The Reversible Lane Network Design Problem (RLNDP) for Smart Cities with Automated Traffic
Abstract
:1. Introduction
2. Background
3. The Reversible Lane Network Design Problem (RLNDP)
3.1. Mathematical Formulation
$\mathit{N}=\left\{1,\text{}\dots ,\text{}i,\text{}\dots ,\text{}I\right\}$:  set of nodes in the network, where $I$ is the number of nodes. 
$\mathit{R}=\left\{\dots ,\left(i,\text{}j\right),\dots \right\}\text{}\forall \text{}\left\{i,j\right\}\in \mathit{N}{\displaystyle \cap}i\ne j$:  set of links of the road network where vehicles move. 
$\mathit{P}=\left\{\dots ,\left(o,d\right),\dots \right\}\text{}\forall \text{}\left\{o,d\right\}\in \mathit{N}{\displaystyle \cap}o\ne d$:  set of origin–destination pairs that represent the travel demand in the network, i.e., ${D}_{od}^{h,h+1}>0$. 
$\mathit{H}=\left\{1,\dots ,h,\dots ,T\right\}$:  set of time periods, where $T$ is the number of time periods (e.g., hours). 
${D}_{od}^{h,h+1}$:  demand trips from an origin node $o$ towards a destination node $d$, of period $h$ to $h+1$, $\forall \left(o,d\right)\in \mathit{P}{\displaystyle \cap}h\in \mathit{H}$. 
${t}_{ij}^{min}$:  minimum driving travel time in freeflow speed at the link $\left(i,j\right)\in \mathit{R}$, expressed in hours. 
${L}_{ij}^{current}$:  the current number of lanes at the link $\left(i,j\right)\in \mathit{R}$. 
${C}_{ij}^{lane}$:  lane capacity of the link $\left(i,j\right)\in \mathit{R}$, expressed in vehicles for the period of analysis. 
$M$:  big number. 
${l}_{ij}^{h,h+1}$:  integer variable equal to the number of lanes of each road link $\left(i,j\right)\in \mathit{R}$, of period $h$ to $h+1,\forall h\in \mathit{H}$ 
${f}_{ijod}^{h,h+1}$:  continuous variable that corresponds to the flow of AVs in each link $\left(i,j\right)\in \mathit{R}$ and each OD pair $\left(o,d\right)\in \mathit{P}{\displaystyle \cap}{D}_{od}^{h,h+1}0$, of period $h$ to $h+1,\forall h\in \mathit{H}.$ 
3.2. Scenarios
Algorithm 1 Scenario O: traffic assignment problem without the reversible lane problem  
1: 2: 3: 4: 5: 6: 7: 8: 9:  $h=1$ While $h\le T$ do ${l}_{ij}^{h,h+1}={L}_{ij}^{current}$ function Objective Function $\mathrm{min}\left(1\right)$ endfunction $h=h+1$ Clear all decision variables enddo 

Algorithm 2 Scenario A: the reversible lane problem without changing the traffic assignment  
1: 2: 3: 4: 5: 6: 7: 8: 9:  ${h}_{i}=1$ While $h\le T$ do read ${f}_{ijod}^{h,h+1}$ variables from scenario O function Objective Function $\mathrm{min}\left(1\right)$ endfunction $h=h+1$ Clear all decision variables enddo 

Algorithm 3 Scenarios B and C: both the reversible lane and traffic assignment problems (UE and SO)  
1: 2: 3: 4: 5: 6: 7: 8:  ${h}_{i}=1$ While $h\le T$ do function Objective Function $\mathrm{min}\left(1)or(2\right)$ endfunction $h=h+1$ Clear all decision variables enddo 

4. Application to the Case Study City of Delft
4.1. Setting up the Case Study
4.2. Experiments
4.3. Impacts at the Traffic Level
4.4. Impacts at the Spatial Level
5. Conclusions and Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Traffic Assignment  Reversible Lanes  Mathematical Model  

