimal operation and services scheduling for an electric vehicle battery swapping station. IEEE Trans. Power Syst. 2014, 30, 901–910. [Google Scholar] [CrossRef]
 Pilot Operation of Electric Battery Bus in Pohang, South Korea. Available online: http://www.molit.go.kr/USR/NEWS/m_71/dtl.jsp?lcmspage=3&id=95072972 (accessed on 20 November 2019).
 Marinakis, Y. Location routing problemLocation Routing Problem. In Encyclopedia of Optimization; Floudas, C.A., Pardalos, P.M., Eds.; Springer US: Boston, MA, USA, 2009; pp. 1919–1925. [Google Scholar] [CrossRef]
 AlbaredaSambola, M. Locationrouting. In Location Science; Springer: Berlin/Heidelberg, Germany, 2015; pp. 399–418. [Google Scholar]
 Ghiani, G.; Laporte, G. Locationarc routing problems. Opsearch 2001, 38, 151–159. [Google Scholar] [CrossRef]
 Corberán, A.; Prins, C. Recent results on arc routing problems: An annotated bibliography. Networks 2010, 56, 50–69. [Google Scholar] [CrossRef]
 Prodhon, C.; Prins, C. A survey of recent research on locationrouting problems. Eur. J. Oper. Res. 2014, 238, 1–17. [Google Scholar] [CrossRef]
 Golumbic, M.C.; Hartman, I.B.A. Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006; Volume 34. [Google Scholar]
 Amaya, A.; Langevin, A.; TréPanier, M. The capacitated arc routing problem with refill points. Oper. Res. Lett. 2007, 35, 45–53. [Google Scholar] [CrossRef]
 Xing, L.; Rohlfshagen, P.; Chen, Y.; Yao, X. An evolutionary approach to the multidepot capacitated arc routing problem. IEEE Trans. Evol. Comput. 2009, 14, 356–374. [Google Scholar] [CrossRef] [Green Version]
 Bektaş, T.; Elmastaş, S. Solving school bus routing problems through integer programming. J. Oper. Res. Soc. 2007, 58, 1599–1604. [Google Scholar] [CrossRef]
 Yang, J.; Sun, H. Battery swap station locationrouting problem with capacitated electric vehicles. Comput. Oper. Res. 2015, 55, 217–232. [Google Scholar] [CrossRef]
 Boccia, M.; Sforza, A.; Sterle, C. Flow intercepting facility location: Problems, models and heuristics. J. Math. Model. Algorithms 2009, 8, 35–79. [Google Scholar] [CrossRef]
 Desaulniers, G.; Desrosiers, J.; Solomon, M.M. Column Generation; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006; Volume 5. [Google Scholar]
 Seoul Open Data Center. Available online: https://data.seoul.go.kr/ (accessed on 20 November 2019).
Sets and Parameters  
$\mathcal{N}$  $=\left\{1,2,\dots ,N\right\}$. Set of bus stops 
$\mathcal{A}$  $=\left\{(i,j)i,j\in \mathcal{N},j(\ne i\left)\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{directly}\phantom{\rule{4.pt}{0ex}}\mathrm{accessible}\phantom{\rule{4.pt}{0ex}}\mathrm{from}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4pt}{0ex}}\right(i.e.,\phantom{\rule{4pt}{0ex}}i\to j)\right\}$. Set of arcs 
${\mathcal{N}}_{s}\subseteq \mathcal{N}$  Set of potential bus stops equipped with battery swapping facility (Quick Change Machine (QCM)) 
$\mathcal{R}$  $=\left\{1,2,\dots ,R\right\}$. Set of bus routes 
${\mathcal{D}}_{o}\subset {\mathcal{N}}_{s}$  Set of origins (depots) for all bus routes 
${\mathcal{D}}_{d}\subset {\mathcal{N}}_{s}$  Set of destinations (depots) for all bus routes 
${\phantom{\rule{3.33333pt}{0ex}}}_{r}$  Total number of bus stops on route $r\in \mathcal{R}$ 
${D}_{max}$  Maximum travel distance per charge 
${\overrightarrow{T}}^{r}$  $\equiv ({n}_{1}^{r},{n}_{2}^{r},\dots ,{n}_{{\phantom{\rule{3.33333pt}{0ex}}}_{r}}^{r})$. Ordered sequence of bus stops on route $r\in \mathcal{R}$ where ${n}_{i}^{r}\in \mathcal{N}$ for all $i=1,2,\dots ,{\phantom{\rule{3.33333pt}{0ex}}}_{r}$. Note that (i) $({n}_{i}^{r},{n}_{i+1}^{r})\in \mathcal{A}$ for $i=1,\dots ,{\phantom{\rule{3.33333pt}{0ex}}}_{r}1$ and (ii) ${n}_{i}^{r}{\prec}_{r}{n}_{j}^{r}$ holds for any i and j such that $i<j$ where the precedence relationship in the ordered sequence is denoted by ${\prec}_{r}$. 
${\mathcal{T}}^{r}$  $=\left\{{n}_{1}^{r},{n}_{2}^{r},\dots ,{n}_{{\phantom{\rule{3.33333pt}{0ex}}}_{r}}^{r}\right\}$. Ordered set of bus stops on route $r\in \mathcal{R}$ 
${D}^{r}(\alpha ,\beta )$  Travel distance from bus stop $\alpha \in \mathcal{N}$ to bus stop $\beta \in \mathcal{N}$ on route $r\in \mathcal{R}$ 
${\mathcal{N}}_{s}^{r}$  $=\left({\mathcal{T}}^{r}\cap {\mathcal{N}}_{s}\right)$. Set of potential QCMs in ${\mathcal{T}}^{r}$ 
${\mathcal{T}}_{\alpha}^{r}$  $=\left\{\alpha \equiv {n}_{i}^{r},{n}_{i+1}^{r},\dots ,{n}_{j}^{r}\right\}\subset {\mathcal{T}}^{r}$. Maximal ordered subset (MOS) of ${\mathcal{T}}^{r}$, starting from a potential QCM $\alpha \in {\mathcal{N}}_{s}^{r}$ such that ${D}^{r}({n}_{i}^{r},{n}_{j}^{r})\le {D}_{max}$ and ${D}^{r}({n}_{i}^{r},{n}_{j+1}^{r})>{D}_{max}$

