Improving Highly Automated Traffic Management Models Using Alternative Graph Structures Simultaneously

Abstract

This research focuses on improving the modelling of highly automated centralized traffic management. Authors use a binary integer modelling for traffic flow optimization. One of the main results of the research is the integration of alternative graph structures simultaneously in the investigated traffic optimization model. This allows to take into account different categories of vehicles, especially large vehicles, and specific traffic directions. The study highlights the need for seamless interoperability between graph structures and demonstrates its feasibility through the investigation of advanced safety indicators. In addition, methods are developed and presented that will allow the future integration of non-automated traffic elements and conventional traffic engineering solutions. During our research, we reviewed the automated traffic management models, focusing on the advantages of both continuous and discrete models. Continuous models provide realism but are computationally intensive, while discrete models are simpler but less realistic. Considering this, our paper proposes solutions that offer a trade-off between these approaches, allowing a balance between computational complexity, accuracy, and realism in vehicle modelling.

1. Introduction

The report of the United Nations (“Our Common Future”) defined sustainable development in 1987 as “development that meets the needs of the present without compromising the ability of future generations to meet their own needs” [1]. Our transport systems should also adopt this approach. In the light of this, our transport systems should be designed not only to meet the needs of the present, but also, where possible, to meet the needs of the future. This appears to be particularly challenging in areas where technology and societal expectations are in such a state of flux and rapid evolution that we can only estimate future needs with a high degree of variance. In such cases, we must try to ensure that the system has the flexibility to adapt easily to needs that may not be known today, or that may be able to be met with little effort, cost or loss.

Based on Winter’s definition [2], in self-organising transport, users and providers communicate in a peer-to-peer way, avoiding the negative effect of a third party acting as a coordinator. Of course, some services still need to be centrally organised (such as trip planning, car sharing, ride sharing, etc.). For those services where the coordinating actor needs to have a more detailed knowledge of the available information (such as the network, available vehicle types, etc.), the role of the central components cannot be questioned. However, when mobility patterns (especially demand and supply) cannot be reliably estimated, centralised systems quickly reach their limits in providing a real-time dynamic representation of the real-world processes and in operating in an automated mode. In this case, self-organising networks can operate more efficiently [2].

So, we can see that the flexibility and adaptability of systems can be significantly improved if we take self-organising transport systems into account in the design. However, by basic definition, they differ from traditional public transport systems mainly in the relationship between user and provider. And this is where this article aims to contribute to the development of methodological tools in the field. It can be assumed that under certain circumstances and to a certain extent, in the case of an autonomous or highly automated transport system, the transport infrastructure can be considered as an adaptive actor among the other components that make up the transport system (e.g., the user or the service provider).

But how can transport infrastructure adapt to changing needs? New access roads can be added to a core network element, new legs can be added to an intersection to meet demand, or a motorway lane can be removed to allow trees to grow quickly in its place when traffic is low. Unfortunately, as science stands today, we are not there yet. But it is no longer science fiction to allocate a lane to different directions throughout the day, depending on changing demand. Reflecting further on this example, we can find effective solutions by treating traffic engineering not as a static tool, but as a dynamic system that can adapt to changing needs. The highest level of flexibility, considering the dynamic nature of traffic engineering, would be achieved if traffic engineering could be instantly adapted to current needs. This can be illustrated with abstract examples such as a lane closure that suddenly disappears during a lane change manoeuvre, or a roundabout that opens to let a vehicle through on the opposite leg.

Even if we do not want to give such abstract examples, we would like to explore how we can make our transport system more flexible by reconsidering the spatial representation models and improving their adaptability.

The main contributions of our paper were highlighted in Section 2. In Section 3, the related literature is reviewed, focusing on the representation methods of transport network in highly automated transport systems. Section 4 introduces the modelling framework together with advances focusing on improving the applicability of discrete transport models and infrastructure representation. The methods are demonstrated and validated through a numerical example in Section 5. Further development directions are summarized in Section 6, highlighting interesting opportunities provided by the simultaneous use of alternative graph structures.

2. Main Contribution, Ambition

During our research, we examined the main directions in the development of automated traffic management models. Models based on continuous and discrete variables offer different advantages. Continuous models provide good solutions for an increasingly realistic representation of real-world processes, but the associated calculations are quite computationally expensive. Discrete models are less complex but not always suitable for realistic modelling of reality.

Accordingly, the most relevant contribution of this paper is to introduce solutions that allow the simultaneous use of alternative representations of infrastructure with different levels of detail, improving the applicability of automated traffic management models. This allows for the modelling of different categories of vehicles and provides the possibility to ensure the desired balance between computational complexity, accuracy and realism of the process.

3. Related Works

Together with the development of cooperative, connected and highly automated mobility solutions (CCAM), reliability and safety requirements have inspired significant research efforts in the transport sector and automotive industry [3,4,5,6,7]. New tasks are emerging at the microscopic and macroscopic levels. Studies on the microscopic level have been applied to vehicle control tasks such as driving or parking [8,9,10,11,12]. At the macroscopic level, the main challenge of the future is the complex and advanced traffic management [13,14,15,16]. Recent research emphasizes that advancements in communication technologies, including 6G, also play a critical role in enhancing the efficiency of traffic management systems, thereby facilitating the integration of autonomous vehicles into existing infrastructures [17,18]. Automation includes vehicle control and service planning as well as functions related to passenger processes. This offers the potential to significantly reduce random decisions influenced by external parameters (e.g., congestion [19] or travel costs [20]).

A part of the related literature focuses on individual vehicle control ([21,22]). Others work on a broader perspective, handling the autonomous transport in the system-of-systems approach ([14,23,24]). In both cases, we can conclude that it is fundamental to assign the elementary regions of the infrastructure used for vehicle manoeuvres to the vehicles. To avoid collisions and conflicts, the principle that a vehicle can only be in one place at a time and only one vehicle can be in one place at a time is generally applied ([25,26]).

With respect to the principal concepts involved, the pivotal role of occupancy grid models (as proposed by Elfes in 1989 [27]) is emphasized. This is one of the most commonly employed approaches for the navigation of autonomous transport systems, and it meets the safety requirements previously mentioned. Grid maps are formed by the cell-based representation of the environment [28]. Several grid map types are used for different research purposes, such as the occupancy grid map, the remission grid map, and the likelihood-field grid map. After the mapping process, there are also several approaches that can be used to assign the vehicles. In the case of occupancy grid maps, each spatial unit of the model includes the occupancy probability related to the given unit square [28]. Occupancy grid mapping approaches have been highlighted, for example, in the paper of Porebski et al. [29].

Numerous studies have used different lattice structures to represent the infrastructure component of highly automated transport systems [30,31]. However, until now, no in-depth research investigated the interoperability between different graph structures or the possibility of simultaneous application.

