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Design and analysis of control strategies for pedestrian flows

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Abstract

Exploiting the full potential of pedestrian infrastructure is becoming critical in many environments which cannot be easily expanded to cope with the increasing pedestrian demand. This is particularly true for train stations as in many dense cities space is limited and expansion is difficult and very costly. In this paper, we investigate how to improve the level-of-service experienced by pedestrians by regulating and controlling their movements with a dynamic traffic management system. Although dynamic traffic management systems have been widely investigated in the last two decades to mitigate vehicular traffic congestion, little attention has been given in the literature to dynamic traffic management systems for pedestrian flows. The objective of this paper is to develop the concept of a dynamic traffic management system for pedestrian flows by building on the experience acquired from vehicular traffic management systems. We first propose a general framework for dynamic traffic management systems which takes into account the specificities of pedestrian traffic. The specificities of pedestrian traffic are discussed and emphasized. Then we illustrate the framework by using a control strategy designed for pedestrian flows that mitigates the issues induced by bidirectional flows. We show the effectiveness of this strategy by simulating a subpart of the train station in Lausanne (Switzerland). The results show a substantial improvement despite the relative simplicity of the method. These results emphasize the under-explored potential of pedestrian control and guidance when integrated into a dynamic pedestrian management system.

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Acknowledgements

This research was performed as part of the TRANS-FORM (Smart transfers through unravelling urban form and travel flow dynamics) project funded by the Swiss Federal Office of Energy SFOE and Federal Office of Transport FOT Grant Agreement SI/501438-01 as part of JPI Urban Europe ERA-NET Cofound Smart Cities and Communities initiative. We thankfully acknowledge both agencies for their financial support.

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All authors contributed to the study conception and design. Numerical simulations and software implementation were performed by NM. The section on dynamic pedestrian management systems was developed by NM and MB. The analysis of the results was performed by NM with guidance from RS and MB. The first draft of the manuscript was written by NM and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Nicholas Molyneaux.

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Appendix: Control strategy: gating

Appendix: Control strategy: gating

The movement of pedestrians can be controlled in a way similar to ramp metering (Papageorgiou et al. 1991) and signalized intersections (Febbraro et al. 2004). With the high temporal variations in pedestrian demand, congestion can occur in some areas of the infrastructure while others are still empty. To mitigate the risk of dangerous situations, the flow of pedestrians can be regulated to prevent high levels of congestion. As in road traffic, intersections where multiple streams of pedestrian join are likely to reach higher congestion. Since pedestrian traffic is not constrained by lanes, each pedestrian can choose her sub-route through the junction. In order to guarantee a good level-of-service through the intersection the pedestrian density must not become excessive. To accomplish this, we propose a reactive gating scheme which can control the flow of pedestrians in real-time.

Spatial and temporal representations The spatial domain L is represented in a hybrid way. Firstly as an open continuous space in which pedestrians can move around, and secondly as a graph used for route choice. Time is considered continuous, although the numerical implementation enforces some level of discretization. The graph is used by the pedestrians to navigate the infrastructure and find the path to their destinations. Each node in the graph is an intermediate destination. The motion model uses these intermediate destinations and makes individuals walk towards them.

Supply For this case study, the supply data is considered static. We dot not use elements such as shops or a public transport schedule. The infrastructure used for this case study is a four-way intersection, presented in Fig. 16. This setup is common in train stations for example.

Demand The demand is composed of pedestrian flows with specific origins and destinations. Each individual \(n\in N\) has a free-flow walking speed \(v_n\) sampled from a normal distribution with a mean of 1.34 m/s (Weidmann 1993). Their origin and destination are sampled inside zones representing the entrance and exit points from the infrastructure (zones abc and d from Fig. 16). The infrastructure used for the case study does not contain multiple paths to the pedestrian’s destinations, therefore route choice is reduced to route following. For a given origin and destination, only one single route exists.

For the sake of simplicity, we assume that the pedestrian demand originates in the extremities of the short corridor sections. The pedestrian demand \(D(T')\) comes from two different sources. The first group comes from pedestrians who are walking along the main corridor (moving between a and c in Fig. 16). The second group of pedestrians are those who disembark from trains (entering through zones b or d and walking towards a or c). The demand patterns are different for both of these groups and are represented in Fig. 17 through their respective arrival rates. A Poisson process is used to generate the individual entrance times based on the arrival rate. On the one hand we assume that the pedestrian demand along the main corridor is uniform (\(q_{unc} = 1.0\)) and on the other we assume that the demand coming from the trains is sine-shaped (the total arrival rate is \(q_{con}(t) = 8.0(0.49(\sin (0.15t)+1)+0.05)\)). We use a sine-shaped demand pattern since this is a rough approximation of the demand pattern induced by trains when passengers are alighting.

