1Department of Industrial Engineering and Management, Jerusalem College of Technology-Machon Lev, Jerusalem 91160, Israel
2Department of Information Technology and Computer Engineering, Vinnytsia National Technical University, Vinnitsia 21021, Ukraine
3Department of Mechanical Engineering, Afeka-Tel Aviv Academic College of Engineering, Tel Aviv 69107, Israel
Copyright © 2012 Alexander Rotshtein et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fuzzy sets membership functions integrated with logistic map as the chaos generator were used to create reliability bifurcations diagrams of the system with redundancy of the components. This paper shows that increasing in the number of redundant components results in a postponement of the moment of the first bifurcation which is considered as most contributing to the loss of the reliability. The increasing of redundancy also provides the shrinkage of the oscillation orbit of the level of the system’s membership to reliable state. The paper includes the problem statement of redundancy optimization under conditions of chaotic behavior of influencing parameters and genetic algorithm of this problem solving. The paper shows the possibility of chaos-tolerant systems design with the required level of reliability.
The classical reliability theory [1, 2] is based on the probabilistic approach. Essential limitations of this approach are connected with “the problem of the source data” which depend on many factors and which may not correspond to the real conditions of the system’s functioning. Besides, the statistical data used in the probabilistic reliability models fix only the facts of real failures and do not contain the information about the causes of these failures. Whereas the causes of these failures are connected with the elements’ variables (temperature, humidity, tension, etc.), which become more (or less) than a certain critical level. So, we can affirm that the probabilistic theory [1, 2] models the reliability in the space of events effects (i.e., failures) and suits badly for the reliability modelling in the space of events causes (i.e., variables).
The alternative for the probabilistic modeling of the reliability is the approach based on the fuzzy logic  and related possibility theory . In this case the classical “failure probability” is replaced by “failure possibility” which is modeled by the membership function of the system (or the element) variables to the reliable state (Figure 1).
Figure 1: Relationship of the probability theory and fuzzy logic in reliability estimation.
The explicit dependence of the membership function on the variables (failure causes) makes convenient the integration of the fuzzy model of reliability with the technique of time series , which allows observing the change of the reliability level in the real time.
The chaos theory is a new approach to the analysis of nonlinear time series . It uses the conceptual apparatus of the theory of nonlinear oscillations  and purposes to study the phase portrait of the dynamical system with its intrinsic states of stability (attractors) and bifurcations, that is, “jumps” between stable states. Unlike the classical oscillation theory  where the phase portrait is formed on the base of the system description by means of differential equations, the chaos theory  offers the methods of the phase-portrait extraction from the experimental data, that is, directly from the time series.
The integration of fuzzy reliability model with the phase portraits of variables creates preconditions for the construction of phase portrait reflecting the system reliability dynamics. The pattern of bifurcations which may be interpreted as failure instants is of particular interest.
The works on the fuzzy reliability theory began in 90s of the last century. The first specialized collection of papers in this field is the edited volume . The first monographs containing the approaches to the construction of fuzzy reliability theory are [9–11]. In the works [12, 13] there is a fuzzy algorithmic approach to reliability modeling based on the algebra of regular algorithms  and expert assessments of the performance correctness of operators and conditions by means of membership functions . The problem of reliability optimization of the control resources based on fuzzy algorithmic approach is solved in .
An approach to the online reliability evaluation based on the integration of fuzzy logic and forecasting methods of time series (exponential smoothing and Kalman filtering) is proposed in .
The idea of chaos theory application in the reliability modeling appears in . Two real data bases about software failures are processed by the methods of chaos theory in the paper . It was shown that the deterministic model of failures is more adequate to the experimental data than the traditional stochastic models, for example, the modified Poisson’s law, and so forth. The results of the work  can be considered as a new approach (alternative to the statistical) to the data processing about the failures on the level of elements. We do not know any publications about the chaos theory applications to the reliability modeling of the system taking into account its structure.
In this connection there are following the questions.(1)How does the system structure influence the phase portrait of reliability?(2)Is it possible to solve the redundancy optimization problem with deterministic (chaotic) order of the occurrences of failures?
As far as the failures are connected with the oscillations of variables, then the answers for these questions should be searched on the base of integration of the fuzzy logic and chaos theory.
In this paper we use the fuzzy algorithmic approach to parametric reliability modeling proposed in [12, 13], and the simple generator of chaotic oscillations of variables in the form of logistic function . The further description is organized in the following way.
Section 2 describes the principles of reliability dynamics modelling by means of the composition of membership function and a generator of chaos.
