2.1 Overview of traffic control system parameters
Ahmed and Hawas (2015) developed an integrated traffic signal control system that estimates some actuation index for a running green phase to determine the next best phase to operate and when to do so, by considering some rule-based boundary conditions. Contrary to the typical logic of actuated controllers, the actuation request is not met in accordance to the minimum, maximum green criteria or the preset unit extension for every detector hit by vehicular arrivals. In the modified logic, the actuation request is met by the realization of some boundary conditions. That is, when some boundary conditions are met, specific parameters in the objective function are flagged to deploy, which in turn affect the value of the objective function for each phase. Four modules are utilized to check these boundary conditions. First, the traffic-regime state module is responsible for estimating the congestion status of the links upstream a signaled intersection. Second, the incident status module is responsible for estimating the likelihood of incidents on the various intersection approaches. Third, the transit priority module is responsible for flagging links for transit priority based on the locations and types of transit vehicles. Fourth, the downstream blockage module is responsible for scanning the links downstream the intersection to account for the recurrent blockage (spillback) conditions. Following the four boundary modules, an actuation module is deployed to estimate the so-called actuation index for each phase and identify the next candidate phase set based on the signal control type (e.g., dual, protected, split). The actuation index \({A}_{i,{\varphi }_{j}}^{t}\) of the individual phase \({\varphi }_{j}\) can be calculated using Eq. (1). The base-congestion indicator \({J}_{i,{\varphi }_{j}}^{/,t}\) refers to the virtual-queue of passengers (on the upstream approach of that individual phase \({\varnothing }_{j}\) at time t) and is estimated using Eq. (2).
$${A}_{i,{\varphi }_{j}}^{t}={J}_{i,{\varphi }_{j}}^{/,t}\times \frac{\left(1+{\beta }_{i,{\varphi }_{j},{u}^{/}}^{N}{\times I}_{i,{\varphi }_{j},{u}^{/}}^{N,t}\right)}{\left[{\left(1+{I}_{i,{\varphi }_{j},{d}^{/}}^{B,t}\right)}^{{\beta }_{i,{\varphi }_{j},{d}^{/}}^{B}}\right]}$$
1
$${J}_{i,{\varphi }_{j}}^{/,t}=\left[\begin{array}{c}\begin{array}{c}\left({C}_{i,{\varphi }_{j},{u}^{/}}^{c,t}{\times O}_{i,{\varphi }_{j},{u}^{/}}^{c}\right)+\left({C}_{i,{\varphi }_{j},{u}^{/}}^{b,t}\times {O}_{i,{\varphi }_{j},{u}^{/}}^{b}\right.\\ \left.\times {\beta }_{i,{\varphi }_{j},{u}^{/}}^{b}\right)+\left({C}_{i,{\varphi }_{j},{u}^{/}}^{p,t}\times {O}_{i,{\varphi }_{j},{u}^{/}}^{p}\times {\beta }_{i,{\varphi }_{j},{u}^{/}}^{p}\right)+\end{array}\\ \left\{\left({C}_{i,{\varphi }_{j},{u}^{/}}^{c,t}\times {O}_{i,{\varphi }_{j},{u}^{/}}^{c}+{C}_{i,{\varphi }_{j},{u}^{/}}^{b,t}\times {O}_{i,{\varphi }_{j},{u}^{/}}^{b}\right.\right.\\ +\left.{C}_{i,{\varphi }_{j},{u}^{/}}^{p,t}\times {O}_{i,{\varphi }_{j},{u}^{/}}^{p} \right){\times r}_{i,{\varphi }_{j},{u}^{/}}^{V,t}\times \left.{\beta }_{i,{\varphi }_{j},{u}^{/}}^{V}\right\}\end{array}\right]$$
2
Here, \({C}_{i,{\varphi }_{j},{u}^{/}}^{c,t}\), \({C}_{i,{\varphi }_{j},{u}^{/}}^{b,t}\), and \({C}_{i,{\varphi }_{j},{u}^{/}}^{p,t}\) are the vehicular counts of cars \(c\), normal priority buses \(b\), and high priority buses \(p\), respectively, at time \(t\) along the upstream approach link \({u}^{/}\) corresponding to phase \({\varphi }_{j}\) of intersection \(i\). \({O}_{i,{\varphi }_{j},{u}^{/}}^{c}\), \({O}_{i,{\varphi }_{j},{u}^{/}}^{b}\), and \({O}_{i,{\varphi }_{j},{u}^{/}}^{p}\) are the average passenger occupancies of cars, normal, and high priority buses, respectively. \({\beta }_{i,{\varphi }_{j},{u}^{/}}^{b}\) and \({\beta }_{i,{\varphi }_{j},{u}^{/}}^{p}\) are transit priority coefficients for normal and high priority buses, respectively. \({r}_{i,{\varphi }_{j},{u}^{/}}^{V,t}\) is the ratio of the vehicular queue length to the physical capacity of the corresponding link length, and \({\beta }_{i,{\varphi }_{j},{u}^{/}}^{V}\) is a coefficient for the virtual queue of vehicles.