Fig. 16
figure 16

Route graph and zones used as origin, destination or intermediate destination

Fig. 17
figure 17

Demand pattern used to evaluate the effectiveness of the gating strategy

Fundamental quantities and data The fundamental quantities of interest \(\Delta\) in this analysis are pedestrian density \(\rho\) and flow q. Density is computed using Voronoi diagrams (Nikolić and Bierlaire 2014). By using Voronoi tessellations, an individual density value is obtained for each pedestrian at a given time snapshot. This definition of density reduces the influence of the physical characteristics of the area in which the density is computed and captures the heterogeneity of the pedestrian dynamics. This method significantly reduces the influence of the size of the zone on the density computation, which is the major drawback of the classical average density. The Voronoi density of pedestrian n at time t is denoted by \(\rho _n(t)\). We recall that the density is computed at regular intervals of 1 s. The density is not measured in the whole environment but in the spatial context \(L'\). In Fig. 16 the grey zone in the center, denoted \(L'\), is the area inside which density is measured. The pedestrian flow is not measured since it is the quantity controlled by the strategy. The real time data \(\Delta ^*\) is used by the controller but no historical data is used. In this simulation environment, the quantities are computed during the simulations, but in a true life scenario, these quantities are accessible thanks to cameras or flow counters.

Control and information Since the objective is to regulate the pedestrian flows entering the intersection we propose the usage of gates to achieve this objective. These control devices place an upper bound on the flow of pedestrians. In Fig. 18, two gates are represented with the symbols \(g_1\) and \(g_2\). The configuration \(C_{g_1}(T')\) and \(C_{g_2}(T')\) of these gates is the sequence of flows allowed through the devices over time. The pedestrian flow is modulated continuously over time.

To avoid radical changes in the device’s configurations, the policy updates the configuration at regular intervals of 1 s. The simulation environment uses a discrete event simulator (DES) to manage the events linked to the control strategy. This DES is combined with the time-based motion model from NOMAD.

The key performance indicator needs to be defined in order to measure excessive congestion inside the intersection. As we have detailed information regarding the current level-of-service that each pedestrian is experiencing, we can define an indicator which takes into account the high spatial variability. As pedestrians who experience low density can still move freely, we wish to define an indicator which focuses on those experiencing high densities. Firstly, we define the difference between “low density” and “high density”. This is done by setting a threshold \({\bar{\rho }}\), below which pedestrians are considered to be in an uncongested environment. Using this threshold, we can define the indicator:

$$\begin{aligned} \kappa _{L'}^{{\bar{\rho }}}(t) = \sum _{n\in {N'}}[\rho _n(t) > {\bar{\rho }}], \end{aligned}$$
(12)

where \(\kappa _{L'}^{{\bar{\rho }}}(t)\) can be read as “the number of people inside intersection \(L'\) at time t whose density exceeds the threshold \({\bar{\rho }}\) ”. \(N'\) is the set of pedestrians inside the intersection \(L'\).

Fig. 18
figure 18

Infrastructure used to simulate the usage of gates to control pedestrian flows

After defining a suitable KPI, we need to specify the control policy linking the KPI to the controlled variable. The objective is to regulate the inflow of pedestrians into the intersection based on the pedestrian density which is occurring in the intersection. Here a reactive scheme is used, hence the density is computed at time t and we then fix the inflow of pedestrians based on \(\kappa _{L'}^{{\bar{\rho }}}(t)\). The strategy configuration is linked to the control policy as:

$$\begin{aligned} C_{g}([t^*, t^+]) = {\mathcal {P}}_g(\kappa _{L'}^{{\bar{\rho }}}(t^*)), \end{aligned}$$
(13)

where \({\mathcal {P}}_g\) is the control policy for gate g. This function must be specified in order to make the control strategy operational.

As stated previously, each gate in the system requires an explicit control policy \({\mathcal {P}}_g\) in order to work. For the present case, an offline simulation-based optimization algorithm has been used to find the best control policy specification given an objective function (Ali et al. 2002). Since the simulation is a stochastic process, multiple replications of each scenario are performed to compute the distributions of the indicators under investigation. For this case study, 500 replications are used. In order to reduce the computational burden of the optimization procedure, the control policy has been constrained to a quadratic function \({\mathcal {P}}_g(\kappa ) = a+b\cdot \kappa + c\cdot \kappa ^2\) (for the sake of readability the indices have been dropped on \(\kappa\)). Furthermore, the density threshold \({\bar{\rho }}\) above which the pedestrians are considered congested is also a decision variable in the optimization procedure. The goal of the optimization is to find the optimal quadruplet \(\{a,b,c,{\bar{\rho }}\}^*\).