The most frequently used graph structure when representing the infrastructure component of a highly automated transport system is certainly the square lattice structure [32,33]. Among others, for example Rui Jiang‘s team used the Nagel–Schreckenberg cellular automaton model to simulate traffic for a network similar to the road structure of Manhattan city [34]. The research showed that the real-time availability of information on the shortest path significantly contributes to network operation reliability. Xin Ruan and his colleagues also used a cellular automaton model based on a square lattice structure. They balanced the temporal distribution of the traffic load on bridges, and estimated vehicle dynamics and safety parameters more accurately [35]. Tian et al. also used a square grid graph structure to test the traffic control algorithm designed based on the reaction of the Escherichia coli bacterium to environmental changes. By testing the developed control model, the researchers were able to confirm that models inspired by biological systems have significant potential for controlling future systems [36]. Inoue et al. (2021) also used a square lattice structure to solve the traffic optimization problem based on quantum annealing [37]. The authors showed that the optimization based on quantum annealing gave a better solution compared to the traditional simulation-based procedure.

The different graph structures are also applied in some transport models. For example, hexagon-based lattices are often used to model pedestrian traffic processes. Antoine Torduex and his colleagues studied pedestrian behaviour and route selection processes in a hexagon-based model [38]. The researchers proved that the simulation can be effectively used to model pedestrian traffic processes, and they recommended the wide application of the model. In addition, some research also uses hexagon lattice-based procedures to solve modelling problems related to vehicle traffic. Ke and his colleagues performed a supply and demand forecast related to ride-sourcing using a hexagon lattice-based model [39]. Their study confirmed that the model gave more accurate estimation results compared to previously used methods.

The subsequent section of the literature review will concentrate on studies that examine the optimization of road transport procedures in relation to both spatial and temporal elements. We pay particular attention to the issue of discretization, with an emphasis on the trade-offs between model simplification and realistic representation. It is important to emphasize that the literature review is not exhaustive, and our primary aim is to present relevant examples from the presented groups.

3.1. Models Applying Continuous Time or Space Based Representation

Chalaki and Malikopoulos have developed a distributed optimization environment for V2X-based intelligent transport systems. The model focuses on intersecting vehicle movements to reduce the energy demand of mobility flows and increase transport efficiency [40]. The developed system is made up of two controlling levels. The higher-level control layer iteratively predicts the minimum travel time for all vehicles at every node and selects the appropriate lane to minimize waiting time. The lower-level control layer is an optimization problem that focuses on energy consumption by manipulating the longitudinal motion of vehicles. As a conclusion, the authors demonstrated the performance of the developed system by simulating different intersection scenarios.

Zhou et al. presented a distributed adaptive sliding mode controller (DASMC) that reliably mitigates the effects of various unspecified perturbations [41]. On the way to an intersection, all vehicles use the movement data of neighboring vehicles to perform DASMC in a decentralized manner, resulting in a low-risk joint solution. To deal with high traffic volumes causing unfavorable spillback, the model uses an iterative feedback algorithm to determine the amount of incoming traffic allowed at the junction. The performance of the developed traffic management concept has been demonstrated in several simulations.

Luo, Zhang, and Zhang [42] analyzed the control of wirelessly communicating vehicles by introducing the collision region method and decoupling the control problem into a high-level integrated timing and a decentralized motion planning task. Their demonstrations showed that the new multi-level control concept can obtain close to optimal solutions compared to system-level approaches. Also, the traffic efficiency achieved by the method is much better.

3.2. Models Applying Discrete Time and Space Based Representation

Marzoug and colleagues have developed a complex cellular automation that can interpret the interaction of the transport process characteristics and the collision probability [43]. The authors have shown that traffic characteristics significantly affect the likelihood of lane changes and conflicts.

Another research direction has looked at the control of transport processes for electric vehicles [44]. The whole concept is based on the complex objective function of the optimization problem, which integrates the optimization of travel time and energy consumed. The system consists of (1) a traffic management module that contains the signaling strategy, (2) a traffic simulation module for travel time prediction, and (3) an electric vehicle simulation module for energy usage prediction.

Li et al. suggest a multilevel platoon control method to manage the crossing of traffic flows using neural networks [45]. Higher-level control uses system-level optimization to achieve flexible platooning processes. The objective function includes components such as energy use, travel time or fairness. Low-level control uses a distributed optimization concept to control several intersecting platoons.

4. Methods

Based on the reviewed literature, a key issue for traffic optimization models is whether the applied spatial representation is continuous or discrete. If a continuous spatial representation is used, there is no need to apply additional solutions between discrete spatial points or domains for the optimized variable to control the system smoothly. On the other hand, if the entire decision space is considered in a continuous model, the complexity of the task increases significantly. Accordingly, this is likely to be a more computationally demanding problem than the use of a discrete spatial representation.

Square grid-based methods are widely used for discrete spatial representation. In this case, the computational complexity of the problem is likely to be more favorable. However, this method will only give the values of the variables that describe the behavior of the system at discrete points or regions in space. Therefore, we need to implement additional control levels that implement the control of the system between the discrete points/regions. A compromise solution would be, for example, to define additional squares of equal size between the elements of the square grid. This would allow us to influence the operation of the system between two discrete domains but limit the increase in complexity of the problem compared to a continuous representation. However, it is difficult to imagine that this spatial model would work, since it would lose the property of the square grid dividing the spatial domain into disjoint sets.

Our research ([46,47,48]) has confirmed that the interoperability between different grid structures can significantly contribute to the applicability of graph-based, temporally and spatially discretized models. Therefore, in this article we present the possible advantages and ways of using different lattice structures in an integrated model. Accordingly, in the next part of this article, we introduce a novel representation making it possible to implement the flexible and adaptive infrastructure mentioned in the introduction.

4.1. Methodological Basis—A Widely Used Concept

The new approach starts by defining the essential requirements of the initial model, which can be modified in the required direction and scope as the development progresses.

A key requirement of models using discrete spatial representation is that only one vehicle can occupy each elementary region. These regions do not overlap and form together the occupancy grid. Another important constraint is that a vehicle can only be in one elementary part of space at a discrete time. Since our goal is to uniquely assign the vehicles to regions, we want to avoid vehicles using more than one region at the same time. Safety and dynamic related constraints, such as speed, acceleration or deceleration limits must also be applied. This avoids high-risk situations where several vehicles arrive at the same elementary region at the same time, or where some vehicles may be travelling above the speed limit.

To be able to support the traffic management processes of highly automated transport systems through the discretization of space and time [49], we described the binary integer optimization problem for traffic flows in our model [46]. By reducing the complexity of the model, we managed to improve the efficiency of the developed method [47], which may later play a key role in the real-time traffic management processes. We also showed that the spatial structure of the applied lattices representing the infrastructure component has a significant impact on the efficiency and safety of the traffic management processes [47].

Considering that the previously presented procedures focused on safety and methodological aspects, we developed the optimization method for model components of unit vehicle size. Based on that, it seems reasonable to further develop the model for vehicles of different sizes, taking into account the typical vehicle categories (trucks, heavy goods vehicles, trailers, etc.). By providing interoperability between the applied lattices, it is possible to use the combination of graphs consisting of different sized regions. For this purpose, the generation of different lattices and the investigation of interoperability and simultaneous application are presented in our article.

