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A Multilane Traffic Flow Model Accounting for Lane Width, Lane-Changing and the Number of Lanes

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Abstract

Multilane widely exists in the urban traffic system and its traffic flow is more complex than the single-lane traffic flow since it may be affected by each lane’s width, the number of lanes and lane-changing. In this paper, we first use empirical data to study the impacts of lane width and the number of lanes on multilane traffic flow, then propose a new multilane traffic flow model and finally use numerical tests to explore the influences of the variation of lane width, closing lane and changing the number of lanes on each lane’s traffic flow. The numerical results show that the new model can qualitatively reproduce the complex traffic phenomena resulted by the variation of lane width, closing lane and changing the number of lanes and that changing each lane’s width, closing lane and changing the number of lanes may reduce the reliability of the multilane traffic system. In addition, some numerical results are qualitatively consistent with the phenomena that are resulted by the heavy rain in Beijing on July 21, 2012, which shows that our model can perfectly reproduce some complex traffic phenomena in multilane system from the qualitative perspective.

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Notes

  1. Note: if any condition in Eqs. (5)–(8) is an equality, then the corresponding lane-changing rate is zero, i.e., the corresponding part is zero.

References

  • Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60:916–938

    Article  Google Scholar 

  • Carey M, Balijepalli C, Watling D (2013) Extending the cell transmission model to multiple lanes and lane-changing. Netw Spat Econ. doi:10.1007/s11067-013-9193-7

    Google Scholar 

  • Castillo JM (2012) Three new models for the flow-density relationship: derivation and testing for freeway and urban data. Transportametrica 8:443–465

    Article  Google Scholar 

  • Chowdhury D, Santen L, Schreckenberg A (2000) Statistics physics of vehicular traffic and some related systems. Phys Rep 329:199–329

    Article  Google Scholar 

  • Daganzo CF (1997) A continuum theory of traffic dynamics for freeways with special lanes. Transp Res B 31:83–102

    Article  Google Scholar 

  • Feng SW (1997) Mathematical modeling, field calibration and numerical simulation of low-speed mixed traffic flow in cities. Ph.D. Dissertation of Shanghai University (in Chinese)

  • Flötteröd G, Chen Y, Nagel K (2012) Behavioral calibration and analysis of a large-scale travel microsimulation. Netw Spat Econ 12:481–502

    Article  Google Scholar 

  • Gupta AK, Katiyar VK (2007) A new multi-class continuum model for traffic flow. Transportmetrica 3:73–85

    Article  Google Scholar 

  • Helbing D (2001) Traffic and related self-driven many-particle systems. Rev Mod Phys 73:1067–1141

    Article  Google Scholar 

  • Hoogendoorn SP, Bovy PHL (2001) Platoon-based multiclass modeling of multilane traffic flow. Netw Spat Econ 1:137–166, http://www.jishinet.com/html/FocusNet/2012/0725/780.html (in Chinese)

    Article  Google Scholar 

  • Jiang R, Wu QS, Zhu ZJ (2002) A new continuum model for traffic flow and numerical tests. Transp Res B 36:405–419

    Article  Google Scholar 

  • Jin WL (2010) A kinematic wave theory of lane-changing traffic flow. Transp Res B 44:1001–1021

    Article  Google Scholar 

  • Jin S, Wang DH, Tao PF, Li PF (2010) Non-lane-based full velocity difference car-following model. Physica A 389:4654–4662

    Article  Google Scholar 

  • Kerner BS (2001) Complexity of synchronized flow and related problems for basic assumptions of traffic flow theories. Netw Spat Econ 1:35–76

    Article  Google Scholar 

  • Kerner BS, Konhäuster P (1993) Cluster effect in initially homogeneous traffic flow. Phys Rev E 48:R2335–R2338

    Article  Google Scholar 

  • Laval JA, Daganzo CF (2006) Lane-changing in traffic streams. Transp Res B 40:251–264

    Article  Google Scholar 

  • Lighthill MJ, Whitham GB (1955) On kinematic waves: II. A theory of traffic flow on long crowed roads. Proc R Soc London 229:317–345

    Article  Google Scholar 

  • Liu GQ, Lyrintzis AS, Michalopoulos PG (1996) Modelling of freeway merging and diverging flow dynamics. Appl Math Model 20:459–469

    Article  Google Scholar 

  • Long JC, Gao Z, Zhao X, Lian A, Orenstein P (2011) Urban traffic jam simulation based on the cell transmission model. Netw Spat Econ 11:43–64

    Article  Google Scholar 

  • Mathew TV, Radhakrishnan P (2010) Calibration of micro simulation models for non-lane-based heterogeneous traffic at signalized intersections. J Urban Plan Dev ASCE 136:59–66

    Article  Google Scholar 

  • Michalopoulos PG, Yi P, Lyrintzis AS (1993) Development of an improved high order continuum traffic flow model. Transp Res B 27:125–132

    Article  Google Scholar 

  • Moridpour S, Rose G, Sarvi M (2012) The effect of surrounding traffic characteristics on lane changing behavior. J Transp Eng 136:1–12