The objective function used for this optimization is a combination of two elements. The first element is the median of the 75th percentile of the travel time distributions divided by the 75th percentile of the travel times. The second element is median of the 75th percentile of the travel times distribution through the area \(L'\) divided by the 75th percentile of the travel times through \(L'\). The division by the reference values gives equal weight to each component. This objective function can be written mathematically as

$$\begin{aligned} \frac{med(TT^{75})}{TT^{75}_{ref}} +\frac{med(TT^{75}_{L'})}{TT^{75}_{L', ref}}, \end{aligned}$$
(14)

where \(TT^{75}\) is the 75th percentile of the travel times distribution from one simulation. The subscript ref refers to the reference scenario before gates where installed and the subscript \(L'\) refers to the travel times through the intersection \(L'\). This definition gives emphasis on improving the travel time through the intersection without neglecting the system as a whole.

The optimal set of parameters is shown in Fig. 19a alongside a visualization of the control policy (Fig. 19b). The optimal value of the density threshold is high in terms of pedestrian level-of-service. A value of 3.49 pax/m\(^2\) gets categorized as LOS F (Fruin 1971). A level-of-service (LOS) of F corresponds to a pedestrian density above 1.66 pax/m\(^2\). Secondly, based on Fig. fig:gating:paramsb it is apparent that the best control policy will nearly close the gates as soon as one pedestrian experiences congestion. When one pedestrian experiences a density equal or higher than 3.49 pax/m\(^2\), the inflow into the intersection is reduced to 1.33 pax/s.

Fig. 19
figure 19

Specification of the control policy for both gates used in the case study presented in Fig. 16

State estimation and prediction The control strategy does not rely on prediction. Furthermore, in this example, state estimation is not necessary as the data is readily accessible.

Control and information configuration generation The pedestrians’ reactions to the control and information strategies should be taken into account by addressing the consistency problem materialized by the fixed point problem (7). Since the present case study does not give pedestrians any choice regarding routes or compliance to information, the consistency problem is neglected. The demand is not affected by the control strategy.

The gate’s configuration is therefore computed as:

$$\begin{aligned} C_{g}([t^*, t^+]) = 5.31-1.94\kappa (t^*) - 2.04\kappa (t^*)^2 \end{aligned}$$

where \(\kappa (t^*)\) is the KPI computed using (12). The length of the interval \([t^*, t^+]\) is 1 s.

Results

For each replication of the simulation, the median travel time of all pedestrians is computed. We then visualize the distribution of these medians by using a box plot. The results for the reference scenario without gates and the case with gates are presented in Fig. 20. It is apparent that the median travel times do not change significantly between both scenarios. Gating slightly decreases the mean of the median travel times. The median of median travel times increases when gating is used. Although the travel times are not significantly improved, a positive effect on travel time variance is observed. This is visible through the reduction in variance in the box plots. Without gating the interquartile range is 2.6 s, which is then reduced to 2.06 s when gating is used.

Fig. 20
figure 20

Distribution (500 replications) of the median travel time for both scenarios. No major difference in terms of travel time is visible between both scenarios

To further understand the influence gating has, we can investigate the median travel time distribution per origin-destination group. The pedestrians are classified into two groups: the first group contains pedestrians who go through the gates while the second group is composed of pedestrians not using the gates. Figure 21 presents the median travel time distribution per group for both scenarios. On one hand, gating slightly increases the travel time for the group which use the gates, the median of median travel time goes from 23.00 to 23.63 s. On the other hand, gating significantly improves the walking times of pedestrian who don’t go through the gates. Multiple reasons can explain this result. When pedestrians travel through the gates their travel time is composed of the walking time and the waiting time. When the waiting time exceeds the reduction in walking time induced by the gates, their trip time will increase compared to the reference scenario. Ideally this waiting time is more than compensated when they are allowed to walk through the gates into the intersection: their journey through the intersection should be faster given the lower density. Since the travel time indicator is higher, the excess travel time induced by the gates is not compensated by the faster travel through the intersection. Concerning the pedestrians who don’t use the gates, their significant gain in travel time is explained by the faster walking speed through the intersection which is not hindered by the gates. Since the flow of pedestrians coming through the gates is “flattened” by the gates, it is easier for the pedestrians to move through the intersection.

As supported by the fundamental diagram concept, high pedestrian densities decrease their walking speeds. To confirm these effects we now consider the distribution of pedestrian density inside the intersection. Figure 22 presents the distribution of mean density computed using Voronoi diagrams. When gating is implemented, the mean density is significantly reduced. The control strategy also reduces the variance in density meaning more consistent situations are experienced by the users. The significant reduction in density confirms that the gain in travel time for the pedestrian who don’t use the gates comes from the reduction in density in the intersection.

Fig. 21
figure 21

Distribution (500 replications) of the median travel time for both OD groups for both scenarios. Gating significantly improves the travel time for pedestrians walking along the main corridor

Fig. 22
figure 22

Distribution (500 replications) of mean individual density for both scenarios. Gating significantly reduces the density that pedestrians experience

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Molyneaux, N., Scarinci, R. & Bierlaire, M. Design and analysis of control strategies for pedestrian flows. Transportation 48, 1767–1807 (2021). https://doi.org/10.1007/s11116-020-10111-1

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