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Edited by: Jack Haddad, Technion Israel Institute of Technology, Israel

Reviewed by: Lukas Ambühl, ETH Zürich, Switzerland; Nan Zheng, Monash University, Australia; Mehmet Yildirimoglu, The University of Queensland, Australia

This article was submitted to Transportation Systems Modeling, a section of the journal Frontiers in Future Transportation

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This work attempts to validate the dynamic outputs of bi-modal MFD-based models, also referred to as 3D-MFD models, using empirical data. A previous study (Loder et al.,

The concept of Macroscopic Fundamental Diagram (MFD) was first proposed by Godfrey (

Most of the approaches proposed in the literature are based on the so-called uni-modal or 2D-MFD, which relates the accumulation of

Two types of MFD-based models are primarily used in the literature, namely, accumulation-based and trip-based models. Accumulation-based model (Daganzo,

The present work is essentially the continuation of the previous work of the authors, where the proposed MFD-based models founded on 3D-MFD are validated using empirical data. The authors of Loder et al. (

The remainder of the manuscript is organized as follows: section 2 presents the empirical data for the 3D-MFD, section 4 details the inflow demand estimation techniques, section 5 illustrates the trip length estimation method, section 6 discusses the results and finally, section 7 presents the conclusions of the work.

_{c}) only and public transport vehicles are excluded. It is discussed later about the rationale of choosing only private cars 3D-MFD.

The partitions of City center and Wiedikon considered in the present work in the city of Zürich, Switzerland.

Empirical 3D MFDs for the considered regions.

Empirical data of City center and Wiedikon regions.

In order to fit the empirical data, the same functional form proposed by Loder et al. (

where subscripts _{c,r} and _{b,r} are the accumulations of private cars and public transport vehicles (in veh) in the region _{c,r} and β_{b,r} are the constants of the fit. As briefed in the stated work, this type of functional form has a clear physical interpretation, i.e., constants of the fit indicate the negative marginal effect of each mode on the mean speed of private cars. ^{2} = 0.954 for the City center, it will be shown later that the fit does not reproduce the traffic dynamics, i.e., the production and accumulation time series, accurately in the context of MFD-based framework. This is due to the constant first-order derivative, which oversimplifies the mean speed evolution. This limitation can be addressed by considering either higher-degree terms in the functional form of mean speed evolution or exponential type of functional form proposed in Geroliminis et al. (

Fit parameters of bi-linear functional form for City center and Wiedikon regions.

_{1,r} |
_{2,r} |
^{2} |
||
---|---|---|---|---|

City Center | 6.4476 | −0.0019 | −0.0164 | 0.95 |

Wiedikon | 6.8912 | −0.0027 | −0.0225 | 0.94 |

Considering higher degree terms in the functional form do not result in a concave-shaped 3D-MFD. At the same time, it was proposed in Geroliminis et al. (

From ^{2} values indicates that relatively tight fits are obtained for both fits.

Estimated fits and empirical data for the considered regions.

As already stated, no OD matrix data is available to the authors for the city of Zurich. Hence, inflow demand for both regions under consideration is unknown and it must be estimated. At the same time, the mean trip lengths inside each region are also unknown. On the one hand, estimating the average trip length and inflow demand of the public transport vehicles is feasible, as they have fixed routes and schedules. On the other hand, estimating the average trip lengths of private cars is far from trivial. It is impossible to estimate the trip lengths of the vehicles using LDD without any additional equipment. Recently, Barmpounakis and Geroliminis (

It is also worth noting that the prediction of the number of public transport vehicles that are circulating in the network is straightforward, as they have fixed schedules and AVL devices can be used to extract this information in real-time. However, predicting the number of private vehicles circulating in the network at any given time is less trivial. Moreover, accurate prediction of this variable is crucial for applications like perimeter control. Hence, the number of private vehicles circulating in the network can be considered as the primary variable of interest. In the present work, the accumulation of public transport vehicles is used as an input to the MFD-based models. Thus, the traffic state evolutions of private vehicles are compared with empirical data to validate the models. This is the reason for choosing empirical 3D-MFD of private cars only and excluding public transport vehicles.