Section 3 examines the fuzzy reliability model of an element with a multiple redundancy. We consider the results of computer experiment on the analysis of reliability level bifurcations depending on the multiplicity of redundancy.
Section 4 considers the redundancy optimization problem under the conditions of chaotic oscillations of the parameters of the elements of systems.
2. Basic Principles
The fuzzy chaotic approach to the reliability dynamics modeling is based on the following principles.
2.1. Fuzzy Correctness
According to this principle introduced in [12, 13], there is not a crisp boundary between “correct” and “incorrect” (0) results of the functioning of a system and its elements. For the formal evaluation of the correctness level it is used the multidimensional (by the number of variables) membership function which depends on the measured parameters (input variables). The correctness of each variable is determined by the membership function of the variable to the correct value.
The function can be interpreted as the correctness distribution of the variable : extreme cases correspond to the maximal (minimal) level of the correctness of the variable . Pay attention that the correctness distribution satisfies the axioms of fuzzy sets theory , in contrast to the probabilistic distributions used in the classical reliability theory [1, 2].
The typical correctness distributions (membership functions) are represented in Figure 2. They correspond to three possible cases of fuzzy boundaries between “correct” and “incorrect” (0):(a)correct —incorrect (0),(b)incorrect (0)—correct —incorrect (0),(c)incorrect (0) correct .
Figure 2: The typical correctness distributions.
2.2. Integration of Membership Functions and Time Series
It is assumed that for the variable it is known the time series of its values in discrete moments of time (). Putting these values in the membership function , we receive the dynamics of the correctness level of the variable in the form of the function , Figure 3.
Figure 3: Integration of membership functions and time series.
2.3. Chaos Generator
Chaos means the oscillations which seem random but in truth they are generated by the deterministic nonlinear model. In  about 40 models of the chaos generators are described. Each model contains variables whose values must be fitted on the base of the experimental data. Algorithm of the chaos generation is explained by means of iterative Lamerey diagram, widely used in the classical theory of nonlinear oscillations .
It is assumed that there is a known function , connecting two neighboring elements of time series: and . Iterative diagram consists of this function and the bisector (Figure 4). Choosing the initial point by means of vertical and horizontal lines we obtain the points on the axis as follows:
Figure 4: Iterative Lamerey diagram.
The most popular generator of chaos is given by the logic map  as follows: where is the control parameter determining the nature of chaotic orbits.
Using the iterative equation (2) we can generate the consequence for the given parameter and initial point .For example, if = 0.25 and , then we find that. If = 0.25 and , then.
With the corresponding values of the parameter it is possible to get different types of attractors (Figure 5) by means of iteration algorithm (Figure 4):
Figure 5: Different types of logistic map attractors: (a) stable focus, (b) stable orbit, (c) double orbit, and (d) chaotic orbit.
(a) stable focus (), (b) stable orbit (), (c) double orbit (), and (d) chaotic orbit ().
Increasing gradually the parameter , it is possible to observe the moments of bifurcations, that is, transitions from one type of the attractor to another. Figure 6 shows that in the moment there is a jump from one stable state to two other stable states. In the moment the number of stable states is doubled, and so forth. The moment corresponds to the complete chaos . The described chaos generator will be used further for the integration with the fuzzy reliability model.
Figure 6: Bifurcation diagram of logistic map.
3. Fuzzy Chaotic Reliability Model
We consider the simple system with redundancy which is modeled by fuzzy algorithmic approach proposed in [12, 13] and logistic function (2).
3.1. Element with Redundancy
The element with redundancy is presented in Figure 7 in the form of the parallel circuit where the primary element () has of redundant elements . All the elements are supposed to be homogeneous.
Figure 7: Element with redundancy.
The quality of the element functioning depends on the variable which varies during the time: . To evaluate the reliability of the element it is used: is a membership function which determines correctness distribution of variable during the functioning of .
The parallel circuit (Figure 7) assumes that the failure of the system requires the failure of all elements similar to . That is why the correctness of the element with redundancy functioning is evaluated by the following formula:
3.2. Reliability Bifurcation
The model (3) allows observing the dynamics of the system reliability level, that is, of the correctness function during the chaotic oscillations of the variable according to the logistic map (2).
The purpose of the computer experiment consisted of the research of bifurcations of the correctness level with different correctness distributions of the element and different redundancy rates ().