4.2. Modelling Framework

The referred model developed by the authors was based on a nonlinear binary integer programming approach [46,47]. The applied 3-dimensional binary decision variable ($x_{k,j,i}$) described accurately the position (region $i$) of each vehicle $\left(k\right)$ at each time moment $\left(j\right)$ in the investigated timeframe. Thus, time-space was discretized, and the road network was partitioned into directed locations (regions) in the model in accordance with the occupancy grid concept [27]. The total number of locations on the network was denoted by $o$, the total number of considered vehicles was denoted by $m$, and the total timeframe covered $t$ time moments.

The aim of the developed model was to manage the travel demands effectively by optimizing the load of the transport network at system-level. Note that system-level processes were managed by the implemented vehicle-level transport management process in our model. The following parameters were considered as predefined constant values [46]:

• The discretization of the road network was carried out using square shaped regions of the same size. The side length of the regions was set at 5 m (based on the size of a typical passenger vehicle). Besides the size, shape, and position of the regions, the directions of possible further travel for each elementary region were also pre-defined.
• The starting and target locations of vehicles were defined in $\mathit{O}\mathit{R}\mathit{I}\mathit{G}\in R^{m\times o}$ and $\mathit{D}\mathit{E}\mathit{S}\mathit{T}\in R^{m\times o}$ binary matrices (${orig}_{k,i}$ and ${dest}_{k,q}$ are represented with a value of 1 if i is the starting and/or q is the ending location of vehicle k, respectively). The origin and destination locations were connected to the network by one-way links and were interpreted as sources and sinks (an unlimited number of vehicles could use them at the same time moment).
• The shortest distance between each location pair was defined in $\mathit{D}\in R^{o\times o}$ matrix ($d_{i,q}$ represents the value of the shortest distance between location i and q, assigning infinite value for non-traversable location pairs).
• $\mathit{R}\_\mathit{C}\mathit{R}\mathit{O}\mathit{S}\mathit{S}\in R^{o^2\times o^2}$ binary relation matrix [46] was created to define the crossing route pairs. The rows and columns of the matrix represented the location pairs (routes) in ascending order, comparing the first location against all others, then the second location against all others, and so on. The value of 1 was assigned for the elements of the matrix representing route pairs that had any common elements (excluding the starting locations). For this purpose, the shortest path between the location pairs were investigated. I.e., element ${r\_cross}_{iq,rs}$ had a value of 1 if the shortest paths between the investigated location pairs (i-q and r-s) existed and had any common location excluding the starting points, otherwise 0.
• The maximum allowed speed ($c1$), acceleration ($c2$) and deceleration ($c3$) limits were also predefined constants.

In accordance with the aim of the optimization, the objective function was defined to sum up and minimize the distances between the current and the destination location of each vehicle within the investigated time frame (Equation (1)).

$$y={\sum }_k{\sum }_j\mathit{x}\left(\mathit{k},\mathit{j}\right)\ast \mathit{D}\ast \mathit{d}\mathit{e}\mathit{s}\mathit{t}\left(\mathit{k}\right)\to min$$

(1)

where

$y$: is the objective function depending on $x_{k,j,i}$;

$\mathit{x}(\mathit{k},\mathit{j})\in R^o$: is a row vector identifying the location of vehicle $k$ at the time step $j$;

$\mathit{d}\mathit{e}\mathit{s}\mathit{t}\left(\mathit{k}\right)\in R^o$: is a column vector from $\mathit{D}\mathit{E}\mathit{S}\mathit{T}$ matrix, identifying the destination location of vehicle $k$.

To ensure safe transport management, the appropriate assignment of time and space elements is a fundamental requirement. This ensures that a vehicle can only be at one place at a time, and only one vehicle can be at one place at a time. The conditions were described in the following expressions (Equations (2) and (3)).

$${\sum }_k{x}_{k,j,i}\le 1 ; \forall j,i$$

(2)

$${\sum }_i{x}_{k,j,i}=1 ; \forall k,j$$

(3)

In addition to the above, it was necessary to define additional conditions for the traffic modelling that describes reality and validates safety aspects. These took into account the limitations of speed, acceleration and deceleration.

To avoid collisions, the crossing vehicle movements at the same time steps were prohibited by the following criterion (Equation (4)).

$$\begin{gathered} \left(x_{k,j,i}{+x}_{k,\left(j+1\right),q}+x_{p,j,r}+x_{p,\left(j+1\right),s}\right)\ast {r\_cross}_{\left(i-1\right)\ast o+q,\left(r-1\right)\ast o+s}\le 3; \\ \forall k,p,j,i,q,r,s \end{gathered}$$

(4)

The expression ensures that two vehicles (k and p) cannot travel routes (i-q and r-s) with common elements at the same time step (j, j + 1), by assigning the appropriate element of $\mathit{R}\_\mathit{C}\mathit{R}\mathit{O}\mathit{S}\mathit{S}$ matrix [46].

As the main limitation of the concept, it has been established that the discretization of the time and space domains and the high number of vehicles encountered in real-world applications may significantly increase the number of model variables. This could result in increasing computational complexity and processing time. To alleviate this problem, authors have developed new methods to reduce the complexity of the presented transport management process [47].

The basic principle of the developed simplification methods was the exclusion of different possible values of the variable from the investigation in the case of the introduced constraints. Based on the specific characteristics of the predefined origin and destination locations (e.g., vehicles could not enter any origin or leave any destination location during their travel), their examination was also omitted for some of the constraining expressions. To highlight an example, we present the approach of excluding variables from the expression prohibiting the crossing movements of the vehicles (see Equation (4)). According to the approach, route pairs that do not have any common elements can be directly excluded from the investigation, resulting in the investigation of only the following cases (Equation (5)) instead of $\forall k,p,j,i,q,r,s$:

$$\forall k,p,j,i,q,r,s; where {r\_cross}_{iq,rs}=1$$

(5)

When implementing the elaborated simplifications, it was crucial not to compromise the feasibility of the solution. Since these methods were intended to modify the model, system safety procedures were applied to mitigate the potential safety risks associated with them [47].

Focusing on the safety aspects of traffic flows in the managed autonomous transport system, indicators were developed characterizing the level of road safety for the obtained results. The different indicators examined different types of risks, considering the spatial and temporal characteristics of the travel processes and the speed data of the vehicles [47]):

• $CM$: the indicator expressed the risks arising from crossing vehicle movements, calculating the number and temporal distance of these manoeuvres, as shown in Equation (6).

  $$M={\sum }_{\begin{array}{c}k,h,j,u,i,q,r,s=1\\ k\ne h\\ u\ne j\end{array}}^{m,m,(t-1),(t-1),o,o,o,o}({r\_cross}_{iq,rs}{\ast (x}_{k,j,i}\ast x_{k,j+1,q}\ast x_{h,u,r}\ast x_{h,u+1,s})\ast {\displaystyle \frac{1}{{\left(u-j\right)}^2}})$$

  (6)
  The value of $CM$ is higher if there are more crossing vehicle movements (meaning more summands in the summation), or if the crossing movements take place in shorter time intervals. Accordingly, a lower value of a $CM$ safety indicator implies a safer flow of traffic [47]. A higher $CM$ value indicates that more vehicle routes are assigned that intersect with each other, and it may also suggest that the intersecting trajectories are traversed by the respective vehicles closer in time, which increases the likelihood of side collisions.
• $\overline{VN}$: the indicator implied the level of safety by the average speed at the network level, calculated as the mean of the average speeds of vehicles in the network.
• ${\sigma }_{VV}$: the indicator characterized the level of safety based on the inhomogeneity of speed values at vehicle level. It was interpreted as the mean of the empirical standard deviation of the individual vehicle speeds in the consecutive time steps [47].
• ${\sigma }_{VN}$: by this indicator, the inhomogeneity of speed values was also examined at network level, as the empirical standard deviation of the average speeds of vehicles in the network.