Decision variables  
${y}_{i}$  1 if a QCM is located at bus stop $i\in {\mathcal{N}}_{s}$, and 0 otherwise 
Sets and Parameters  
${L}^{r}\left({n}_{i}^{r}\right)$  Last bus stop where a QCM is installed in ${\mathcal{T}}_{{n}_{i}^{r}}^{r}$ (i.e., ${L}^{r}\left({n}_{i}^{r}\right)={n}_{j}^{r}$ if the ordered set of ${\mathcal{T}}_{{n}_{i}^{r}}^{r}\cap {\mathcal{N}}_{s}$ with respect to ${\prec}_{r}$ is equal to $\left\{{n}_{i}^{r},\dots ,{n}_{j}^{r}\right\}$). 
${\mathcal{I}}^{r}\left(\alpha \right)$  $\equiv \left({\mathcal{T}}_{\kappa}^{r}\setminus \left\{\alpha \right\}\right)\cap {\mathcal{N}}_{s}^{r}$ where ${L}^{r}\left(\kappa \right)=\alpha $. Note that ${\mathcal{I}}^{r}\left(\alpha \right)=\varphi $ if $\alpha \in {\mathcal{D}}_{o}$. 
${\mathcal{O}}^{r}\left(\alpha \right)$  $\equiv \left({\mathcal{T}}_{\alpha}^{r}\setminus \left\{\alpha \right\}\right)\cap {\mathcal{N}}_{s}^{r}$. Note that ${\mathcal{O}}^{r}\left(\alpha \right)=\varphi $ if $\alpha \in {\mathcal{D}}_{d}$. 
Decision variables  
${x}_{\alpha \kappa}^{r}$  1 if a bus on the route r performs a battery swapping at QCM $\alpha $ and the next immediate battery swapping is performed at QCM $\kappa $ where $\alpha ,\kappa \in {\mathcal{N}}_{s}^{r}$ and $\alpha {\prec}_{r}\kappa $, and 0 otherwise. 
Sets and Parameters  
$\mathcal{P}\left(r\right)$  Set of paths for the route $r\in \mathcal{R}$ where a path is a set of consecutive subroutes where connecting points correspond to QCMs 
${\mathcal{N}}_{r}\left(l\right)$  Set of stations which the path l for the route r visits. 
Decision variables  
${x}_{l}^{r}$  1 if the path $l\in \mathcal{P}\left(r\right)$ is selected, and 0 otherwise. 
# of Routes (# of Stations)  Optimal # of QCM Stations  Avg. CPU Time (sec)  Max Gap between CPU Times (sec)  # of Variables  # of Constraints 