If an incident is detected, the value of the base congestion indicator \({J}_{i,{\varphi }_{j}}^{/,t}\) is adjusted (increased) by the incident coefficient \({\beta }_{i,{\varphi }_{j},{u}^{/}}^{N}\) to account for the potential incidents on the upstream approach, \({u}^{/}\), as shown in Eq. (1). The value (in Eq. (1)) is also adjusted (decreased) by applying a downstream blockage coefficient \({\beta }_{i,{\varphi }_{j},{d}^{/}}^{B}\) to account for blockage on the downstream exit link of phase \({\varphi }_{j}\). The congestion on the downstream link is estimated using the information extracted from the downstream (exit) link detectors. The system penalizes the links that have full or partial blockage; for instance, if one link is entirely blocked, the upstream phases of this particular link will be “penalized” and, as such, lesser green times will be allocated to the phases that feed vehicles to the blocked link. This action will prevent any further blockage of the incident links, reducing the likelihood of spill-backs from and along entirely blocked incident links. The transit priority parameters in Eq. (2) account for priority buses. If a priority bus is detected, the base congestion indicator is increased compared to the case in which no priority buses are detected. In brief, Eq. (1) estimates the congestion indicator (base and adjusted) depending on the conditions identified by the four boundary modules.
We define the so-called actuation index of a candidate phase set \({Z}_{i,{{\Phi }}_{k}}^{t}\) as the sum of the actuation indices of the two concurrent individual phases of the candidate phase set\({{\Phi }}_{k}\), where \({{\Phi }}_{k}={\{\varphi }^{k,1}\cup {\varphi }^{k,2}\}\). \({Z}_{i,{{\Phi }}_{k}}^{t}\) represents the final adjusted virtual queue of passengers considering the estimated impact of all of the relevant boundary conditions. The most deserving candidate phase set (to follow the current one) is the one with the maximum \({Z}_{i,{{\Phi }}_{k}}^{t}\)value.