4.3. Upgrading the Formerly Developed Safety Indicators

In the developed modeling framework, traffic safety can be characterized using additional methods and indicators. In this subsection, we present the further development of our previous indicators ($\overline{VN}$, ${\sigma }_{VN}$) and the development of a new safety assessment concept ($CF$). It should be noted that during the evaluation of the numerical examples, we also applied the previously introduced $CM$ and ${\sigma }_{VV}$ indicators in their unchanged form.

The network level average speed ($\overline{VN}$) was calculated as the mean, and the inhomogeneity of speed values (${\sigma }_{VN}$) was calculated as the empirical standard deviation of the average speeds of vehicles in the network [47]. With a different approach, similar values can be expressed based on the speed data of all vehicles in the consecutive time steps over the investigated time interval. By applying the instantaneous speed values, the distance travelled by the vehicles in the network and the length of time spent there can be taken into account. As a result, the individual vehicles are not considered with the same weight and the estimation can be made more precise.

Based on the above presented considerations, the modified formulas of $\overline{VN}$ and ${\sigma }_{VN}$ are presented in Equations (7) and (8). Notation $orig\left(k\right)$ and $dest\left(k\right)$ indicate the origin and destination locations of vehicle $k$ assigned from matrices $\mathit{O}\mathit{R}\mathit{I}\mathit{G}\in R^{m\times o}$ and $\mathit{D}\mathit{E}\mathit{S}\mathit{T}\in R^{m\times o}$, respectively.

$$\overline{VN}={\displaystyle \frac{{\sum }_{k,j,i,q=1}^{m,\left(t-1\right),o,o}{x}_{k,j,i}\ast x_{k,j+1,q}\ast d_{i,q}}{{\sum }_{k=1}^m(t-1-{\sum }_{j=1}^{(t-1)}{x}_{k,j,dest\left(k\right)}-{\sum }_{j=1}^{(t-1)}{x}_{k,j+1,orig\left(k\right)})}}$$

(7)

That is, the lengths of the road sections that each vehicle travelled was added in the numerator. Then it was divided by a sum that added the number of time steps elapsed between the start of each vehicle from its origin and the arrival to its destination.

To determine the correct number of the considered time steps, the sum of time steps spent in the origin location, and the sum of time steps spent in the destination location were subtracted from the total number of the investigated ($t-1$) time steps inside the summation. In this way, the average speed at the network level was undefined if none of the vehicles left its origin location for the entire investigated timeframe.

A lower value of indicator $\overline{VN}$ implies a safer flow of traffic in the case of equally efficient transport management processes by the lower average speeds of the vehicles.

The indicator ${\sigma }_{VN}$ characterized the level of safety based on the differences in the speed data of the vehicles in the different time steps.

$${\sigma }_{VN}=\sqrt{{\displaystyle \frac{{\sum }_{\begin{array}{c}k,j,i,q=1\\ x_{k,j,i}=1\\ x_{k,j+1,q}=1\\ x_{k,j,dest\left(k\right)}\ne 1\\ x_{k,j+1,orig\left(k\right)}\ne 1\end{array}}^{m,\left(t-1\right),o,o}{\left(x_{k,j,i}\ast x_{k,j+1,q}\ast d_{i,q}-\overline{VN}\right)}^2}{{\sum }_{k=1}^m\left(t-1-{\sum }_{j=1}^{\left(t-1\right)}{x}_{k,j,dest\left(k\right)}-{\sum }_{j=1}^{\left(t-1\right)}{x}_{k,j+1,orig\left(k\right)}\right)}}}$$

(8)

While using the regular formula of standard deviation, the time steps spent in the origin/destination locations were excluded (only the active vehicles were considered).

The indicator refers to the similarity of the speed data of the different vehicles over the considered time steps. Thus, a lower value of indicator ${\sigma }_{VN}$ implies a safer flow of traffic due to a smaller difference between the speed data of the individual vehicles.

As a new method for determining road safety in the autonomous transport system, we propose the investigation of close following vehicle manoeuvres on the network.

According to the concept of the developed binary integer model, it was allowed for a vehicle to arrive at a location that was left by another vehicle at the same time step. The structure of $\mathit{R}\_\mathit{C}\mathit{R}\mathit{O}\mathit{S}\mathit{S}$ matrix made this possible: the value of 1 was not assigned to the compared route pairs in the matrix if the only common element of the shortest routes between them was the starting location of one of the routes. Note that this consideration also made the exclusion of those cases from Equation (4) possible, when any of the compared vehicles was staying in an origin or destination location during the investigated time step. The efficiency of the transport management was facilitated by providing the possibility of such close following of vehicles without harming the defined safety constraints.

However, in such cases, it is rather safety-critical to ensure that the movement process of the followed vehicle actually takes place during the time step between the examined time moments. Otherwise, the basic safety criterion of the introduced traffic management (namely, only one vehicle could be in a given location at a given time moment) concept would be violated. Based on this, the safety indicator related to close following vehicles was defined as the total number of vehicle movements ending at a location that was left by another vehicle at the same time step (Equation (9)).

$$\begin{array}{l}CF={\displaystyle \sum _{\begin{array}{c}k,h,j,i,q,r,s=1\\ k\ne h\end{array}}^{m,m,\left(t-1\right),o,o,o,o}}\left[\right(x_{k,j,i}\ast x_{k,j+1,q}\ast x_{h,j,r}\ast x_{h,j+1,i}+x_{k,j,i}\ast x_{k,j+1,q}\ast x_{h,j,q}\ast x_{h,j+1,s})\\ -\left(x_{k,j,orig\left(k\right)}\ast x_{k,j+1,q}\ast x_{h,j,orig\left(k\right)}\ast x_{h,j+1,orig\left(k\right)}+x_{k,j,orig\left(h\right)}\ast x_{k,j+1,orig\left(h\right)}\ast x_{h,j,orig\left(h\right)}\ast x_{h,j+1,s}\right)\\ -{(x}_{k,j,dest\left(k\right)}\ast x_{k,j+1,dest\left(k\right)}\ast x_{h,j,r}\ast x_{h,j+1,dest\left(k\right)}+x_{k,j,i}\ast x_{k,j+1,dest\left(h\right)}\ast x_{h,j,dest\left(h\right)}\ast x_{h,j+1,dest\left(h\right)}\left)\right] \end{array}$$

(9)

The sum in the first bracket identified those cases when any of the compared vehicles were located at a cell at a time moment ($j$), that was the arriving location of the other compared vehicle in the following time moment ($j+1$). From these cases, those instances were subtracted when any of the compared vehicles stayed in their origin or destination location (sums in the second and third brackets, respectively).