    Google Scholar 

  • Ngoduy D (2010) Multiclass first order modelling of traffic networks using discontinuous flow-density relationships. Transportmetrica 6:121–141

    Article  Google Scholar 

  • Ngoduy D (2011) Multiclass first-order traffic model using stochastic fundamental diagrams. Transportmetrica 7:111–125

    Article  Google Scholar 

  • Payne HJ (1971) Models of freeway traffic and control. Math Models Publ Syst Simul Counc Proc Ser 1:51–61

    Google Scholar 

  • Qi HS, Wang DH, Chen P, Bie YM (2013) Location-dependent lane-changing behavior for arterial road traffic. Netw Spat Econ. doi:10.1007/s11067-013-9202-x

    Google Scholar 

  • Richards PI (1956) Shock waves on the highway. Oper Res 4:42–51

    Article  Google Scholar 

  • Tang CF, Jiang R, Wu QS (2007) Extended speed gradient model for traffic flow on two-lane freeways. Chin Phys B 16:1570–1575

    Article  Google Scholar 

  • Tang TQ, Huang HJ, Xu G (2008) A new macro model with consideration of the traffic interruption probability. Physica A 387:6845–6856

    Article  Google Scholar 

  • Tang TQ, Huang HJ, Shang HY (2009a) A new dynamic model for heterogeneous traffic flow. Phys Lett A 373:2461–2466

    Article  Google Scholar 

  • Tang TQ, Huang HJ, Wong SC, Gao ZY, Zhang Y (2009b) A new macro model for traffic flow on a highway with ramps and numerical tests. Commun Theor Phys 51:71–78

    Article  Google Scholar 

  • Tang TQ, Li CY, Huang HJ, Shang HY (2011) Macro modeling and analysis of traffic flow with road width. J Cent S Univ Technol 18:1757–1764

    Article  Google Scholar 

  • Tian C, Sun DH (2010) Continuum modeling for two-lane traffic flow with consideration of the traffic interruption probability. Chin Phys B 19:120501

    Article  Google Scholar 

  • Wong GCK, Wong SC (2002) A multi-class traffic flow model-an extension of LWR model with heterogeneous drivers. Transp Res A 36:827–841

    Google Scholar 

  • Wu Z (1994) A fluid dynamics model for low speed traffic system. Acta Mech Sinica 26:149–157 (in Chinese)

    Google Scholar 

  • Yang XB, Zhang N, Gao ZY (2008) Changes in traffic characteristics affected by number of lanes on freeways. Proceedings of The Transportation Research Board (TRB) 87th Annual Meeting, Washington, D.C. January 13–17

  • Yu L, Shi ZK (2009) Density wave in a new anisotropic continuum model for traffic flow. Int J Mod Phys C 20:1849–1859

    Article  Google Scholar 

  • Zhang HM (1998) A theory of nonequilibrium traffic flow. Transp Res B 32:485–498

    Article  Google Scholar 

  • Zhang HM (2001) New perspectives on continuum traffic flow models. Netw Spat Econ 1:9–23

    Article  Google Scholar 

  • Zhang HM (2002) A non-equilibrium traffic model devoid of gas-like behavior. Transp Res B 36:275–290

    Article  Google Scholar 

  • Zhang HM, Shen W (2009) Numerical investigation of stop-and-go traffic patterns upstream of freeway lane drop. Transp Res Rec 2124:3–17

    Article  Google Scholar 

  • Zhang P, Wong SC (2006) Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models. Phys Rev E 74:026109

    Article  Google Scholar 

  • Zhang P, Liu RX, Wong SC (2005a) High-resolution numerical approximation of traffic flow problems with variable lanes and free-flow velocities. Phys Rev E 71:056704

    Article  Google Scholar 

  • Zhang P, Wong SC, Shu CW (2005b) A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway. J Comput Phys 212:739–756

    Article  Google Scholar 

  • Zhang P, Wong SC, Dai SQ (2006) Characteristic parameters of a wide cluster in a higher-order traffic flow model. Chin Phys Lett 23:516–519

    Article  Google Scholar 

  • Zhang J, Lam William HK, Chen BY (2013) A stochastic vehicle routing problem with travel time uncertainty: trade-off between cost and customer service. Netw Spat Econ. doi:10.1007/s11067-013-9190-x

    Google Scholar 

Download references

Acknowledgments

This study has been supported by the National Natural Science Foundation of China (70971007 and 71271016). The authors would like to thank the two anonymous referees, area editor and editor-in-chief for their helpful comments and valuable suggestions which have improved the paper substantially.

Author information

Authors and Affiliations

  1. School of Transportation Science and Engineering, Beihang University, Beijing, 100191, China

    Tie-Qiao Tang & Yun-Peng Wang

  2. School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China

    Xiao-Bao Yang

  3. School of Economics and Management, Beihang University, Beijing, 100191, China

    Hai-Jun Huang

Authors
  1. Tie-Qiao Tang
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  2. Yun-Peng Wang
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  3. Xiao-Bao Yang
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  4. Hai-Jun Huang
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Corresponding author

Correspondence to Tie-Qiao Tang.