In this work three MFD-based models are used namely: accumulation-based, trip-based and accumulation-based with outflow delay. Accumulation-based model uses conservation equation to resolve the reservoir dynamics given as follows,

where _{c,r} is accumulation of private cars in region _{in,c,r} and _{out,c,r} are inflow and outflow, respectively. Forward Euler scheme is used to discretize this Ordinary Differential Equation (ODE), which yields as follows,

where Δ_{c,r} is the mean trip length for private cars inside the region

Now considering the trip-based formulation, mathematically it can be expressed as,

where τ_{c,r}(_{c,r}. The trip starting times of all vehicles are computed based on the inflow demand. The vehicles leave the reservoir once they travel their assigned trip lengths _{c,r}. During the course of the trip, the mean speed is updated whenever there is an entry or exit of a vehicle.

The outflow is delayed by the order of travel time at any time instance,

where,

The travel time function can be obtained from the velocity 3D-MFD, i.e.,

As explained earlier the bi-modal model is solved for the accumulation of private cars only and therefore, it is enough to re-construct the inflow demand for the private cars. Note that this does not mean that the effect of public transport vehicles on private car traffic is neglected. Both traffic modes are indeed considered but, number of circulating public transport vehicles is given as input to the model as illustrated in section 3. This differs from the general expression of bi-modal MFD models, where accumulations of both modes are simultaneously computed using the bi-modal inflow demand (Paipuri and Leclercq,

where _{in,c,r} and _{out,c,r} are inflow and outflow for region _{in,c,r}(_{out,c,r}(_{c,r}, as follows,

where _{c,r} is the mean trip length of all trajectories of private cars in region _{out,c,r} at time _{in,c,r} at time _{in,c,r}. Once _{in,c,r} is computed, _{in,c,r} can be estimated using Equation (7). Since _{out,c,r} is known at every time step _{in,c,r} is _{i} is the travel time. For sake of simplicity, the subscripts (

Empirical data does not contain the travel time of each vehicle. However, data has the evolution of mean speed with time inside each region (City center and Wiedikon). According to the hypothesis of the MFD-based framework, all vehicles travel at the same speed given by the MFD at any given time. Hence, it is possible to extract the travel time of each vehicle from the mean speed evolution graphically. Two different approaches can be adopted in this context.

Approximation methods to calibrate inflow cumulative curve.

_{i,1}δ_{i,2}δ

where

Once the inflow cumulative curve is estimated, the inflow demand can be computed by taking the first-order derivative of the cumulative curve. It should be noted that both of the presented approaches are only approximation of inflow demand and it is not possible to segregate the internal and transfer demands for each region under consideration. However, it is noticed from the results that in the absence of relevant data, this approach gives a satisfactory inflow demand for the bi-modal MFD-based model.

Estimated inflow demand patterns.

It can be argued that using the same data to estimate the inflow demand profile and assessing the accuracy of MFD-based simulation can result in a degree of over-fitting. However, it will be shown in section 6 that an improper calibration of 3D-MFD fit can result in erroneous traffic state dynamics, even with well-estimated inflow demand. This study highlights the importance of calibration of 3D-MFD fit and it is only possible if the errors from inflow and trip length estimation are minimized. Moreover, different demand scenarios can be studied in the context of implementing demand management strategies with an accurately calibrated 3D-MFD fit. The proposed inflow estimation method can be compared with loop detector data at the perimeter in-bounds of the region, if available. Total inflow traffic counts can be estimated; however, they cover only a fraction of links in the region. Hence, the traffic counts on these selected links need to be expanded to the entire region by using some metric of expansion factor. Estimating this expansion or scaling factor is not straightforward and can introduce bias in the macroscopic traffic variables. Yet, this study cannot be realized in the current context as data is available only in the form of aggregated loop observations and not individual ones.

As already mentioned, there is no available empirical data on the trip lengths inside each region. Therefore for estimating trip lengths, a network exploration technique (Batista et al.,

Origins are randomly generated, 2,000 in the present work, for the whole city of Zurich.

Network exploration technique.

Distribution trip lengths.

It is worth noting that this method yields an approximate trip length distribution. In reality, users do not always take network shortest path in distance. Paipuri et al. (

As presented in section 2, the evolutions of accumulation and mean speeds vary widely from day-to-day for both regions under study. Hence, only 3 weekdays, i.e., Monday, Tuesday, and Wednesday, are selected to present the validation results. It is noticed that the remaining 2 weekdays have very different evolutions, especially during unloading periods. Consequently, these 2 days require a separate 3D-MFD fit in order to have an accurate prediction of traffic states. For instance, in the case of City center, it can be noticed from

_{1}–A_{3}_{1}–C_{3}_{1}–B_{3}^{2} is obtained using a single 3D-MFD, the scatter in the empirical data has a significant influence on the simulation results. It can be argued that the 3D-MFD fit can be adjusted to improve the evolution of traffic states in the present case. However, the accuracy gained in a specific time period by refining the fit will result in a higher discrepancy in another time period. In the present case, the fit yields reasonable accuracy at the peak periods, where off-peak periods show significant differences.