The experiment is carried out with two correctness distributions shown in Figure 8: triangle (a) and threshold (b). During the chaos generation the parameter of the logistic map (2) was changed in the range from 2.5 to 4. For each distribution (Figure 8) we obtained 4 bifurcation diagrams, each of them corresponds to different redundancy rates . The results are represented in Figures 9 and 10, where the horizontal axis is the chaos parameter (), and vertical axis is the reliability level .
Figure 8: Correctness distributions of variable in the experiment.
Figure 9: Reliability bifurcations for triangle correctness distribution.
Figure 10: Reliability bifurcations for threshold correctness distribution.
Figures 9 and 10 show that in spite of the chaos growth (parameter ) by the increasing of redundancy rate (), it is possible(a)to postpone the moment of the first bifurcation which is associated with the reliability loss and(b)to decrease the diameter of an orbit around which there are oscillations of the level of system’s membership to the stable state.
That is why we can consider a redundancy optimization problem under chaotic oscillations of the parameters of elements.
4. Redundancy Optimization under Chaosof Parameters
We consider a sequential system where each element has some level of redundancy. This system is described by the series-parallel structure (Figure 11), where is a component which depends on the variable , is the redundancy rate of the component , and is the vector of the redundancy rates , .
Figure 11: Sequential system with redundancy of elements.
It is supposed to be known that is correctness distribution of variable during the element functioning, is the range of possible values of the variable and is mean cost of one redundant component like .
For the system in the Figure 11 taking into account (3) we have
A fuzzy logic-based variable speed limit controller
Duo Li,Corresponding author
- School of Highway, Chang'an University, Xi'an, Shanxi, China
Correspondence to: Duo Li, School of Highway, Chang'an University, Xi'an, Shanxi 710064, China. E-mail: firstname.lastname@example.org
- Department of Civil and Environmental Engineering, the University of Auckland, Auckland, New Zealand
Variable speed limit (VSL) is an emerging intelligent transportation system (ITS) measure to improve operational and safety performance of motorway systems. Rule-based algorithms have been widely used in VSL applications because of their comprehensibility and ease of application. However, most of the algorithms proposed in the literature under this category are rather rough for the speed control. Pre-specified rules show some difficulties in appropriately activating/deactivating control actions in real time because of non-stationary and nonlinear nature of the traffic system. This paper proposes a fuzzy logic-based VSL control algorithm as an alternative to the existing VSL control algorithms. The proposed algorithm uses fuzzy sets instead of crisp sets to allow the separation of attribute domains into several overlapping intervals. The discretization using fuzzy sets can help to overcome the sensitivity problem caused by crisp discretization used in the existing VSL algorithms. The proposed algorithm is assessed for a test bed in Auckland using AIMSUN micro-simulator and verified against a well-known VSL algorithm. The simulation results show that the proposed algorithm outperforms the existing one to improve the efficiency performance of the motorway system with the critical bottleneck capacity increased by 6.42% and total travel time reduced by 12.39% when compared to a no-control scenario. Copyright © 2015 John Wiley & Sons, Ltd.
Variable speed limit (VSL) is an emerging ITS measure for motorway traffic management where the speed limit of motorway sections is determined based on real-time traffic conditions with an attempt to improve safety and harmonize the traffic flow by decreasing speed variation among vehicles across lanes, within the same lane and also between upstream and downstream traffic flows . A well-established consensus on safety benefits because of VSL has been reached in the literatures [2-4], which is obviously attributed to the strong relationship between speed behavior and traffic safety. Nevertheless, there remains a considerable discrepancy with regard to efficiency benefits of VSL. Hegyi et al.  conducted a simulation-based study to assess their proposed VSL algorithm and reported 21% decrease in total time spent (TTS) on the network; while Long et al.  performed a similar simulation-based study using the same algorithm as proposed by Hegyi et al.  finding no significant improvement in TTS. Similarly, Hoogendoorn et al.  observed a 4% increase in the capacity of bottleneck when applying VSL, which is contradicting with the finding of Van den Hoogen and Smulders  that observed no capacity increase because of VSL.
Over the years, a number of VSL algorithms have been developed, which can be classified into two groups, namely, optimal control algorithms and rule-based algorithms. In the former, a better utilization of the road infrastructure is realized by optimizing objective functions. Model predictive control (MPC) method proposed by Hegyi et al.  is the most prominent example of optimal control algorithms, which targets to minimize the travel time of each vehicle in the network. Although the effectiveness of optimal control algorithms has been evidenced by simulation studies [9-14], only few of them are actually implemented or in operation. There are two main reasons for this situation: (i) most of the optimal control methods proposed in the literature are relatively complex for motorway operators to implement and (ii) nonlinear optimization may require frequent changes in speed limits to be displayed in VSL display units, which is unlikely to be acceptable to public.
