Urban traffic simulation using credible driver modeling method

Abstract

In the paper we describe the process of development and verification of the novel traffic simulation method based on a credible driver model. The model is based on the actions of real drivers recorded in series of experiments. The developed simulation method reflects the unpredictable and totally individual characteristics of individual drivers. We show that the characteristics is a very significant factor in overall traffic efficiency. We believe that our credible driver model may become a basis for novel traffic management systems and active-safety mechanisms installed in the vehicles.

Keywords

Driver model simulation traffic management

1 Introduction

Growing demands concerning urban traffic efficiency pose significant challenges in all big agglomerations worldwide. Continuous migration of people towards big metropolises causes tremendous problems in traffic management in city centers, where majority of people arrive every day. These problems are addressed with various measures, focusing mostly on limiting the number of cars and increasing the capacity of the road system. However it is clearly visible, that current assumptions of the traffic systems, based on individual and autonomous human drivers, become insufficient.

The automotive industry is currently approaching a very important breakthrough, which is the introduction of fully autonomous cars. Introduction of automatically controlled cars can form a completely new basis for the very assumptions of the traffic systems in cities, significantly improving its efficiency and safety. This clearly visible aim poses a great challenge for the most important players on the automotive market. The race for the autonomous car that works has started a few years ago, and still there is no obvious leader. Several companies continue testing their solutions, reporting significant progress but also hundreds of disengagement events [16].

It seems that at least several years of intensive research and tests must pass before the race ends. What is even more important, creation of the fully autonomous car that works will not cause any sudden changes in the traffic systems we know now. Solving formal issues in different countries, adopting the algorithms to different regulation and removing manually-driven cars from the traffic systems will definitely take decades. In the meantime the automotive industry will continue to introduce new improvements to the cars and traffic management systems, which will fluently limit the amount of decisions made by human drivers (cf., e.g. [28]). This trend can be already observed – active-safety systems, which can take over the control over selected features of a car, are already installed in mass-produced vehicles.

The changes in the equipment installed in cars create new possibilities for collecting and analyzing data concerning the decisions made by drivers in particular road situations. Modern cars are nowadays equipped with GPS receivers, proximity sensors, parking cameras and even front and rear radars. The information from these devices can be used for building comprehensive picture of a road situation. Additional internal sensors, which measure engine state (current rpm and power, selected gear, sate of pedals, indicators, lights, radio etc.) give sufficient information for observing and recording driver’s decisions in particular situation.

We believe, that this information can be successfully used for improving the efficiency of traffic management systems. The concept of using this information is based on the idea of building individual drivers models and using them for predicting drivers decisions. The ability of predicting how fast a particular driver would leave a crossroad or what would be her/his decision when traffic light change in front can be used for altering traffic lights sequences or for calculating efficient alignment of cars on lanes.

The researchers working in the domain of urban traffic modeling have defined a very wide variety of driver modeling methods. In fact, it is really hard to name any well-recognized and popular methods. Rather than that, researchers tend to design new methods of driver modeling for each considered problem [8]. One common feature of the vast majority of existing models is generalization – it is often assumed, that each driver can be modeled with the same set of features or that only several types of drives exist.

Our research show that these assumptions are false [5]. The experiments with drivers in real traffic show that individual features of drivers are significant. Drivers perceive similar road situations differently and act differently, which has a crucial impact on overall traffic efficiency.

In this paper we present further conclusions from the experiments and propose a novel method for modeling individual drivers behaviors. Moreover, we present a traffic simulation method based on the model. The evaluation of the simulation method is described, followed by results of its application for testing new traffic management methods.

2 From urban traffic simulation to driver modeling

In this section the most important aspects of the background for the presented model and simulation are sketched out. The general overview of the urban traffic simulation and driver modeling methods are presented.

2.1 Urban traffic simulation

Simulation in transport field is a very important issue, as it may make possible analysis of complex behaviors, being a combination of many interactions of particular objects, especially when the analyzed case is too complex for using formal methods and models. Simulation allows analyzing many different variants of the modeled reality, e.g. detailed analysis of the behavior of particular objects, detailed analysis of the relations among the objects or groups of objects and for graphically-attractive (usually 3D) visualization of the whole environment. Analysis of the simulation of particular configurations of the system makes possible optimization of selected transport system parameters, e.g. optimization of traffic light management can help in increasing of the traffic flow. Simulations are also often used for considering new communication methods and complex traffic scenarios, as other methods of analysis of such cases are expensive, dangerously uncertain or simply impossible.