5 (441)  12  0.086  0.012  441  595 
10 (822)  20  0.301  0.076  822  1185 
20 (1611)  37  0.708  0.071  1611  2422 
30 (2286)  53  0.968  0.066  2286  3631 
40 (2681)  65  1.326  0.411  2681  4710 
50 (3058)  73  2.001  0.444  3058  5822 
60 (3597)  82  2.493  0.558  3597  7074 
70 (3929)  102  2.561  0.329  3929  7880 
80 (4235)  121  2.848  0.186  4235  8537 
90 (4435)  137  3.034  0.11  4435  9282 
100 (4732)  152  3.378  0.121  4732  10,287 
200 (7008)  256  5.614  0.233  7008  18,194 
300 (8059)  315  7.725  1.17  8059  25,352 
400 (8937)  382  9.246  0.377  8937  31,492 
500 (10,934)  549  9.87  0.4  10,934  34,908 
600 (12,650)  721  9.793  0.459  12,650  37,799 
635 (13,191)  782  10.313  1.357  13,191  38,636 
# of Routes (# of Stations)  Optimal # of QCM Stations  Avg. CPU Time (sec)  Max Gap between CPU Times (sec)  # of Variables  # of Constraints 

5 (441)  12  16.835  0.202  972,846  240,434 
10 (822)  21  35.891  1.223  2,205,658  681,564 
15 (1230)  28  122.667  1.658  6,757,662  1,816,809 
18 (1552)  32  687.125  5.805  10,757,662  4,816,809 
20 (1611)  37  2356.156  15.425  34,541,551  12,426,973 
30 (2286)           
40 (2681)           
50 (3058)           
# of Routes (# of Stations)  Optimal # of QCM Stations  Avg. CPU Time (sec)  Max Gap between CPU Times (sec)  # of Variables  # of Constraints  Avg. # of Iterations 

5 (441)  12  15.781  0.042  515  202  86 
10 (822)  21  60.418  0.084  956  422  215 
15 (1230)  28  285.118  1.584  1355  569  348 
20 (1611)  37  1296.621  3.584  1950  744  586 
30 (2286)  53  2874.516  6.871  3125  968  863 
40 (2681)  65  4225.889  10.668  4857  1268  1153 
50 (3058)  73  5826.146  15.139  22,618  1605  1868 
60 (3597)  82  7361.411  23.584  30,520  2841  2735 
70 (3929)  102  8165.156  43.458  37,211  3869  3868 
80 (4235)  121  10,041.453  167.165  43,584  5412  5066 
90 (4435)  137  14,025.584  304.545  51,251  8169  6166 
100 (4732)  152  23,218.975  518.107  58,518  9887  7259 
200 (7008)  256  81,540.145  682.618  135,685  34,815  25,015 
300 (8059)  315  345,154.587  822.123  209,871  65,036  68,121 
SetCovering Model  FlowBased Model  PathBased Model  