In rule-based algorithms [15-21], speed limits are determined based on some pre-specified rules. They are widely used in the existing motorway traffic management systems because of their comprehensibility and ease of application. The pre-specified rules show some difficulties in appropriately activating/deactivating control actions in real time because of non-stationary and nonlinear nature of the traffic system, which leads to the algorithms being rather rough for the speed control. When an attribute is not discrete but continuous, e.g. traffic flow, a so-called “cut-off point” over the domain of attribute values is used to discretize the attributes. This feature makes a rule-based algorithm too sensitive, resulting in misclassifications . Furthermore, rule-based algorithms dynamically change the speed limit whenever the traffic measurement exceeds or drops below a certain threshold value (or “sharp cut-off point”), which may lead to oscillations in traffic flow. A fuzzy logic control (FLC)-based approach can be useful to deal with the sensitivity issues faced by the existing rule-based algorithms, which uses fuzzy sets instead of crisp sets to allow the separation of attribute domains into several overlapping intervals . The FLC approach can be more suitable for VSL controller for the following reasons:
- it does not require a mathematical model; thus it is not limited by the accuracy of the system model;
- it can utilize incomplete or imprecise information, thereby decreasing sensitivity to missing input data;
- it generates more smooth outputs rather than oscillatory speed limits;
- it is extremely suitable to combine objective knowledge (formulae and equations) and subjective knowledge (linguistic information); and
- it is easy to tune by changing weight factor and the parameters of membership functions.
This paper proposes a fuzzy logic-based VSL control algorithm to overcome the aforementioned weaknesses of the existing rule-based VSL control algorithms. The proposed algorithm is assessed for a test bed in Auckland using AIMSUN micro-simulator and verified against a well-known VSL algorithm, namely, logic tree-based VSL algorithm proposed by Allaby et al. .
2 Rule-Based VSL Algorithm
In rule-based VSL algorithms, speed limits to be displayed in VSL controllers are determined based on pre-specified threshold values for flow, occupancy, mean speed or their combinations. A great deal of research works has been performed in recent years to develop and evaluate rule-based VSL algorithms to improve motorway performances. Following paragraphs provides a brief review of some of those strategies relevant to this study.
2.1 Field-based studies
Van den Hoogen and Shmulders  reported on the implementation of a multi-parameter-based algorithm on a 20-km stretch of A2 motorway in Netherland. They reported that VSL is not suitable to reduce congestion at bottlenecks because it does not increase the capacity of bottlenecks. Hoogendoorn et al.  reported on a trial conducted on A20 near Rotterdam, in Netherland starting from 28 June 2011. It was revealed that the number of vehicle loss hours reduced by 20% with 4% increase in the capacity at the main bottleneck. Papageorgiou et al.  analyzed the traffic data before and after implementation of VSL on a European motorway. The control system used flow as well as mean speed thresholds. It was reported that VSL declined the slope of flow–occupancy diagram at under-critical conditions and shifted the critical occupancy to a higher value. However, the results regarding the impact of VSL on the capacity were not conclusive. A flow-based algorithm is in operation on M25 motorway in United Kingdom since 1995. UK Highways Agency  conducted a business case to evaluate its effectiveness and reported that there was a little change in weekday journey times while off-peak journey times increased slightly compared to the previous year. Heydecker and Addison  conducted a statistical analysis of the field data collected from the same motorway M25 in UK. They investigate the relationship between speed and occupancy with VSL in operation and reported that the Underwood's exponential form best explains the speed–occupancy relationship. Their results also showed that the relationship can differ depending on VSL control status. Soriguera et al.  assessed a speed-based dynamic speed limits (DSL) system implemented on a test section of a freeway in Barcelona, Spain. They showed the effectiveness of the proposed DSL in reducing accident risk, emissions and fuel consumption at the expense of increased free-flow travel times.
2.2 Simulation-based studies
Lee et al.  used a real-time crash prediction model integrated with PARAMICS micro-simulator to assess a speed-based algorithm. It was revealed that total crash potential was decreased at the expense of increase in total travel time. Elefteriadou et al.  reported on the implementation of an occupancy-based VSL algorithm for a 9-mile long section of I-4 in United States. Based on the investigation performed using CORSIM micro-simulator for eastbound direction of the road segment, they reported that there were no considerable differences between the existing operations and no-control scenario in terms of traffic throughput and total travel time. Habtemichael and Picado-Santos  assessed the performance of a flow-based VSL algorithm under different levels of driver compliance and traffic conditions. They reported that the mobility benefit because of VSL algorithm under heavily congested scenario was not statistically significant. While for non-congested and lightly congested scenarios, travel time savings were improved by 6% and 16%, respectively.