The importance of the discussed issue has pushed researchers to develop various simulation methods over the last few decades. These methods were classified according to different criteria:

Phenomena description method: Black-box models (modeling the traffic phenomena using general purpose models, as e.g. neural network and focusing on analysis of its inputs and outputs) [18]. White-box models describing in detail the simulated phenomena, using e.g. laws of physics. Gray-box models leveraging partial models of the phenomena [22].

Determinism: deterministic [11] and stochastic models, their results are repeatable or non-repeatable [19].

Parameter representation method: discrete [40, 41] and continuous [38] (both in time and space).

Detail level: micro, meso and macro [20].

The most popular classification of the traffic models is based on the detail level. In this classification, simulation in macro-scale considers only aggregated traffic flow of the modeled road [23]. It is usually described using average velocity and traffic density, but without considering particular vehicles. This approach is the oldest, still very popular, researched and modified [36], and because of its simplicity—very efficient (large traffic networks may be simulated using mediocre hardware).

Meso-scale simulation adds certain features of particular vehicles to the model, however the whole simulation is computed for a vehicle group moving along some path. For examples of such approach one can refer to gas kinetic models [35] and queuing network models [4].

Micro-scale simulation treats each vehicle as single object (cf. e.g., [2 , 27]), considers traffic lanes, directions, lane changing and interactions among the vehicles. Each single vehicle has its own individual position, velocity, acceleration and many other parameters. This approach considers a more accurate environmental state, traffic lights on crossroads, distance among the vehicles etc.

As this paper focuses on features of a single driver, searching for his/hers individual profile and planning to leverage it in simulation, it is obvious that macro- and meso-scale models cannot be used, as aggregating individual features of the drivers in order to form a group of the drivers does not make sense. Thus, the micro-scale model is of course a simple choice.

Micro-scale models have high computational demands and often require supercomputing-level hardware to be completed in reasonable time. The most popular micro-scale models are based on:

Cellular Automata [34]: uses a discrete, n-dimensional grid. During the discrete time steps the states of the cells are modified according to the previously set rules for interactions among them. Usually each of the cells models a part of the road with constant dimensions, and the state of the cell represents the position of the vehicle.

Car Following (a continuous time model): leverages differential equations modeling the movement of the vehicles. Based on them the location and velocity of each vehicle is computed. Their input parameters are velocity, distance to the preceding vehicle and the velocity of the preceding vehicle [9]. The most popular ones are Optimal Velocity Model [3], Gazis-Herman-Rothery(GHR) [9], Wiedemann’s model [9] and Gipps’ model [42].

Physical micro-scale models: they are the most accurate and the most computationally complex ones. Their goal is to imitate the dynamics of the real world by mimicking different traffic situations, e.g. car driving, lane changing, simulation of dynamics of different types of crossroads. The space is often continuous and the time is discrete (but the time steps are relatively very small) [37]. Examples of such models are:AIM-SUN 1 , MITSIMLab 2 , PARAMICS 3 , VISSIM 4 , and RoBOSS [43]. The computational power demand is often decreased by lowering the accuracy or increasing the time step.

Recalling again the goal of the research presented in this paper—modelling of individual behavior profile of the driver, and keeping in mind that this profile affects all possible ways of interaction between the driver and the vehicle, there seems to be only one possibility of constructing an appropriate simulation model for this case, namely the physical micro-scale model.

2.2 Driver modeling

Complexity of the human nature and behavior, even constrained to observation of human-vehicle interactions, makes the proper modeling of such phenomenon a dream, that if realized properly, might lead to its better understanding, predicting different unplanned situations and designing the appliances that the human uses (e.g. vehicle controls) much more human-friendly and much more safe. The task of modeling of driver was firstly attempted in 1938 [17]. In the 60s., dawn of the computer era made possible development of more advanced drive modeling methods [32, 44]. First goals of the approaches were to improve the safety and enhance the driving education methods. These first approaches were based on aggregation of the behaviors focusing on the maneuvers realized. The main constraint of such approach was that they might have been used only for description purposes, and they were hampered by a very low predictability.