Based on the elaborated formula, a lower value of $CF$ safety indicator implies a safer flow of traffic by the lower number of close following vehicle movements. Accordingly, a higher $CF$ value shows the increased likelihood of rear collisions.

4.4. Relevance of Different Size and Shape Occupancy Grids

In addition to interpreting the basic context, it is also worth considering how to define the size of the elementary regions of space. Of course, increasing the resolution reduces the negative effects of discretisation, such as estimation error and degradation of control performance. On the other hand, the use of grids of larger elementary regions offers certain advantages, such as a reduction in computational complexity and the closely related reduction in computational resources and time. Considering all these, it seems reasonable to introduce a method that allows the interconnection and interoperability of different sized occupancy grids. This would make it possible to adapt the modelling framework for the different sizes of traffic actors, representing an important research advance in the modelling and management of complex traffic situations. In addition, providing interoperability between different occupancy grids also makes it possible to assign separate grids to different traffic directions, which may be beneficial in reducing the complexity of the traffic management.

To examine the computational demands, it is useful to introduce the concepts used to describe the complexity of occupancy grids.

When creating a grid, the plane is filled with polygons such that adjacent polygons cover the plane without gaps and there is no overlap between the adjacent polygons. In most cases, regular filling is used, so that the polygons concerned are congruent regular polygons. We will call these elementary polygons.

In line with the above, we define the sequence of operations to create a grid as covering of the plane without gaps and overlaps. We will refer to each step of the covering process as modelling. We perform the covering process in steps such that each bounding edge of the constrained two-dimensional space domain modeled up to step (r − 1) is adjacent to an elementary polygon modeled at step r.

The constrained two-dimensional space modelled up to the (r − 1)-th step is collectively referred to as the kernel. We will refer to the set of elementary polygons modelled in the r-th step as the r-th layer. By extension of the kernel, we mean the addition of the layer modelled in the r-th step to the constrained two-dimensional space modelled up to the (r − 1)-th step of the covering process. In line with this, we form the kernel modelled in step (r + 1) as the last operation of the r-th step of the covering process by merging the kernel and the layer.

The matching edges create a connection between the elementary polygons of the occupancy grid. The internal connection is the connection between the elementary polygons forming a layer. We describe the complexity of the occupancy grid ($UC$) as the limit of the number of connections per unit area (A) as a function of the number of steps of the covering process (r) [48].

Square grid

The most commonly used regular elementary polygon forming occupancy grids is the square. In this case, the elementary polygon can have a maximum of 4 neighbours, so we do not classify it as one of the more complex grids. It cannot handle diagonal vehicle movements because the polygons that do not touch at the sides are not adjacent. Similarly, there can be no vehicle movement between polygons that only touch at their vertices.

For the occupancy grid under consideration, we express the total number of connections between adjacent squares (N) as a function of the covering steps (r) as described in Equation (10).

$$N_1=0 and N_r=N_{r-1}+2\ast \left(8\ast \left(r-1\right)-4\right)+2\ast \left(8\ast \left(r-1\right)\right), if r>1$$

(10)

where the second term of the summation is the number of connections between the kernel and the layer, and the third term is the number of connections between the elementary polygons within the layer.

The number of elementary polygons (M) forming the occupancy grid can be written as a function of r as follows.

$$M_1=1 and M_r=M_{r-1}+8\ast \left(r-1\right), if r>1$$

(11)

We then describe the area covered by the occupancy grid also as a function of r, based on the area of the elementary regions (polygon) ($A_e$) of the investigated grid structure.

$$A_r=A_e\ast M_r , \forall r$$

(12)

In light of the above, we describe the complexity of the square occupancy grid as follows in Equation (13).

$$\underset{r\to \infty }{\lim }{UC}_r=\underset{r\to \infty }{\lim }{\displaystyle \frac{N_r}{A_r}}$$

(13)

Hexagon grid

For the hexagon-based occupancy grid, the total number of connections between the elementary polygons as a function of the covering steps (r) has been described in Equation (14).

$$N_1=0 \mathrm{a}\mathrm{n}\mathrm{d} N_r=N_{r-1}+\left[2\ast 2\ast \left(6\ast \left(r-1\right)-6\right)+2\ast 6\right]+2\ast \left(6\ast \left(r-1\right)\right), if r>1$$

(14)

where the expression in the square bracket expresses the number of connections between the kernel and the layer, distinguishing the elementary polygons of the layer according to whether they are adjacent to one (first member of the square bracket) or two (second member of the square bracket) of the elementary polygons of the kernel. The third term of the formula expresses the number of connections between the elementary polygons within the layer.

The number of elementary polygons forming the occupancy grid is described in Equation (15).

$$M_1=1 \mathrm{a}\mathrm{n}\mathrm{d} M_r=M_{r-1}+6\ast \left(r-1\right), if r>1$$

(15)

The area covered by the occupancy grid ($A_r$), and the related computational complexity ($\underset{r\to \infty }{\lim }{UC}_r$) then can be calculated as presented in Equations (12) and (13), in this case using the data of the hexagon-based structure.

If the size of the elementary regions in the examined square and hexagonal grid structures is determined such that the diameter of their inscribed circles is uniformly 5 units [48], then based on the presented equations, the value of $\underset{r\to \mathrm{\infty }}{\lim }{UC}_r$ is 1.6 for the square grid, while for the hexagonal grid it is 0.277. This clearly demonstrates the higher computational complexity associated with the hexagonal grid compared to the square grid.

Based on the presented considerations, the modelling process and the calculations related to computational complexity can be carried out according to the same principle using any regular polygon.

Computational complexity analysis

Below we examine the effect of increasing resolution on computational complexity. Accordingly, we investigate what happens when the model is built up from smaller and smaller elementary regions (polygons), gradually reaching higher and higher resolutions. The number of connections between elementary regions characterizes the evolution of computational complexity as a function of the number of elementary regions included in the model. In line with the above, the x-axis of Figure 1 shows the number of elementary regions in the case of the hexagon- and square-based grids.

The number of connections increases slower for the square grid reaching 160 k connections for 40 k elementary regions. In comparison, the complexity of the hexagonal grid (blue circle) grows faster, reaching 180 k connections for 30 k elementary regions.

4.5. Connecting Alternative Networks–Through the Example of Large Vehicle Modelling

In the developed model, the unit regions of the lattice representing the road network were sized for passenger vehicles. In order to handle larger vehicles in the model, there are two basic ways:

• the vehicles are still treated as a unit, therefore cell size has to be increased, or
• the cell size remains the same, then a larger vehicle occupies more cells at a given time.

In our case, the first method seems more reasonable, considering the constraints describing the model (e.g., the system of equations determining that a vehicle can only be in one location at a time). Furthermore, this method supports more realistic modelling: the central system defines the trajectories of the vehicles; however, the realization of real vehicle movements within the cells relies on the vehicle’s sensors. In the case of the second method (a vehicle can occupy several cells simultaneously), a more precise definition of vehicle movements would be necessary, which can be a particularly difficult task in the case of large vehicles without pivot joints.