Albania  evaluated a speed-based VSL algorithm for a section of E4 motorway in Sweden using VISSIM micro-simulator. The author concluded that application of VSL led to higher overall speeds, a delayed onset of congestion and lower average travel times compared to no-control scenario. Kianfar et al.  employed several statistical methods to examine the flow changes between before and after application of VSL. They assessed the VSL algorithm implemented by Missouri of Department of Transportation (MoDOT) that used a combination of average speed, occupancy and volume. They reported that the flow changes because of the VSL algorithm were inconsistent across eight test sites. The maximum flows prior to and after breakdown increased at some locations and decreased at other locations after using the VSL.
Allaby et al.  proposed a logic tree-based VSL algorithm and assessed it for an 8-km-long road section in Toronto, Canada. This algorithm used threshold values for mean speed, flow and occupancy. Simulation results revealed that although noticeable safety benefits because of the VSL were achieved, the implementation of VSL increased the travel time for all tested cases. In this algorithm, VSL display units are linked with detectors installed on the upstream side of the known bottlenecks. The speed limits to be displayed on VSL display units were calculated based on the mean speed, occupancy and volume measurements from the loop detectors and a set of pre-specified threshold values . Li and Ranjitkar proposed a modified logic tree-based VSL algorithm and assessed it individually and in combination with ramp metering for a critical bottleneck section of Auckland Motorway using AIMSUN micro-simulator [25-28].
The findings reported in the literatures show that the traditional rule-based VSL algorithms are facing some difficulties in effectively improving mobility gains. The application of fuzzy logic control in intelligent transportation systems (ITS) has been considered in previous studies to combine the mainline VSL controller with ramp metering. Lin and Lin  proposed a VSL control algorithm for middle-low traffic density section, and a VSL together with an on-ramp control algorithm for middle-high traffic density section based on the principle of fuzzy logic. The ability of the proposed algorithms to improve motorway efficiency was not assessed in the paper. Ghods et al.  developed a genetic-fuzzy ramp metering and VSL control algorithm to reduce the peak hour congestion. The authors reported that the proposed method showed a superior performance in comparison with the traditional ALINEA controller and the genetic-fuzzy ramp metering only case. Huang et al.  proposed a fuzzy control method to combine ramp metering and VSL. The proposed method aimed to deal with traffic systems effectively, while keeping computational simplicity. However, no case study was performed to show the effectiveness of the proposed method to improve motorway efficiency in the paper.
3 Fuzzy Logic Control-Based VSL Algorithm
The term “fuzzy logic” was introduced in 1965 by Zadeh . Compared to traditional binary sets fuzzy logic variables have a truth value that ranges in degree between 0 and 1. In a crisp classification, the membership degree of the example equals a value of 1 as shown in Figure 1-(a). Thus, the size of a subset equals counts of examples belonging to the subset. In a fuzzy classification, the number of examples of a subset is computed by aggregating the membership degree of examples that belong to a fuzzy set. Figure 1-(b) illustrates that the degree of class membership of the example, Occ indicates the extent to what an example belongs to the fuzzy set. The basic structure of a fuzzy logic VSL controller consists of three components, namely, fuzzification of the input variables, construct rule base and defuzzification of the output variable membership function.
The first stage inside the fuzzy VSL controller is fuzzification which converts crisp input values into a set of fuzzy variables defined by membership functions. Fuzzification determines how well the condition of each rule matches the particular input. There are two inputs into the fuzzy VSL controller, namely, downstream occupancy and flow which are measured immediately downstream of the on-ramp merge. The fuzzy sets for downstream flow are determined based on the results of the preliminary data analysis which can be found in . We collected three months data for the motorway starting from 5 March 2012 to 27 May 2012. After removing some faulty or irrelevant data including weekends and days with adverse weather conditions, we extracted a total of 53 flow breakdown points for the bottleneck section. Downstream flow is from 1900 to 2500 veh/h/l and described as three Gaussian set “low”, “medium” and “high” as depicted in Figure 2. The Product Limit Method (PLM) is employed to generate a plot of the probability of breakdown under different occupancy levels based on the forementioned data analysis results. After examining different distributions to determine which one best fits the PLM plot, a polynomial regression has been applied to the PLM plot allowing the data to be more readily usable. Speed limits as the output of fuzzy algorithm have also been converted to fuzzy set. Three triangular set “low”, “medium” and “high” are assigned to speed limits as shown in Figure 2. Note that we fine-tuned above described membership functions based on simulation results. Different combinations of centroid values in traffic flow and speed limit membership functions were tested in Micro-simulator AIMSUN; the one that yield the lowest total travel time (TTT) was selected for this study.