Throughout the years of research many different models have been created. Each subsequent one was enhanced with new capabilities. The complexity of the modeled phenomenon resulted in a situation, where each researcher created his own model depending on his/hers individual demands. As the time passed, at least several means for classification of such models were proposed. Several classifications, e.g. [12] define the following groups:

Descriptive models: based on the driver’s actions, describing the whole driving process or its part from the perspective of the maneuvers the driver has realized or is about to realize. The accuracy of the prediction in this model was limited and it did not consider the capabilities of the driver nor different traffic situations. Nevertheless this model is still widely used in the domain of traffic safety [25, 33]. Descriptive models can be further divided into hierarchical ones [33] (showing the driver’s behavior on different levels, of e.g. capabilities, rules, knowledge) and “in loop” ones (showing the driving process based on inputs, outputs and observations and are used for control in order to follow a certain planned route [15].

Functional models: they focus on the cognitive state of the driver, considering also his/her motivation or risk assessment capability. These models can be further divided into: information processing models modeling the mind activities of the driver, explaining the reasons for undertaking certain actions [21] and motivation models, using the individual risk assessment of each driver [14].

Another more recent categorization proposed in [31] in the domain of micro-scale traffic models identifies the following approaches:

Simulation based on intelligent transport systems, e.g.: modeling of the traffic in order to examine the dependencies among the sub-systems (driver, vehicle and infrastructure) (cf. e.g., [13 , 39]); development of a crowd-based data acquisition system and implementation of real-world traffic network along with simulation of traffic [26].

Modeling using micro-scale models of driving along with lane changing e.g.: modeling undertaking of the decisions by the drivers based on the parameters connected to safe distance to the preceding vehicle [29]; research on the car following models and the lane changing models using real-world data in order to simulate the traffic in micro-scale [30].

Among many approaches one very general architecture of a driver model is widely recognized and used as a basis for particular solutions. The 3-layer hierarchical model proposed by J.A. Michon 1985 (see Fig. 1) defines the following levels of decision-making process:

Strategic level: responsible for such actions as route and stops planning and any other long-term plans.

Tactical level (maneuver level): responsible for undertaking short-term decisions such as: lane changing, curve passing, overtaking or operations that may decrease the estimated time of arrival.

Operational level: responsible for undertaking very short-term decisions, such as steering of wheels, changing of the velocity by acceleration, deceleration etc.

It is easy to see that there are many available classifications of the driver models, each of them has its own specific approach and refers to different aspects of driver behavior. Each of them tries to simplify the modeled reality by e.g. averaging different parameters (e.g. acceleration, maneuver velocity etc.). Other possibilities are using linear models or fuzzification of the data, however they can hamper the overall quality of the simulation and the accuracy of the analysis of the traffic and driver phenomena.

Taking into consideration the advantages and disadvantages of the presented models, a novel driver modeling method was proposed, focusing on individual characteristics of the particular drivers. The method is not based on any arbitrary assumptions or parameters values. Rather than that it allows representing real actions of real drivers recorded in the course of the experiments conducted in real traffic. Such an approach makes possible attaining high reliability of the simulation results, which allows to draw reliable conclusions.

3 Individual driver behavior modeling

The majority of research undertaking the problem of driver modeling is in fact focused on a different problem, which requires a driver model as a necessary element or tool. Various models have been created in order to verify the efficiency of particular road structures, traffic management systems, algorithms for car-following or active-safety. The driver models required in such research, which can be reduced to a decision function transforming the situation into car control, represent arbitrary assumptions and observations of typical behavior of cars in the considered situations. In many cases there is only one, common model with fixed decision function. In other approaches several classes of drivers are identified, where each class has a different decision function. Such a solution assumes the existence of a fixed set of driving styles. There were few attempts to capture the actual parameters of the decision functions from reality. These simplifications and generalizations are justified by the assumption, that the individual differences between drivers are insignificant when a large scale simulation of a complex road systems is considered.

We believe that making such arbitrary assumptions concerning the decision function cannot lead to reliable results. In reality, it is a very rare situation to identify two drivers who drive a car in the same manner. Even very subjective and superficial observations of a single driver in similar situation lead to the conclusion, that some decisions are different than others, that there are some immeasurable factors which result in observable differences in behavior. When a large number of such individual drivers coexist in a single road system, their decisions influence one another, which certainly boosts the effect of unpredictability. Therefore, we believe that neglecting the influence of individual driving style on the overall features and effectiveness of the road system, is unjustified.