Scenario O  Current traffic situation without reversible lanes  UE  No  NLP 
Scenario A  First days after implementing reversible lanes, AVs follow previous paths (scenario O)  Not performed  Yes  MINLP 
Scenario B  Longterm scenario with reversible lanes and UE traffic conditions. AVs choose their paths (selfish behavior)  UE  Yes  MINLP 
Scenario C  Longterm scenario with reversible lanes and SO traffic conditions. The system chooses AV paths (unselfish behavior)  SO  Yes  MINLP 
Period (hh)  Scenario O  Scenario A  Scenario B  Scenario C  

OF (1) (h veh)  Calculus (s)  OF (1) (h veh)  Calculus (s)  OF (1) (h veh)  Calculus (s)  OF (2) (h veh)  Calculus (s)  
6  7  105  0.1  105  0.3  105  0.4  105  0.4 
7  8  729  0.2  721  0.4  721  0.9  907  2.1 
8  9  1353  0.2  1338  0.4  1325  0.8  2001  4.0 
9  10  2541  0.3  2528  1.4  2523  219.6  4404  1768.6 
10  11  1733  0.2  1711  1.0  1673  21.6  1900  113.7 
11  12  2220  0.2  2217  0.7  2193  14.7  2802  433.3 
12  13  1831  0.2  1826  0.9  1825  395.6  2183  764.1 
13  14  353  0.1  345  0.4  345  0.5  353  0.5 
14  15  2046  0.2  2016  0.6  1934  8.8  4988  39.9 
15  16  843  0.1  841  0.6  841  6.8  858  8.0 
16  17  2194  0.2  2124  0.5  2078  5.8  3191  42.7 
17  18  374  0.1  370  0.4  370  0.5  373  0.5 
18  19  1120  0.2  1117  0.4  1117  6.5  1291  38.6 
19  20  247  0.1  247  0.7  247  0.4  250  0.4 
20  21  33  0.1  33  0.3  33  0.3  33  0.4 
21  22  638  0.2  627  0.3  615  1.4  658  4.4 
22  23  594  0.1  544  0.5  537  0.7  569  1.0 
23  24  404  0.1  402  0.4  402  0.4  406  0.4 
24  1  404  0.1  353  0.3  346  0.4  375  0.4 
Total  19,761  00:00:03  19,466  00:00:11  192,300  00:11:26  27,648  00:53:43  
(h veh)  (h:m:s)  (h veh)  (h:m:s)  (h veh)  (h:m:s)  (h veh)  (h:m:s) 
Period  Average Degree of Saturation (%)  Average Congestion (%)  Congested roads (Degree of Saturation≥100%) (km)  Total Travel Distance (km veh)  Total Travel Times (h veh)  Total Delay (h veh)  