At the heart of the controller, the rules, sometimes called the knowledge base, are designed based on operator experience, expert opinions and system knowledge. Basically the rules that belong to a linguistic controller are expressed in the following format.
IF <premise> THEN<consequent>
There is also the possibility to combine several premises with operators.
IF <premise 1> AND/OR <premise 2> AND/OR<premise 3> .....
All rules are evaluated in parallel based on fuzzy set theory that describes interpretation of the logical operations such as the complement, intersection and union of sets. The consequent of each rule assigns an entire fuzzy set to the outputs. The fuzzy set is represented by a membership function to indicate the qualities of the consequent. Thus every rule has a nonzero degree overlapping with other rules. The aggregation method is chosen to combine the inference results of these rules. The rule base to determine speed limit for the proposed fuzzy logic-based VSL controller is stated in Table 1, which is determined based on the preliminary data analysis results as discussed in the previous section. The rule number 1 through 3 cover all traffic flow conditions from light traffic to heavy congestion; hence one of these rules is always active. These rules are to prevent the formation of downstream congestion rather than simply react to it. The fuzzy sets for the downstream volume that are computed based on the historical measured maximum flow rate of downstream can be seen as a prediction of the downstream bottleneck behavior. The rule number 4 uses the premise that high downstream occupancy indicates bottleneck formation, calling for a more restrictive speed control. The weight assigned to each rule indicates the priority given to the respective rule. The higher weight given to rule number 3 and 4 emphasizes that these rules are given higher priority to achieve an effective VSL control. We optimized these weight values based on simulation results where a combination producing the lowest TTT was chosen.
|1||1.0||IF downstream flow is low||THEN speed limit is high|
|2||1.0||IF downstream flow is medium||THEN speed limit is medium|
|3||2.0||IF downstream flow is high||THEN speed limit is low|
|4||2.0||IF downstream occupancy is high||THEN speed limit is low|
The defuzzification process is to convert each fuzzy output variable into a crisp (non-fuzzy) form (speed limit). The centroid method is commonly used in the defuzzification process. The equation of centroid gravity method shown below:
where f(x) is the membership function for input x. In practice, a discrete fuzzy centroid equation is used because it is easier to calculate than the above continuous centroid equation. The discrete centroid equation is expressed as follows:
is the number of the output classes,
is the centroid of the ith output class,
is the results of the aggregation of rules at the ith output class and
is the area of the ith output class.
The speed limits produced by Equation (2) are directly applied to VSL display units without further discretization. Figure 3 shows the procedure of how sample inputs convert to the outputs in fuzzy speed limit control through the above described steps.
4 Modeling of Test Bed in Aimsun
A critical bottleneck section on State Highway 1 of Auckland Motorway connecting Central Auckland with Northern Auckland was selected for this study. Figure 4 presents a layout of the study section, which consists of five on-ramps and four off-ramps in a direction towards Auckland city center. Here O1 represents on-ramp from Esmonde Road while O5 represents on-ramp from Greville Road. The network data used in this study was provided by New Zealand Transport Agency (NZTA) which provides loop detector measurements from the on-ramps, off-ramps and mainline accumulated over a 30-s time period.
A model of the study section of the motorway was developed in AIMSUN micro-simulator . We developed an Application Programming Interface (API) program in AIMSUN to implement and verify the performance of the FLC algorithm for the critical section of Auckland Motorway. Figure 5 illustrates the proposed algorithm implemented in AIMSUN. During each control interval, FLC API receives traffic information including traffic flow and occupancy data collected from the mainline detectors. Then, FLC API converts real-time traffic information to speed limit values through the three fuzzy steps, namely, fuzzification, inference and defuzzification. Finally, FLC API assigns the determined speed limit values to each VSL controller. The control interval in the proposed FLC API is flexible for different applications. In this case, FLC API collects traffic information and determines speed limits at every 2-min interval. The modeled traffic flow consists of 90% cars and 10 heavy vehicles.