The proposed, credible driver modeling method has been developed without making arbitrary assumptions concerning the features of the decision function of a particular driver. Instead we decided to perform a set of experiments, involving real drivers, in order to verify several, commonsense hypotheses. The results and conclusions were used for forming the assumptions of the credible driver modeling method. The hypotheses verified within experiments are:

behavior of a single driver is never identical, even in similar situation,

behavior of a single driver in similar situation is similar,

behavior of different drivers in similar situation can be similar, but can also be significantly different.

In order to verify these, very imprecise, hypotheses, several notions must be clarified. The behavior of a single driver is defined as a her/his decision in particular situation, expressed by setting specific control to the car. The driver can press pedals, select a gear, switch lights or indicators – the values in the decision vector can be continuous or discrete and the length of the vector is constant.

The situation is defined as a set of observable and measurable factors influencing the driver’s decisions. The similar situations are defined as situations in which all observable and measurable factors are the same. Obviously, the method of verifying the hypotheses and building the driver model must use a specific set of factors, which are considered observable and measurable. We believe that it is not possible to list all factors, which might influence the driver – these could include humor, blood pressure, events experienced or the color of the sky. Therefore we do not assume that the behavior is a function transforming a situation into a decision vector, because some immeasurable factors can influence the decision in the same situation.

The considered set of observable and measurable factors includes the detailed information about the road system structure and the state of traffic lights and all other cars. The information about the road structure is described according to the formal model proposed in [6]. The model represents multi-lane roads and crossroads with details concerning horizontal and vertical road signs, width of lanes and even location of curbs. State of other cars is described by location (lane, position on lane), velocity and state of indicators.

The behaviors in particular situations are considered similar when the range of corresponding values of decisions is significantly smaller than the range of all values of decisions observed in similar situations. Two sets of behaviors are considered significantly different when the range of corresponding values of decisions in the first set do not overlap with analogous range in the second set.

3.1 Experiments with real drivers

Collecting data on the decisions of real drivers made in particular road situations required building a laboratory, which has been installed in a car. The elements installed in the car are presented in Fig. 2.

The devices installed in the car collected the following information every 200 ms:

time stamp,

longitude and latitude form GPS,

throttle position,

engine RPM,

trip distance from OBDII,

speed from OBDII,

brake pedal status,

clutch pedal status,

turn indicators status,

video from 8 cameras used mostly for estimating distance from the preceding car.

More details on the architecture, integration and components used in the mobile laboratory can be found in [7] and in [5].

The experiments involved 15 people of different age, sex and driving experience. Each driver covered 5 times exactly the same, cyclic route, which took around 30 minutes. The route, presented in Fig. 3, consisted of 6 crossroads. Three of them were right-turns with traffic lights (3, 4 and 5), other three were without right of way.

The collected dataset is, to our best knowledge, unique. In contains information about situation in the environment and drivers decisions, collected in real traffic, which gives the opportunity to extract features and values describing real drivers behavior and to form correct assumptions for modeling this phenomenon.

3.2 Credible Driver Model

The collected data has been converted to time series of state of the car and decision of the driver. The most significant of these information are used here to verify the formulated hypotheses.

The first set of results, presented in Fig. 4 provides comparison of two drivers accelerating on second gear (clutch released till clutch pressed again) with no obstacles ahead. We can observe, that Driver 8 increases power during accelerating, while Driver 11 keeps it almost constant. The charts in this section present the average value and standard deviation based on at least five repetitions of a maneuver.

The first hypothesis is the simplest to verify, and rather obvious in fact. No matter how many repetitions of a maneuver occurred, The values in the time series of a single driver were different each time.

The results also allow drawing conclusion regarding the second hypothesis. The range of values used by each of the drivers in parts of this characteristics (0 - 0.2; 0.65 - 0.85) is significantly smaller that all observed values. This means that driver behavior in similar situation is similar, which forms the very basis for any further attempts to create a credible driver model.

Another interesting set of result is presented in Fig. 5. The braking characteristics is presented as the time-to-collision value in the domain of distance to the red light. The two drivers were approaching crossroads 3, 4 and 5; the situation on different crossroads cannot be considered similar.