To ensure that the efficiency of the basic model does not decrease due to the use of a larger cell size, we propose to create one (or more) alternative representation(s) of the same road network. Following this approach, several structures can be created consisting of different-sized and shaped regions for different vehicle categories and special traffic directions. Following this:

• normal size vehicles are handled on the original graph (lattice);
• trucks are moved along the same principles, but on an alternative graph(s) with a larger cell size structure;
• the relationship between the original and alternative graphs is to be defined. To exclude collisions, no more than one vehicle may be present at a spatial point at the same time, regardless of the type of vehicle.

Based on this approach, it is necessary to model the road network using larger size cells considering:

• the characteristics of the developed model: the values of speed, acceleration, and deceleration are derived from the size of the cells travelled during the time units, so larger cell sizes could result in high speeds of the trucks;
• aspects of efficient traffic management: large vehicles traveling on larger cells occupy a large spatial area during the time steps (moving from one cell to another), while during each time step, a part of the large cells is free (and could even be passed through by smaller vehicles).

We can effectively implement the above-mentioned aspects by using overlapping cells in the case of alternative graphs (e.g., for describing the movements of trucks or large vehicles).

Overlapping cells allow the speed to be adjusted realistically and make the system more efficient. In the case illustrated in Figure 2, for example, the size of the cells of the alternative lattice structures (middle and right-side figures) is twice the size of the original cells, which were shifted with the original cell sizes.

Several alternative representations of the same road network (with sufficiently large cells) can be used to manage different vehicle categories, emergency vehicles, and special traffic directions. As Figure 2 illustrates, the efficiency can be improved even further in this way, since it is not necessary to increase the lateral size of the cells for a realistic description of the traffic movements. For example, the truck moving from location i = 17 to i = 34 covers route (17-21-25-43-30-34).

The alternative graphs increase the number of considered locations (o). In the following, the total number of locations of the original representation is denoted by $o_O$, while the number of all locations of the alternative mappings is denoted by $o_A$. The number of considered passenger cars is denoted by $m_p$, and the number of trucks by $m_h$.

Accordingly, the predefined parameters of the model have to be adjusted as follows:

• It is necessary to extend the $\mathit{O}\mathit{R}\mathit{I}\mathit{G}$ and $\mathit{D}\mathit{E}\mathit{S}\mathit{T}$ matrices with the origin and destination locations of trucks defined on the alternative graphs ($\mathit{O}\mathit{R}\mathit{I}\mathit{G}\in R^{(m_p+m_h)\times (o_O+o_A)}$ and $\mathit{D}\mathit{E}\mathit{S}\mathit{T}\in R^{(m_p+m_h)\times (o_O+o_A)}$ matrices);
• It is necessary to extend the $\mathit{D}$ matrix with the shortest distances of the cells of the alternative graphs ($\mathit{D}\in R^{(o_O+o_A)\times (o_O+o_A)}$ matrix);
• In addition to the size, shape, and position of the individual locations (cells), the possible directions of leaving the locations must be defined in the case of all representations. Transitions between the original and alternative representations cannot be allowed (passenger vehicles can only use the original, and trucks can only use the alternative graphs).

For the proper handling of trucks and other large vehicles, the relationship between the created alternative graphs and the original graph must be specified. Equation (2) ensures that only one vehicle is present at a given location at a given time moment. It is necessary to extend the range of cases examined in the expression. Accordingly, all locations from the alternative graphs that overlap the examined location of the original representation must be taken into account in the expression, and vice versa.

Let $i_{AO}$ denote the locations of the alternative graphs that have a common spatial element (overlap) with the $i_O$ location of the original representation, and $i_{OA}$ the locations of the original representation that have a common spatial element (overlap) with the $i_A$ location of the alternative graphs. Then, considering that one vehicle can be in a place at a time, and this must also be ensured in the case of overlapping model structures, the original Equation (2) should be replaced by the following two relations (Equations (16) and (17)).

$${\sum }_{k_p}{x}_{k_p,j,i_O}+{\sum }_{k_h,i_{AO}}{x}_{k_h,j,i_{AO}}\le 1 ; \forall j,i_O$$

(16)

$${\sum }_{k_p,i_{OA}}{x}_{k_p,j,i_{OA}}+{\sum }_{k_h}{x}_{k_h,j,i_A}\le 1 ; \forall j,i_A$$

(17)

For example, when examining the location i = 5 at a given time moment, the locations i = 21, i = 25 and i = 40 must also be included on the left side of the inequality. Accordingly, we must ensure that no more than one vehicle (regardless of whether it is a car or a truck) is present at locations 5, 21, 25, and 40 at a given time moment, since these cells have a common spatial element.

Ensuring the connection between the graphs is also important to avoid collisions resulting from crossing vehicle movements (see Equation (4)). We also need to extend the $\mathit{R}\_\mathit{C}\mathit{R}\mathit{O}\mathit{S}\mathit{S}$ matrix for specifying intersecting routes due to the increased number of considered locations ($\mathit{R}\_\mathit{C}\mathit{R}\mathit{O}\mathit{S}\mathit{S}\in R^{({o_O+o_A)}^2\times ({o_O+o_A)}^2}$). In doing so, we must also pay special attention to the appropriate comparison of the routes marked in the original and alternative representations (routes with a common spatial element are also considered intersections in different graphs).

In conclusion, if we want to coordinate the movement of small and large vehicles on the same occupancy grid whose elementary region is adapted to the size of the large vehicles, we are always forced to reserve a large region for the smaller vehicles. For many manoeuvres, it would be sufficient to reserve a smaller region. This is why it is so important that we have introduced the approach required to use alternative occupancy grids that can be applied simultaneously in an integrated manner. This approach could eventually lead to a reduction in resolution for smaller vehicles, which could affect accuracy in dense traffic scenarios or complex junctions.

For seamless interoperability between different occupancy grids, it is essential that the elementary regions of any alternative grid are considered occupied if any of the overlapping elementary regions of any other grid are occupied. Therefore, if the above conditions are met, it can be assumed that there is no reason to expect a violation of safety requirements. Accordingly, it is unlikely that any safety related factor will have a negative impact on seamless interoperability. However, in the case of alternative occupancy grids consisting of elementary regions of significantly different sizes, the occupancy of overlapping larger regions may in some cases prove to be overly cautious, resulting in an unnecessarily large reduction in efficiency. It is therefore important to consider interoperability issues when selecting suitable alternative occupancy grids.

4.6. How to Evaluate and Select Occupancy Grid Structures for the Efficient Integration of Conventional Vehicles

The introduction of alternative occupancy grids provides an opportunity to consider additional aspects beyond the management of HGVs and large vehicles. Discretization of the roadway using different occupancy grids is also possible for conventional transport systems.

Effective representation and the careful selection of occupancy grids are crucial for overcoming the integration challenges faced by conventional vehicles in highly automated transport systems. In the case of non-automated traffic, vehicles do not necessarily move according to the rules prescribed by the model. In this case, the fit of a given occupancy grid to real-world processes can be evaluated as a function of the area required to model vehicle movements and traffic flows.