Evolution of accumulation, outflow, and mean speed in the City center using single bi-linear 3D-MFD for 3 weekdays. _{i}_{i}_{i}

This phenomenon can be noticed in the case of 2D-MFD as well, where using a single functional fit cannot describe the relationship between travel production and accumulation for the whole range of observed accumulations. Consequently, inconsistencies in the evolution of traffic state variables can be observed for a certain range of accumulations. As in the case of 2D-MFD, this limitation can be addressed using piece-wise bi-linear 3D-MFDs instead of single 3D-MFD. Estimating piece-wise linear or quadratic 2D-MFD is a straight forward whereas, computing a piece-wise bi-linear 3D-MFD is more complicated. At least three different approaches can be proposed here: (i) the range of accumulation of private cars can be divided into several bins and a 3D-MFD fit can be defined within each bin, (ii) the range of accumulations of public transport vehicles can be binned, where different 3D-MFDs can be defined for different bins and (iii) both accumulations of private cars and public transport vehicles can be divided into several bins. The third approach is not feasible using empirical data, as private cars and public transport vehicles tend to operate within a particular region in the accumulation space. Thus, 3D-MFD fits cannot be defined for the ranges of accumulations, where empirical data is not available. However, this approach can be used in micro-simulation studies, where simulations with different demand patterns can be used to cover the entire accumulation space. Either the first or second approach can be used with the current data set.

Another critical factor to consider while defining the functional fit is the scatter in the data. It was already shown the existence of the hysteresis phenomenon in the case of 2D-MFD due to various factors like network heterogeneity, demand pattern (Buisson and Ladier,

_{1}–A_{3},C_{1}–C_{3}_{1}–B_{3}

Fit parameters of multi bi-linear 3D-MFD fits for City center region.

_{1,d} |
_{2,d} |
^{2} |
|||
---|---|---|---|---|---|

I | 00:00 a.m. to 08:15 a.m. | 8.0607 | −0.0024 | −0.0411 | 0.99 |

II | 08:30 a.m. to 13:00 p.m. | 6.1729 | −0.0024 | −0.0053 | 0.67 |

III | 13:15 a.m. to 16:00 p.m. | 5.7709 | −0.0019 | −0.0046 | 0.83 |

IV | 16:15 p.m. to 23:45 p.m. | 7.1409 | −0.0018 | −0.0346 | 0.98 |

Evolution of accumulation, outflow, and mean speed in the City center using multi bi-linear 3D-MFDs for 3 weekdays. _{i}_{i}_{i}

In order to present the differences between different MFD-based models, day 1 is selected.

Evolution of accumulation and mean speed in the City center using multi bi-linear 3D-MFDs for day 1.

A similar approach is used for the Wiedikon region to validate the MFD models with empirical data. _{2,r}, which quantifies the effect of public transport vehicles on private cars mean speed, is positive for time periods II and III. It signifies that the presence of public transport vehicles is increasing the mean speed of private cars, which is counter-intuitive. However, this coefficient in

Fit parameters of multi bi-linear 3D-MFD fits for Wiedikon region.

_{1,w} |
_{2,w} |
^{2} |
|||
---|---|---|---|---|---|

I | 06:00 a.m. to 08:45 a.m. | 8.2696 | −0.0046 | −0.0047 | 0.92 |

II | 09:00 a.m. to 12:45 p.m. | 7.1347 | −0.0039 | 0.0105 | 0.55 |

III | 13:00 p.m. to 15:30 p.m. | 6.3569 | −0.0031 | 0.0244 | 0.75 |

IV | 15:45 p.m. to 23:45 p.m. | 7.7508 | −0.0025 | −0.0748 | 0.93 |

Another important observation in empirical data of the Wiedikon region is the traffic states of evening peak hour. It can be noticed from

Evolution of accumulation, outflow, and mean speed in the Wiedikon using multi bi-linear 3D-MFDs for 3 weekdays. _{i}_{i}_{i}