Figure 1 shows the dual-phase setting for an actuated controller, including the possible sequencing of individual phases and concurrent or combined phase sets. \({\varphi }_{\text{k}}\) refers to the lane group movement for a typical four-approach intersection, where \(k\) = [1,8] based on the standard National Electrical Manufacturers Association phasing (Holm et al., 2007). The notation \({{\Psi }}_{\text{c}}\) is used in this paper to refer to the set of candidate phase sets, if the current green phase set is \({{\Phi }}_{\text{c}}\). Let us assume that the current green phase set is \({{\Phi }}_{1}\), indicating that \({\varphi }_{1} \text{a}\text{n}\text{d} {\varphi }_{5}\) are concurrently green, while the other competing phase sets \({{\Phi }}_{2}\){\({\varphi }_{1}\cup {\varphi }_{6}\)}, \({{\Phi }}_{3}\){\({\varphi }_{2}\cup {\varphi }_{5}\)}, and \({{\Phi }}_{4}\) {\({\varphi }_{2}\cup {\varphi }_{6}\)}) are kept red. On deciding whether to extend the green of \({{\Phi }}_{1}\) or switching green to any of the other potential sets (\({{\Phi }}_{2}\), \({{\Phi }}_{3}\), or \({{\Phi }}_{4}\)), the logic scans all of the sets included in the set denoted by\({{\Psi }}_{1}=\left\{{{\Phi }}_{1},{{\Phi }}_{2},{{\Phi }}_{3},{{\Phi }}_{4}\right\}.\)
The controller parameters should be calibrated for optimal performance, as they are likely to affect the network performance. Typically, there are four parameters to calibrate in Eq. (1). To downsize the problem and facilitate demonstration of the adopted procedure and results, only non-incident traffic conditions are discussed in this paper. Thus, there is no need to calibrate the incident-related parameter, and it is set to unity herein. By dropping the incident-related parameter, three primary parameters or coefficients remain to calibrate for the integrated complex controller with different objective functions. The first parameter is the coefficient for a virtual queue of vehicles on the upstream approach link (called \({\beta }^{V}\)herein for simplicity). Second are the transit priority coefficients \({\beta }^{b}\) and \({\beta }^{p}\), corresponding to regular and high priority buses, respectively. For simplicity, \({\beta }^{b}\)and\({ \beta }^{p}\) were assumed to be equal in this study. Third is the downstream blockage coefficient \({\beta }^{B}\).
The traffic network performance is represented herein by three measures of effectiveness (MOEs): the total number of bus trips \({N}_{bus}\) served during a specific analysis period, total network travel time in hours \({T}_{t}\), and mean trip travel time in seconds \({t}_{m}\). A simulation-based optimization approach was adopted to model the relationships among the above three control parameters and the resulting MOEs. The calibration of the control parameters was formulated as an optimization problem of multiple objective functions; finding the values of \({\beta }^{V}, {\beta }^{b}\)or\({ \beta }^{p}\), and\({\beta }^{B}\) to maximize \({N}_{bus}\)and minimize \({T}_{t}\)and \({t}_{m}\).
2.2 RSM with calibration framework
In simulation-based optimization, finding the best values of the control parameters is an extremely cumbersome process. The best parameter values (of \({\beta }^{V}, {\beta }^{b}\)or\({ \beta }^{p}\), and\({\beta }^{B}\) combination) are chosen from the different domains of these individual parameters. What adds to the challenge is the fact that these individual parameter domains are not known in advance. Furthermore, the domain ranges of the individual parameters may significantly differ in accordance to the targeted objective function as well. For instance, the domain range of \({\beta }^{V}\)that result in maximum \({N}_{bus}\) may significantly differ from the one that minimizes \({T}_{t}\)and \({t}_{m}\).
The RSM was used in this study to determine the best parameter configurations, as the RSM requires fewer simulation experiments than the gradient-based method (Carson and Maria, 1997). The RSM involves constructing a surrogate mathematical model or multiple models to approximate the underlying function (Deng, 2007). The RSM can be divided into two general methods: central composite design (CCD) and Box–Behnken design (BBD) (Fu, 2015). In this study, BBD was used to obtain the optimum solutions (of the parameters vis-à-vis the specified MOEs), as BBD is slightly more efficient than CCD (Ferreira et al., 2007). BBD is also more efficient and economical than the similar three-level full factorial designs (Bezerra et al., 2008). The details of RSM application will be described in the subsequent discussion of the simulation-based experiments.
The RSM was proposed and described by Box and Wilson (1951). It involves the use of techniques (mathematical and statistical) to define the relationships between the response and independent variables (inputs) and determines the effects (alone or in combination) of the independent variables on the processes. To analyze the consequences of the independent variables, a meta-model is generated. The graphical aspect of this meta-model led to the term “response surface methodology.” The RSM can also be applied to computer simulation models of physical systems (Myers et al., 2009).