Scenario O  Scenario A  Scenario B  Scenario C  Scenario O  Scenario A  Scenario B  Scenario C  Scenario O  Scenario A  Scenario B  Scenario C  Scenario O  Scenario A  Scenario B  Scenario C  Scenario O  Scenario A  Scenario B  Scenario C  Scenario O  Scenario A  Scenario B  Scenario C  
6  7  39.3%  24.1%  24.1%  24.1%  1.8%  1.2%  1.2%  1.2%  0.00  0.00  0.00  0.00  5975  5975  5975  5975  106  105  105  105  1  0  0  0 
7  8  73.6%  51.2%  51.3%  48.6%  11.1%  6.0%  6.0%  5.7%  3.68  0.87  0.87  0.87  40,752  40,752  40,694  40,355  945  908  908  907  271  233  234  229 
8  9  71.6%  45.5%  57.4%  47.0%  20.4%  14.5%  13.9%  15.0%  15.06  8.77  8.77  7.88  66,066  66,066  65,826  68,944  2094  2021  2018  2001  927  854  867  784 
9  10  94.8%  81.4%  82.3%  70.1%  35.3%  30.4%  29.7%  30.4%  29.60  27.19  25.94  21.21  102,330  102,330  102,699  109,752  4833  4764  4690  4404  2865  2796  2708  2271 
10  11  85.1%  67.9%  65.8%  58.8%  27.6%  21.2%  19.7%  19.9%  17.90  15.06  8.82  8.98  86,097  86,097  86,171  86,407  2167  2059  1903  1900  543  435  287  272 
11  12  82.6%  71.8%  70.7%  63.4%  36.7%  31.9%  30.0%  31.1%  28.96  26.66  23.62  25.93  108,008  108,008  107,118  108,298  2973  2957  2880  2802  941  925  858  728 
12  13  82.9%  68.3%  67.7%  62.3%  29.1%  23.5%  23.3%  21.7%  20.11  16.44  16.44  12.91  98,178  98,178  98,190  99,949  2279  2256  2266  2183  560  538  551  415 
13  14  57.3%  36.9%  36.9%  36.9%  5.9%  3.7%  3.7%  3.7%  2.41  0.15  0.15  0.15  19,777  19,777  19,777  19,777  391  353  353  353  48  9  9  9 
14  15  89.2%  77.3%  70.8%  67.0%  21.3%  17.9%  14.2%  17.0%  13.53  10.64  5.28  10.09  60,916  60,916  59,198  64,241  5581  5432  5161  4988  4419  4270  4034  3755 
15  16  56.4%  44.5%  44.5%  38.4%  15.4%  12.5%  12.5%  12.6%  0.15  0.15  0.15  0.15  44,758  44,758  44,758  45,063  870  860  860  858  34  24  24  20 
16  17  100.4%  69.7%  68.5%  63.0%  32.3%  23.2%  21.0%  20.1%  26.68  15.43  11.89  10.84  101,387  101,387  100,481  101,527  3690  3344  3203  3191  1871  1524  1406  1383 
17  18  63.6%  42.3%  42.3%  42.3%  6.9%  4.3%  4.3%  4.3%  0.00  0.00  0.00  0.00  19,388  19,388  19,388  19,388  391  373  373  373  22  4  4  4 
18  19  72.5%  52.8%  52.7%  46.8%  18.1%  12.9%  12.9%  13.2%  1.92  1.15  1.15  1.15  63,122  63,122  63,118  62,779  1310  1294  1294  1291  238  222  222  213 
19  20  42.7%  27.2%  27.2%  27.2%  4.5%  3.6%  3.6%  3.6%  0.00  0.00  0.00  0.00  13,500  13,500  13,500  13,500  250  250  250  250  4  3  3  3 
20  21  36.1%  27.1%  27.1%  27.1%  0.6%  0.5%  0.5%  0.5%  0.00  0.00  0.00  0.00  1670  1670  1670  1670  33  33  33  33  0  0  0  0 
21  22  63.6%  38.5%  40.2%  35.4%  10.2%  5.7%  5.0%  5.3%  4.02  0.57  0.57  0.57  37,254  37,254  36,966  37,056  719  662  659  658  101  44  56  50 
22  23  83.4%  47.6%  55.6%  55.6%  9.2%  5.7%  5.6%  5.6%  3.31  0.17  0.17  0.17  31,169  31,169  30,822  30,822  826  574  569  569  289  37  41  41 
23  24  53.8%  28.1%  28.1%  28.1%  6.8%  3.4%  3.4%  3.4%  0.00  0.00  0.00  0.00  23,520  23,520  23,520  23,520  417  406  406  406  16  5  5  5 
24  1  108.1%  52.5%  67.4%  67.4%  5.8%  2.4%  2.4%  2.4%  3.31  0.17  0.17  0.17  20,495  20,495  20,148  20,148  633  379  375  375  287  33  36  36 
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Conceição, L.; Correia, G.H.d.A.; Tavares, J.P. The Reversible Lane Network Design Problem (RLNDP) for Smart Cities with Automated Traffic. Sustainability 2020, 12, 1226. https://doi.org/10.3390/su12031226
Conceição L, Correia GHdA, Tavares JP. The Reversible Lane Network Design Problem (RLNDP) for Smart Cities with Automated Traffic. Sustainability. 2020; 12(3):1226. https://doi.org/10.3390/su12031226
Chicago/Turabian StyleConceição, Lígia, Gonçalo Homem de Almeida Correia, and José Pedro Tavares. 2020. "The Reversible Lane Network Design Problem (RLNDP) for Smart Cities with Automated Traffic" Sustainability 12, no. 3: 1226. https://doi.org/10.3390/su12031226