The first phase of the calibration procedure involves the adjustment of global parameters such as simulation step, reaction time, acceleration and deceleration. After calibrating the global parameters, local parameters that are defined at the section level and applied locally to vehicles are also calibrated. The model is calibrated against the data collected on Monday, 12 March 2012 and then validated against the data collected on Friday 9 March 2012. GEH statistic  is used to calibrate and validate the simulation model based on flow data collected from 14 loop detectors. The GEH statistic can be obtained as follows:
is estimated count using AIMSUN model, and
is field count.
The developed model is considered to be acceptable if the GEH values for more than 85% of the observed detectors remain below 5 and that was the case in this study for both calibration and validation. Hence the model was accepted for further analysis to test different scenarios described in the next section.
Total travel time (TTT) is commonly used to reflect overall performance of the network. A lower TTT represents lower delay and a higher outflow and therefore better traffic conditions. TTT can be expressed as follows in vehicles times hours (veh*h):
is density of a segment i,
is measurement duration,
is the distance between two measured stations i − 1 and i,
is a number of measurement stations and
is a time horizon.
For vehicular emissions, we used the QUARTET pollution emission model embedded in the AIMSUN micro-simulator. This model computes vehicular emissions at each simulation step based on vehicle state, speed and acceleration .
5 Analysis Results
We systematically assessed three different control scenarios for the study section including no control, logic tree-based algorithm and FLC algorithm. The “no-control scenario” is chosen as a reference to measure improvements offered by other control scenarios. In the following tables, the highest improvements are highlighted with a dark background.
5.1 Traffic conditions around merging areas
Figure 6 presents scatter plots of occupancy versus flow data points collected from the bottleneck section for different scenarios. The initial trend of increase in occupancy with increase in flow was observed while this trend reversed after the capacity was reached. It can be observed that this change in trend occurred within occupancy ranging from 15% to 18% for all scenarios. The critical occupancy values before flow breakdown are located around 16% for the first two cases, while for FLC case, it is located around 17%. Nevertheless, the exact critical occupancy and capacity values were not clear in the scatter plots.
We used a method proposed by Kianfar et al.  to compute the capacity and critical occupancy at the bottleneck section. Critical occupancies were determined using a two-step procedure. First, the scatter plots of flow versus occupancy data points were observed for any clear change in trends. The initial trend of increase in flow with increase in occupancy was observed. This trend reversed after the critical occupancy was reached. Nevertheless, the exact critical occupancy value was not easily discernible from the scatter plot itself. Thus, regression lines were fitted for uncongested and congested regions for different possible critical occupancy values, and then root mean square error (RMSE) values were computed for each case. The critical occupancy value that produced the least RMSE value was selected. The capacity values were obtained from best fit-lines at the critical occupancy points. As presented in Table 2, there is not much difference in the critical occupancy and bottleneck capacity values for the first two scenarios that are no-control case and logic tree algorithm case. For FLC case, the capacity is increased to 2318 veh//h/l, which is around 6% improvement compared to no-control case with a capacity value of 2178 veh/h/l. Meanwhile the critical occupancy is also shifted from 16.13% for no-control case to 17.38% for FLC case.
|Critical occupancy (%)||16.13||16.25||+0.77||17.38||+7.75|
Figure 7 presents flow contour plots along the mainline of the bottleneck section under different control scenarios. It can be observed that both of the VSL control scenarios produced relatively higher traffic flow near the detector zone D1 and D2 when compared against no-control case during the peak period. This might be because of improvement in the bottleneck capacity as shown in Table 2. Figure 8 presents occupancy contour plots along the mainline of the bottleneck section under different control scenarios. For all of the three scenarios, the bottleneck is formed on the mainline near the first detector zone D1. For no-control scenario, large variations can be observed in occupancy measurements over space and time indicating formation of strong shockwaves in this scenario. Relatively less variation can be observed for the two VSL control scenarios. For FLC algorithm, occupancies are relatively lower than those for logic tree-based algorithm in the first three detector zones D1 to D3. Hence, VSL can reduce formation of shockwaves by harmonizing the traffic flow at the upstream segments. Figure 9 presents speed contour plots along the mainline of the bottleneck section under different control scenarios. Large variations in speed measurements can be observed for the no-control scenario. The variations are reduced for both of the VSL control scenarios. FLC scenario has yielded the minimum variations in speed. Here only the results with 100% driver compliance are presented. That is a case in which the vehicles strictly follow the speed limits displayed in VSL display units.