Appendix

Appendix

In this section, we prove \( {\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{N}}{s}_{\mathrm{i}}\left(x,t\right)}=0 \). When N = 2, Eqs. (5)–(7) can be rewritten as follows:

$$ {s}_1=\left\{\begin{array}{ll}-\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{1\mathrm{e}}\left({\rho}_1\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{1\mathrm{e}}\left({\rho}_1\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \end{array}\right., $$
(A1)
$$ {s}_2=\left\{\begin{array}{cc}\hfill -\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right),\hfill & \hfill \mathrm{if}\ {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill \frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \hfill \mathrm{if}\ {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \end{array}\right.. $$
(A2)

From Eqs. (A1) and (A2), we can obtain s 1(x, t) + s 2(x, t) = 0.

When N = 3, Eqs. (5)–(7) can be rewritten as follows:

$$ {s}_1=\left\{\begin{array}{ll}-\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{1\mathrm{e}}\left({\rho}_1\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{1\mathrm{e}}\left({\rho}_1\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \end{array}\right., $$
(A3)
$$ {s}_2=\left\{\begin{array}{ll}-\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right)-\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}-\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right)+\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right)-\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}\left(\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta }+\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\right)\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \end{array}\right., $$
(A4)
$$ {s}_3=\left\{\begin{array}{ll}-\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{3\mathrm{e}}\left({\rho}_3\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em {v}_{3\mathrm{e}}\left({\rho}_3\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \end{array}\right.. $$
(A5)

From Eqs. (A3)–(A5), we can obtain s 1(x, t) + s 2(x, t) + s 3(x, t) = 0.

When N = 4, Eqs. (5)–(7) can be rewritten as follows:

$$ {s}_1=\left\{\begin{array}{ll}-\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\ {v}_{1\mathrm{e}}\left({\rho}_1\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right),\hfill & \mathrm{if}\ {v}_{1\mathrm{e}}\left({\rho}_1\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \end{array}\right., $$
(A6)
$$ {s}_2=\left\{\begin{array}{ll}-\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right)-\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}-\frac{\gamma_{2,1}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_1}{\rho_{1,\mathrm{j}}}\right)+\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right)-\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)<{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \\ {}\left(\frac{\gamma_{1,2}{\rho}_1{v}_1}{\varDelta }+\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\right)\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{1\mathrm{e}}\left({\rho}_1\right)\hfill \\ {}\hfill {v}_{2\mathrm{e}}\left({\rho}_2\right)>{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.\hfill \end{array}\right., $$
(A7)
$$ {s}_3=\left\{\begin{array}{ll}-\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right)-\frac{\gamma_{3,4}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_4}{\rho_{4,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)<{v}_{4\mathrm{e}}\left({\rho}_4\right)\hfill \end{array}\right.\hfill \\ {}-\frac{\gamma_{3,2}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_2}{\rho_{2,\mathrm{j}}}\right)+\frac{\gamma_{4,3}{\rho}_4{v}_4}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)<{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)>{v}_{4\mathrm{e}}\left({\rho}_4\right)\hfill \end{array}\right.\hfill \\ {}\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right)-\frac{\gamma_{3,4}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_4}{\rho_{4,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)<{v}_{4\mathrm{e}}\left({\rho}_4\right)\hfill \end{array}\right.\hfill \\ {}\left(\frac{\gamma_{2,3}{\rho}_2{v}_2}{\varDelta }+\frac{\gamma_{4,3}{\rho}_4{v}_4}{\varDelta}\right)\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\kern0.5em \left\{\begin{array}{c}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)>{v}_{2\mathrm{e}}\left({\rho}_2\right)\hfill \\ {}\hfill {v}_{3\mathrm{e}}\left({\rho}_3\right)>{v}_{4\mathrm{e}}\left({\rho}_4\right)\hfill \end{array}\right.\hfill \end{array}\right., $$
(A8)
$$ {s}_4=\left\{\begin{array}{ll}-\frac{\gamma_{4,3}{\rho}_4{v}_4}{\varDelta}\left(1-\frac{\rho_3}{\rho_{3,\mathrm{j}}}\right),\hfill & \mathrm{if}\ {v}_{4\mathrm{e}}\left({\rho}_4\right)<{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \\ {}\frac{\gamma_{3,4}{\rho}_3{v}_3}{\varDelta}\left(1-\frac{\rho_4}{\rho_{4,\mathrm{j}}}\right),\hfill & \mathrm{if}\ {v}_{4\mathrm{e}}\left({\rho}_4\right)>{v}_{3\mathrm{e}}\left({\rho}_3\right)\hfill \end{array}\right.. $$
(A9)

From Eqs. (A6)–(A9), we can obtain s 1(x, t) + s 2(x, t) + s 3(x, t) + s 4(x, t) = 0.

Using the same method, we can prove \( {\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{N}}{s}_{\mathrm{i}}\left(x,t\right)}=0 \) when N > 4.

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Tang, TQ., Wang, YP., Yang, XB. et al. A Multilane Traffic Flow Model Accounting for Lane Width, Lane-Changing and the Number of Lanes. Netw Spat Econ 14, 465–483 (2014). https://doi.org/10.1007/s11067-014-9244-8

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