The first step in the application of the RSM as an optimization technique is selection of independent variables according to the objective function. In this study, three independent variables were selected (\({\beta }^{V}, {\beta }^{b}\)or\({ \beta }^{p}\), and\({\beta }^{B}\)), as explained in the previous section. The second step is to choose the experimental design method; in this research, the BBD was used. Afterwards, mathematical and/or statistical treatment is applied to fit a polynomial function using a \(p\)-value of 0.1, followed by evaluation of the fitness of the model. Finally, the necessity and possibility of performing a displacement toward the optimal area is verified to find the optimum value of each studied variable (Bezerra et al., 2008). The regions of the parameters were initially chosen arbitrarily. The RSM (BBD) was then applied to generate the input settings for the various parameter combinations. The control system was simulated using the values of each combination, and the MOEs were extracted from the simulation results.
The so-called desirability function approach, as outlined by Derringer and Suich (1980), was used to account for the multi-objective simultaneous considerations of the responses. Initially, each response was converted into a so-called desirability varying from zero to one on a dimensionless scale. The individual desirability parameters were then used to estimate the composite desirability \(D\) using the geometric mean formula. The estimated D depends on the specific preset objective (lower, target, upper) of each individual desirability element (response); the weight \(r\), which defines the form shape of the desirability function for each response; and the importance parameters \(w\) of the various desirability items that are combined into a single composite desirability.
Given the objectives of interest (e.g., to maximize \({N}_{bus}\) and minimize \({T}_{t}\) and \({t}_{m}\) simultaneously), the individual desirability parameters were stated, and the problem was transformed into one of maximizing D. D unifies the individual desirability parameters of all of the response variables into a single measure, and emphasis is placed on the response variables using the importance parameter \(w\). The importance parameters reflect the relative importances of the individual desirability parameters in estimating D, as shown in Eq. (3) (weighted geometric mean).
Here, \({d}_{{\text{N}}_{bus}}, {d}_{{T}_{t}}\), and \({d}_{{t}_{m}}\)are the individual desirability parameters of \({N}_{bus}\), \({T}_{t}\), and \({t}_{m}\), respectively, and \({w}_{{\text{N}}_{bus}}, {w}_{{\text{T}}_{t}}\), and \({w}_{{\text{t}}_{m}}\) are the importance parameter of \({N}_{bus}\), \({T}_{t}\), and \({t}_{m}\), respectively. Each importance parameter determines the influence of the respective response on D. If equal importance is placed on the responses, then \({w}_{{\text{N}}_{bus}}, {w}_{{\text{T}}_{t}}\), and \({w}_{{\text{t}}_{m}}\) are set equal to the same value. In this study, all of the importance parameters were set equal to one, i.e., \({w}_{{\text{N}}_{bus}}{=w}_{{T}_{t}}={w}_{{t}_{m}}=1\).
It is to be noted that the type of the objective function (maximum or minimum) and the target values of response(s) are key factors in setting the desirability functions. For instance, if the objective is to minimize a response, the desirability is set to one for all response values less than or equal to a specific lower bound target. Alternatively, if the objective is to maximize a response, then the desirability is set to one for all values equal to or above a specific upper bound target. As an example, if the objective for the response of the total number of bus trips\({ y}_{{N}_{bus}}\) is a maximum, then the individual desirability function \({d}_{{N}_{bus}}\)is defined as follows:
$${d}_{{N}_{bus}}=\left\{\begin{array}{cc}0& { y}_{{N}_{bus}}<{ L}_{{N}_{bus}}\\ {\left(\frac{{ y}_{{N}_{bus}}-{ L}_{{N}_{bus}}}{{ T}_{{N}_{bus}}-{ L}_{{N}_{bus}}}\right)}^{r}& { L}_{{N}_{bus}}\le { y}_{{N}_{bus}}\le { T}_{{N}_{bus}}\\ 1& { y}_{{N}_{bus}}>{ T}_{{N}_{bus}}\end{array}\right.$$
4,
where \(r\) is the weight that defines the functional form of the desirability function and \({ L}_{{N}_{bus}}\) and \({ T}_{{N}_{bus}}\)are the lower bound and target values of the response (\({N}_{bus}\)), respectively. Furthermore, if the objective for the response is a minimum (e.g., network total travel time \({y}_{{T}_{t}}\)), then the individual desirability for the response \({d}_{{T}_{t}}\)is defined as follows:
$${d}_{{T}_{t}}=\left\{\begin{array}{cc}1& {y}_{{T}_{t}}<{T}_{{T}_{t}}\\ {\left(\frac{{U}_{{T}_{t}}-{y}_{{T}_{t}}}{{U}_{{T}_{t}}-{T}_{{T}_{t}}}\right)}^{r}& {T}_{{T}_{t}}\le {y}_{{T}_{t}}\le {U}_{{T}_{t}}\\ 0& {y}_{{T}_{t}}>{U}_{{T}_{t}}\end{array}\right.$$
5,
where\({ U}_{{T}_{t}}\) and \({T}_{{T}_{t}}\) are the upper bound and target of the response (\({T}_{t}\)), respectively. The weights of the desirability function r define the shape of the individual desirability function. If \(r\) > 1, more emphasis is placed on being close to the target response, and if 0 < \(r\) < 1, less emphasis is placed on this factor (Myers et al., 2009). If \(r\) = 1, which was used in this study, the desirability function is linear.