# of Routes (# of Stations)  CPU Time (sec)  Optimal # of QCM Stations  CPU Time (sec)  Optimal # of QCM Stations  CPU Time (sec)  Optimal # of QCM Stations 
5 (441)  0.086  12  16.835  12  15.781  12 
10 (822)  0.301  21  35.891  21  60.418  21 
15 (1230)  0.514  28  122.667  28  285.118  28 
20 (1611)  0.708  37  2356.156  37  1296.621  37 
30 (2286)  0.968  53      2874.516  53 
40 (2681)  1.326  65      4225.889  65 
50 (3058)  2.001  73      5826.146  73 
60 (3597)  2.493  82      7361.411  82 
70 (3929)  2.561  102      8165.156  102 
80 (4235)  2.848  121      10,041.453  121 
90 (4435)  3.034  137      14,025.584  137 
100 (4732)  3.378  152      23,218.975  152 
200 (7008)  5.614  256      81,540.145  256 
300 (8059)  7.725  315      345,154.587  315 
400 (8937)  9.246  382         
500 (10,934)  9.87  549         
600 (12,650)  9.793  721         
635 (13,191)  10.313  782         
Additional Parameter  
$\gamma $  Maximally allowable number of bus routes stopping at a QCM station for battery swapping 
Original PathCovering Model  Extended PathBased Model  

# of Routes (# of Stations)  # of QCM Stations  Maximum Flock ($\gamma $)  Variance of Flocking  CPU Time (sec)  # of QCM Stations  Maximum Flock ($\gamma $)  Variance of Flocking  CPU Time (sec) 
5 (441)  12  4  1.401  15.781  14  2  0.863  16.115 
10 (822)  21  7  1.987  60.418  22  4  1.478  69.054 
15 (1230)  28  9  2.081  285.118  29  5  1.975  308.487 
20 (1611)  37  9  1.993  1296.621  39  5  1.072  1583.142 
30 (2286)  53  9  1.974  2874.516  55  5  1.882  3593.145 
40 (2681)  65  9  2.013  4225.889  67  5  1.342  5451.396 
50 (3058)  73  9  1.955  5826.146  76  5  1.277  9828.708 
60 (3597)  82  11  2.154  7361.411  85  7  1.270  10,453.203 
70 (3929)  102  15  3.377  8165.156  104  8  1.483  13,717.462 
80 (4235)  121  21  3.670  10,041.453  125  10  2.495  12,903.267 
90 (4435)  137  24  3.823  14,025.584  141  10  2.478  19,032.717 
100 (4732)  152  25  4.284  23,218.975  158  12  2.876  34,317.645 
Extended PathBased Model Experimented with 60 Routes  

$\gamma $  # of QCM Stations  Maximum Flock  Variance of Flocking  CPU Time (sec) 
6  infeasible       
7  85  7  1.270  10,453.204 
8  84  8  1.621  9358.721 
9  84  9  1.819  8715.780 
10  82  10  2.084  7521.184 
11 (no restict)  82  11  2.154  7361.411 
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Moon, J.; Kim, Y.J.; Cheong, T.; Song, S.H. Locating Battery Swapping Stations for a Smart eBus System. Sustainability 2020, 12, 1142. https://doi.org/10.3390/su12031142
Moon J, Kim YJ, Cheong T, Song SH. Locating Battery Swapping Stations for a Smart eBus System. Sustainability. 2020; 12(3):1142. https://doi.org/10.3390/su12031142
Chicago/Turabian StyleMoon, Joon, Young Joo Kim, Taesu Cheong, and Sang Hwa Song. 2020. "Locating Battery Swapping Stations for a Smart eBus System" Sustainability 12, no. 3: 1142. https://doi.org/10.3390/su12031142