In the light of the above, the ratio ($\rho$) of the area occupied by a vehicle during its movement at each discrete time instant ($A_k$) to the area reserved by the system for that vehicle ($A_k^S$, the total area of the elementary regions involved in the route of the examined vehicle) allows us to select the grid best suited to support the typical vehicle movements. For a given occupancy grid, the closer the value of the indicator ($\rho$) is to one, the more efficient the modelling of the road network is.

$$A_k=t·A_v; A_k^S={\sum }_j{f}_{kj}·A_e; \rho ={\displaystyle \frac{A_k}{A_k^S}}$$

(18)

where

$t$: is the number of investigated time steps;

$A_v$: is the area of the modelled vehicle;

$f_{kj}$: is the number of elementary regions occupied by the vehicle k at the time step j;

$A_e$: is the area of the investigated elementary region.

Figure 3 illustrates the movements of a vehicle during four time steps in the case of two different lattice structures. In this time period, the value of the introduced indicator is four times the area of the vehicle (sum of the purple areas) divided by the sum of the areas of the elementary regions occupied in the given time steps (grey fields). Suppose that the size of the passenger car is 2.5 m × 5 m, while for both the square grid and the hexagon-based structures, we define the diameter of the circle inscribed within the elementary polygon (region) to be 5 m. In this case, $A_v$ is 12.5 m², while $A_e$ is 25 m² in the case of the square lattice (SL), and 21.65m² in the case of the hexagon lattice (HL). As shown in Figure 2, the value of $(\sum _j{f}_{kj})$ is 8 for both lattice structures; thus, the value of ${\rho }_{SL}$ is 0.25, while ${\rho }_{HL}$ is 0.29 based on Equation (18). This indicates the hexagon-based representation to be more efficient for the modelling in this case.

However, in addition to the individual vehicle movements, the fit of a given grid should be examined in terms of the relevant traffic directions/traffic flows as well. To characterize the fit between a given traffic flow and the grid under consideration, we define the ratio ($\theta$) of the nominal area demand of the given traffic flow ($A_F$− area used by the examined traffic flow according to the classical traffic engineering approaches) to the total area of the elementary regions actually assigned to the traffic flow by the system (${Sum\_A}_e^S$), as described in Equation (19).

$$\theta ={\displaystyle \frac{A_F}{{Sum\_A}_e^S}}$$

(19)

We interpret the indicator θ using Figure 4. The grey fields in the left side figure show the nominal area demand ($A_F$) for a square lattice-based occupancy grid in the case of traffic flows coming from the south and making a right turn at the intersection. In this example, $A_F$ is equal to the sum of the areas of the elementary regions used to model the given turning traffic direction (${Sum\_A}_e^S$), so the value of θ is 1.

The value of the indicator may of course differ from 1 if the elementary regions of the applied occupancy grid do not precisely overlap the area used according to the classical traffic engineering approaches. This is illustrated in the figure on the right, which still considers the right-turning traffic flow from the south. In this case, the nominal area demand is indicated by the blue area in the background, which is practically the same (and of the same size) as in the previous example. In yellow, we indicate the elementary regions of the hexagonal alternative occupancy grid that are used by the considered traffic flow. Using the dimensions of the elementary polygons described in the case of Figure 3, the value of θ in this case is calculated to be 0.92.

The closer the value of the indicator is to one, the better the conceptual grid covers the road surface along the conventional traffic engineering principles. However, if the value of the indicator is greater than one, the traffic flows can potentially be accommodated with a smaller area occupancy than the nominal area demand.

The use of alternative grids provides the opportunity to model both priority traffic directions, infrequent, less typical vehicle movements, large vehicles, and emergency vehicles. Our aim in modelling emergency vehicles is to ensure priority. An occupancy grid that matches the intended route of the emergency vehicle ensures safe and efficient passage of the vehicle through the infrastructure element. In line with this, the area-dependent $\rho$ indicator can help to select the occupancy grid that best fits the planned route.

In this case, determining the movements (route) of the emergency vehicle is not part of the optimization task. The objective is to get the vehicle through the infrastructure element as quickly as possible. Therefore, it is appropriate to assume that the movement (and therefore the occupied route) of the emergency vehicle is a given constraint and to adapt the movement of the other vehicles to it during the optimization. For example, in a more complex junction modelled with a square occupancy grid, an alternative grid may be appropriate to implement diagonal vehicle movements. The alternative grid may even be designed from a different shaped and sized elementary region, such as a hexagon, where a greater number of connections between elements may result in a more efficient coordination of traffic flows. In this case, we can ensure the safety and reliability of traffic flows by defining the relationship between the original and alternative grids as described earlier in our concept. In order to compare the different scenarios and select the occupancy grid that best fits the planned vehicle movements and traffic directions, it is useful to consider the fit indices $\rho$ and $\theta$ in addition to the safety related indicators studied previously.

Our results show that interoperability between different occupancy grids can significantly contribute to the applicability of time and space discretised models. In addition, the presented approach opens up the possibility to model specific vehicles (such as HGVs and emergency vehicles) as well as conventional passenger vehicle traffic flows.

5. Results and Discussion

The considerations described in Chapter 4 were adapted in the modelling environment. To demonstrate the applicability of the elaborated concept, we present numerical examples using the network representation introduced in Figure 1.

Regarding the applied square lattice, we considered the side of the elementary regions to be 5 m long (based on the dimensions of an average passenger car). We considered the size of the trucks to be twice the size of the passenger car. Accordingly, the length of the cells in the alternative representations was twice the length of the cells used in the original model. The value of the offset was the size of a region (5 m). The total number of locations of the original representation ($o_O$) is 16, while the number of all locations of the alternative mappings ($o_A$) is 32.

The initial data are presented in

Table 1. Initial constant data.

	DS1		DS2		DS3
$m_p$	10		8		6
$m_h$	2		4		6
	Origin	Destin.	Origin	Destin.	Origin	Destin.
k = 1	1	13	1	13	1	13
k = 2	1	15	2	13	2	14
k = 3	2	13	2	14	2	15
k = 4	2	14	2	15	3	15
k = 5	2	15	3	14	3	16
k = 6	3	14	3	15	4	16
k = 7	3	15	3	16	17 (1)	33 (13)
k = 8	3	16	4	16	17 (1)	34 (14)
k = 9	4	14	17 (1)	33 (13)	17 (1)	33 (13)
k = 10	4	16	17 (1)	34 (14)	17 (1)	34 (14)
k = 11	17 (1)	33 (13)	17 (1)	33 (13)	17 (1)	33 (13)
k = 12	17 (1)	34 (14)	17 (1)	34 (14)	17 (1)	34 (14)

. In addition, the value of the speed limit was chosen to be 15 m/s, the limit of both the acceleration and deceleration was set to 10 m/s².

The calculations were performed applying three different traffic demand structures. In the case of the first demand structure (DS1), we optimized the routes of 10 passenger vehicles and 2 trucks, with randomly selected origin and destination locations within the network. Subsequently, in the second and third demand structures, the proportion of trucks within the total traffic volume was gradually increased (in DS2, 4 out of the 12 examined vehicles were trucks, while in DS3, this number increased to 6).