Similarly as in case of previous results, the characteristics of different drives can overlap and the range of control values can be very similar. The same pair of drivers can have completely different decisions in other situations, which confirms the third hypothesis.

However, there are other conclusions from the results, which significantly modify the hypothesis and lead to defining different approach to the problem of credible driver modeling. The result show clearly that even if the behavior of two drivers is significantly different in some situations, it can be similar in others. What is more, there are no fixed relations between the values of control used by the drivers. We cannot assume, that one driver always approaches a crossroad faster than the other. This phenomenon is visible in Fig. 5, where Driver 9 is more aggressive approaching the red light on crossroads 3 and 4, while on crossroad 5 the situation is exactly the opposite.

These observations show, that making assumptions concerning the behavior of a driver in a new situation, having some information about her/his actions in different situations is unjustified. Therefore the method of simulating the behavior of drivers, which is presented further in this paper, is valid only in situations, which have been observed in reality. Conducted research show that this is the only credible way of simulating the considered phenomenons. Nevertheless, the method allows us to draw quantitative conclusions concerning the influence of particular driving characteristics on the efficiency of traffic in particular road situation.

The proposed model is based on a sets of series of decisions recorded during performing a particular maneuver in particular situations. In simulation of the model we are not converting the data into any form of average or aggregated values. The series of decisions are directly used for controlling the simulated car.

Summing up, the proposed credible driver model DM is constructed as the following set: $DM = {(m_{1}, s_{1}, {ch}_{1}), . . .}$ where m specifies a maneuver (eg. braking before traffic lights, accelerating from stop position, following another car,...), s is a situation, and ch is a set of relations representing measured behaviors at certain situation. Particular format of elements of ch depends on the value of m, for example:

for braking maneuver $ch = {b_{s}^{1}, b_{s}^{2}, . . .}$ , $b_{s}^{i} : D \to V$ , where $V = ℝ_{+}, D = ℝ_{-}$ are velocity and distance from stopping pointrespectively,

for accelerating maneuver $ch = {a_{s}^{1}, a_{s}^{2}, . . .}$ , $a_{s}^{i} : T \to V$ , where $V = ℝ_{+}, T = ℝ_{+}$ are velocity and time since departure respectively,

for car following maneuver $ch = {f_{s}^{1}, f_{s}^{2}, . . .}$ , where $f_{s}^{i} : V \times D \to V_{r}$ , $V = ℝ, D = ℝ_{+}, V_{r} = ℝ$ are velocity of the followed car, distance from the followed car and relative velocity of the cars respectively.

Each of gathered characteristics reflects behavior of particular driver during performing a single maneuver. Each of them can be used to calculate control of vehicles in the simulation because it represents real driver behavior.

4 Credible urban traffic simulation

In order to develop a simulation model that would express individual features of the driver, and retain flexibility, we decided to leverage three-level architecture. Architecture of the proposed model is presented in Fig. 6.

The proposed architecture of the driver model consists of three levels:

First level (strategic): the route planning consists of the mechanism searching for the path between the origin and target points in a graph. As a result the sequence of points is returned that should be followed in order to reach the destination. In the case of simple simulations such procedure is realized only at the beginning, while in the case of more complex simulations (covering lane changing, avoiding of the traffic jams etc.) the procedure may be repeated multiple times.

Second level (tactical): responsible for reaction to obstacles that may appear during the ride. The Detection module encompasses the following submodules: environmental state submodule, collision detection submodule and traffic lights submodule. The final decisions undertaken by the driver are selected by the Decision merging module.

Third level (vehicle control): encompasses mechanisms for velocity and turning control.

4.1 Tactical level

The Detection modules continuously feeds the Decision merging module with suggestions regarding velocity change.

Environmental state submodule is responsible for detection of environmental objects such as curves and straight fragments of lanes, calculating the distance to the spotted objects, and proposing a prefered velocity based on the driver profile.

Collision detection submodule is responsible for detection of vehicles in front of the actual vehicle: based on their distance and the difference of their velocities an information about current distance and current velocity of the preceding vehicle is calculated.

Traffic lights and signs detection submodule is responsible for detection of the traffic infrastructure, such as signs and lights in the current field of view (in the case of this simulation, not farther than 300 m in front of the vehicle). It calculates the distance to the traffic infrastructure in front of the vehicle and detects their state (e.g. color of light). It proposes also the preferred velocity when reaching the infrastructure elements (e.g. 0 km/h in the case of red light).