Figure 2 shows the framework of control system calibration using the RSM. The integrated traffic control system (Ahmed and Hawas, 2015) was chosen in this investigation for the case study. First, the parameters to calibrate were selected (herein, \({\beta }^{V}\), \({\beta }^{b}\)or\({ \beta }^{p}\), and \({\beta }^{B}\)). The regions of the parameters were initially chosen arbitrarily. The RSM (BBD) was applied to generate the input settings for the various parameter combinations. According to BBD, 15 combinations of parameter values were created. The control system was simulated using the values of each combination, and the MOEs were extracted from the simulation results. The MOEs used in this study were \({N}_{bus}\), \({T}_{t}\), and \({t}_{m}\). Mathematical and statistical treatment was then applied to fit a polynomial function using a \(p\)-value of 0.1.
The parameter values were checked to determine whether they were within the specified regions. If the value of a parameter was on the border (of the specified region), then the region of the parameter was adjusted, and a second model was generated (with a new region). The whole process was repeated until the optimal setting, for which the values of all of the parameters were within the specified regions (not on borders of the regions), was identified. The optimal setting was then verified by simulating the control system using 10 runs. The MOEs resulting from the multiple runs were then averaged. Each “average” MOE was checked to verify whether it was within the 95% confidence interval of the corresponding values extracted from the response surface model.
In this study, three objective functions were considered first (maximize \({N}_{bus}\) while simultaneously minimizing \({T}_{t}\) and\({t}_{m}\)). The calibration was also performed considering only one objective function (maximize\({N}_{bus}\)) and two objective functions (maximize \({N}_{bus}\) and minimize\({ t}_{m}\)). D was estimated using the individual desirability of each objective function.
2.3 Experimental traffic network
A grid-type network of 49 intersections is used in this study, as indicated in Fig. 3. Each intersection has four approach links (from the east, west, north, and south) and four exit links with three continuous lanes and two additional left-turn pockets with 80 m storage length each. Due to the extensive set of simulation-based runs and the corresponding RSM optimization in this study, it was decided to focus on only the networks exhibiting high (E) to very high (F) traffic demand levels.
The network consists of short links (300 m) and relatively longer links (600 m) side by side, alternately in the vertical and horizontal dimensions. This network has seven horizontal and vertical arterials, and the origin and destination nodes are chosen from the eastern, western, northern, and southern boundary link entrances and exits, respectively. The “car” OD trip distribution is as follows: from any origin on the eastern boundary, 60% of the originating trips are split equally among the destinations on the western boundary, 20% among the destinations on the northern boundary, and the remaining 20% on the southern boundary. Similar directional distributions are followed for any origin on the western, northern, and southern boundaries.
Two different levels of traffic demand are used; E and F corresponding to high and very high car traffic volumes of 1000 and 1500 per hour. Mean bus route headways of 10 and 5 min are used with the E and F demand levels, respectively. Both demand cases are tested with a maximum green time (of any individual phase or phase set) of 45 s. The test network includes 18 directional bus routes. Origins and destinations on the eastern and western boundaries are considered for bus routes. Some of the bus routes overlap on some of the links, and some intersections have both left- and right-turning bus trips on their associated approach links.