Table 1 provides an overview of the vehicles’ origins and intended destinations. The starting and destination locations of the trucks were defined on the alternative representations (the corresponding regions of the original representation are also indicated in brackets in these cases).

The optimization processes were performed using the MATLAB R2023a software, the number of the modelled time moments ($t$) was set to 10 s in all three cases.

Based on the results of the optimization, the modelled junction was emptied during the considered timeframe for all demand structures. The resulting routes per vehicle were summarized in the columns of

Table 2. Routes of certain vehicles produced as a result of the optimization (DS1).

Vehicle ($\mathit{k}$)	1	2	3	4	5	6	7	8	9	10	11	12
j = 1	1	1	2	2	2	3	3	3	4	4	17	17
j = 2	1	1	2	6	2	3	11	3	12	4	25	17
j = 3	1	1	5	14	2	3	15	7	11	12	33	17
j = 4	1	6	13	-	2	11	-	12	14	16	-	17
j = 5	1	11	-	-	6	14	-	16	-	-	-	25
j = 6	5	15	-	-	11	-	-	-	-	-	-	30
j = 7	13	-	-	-	15	-	-	-	-	-	-	34
j = 8	-	-	-	-	-	-	-	-	-	-	-	-
j = 9	-	-	-	-	-	-	-	-	-	-	-	-
j = 10	-	-	-	-	-	-	-	-	-	-	-	-

,

Table 3. Routes of certain vehicles produced as a result of the optimization (DS2).

Vehicle ($\mathit{k}$)	1	2	3	4	5	6	7	8	9	10	11	12
j = 1	1	2	2	2	3	3	3	4	17	17	17	17
j = 2	1	2	6	2	3	3	11	12	17	17	25	17
j = 3	1	2	14	6	11	3	12	16	25	17	33	17
j = 4	5	6	-	11	14	3	16	-	33	17	-	17
j = 5	13	10	-	15	-	11	-	-	-	17	-	21
j = 6	-	13	-	-	-	15	-	-	-	21	-	30
j = 7	-	-	-	-	-	-	-	-	-	30	-	34
j = 8	-	-	-	-	-	-	-	-	-	34	-	-
j = 9	-	-	-	-	-	-	-	-	-	-	-	-
j = 10	-	-	-	-	-	-	-	-	-	-	-	-

and

Table 4. Routes of certain vehicles produced as a result of the optimization (DS3).

Vehicle ($\mathit{k}$)	1	2	3	4	5	6	7	8	9	10	11	12
j = 1	1	2	2	3	3	4	17	17	17	17	17	17
j = 2	1	6	2	11	3	12	17	17	17	17	25	17
j = 3	1	14	6	15	8	16	17	17	25	17	33	17
j = 4	1	-	11	-	16	-	17	25	33	17	-	17
j = 5	1	-	15	-	-	-	17	30	-	21	-	17
j = 6	1	-	-	-	-	-	17	34	-	30	-	21
j = 7	1	-	-	-	-	-	21	-	-	34	-	30
j = 8	1	-	-	-	-	-	29	-	-	-	-	34
j = 9	9	-	-	-	-	-	33	-	-	-	-	-
j = 10	13	-	-	-	-	-	-	-	-	-	-	-

.

In each of the three defined traffic demand structures, we optimized the routes of 12 vehicles. In the first case, we modeled 10 passenger cars with varying origins and destinations, along with 2 trucks departing from the same location. In the subsequent scenarios, the proportion of trucks was increased, with these vehicles always starting from the same location.

The simulation results demonstrated that as the number of trucks departing from the same location increased, the efficiency of traffic flow decreased. In the case of DS1, the traffic demands were resolved in 7 s (the value of the objective function was 740), while in DS2 and DS3, the process took 8 and 10 s, respectively (objective function values were 750 and 890). The results confirmed that none of the vehicles violated the pre-defined safety constraints. The values of the safety indicators are presented in

Table 5. The value of the safety indicators.

	DS1	DS2	DS3
$CM$	27.627	25.627	24.449
$CF$	7	9	7
$\overline{VN}$	8.190	8.056	8.194
${\sigma }_{VV}$	1.815	2.208	2.208
${\sigma }_{VN}$	1.222	1.297	1.222

.

The comparison of the safety indicators presents a mixed picture. The risk associated with crossing vehicle movements ($CM$) were lowest in the case of DS3, as this scenario involved the highest number of trucks traveling on the same routes and the fewest passenger cars being directed to lateral movements. Despite involving four trucks, the traffic was nearly as efficiently managed in DS2 as in DS1 (with only minor differences in their objective function values). However, this efficiency came at the cost of a high number of vehicles following closely behind each other ($CF$).

The indicators derived from the speed data showed no significant differences between the solutions, which is unsurprising given that the area modeled in the sample was small, and identical speed and dynamic constraints were defined for both passenger cars and trucks. Because of the defined dimensions and constraints, each vehicle could move 1, 2, or in some cases, 3 locations (squares) per time step. DS2 resulted in the lowest network level average speed ($\overline{VN}$), and in parallel, this scenario exhibited the highest standard deviation of the average speeds of the vehicles over the entire examined time interval (${\sigma }_{VN}$).

The example has demonstrated the applicability of the presented concept to manage the traffic of large vehicles, taking into account the relevant safety constraints, with the use of several alternative grids simultaneously.

6. Conclusions

Based on our literature review, automated traffic management models can be characterized by different advantages and disadvantages. The advantages of the continuous variable solution are high resolution and realistic description of the processes, while the discrete models are more effective and characterized by lower computational requirements.

Our goal was to develop a model that allow us to combine the advantages of discrete and continuous models. The simultaneous use of discrete alternative grids with different resolutions creates the potential for efficient traffic management, while at the same time it is possible to simultaneously integrate higher resolution alternative grids when process alignment justifies it.

Consistent with the above, the binary integer linear model developed in our previous research [46,47] has provided an opportunity to develop a framework for the integrated, simultaneous application of alternative occupancy grids.

While we have seen examples in various literatures of different occupancy grid structures being used to represent the road network, the results showed that there is a lack of comprehensive research exploring the interoperability between the grids and the possibilities of simultaneous implementation. And this is where our results offer promising added value for the management of different modes and types of vehicles in the road network. The simultaneous integration of alternative graph structures provides a flexible way to consider different vehicle categories and specific traffic directions in the traffic optimization problem.

In our work, special attention has been paid to the precise definition of the relationship between the original and alternative representations, to guarantee the safety-critical and realistic modelling of the network.

During the analysis, we investigated the computational complexity and presented indicators characterizing traffic safety. The presented numerical examples illustrated the applicability of the novel concept to manage the traffic of different size vehicles, considering the relevant safety constraints, with the simultaneous application of alternative grids.

Our research has demonstrated that the interoperability between different grids can considerably enhance the applicability of graph-based, temporally, and spatially discretized models. On the one hand, this permitted the inclusion of trucks and other large vehicles in the traffic management of highly automated systems. On the other hand, it also afforded several opportunities for further development, as detailed in our paper. In this regard, the discretization of traffic processes pertaining to non-autonomous traffic components and the simultaneous application of alternative occupancy grids tailored to specific traffic directions and manoeuvres represent the most promising avenues for future development.