All the above-described modules check the distance required for realization of the selected decision using the predefined function calculateRealizationDistance, that based on the selected driver profile returns the distance required for changing of the velocity from the actual speed to the desired one. If such realization of the velocity change may be postponed into farther future (e.g. red lights are in 100 m distance, while the distance required for the full stop is 20 m), it is possible to neglect the current suggestion. However if the realization is to be done in the near future, the suggestion that must be realized in the closest future should be chosen.

The function calculateRealizationDistance, in order to calculate the distance needed either to brake or to accelerate, finds appropriate characteristics of braking or acceleration in the current driver profile related to the current situation, localizes the values of the current and desired speed and computes based on the characteristics, the distance required to realize this maneuver.

The decision merging module is responsible for choosing among particular suggestions regarding velocity change, fed by particular submodules. In Fig. 7 one can see a simple example of the work of this module. In the observed case, three suggestions are available and the decision merging module selects the first one: acceleration.

During realization of the braking or acceleration mechanisms, the submodules keep raising their suggestions, therefore we can be sure that if our current situation changes rapidly (e.g. we suddenly notice the animal bursting into the road), the currently realized maneuver may be interrupted and the decision about beginning of realization of another maneuver may be undertaken (rapid braking in this case). Now, the decision is fed into the control module residing on the operational level.

4.2 Operational level

The main goal of the operational level modules is to control the speed of the car at certain conditions (e.g. in particular situation). Let us firs focus on the deceleration maneuver. The characteristics of the deceleration in the simulation should follow the observation taken from the real driver. Considering different braking maneuvers observed in reality, we have at our disposal a family of functions $b_{s}^{i}$ describing the driver’s behavior when braking at a certain situation s. The values of the functions between the measurement points are calculated using linear interpolation. Each $b_{s}^{i}$ function is monotonic and decreasing. A graph of an exemplary braking function is presented in Fig. 8.

The aim of the control is to repeatedly calculate the deceleration value of the simulated car, following the characteristics, until the target velocity v_k is reached. Any of the $b_{s}^{i}$ functions is valid, because they all represent possible behavior of the driver in the situation. During a single braking maneuver one, arbitrary selected characteristics is used.

The simulation continuously computes system state changes using a fixed time-step Δt. In each time-step the car controller repeats the following steps:

finds the current speed v₀ and the corresponding d₀ using the selected function $b_{s}^{i}$ ,

calculates the distance covered by the car during Δt, while braking according to $b_{s}^{i}$ ,

calculates the target velocity v₁ and the acceleration a_0-1, which should be used in the next time-step.

The second of these steps is non-trivial, because the $b_{s}^{i}$ does not reflect time directly. We assume, that the acceleration a_0-1 is constant during a time-step, therefore, the following statement is valid: $\frac{v_{0} + v_{1}}{2} = \frac{d_{1} - d_{0}}{Δ t}$ Knowing, that $b_{s}^{i} (d_{0}) = v_{0}$ we can conclude, that: $b_{s}^{i} (d_{1}) = \frac{2}{Δ t} d_{1} - \frac{2 d_{0}}{Δ t} - v_{0}$

At the same time d₁ ∈ [d₀, d₀ + v₀Δt] and v₁ < v₀, which makes it possible to find the value of d₁ and the corresponding value of v₁ using a simple gradient method.

The whole procedure is repeated until the final speed v_k is reached. Acceleration procedure is carried out analogically – the calculations are a little simpler, as the characteristics is stored in the domain of time.

The presented algorithm for calculating decisions of the simulated driver is implemented in one of four applications, which form the designed simulation system. The central part of the system is the RoBOSS simulator [43], which loads the model and continuously calculates state changes, simultaneously introducing changes in car control received from car controller application. Two other applications are: simple traffic lights controller and 3D visualization.

The RoBOSS is a simulation system of rigid bodies making possible of simulation of mechanical appliances, vehicles and robots. It supports modeling of complex test environments encompassing robots and static elements. Calculated changes in the model state are based on 3D kinematics and dynamics of rigid bodies, collisions detection, friction and bounciness. Modeling of all of these details made it possible to create a reliable simulation of individual driver’s profile, which has been confirmed by the evaluation presented in the next section.