5.2 System-wide performance
Table 3 presents the values of MOEs computed for the entire study area of the motorway network under different control scenarios. The logic tree-based algorithm recorded an improvement of 5.6% in TTT when compared with no-control scenario while FLC scenario witnessed an improvement of 12.4% compared to no-control scenario. FLC scenario witnessed the minimum average stop time duration per vehicle (improved by 85%) and the minimum average number of stops per vehicle (improved by 72%). Meanwhile, emission levels also reduced reasonably. These improvements might be because of improved driving conditions for FLC scenario compared to no-control scenario that witness stop-start driving conditions. Figure 10 presents a time series plot of TTT for the entire study area under different control scenarios. It can be observed that improvements contributed by VSL mainly occurred during the peak period from 6:45 to 8:15 am. Here also the proposed FLC algorithm produced the lowest TTT for most of the time.
|TTT (in veh-h)||1719||1622||−5.64||1506||−12.39|
|Stop time/veh. (in sec/km)||27.94||14.04||−49.75||4.20||−85.0|
|# of stops/veh||0.71||0.42||−40.85||0.19||−72.2|
|CO2 (in gm/km)||1.37 × 106||1.35 × 106||−1.46||1.32×106||−3.74|
|Carbon monoxide (CO) (in gm)||786.86||752.89||−4.32||714.62||−9.18|
|NOx (in gm/km)||3060||3010||−1.63||2950||−3.59|
To assess the impact of driver compliance on mobility benefits of the system for studied measures, each scenario is further analyzed by considering three levels of driver compliances including 60%, 80% and 100% as presented in Table 4. It can be observed that the mobility benefit of VSL is at its highest level with 100% driver compliance (that is a case of strictly enforced VSL). Then TTT values increased with reduction in driver compliance rates. We conducted T-test to check how significant the differences in the TTT values are under different driver compliance levels. For FLC case, T-values computed using 100% driver compliance rate exceed the critical T-value when compared with no-control case, while for all other cases no statistically significant difference was observed among them.
6 Concluding Remarks
Traditional rule-based VSL algorithms are rather rough for controlling non-linear, complex motorway systems whereas pre-fixed rules show some difficulty in appropriately activating/deactivating control actions. This paper proposes a fuzzy logic approach to enhance the performance of the existing rule-based VSL controllers. The algorithm developed is assessed using a case study in Auckland, New Zealand and compared with a known algorithm proposed in the literature. The performance of studied algorithms is measured by total travel time, average number and duration of stops and emission levels with different driver compliance rates. The main conclusions drawn of this study are as follows.
- VSL offers an effective solution to improve mobility gains as well as emission reductions.
- The mobility benefit of VSL is at its highest level with 100% drivel compliance. With an increase in rates of driver compliance, travel time savings increase.
- The proposed algorithm dramatically improves the efficiency of the infrastructure and outperforms all other studied measures in terms of total travel time and environmental-related MOEs.
The findings in this research are limited in scope as they are based on a particular bottleneck section of Auckland motorway modeled in AIMSUN micro-simulation. A model can have its own limitations to represent real-world traffic conditions. It is recommended to conduct similar investigation under a range of different traffic conditions and for a range of motorway networks in the field before generalizing any such findings. In this study, all the parameters of the proposed fuzzy logic-based VSL controller are optimized manually for the test bed in Auckland. However, the proposed fuzzy algorithm requires tedious and time-consuming retuning if applied to other locations. Besides, the fuzzy controller with the pre-specified setting parameters cannot adequately deal with inconsistent traffic demand pattern and sudden change in traffic conditions because of bad weather conditions, incidents or road maintenance. It shall be useful to apply more robust optimization methods using artificial intelligence techniques such as Genetic Algorithm (GA) to automatically and adaptively tune the fuzzy set parameters for real world application in other motorway networks. Speed limit values computed by the proposed algorithm are not discretized in this paper; this enables the proposed algorithm to yield the best achievable results. However, for practical applications speed limits are generally discretized to make it acceptable to drivers, such as 100/90/80/70, which will reduce the efficiency of the proposed algorithm. It shall be noted that 100% drivers' compliance cannot be achieved using non-discretized speed limits. Therefore, we provided the performance of the proposed algorithm under different compliance levels (in Table 4). The effect of discretized speed limits on the efficiency performance will be considered our future research.
7 List of Symbols and Abbreviations
the centroid of the ith output class
estimated count using AIMSUN model
the membership function for input x
a time horizon