The final part of the current work demonstrates the importance of considering bi-modal MFD in the MFD-based simulations for the networks with an ample number of circulating public transport vehicles. Instead of separating the modes of vehicles, it is assumed that the network consists of a single population of vehicles. The empirical data of private cars and public transport vehicles are treated alike and an MFD is established between total network production and total network accumulation. In other words, this method yields a classical 2D-MFD relating the total accumulation to the total production in the network. This 2D-MFD can be used with regular MFD-based models (Mariotte et al.,

where _{d} is accumulation of vehicles in the City center. In order to be consistent with the 3D-MFD approach, multiple 2D-MFDs based on different time periods are considered in the present context as well.

Fit parameters of multi linear 2D-MFD fits for City center region.

_{1,d} |
^{2} |
|||
---|---|---|---|---|

I | 00:00 a.m. to 08:15 a.m. | 6.8128 | −0.0028 | 0.99 |

II | 08:30 a.m. to 13:00 p.m. | 6.0237 | −0.0022 | 0.63 |

III | 13:15 p.m. to 16:00 p.m. | 5.6202 | −0.0018 | 0.79 |

IV | 16:15 p.m. to 23:45 p.m. | 6.0580 | −0.0021 | 0.97 |

Evolution of accumulation and mean speed in the City center using multi linear 2D-MFDs for day 1.

The primary objective of this work is to validate the bi-modal MFD-based models using empirical data of the city of Zurich. The data contains the time series values of accumulations and productions for private cars and public transport vehicles for 7 days for two different regions, namely, City center and Wiedikon. However, crucial information about the mean trip lengths, inflow demand or OD matrix is missing to perform a representative MFD simulation.

A single bi-linear functional form is considered to fit the empirical data as the first step toward validation. A simple yet, efficient method to estimate the inflow demand is proposed. Two different approaches can be used in the context of inflow demand estimation. The first approach assumes the travel time to be constant throughout the trip. In contrast, a more accurate second approach accounts for the variability in travel time by discretizing the time by a finite time step. The second approach relies on the finer resolution of the mean speed data. Two approaches yield very similar inflow demand profiles in the present work as the mean speed is aggregated for every 15 min and travel times are typically lower than the aggregation interval. In order to estimate the mean trip lengths, a network exploration technique is used. The whole Zurich network is randomly sampled and the shortest paths between OD pairs are computed. The trips that transverse across the regions under consideration are extracted to estimate the trip lengths distribution and eventually, mean trip lengths.

It is noticed from the simulation results that using a single bi-linear 3D-MFD is a too crude approximation for predicting the evolution of traffic states accurately. Hence, the approach of multi 3D-MFD fits based on certain time periods is proposed. Four different 3D-MFDs are considered covering peak and off-peak periods during morning and evening times. It is noticed that the simulation results with the proposed multi bi-linear 3D-MFDs are relatively in good agreement for both the City center and Wiedikon regions. The improvement from using single 3D-MFD to multi 3D-MFDs in the simulation results can be attributed to a better representation of empirical data. The current framework can be extended to validate the passenger flow instead of the flow of vehicles. Estimating an empirical passenger 3D-MFD is more challenging as the occupancy data of private cars and public transport vehicles is difficult to gather. If the data is available, the current validation method can be used with the accumulation of passengers as the primary unknown variable instead of the accumulation of cars. Finally, the paper is concluded by showing that MFD-based models based on 2D-MFD have limitations, especially when the networks with public transport are considered. The shortcomings are demonstrated by performing an MFD-based simulation using 2D-MFD instead of 3D-MFD.

The data analyzed in this study is subject to the following licenses/restrictions: we access the data on the courtesy of their owner at ETHZ. The data should be asked to the owners. Requests to access these datasets should be directed to Loder Allister,

LL came up with the idea, participated in the discussion of the results, edited the manuscript, and responsible for funding acquisition. MP did the data curation, implemented the models, performed the formal analysis, investigation and visualization of results, and preparation of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer LA declared a past co-authorship with one of the authors LL to the handling editor.

The authors are very grateful to Dr. Allister Loder and all the ETHZ transportation groups lead by Prof. Axhausen and Prof. Menendez for sharing the Zurich city data from their previously published work (Loder et al.,

_{∞}robust perimeter flow control in urban networks with partial information feedback