5 Evaluation of the credible simulation method

In order to evaluate the appropriateness of the constructed model to micro-scale simulation, validation was performed. A simulation of the drive of one vehicle on a rectangular track was prepared. In the course of the driving, the vehicle accelerated to 50 km/h on straightaways and decelerated to 0.5 km/h on curves. Such significant spread of the velocities allowed for testing almost all possible characteristics, on which the model has been built.

In order to assess the accurateness of the mapping of velocity characteristics for each simulation step, a relative error of the actual and expected velocities in particular time step was calculated: $δ_{v} = \frac{| v_{e} - v_{a} |}{v_{a}} \cdot 100 %$ where: δ_v is a relative percent error, v_a is an actual velocity and v_e is an expected velocity. After carrying out analysis of the error data series δv that were obtained after the validation, we stated, that the error is different for different ranges of velocities. In order to better show the distribution of the error, it was computed for the following velocity ranges: 0–2 m/s, 2–4 m/s, 4–6 m/s, 6–8 m/s, 8–10 m/s, 10-12 m/s, over 12 m/s.

Validation of the accuracy of the model was carried out for the braking (Fig. 9) and acceleration (Fig. 10) maneuvers. The graphs show the distributions of the errors δv in particular velocity ranges (displayed as box-and-whiskers plots).

Analyzing the results one can easily see, that the high error were obtained for low velocities and it decreased with increase of the velocity. Even small error causes the relative error to become significant, if the velocity is low, however, in such case the position of a vehicle also changes slowly, which reduces the influence of the error on the results of the simulation. Nevertheless, having the average relative error below 2% is definitely a good result.

6 Applications of the simulation method

This section presents an application of the developed credible simulation method. The general aim of the experiments presented here is to estimate the influence of driving style of individual drivers on the traffic. The considered is the evaluation of the influence of drivers alignment on time required to leave the crossroads, right after the change of the light from red to green. The simulation considers single lanes. There are two completely different types of the drivers in the simulation, called “lazy” and “aggressive”. Their characteristics are, of course, selected from behaviors recorded in reality (Fig. 11).

Initial tests were performed for separated groups of 40 lazy and 40 aggressive drivers. The time of leaving the crossroad were 45 s for aggressive and 74 s for lazy drivers.

A hypothesis can be formulated, that mixing the drivers will give worse results than dividing them by type on different lanes: $2 \cdot T_{mixed} > T_{lazy} + T_{aggressive}$ where T_mixed is the time of passing of the mixed group (lazy and aggressive together); T_lazy is the passing time of the lazy drivers and T_aggressive is the passing time of the aggressive drivers.

Total time (T_total = T_lazy + T_aggressive) of the passing of both lazy and aggressive was 120, 652 s. In order to prove the formulated hypothesis, a measure of improvement was formulated: $Enhancement = (\frac{2 \cdot T_{mixed} - T_{total}}{T_{total}}) \cdot 100 %$ where T_mixed is the passing time of the mixed drivers, and T_total is the passing time of subsequently one type of the drivers, then another one.

Analyzing the results of different alignments of cars in the considered situation (Table 1) one can see that the above-formulated hypothesis is true in all the conducted experiments (Acceleration > 0). The more important conclusion is that the influence of the individual driving style on traffic efficiency is very significant (up to 29,8% in this simple case). Therefore, it cannot be neglected if the results of traffic simulations are to be credible.

7 Conclusions

Following the observation presented in our previous paper [8], that there is no such thing as an average driver, as it was assumed during many of the previous publication in the field, we propose a simulation model based on the characteristics observed during real-world experiments (with the real drivers).

In the paper we have shown that the proposed driver model may be successfully used for simulation purposes, giving credible results. Moreover, it can pave the way for more sophisticated methods useful for the traffic control, like assigning drivers to the proper lanes based on their behavioral profile.

It is to note, that the proposed model can be further enhanced, using different AI techniques, e.g. Fuzzy Systems, as a basis for decision making, considering such noisy data as gathered in the experiments shown in this paper [1].

In the future we plan to further explore this research field, by proposing the above-mentioned traffic management methods and testing them in a simulated environment. We are also planning to strive for getting financing by research grants in order to perform more comprehensive experiments with real drivers.

Footnotes

Acknowledgments

The research presented in this paper was partially supported by the AGH University of Science and Technology Statutory Fund no. 11.11.230.124.

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