**2. Integration of ramp metering and route guidance for HDVs**

This section will focus on providing a review on the combined RM and RG control as two of the most common traffic control management techniques. To do so, first, a review on traffic flow models will be provided and then, the most common RM and RG strategies will be explained, respectively. At the end of this section, a review on the studies with the focus on the integration of RM and RG will be presented.

#### **2.1 Traffic flow models**

Traffic flow models can be categorized into first order and second order models. The most frequently used models are first order models, such as Lighthill-Whitham-Richards (LWR) model [1], which is a continuous model, and the celltransmission model (CTM) [2], which is a discretized version of the LWR model. The second-order traffic flow models, besides considering the dynamics of the traffic density, introduce a dynamic equation for the mean velocity. The most famous second order model is the Modèle d'Écoulement de Trafic sur Autoroute NETworks (METANET) model [3, 4]. In this section, a review on the CTM and METANET model, as the two most used discrete traffic flow models in the literature, will be provided. The notations adopted in this section are adopted from [5]. **Tables 1** and **2** describe the model variables and parameters of these two models with their symbols, definitions, and units.

#### *2.1.1 The cell transmission model (CTM)*

The CTM was first developed by Daganzo [2] in 1992 and then, through out the following years, many other extensions of it were developed. The following version is the original version of the CTM with some minor modifications from [6] and

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected… DOI: http://dx.doi.org/10.5772/intechopen.94332*

the notations are borrowed from [5]. The CTM is characterised by the following equations:

$$
\rho\_i(k+1) = \rho\_i(k) + \frac{T}{L} \left( \Phi\_i^+(k) - \Phi\_i^-(k) \right) \tag{1}
$$

$$
\Phi\_i^+(k) = \phi\_i(k) + r\_i(k) \tag{2}
$$

$$
\Phi\_i^-(k) = \phi\_{i+1}(k) + s\_i(k) \tag{3}
$$

$$s\_i(k) = \frac{\beta\_i(k)}{1 - \beta\_i(k)} \phi\_{i+1}(k) \tag{4}$$

The dynamic equation of the on-ramp queue length is:

$$l\_i(k+1) = l\_i(k) + T(d\_i(k) - r\_i(k)).\tag{5}$$

The mainline flows and on-ramp flows are:

$$\phi\_i(k) = \min\left\{ (1 - \beta\_{i-1}(k))v\_{i-1}(\rho\_{i-1}(k) + r\_{i-1}(k)), w\_i(\rho\_i^{\max} - \rho\_i(k) - r\_i(k)), q\_i^{\max} \right\} \tag{6}$$


**Table 1.**

*CTM model variables and parameters of cell i during interval [kT,(k+1)T).*

Moreover, it seems unrealistic that all the HDVs will suddenly be replaced by the AVs in the near future. Rather, what seems more plausible is that the AVs will be introduced onto the roads in the presence of the HDVs. Therefore, there is a need to consider cases where it becomes necessary to model the interactions between AVs and HDVs. Delays caused at on-ramps and off-ramps are some of the major contributors to overall system efficiency degradation. In addition to the increase of congestion in the merge lane and outer freeway lanes, merging lanes can have an overflow effect which causes the entire freeway to become congested. However, with the advent of CAVs, a lot more information has been made available for

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

Moving on to mixed–autonomy highway networks, as a specific example of the interaction between HDVs and CAVs, the overtaking behavior performed by a CAV is chosen as the target driving behavior for the last section of this chapter. The reason for this choice is that it is one of the more challenging driving behaviors when compared to car following and lane changing as it encompasses the

This chapter is organized as follows: Section 2 reviews the integration of RM and RG, that have shown significant improvements on different control measures for highway networks with HDVs. Section 3 addresses the specific problem of improving freeway on-ramp merging efficiency by optimally coordinating CAVs. Finally, Section 4 explores the overtaking behavior accomplished by a CAV in the presence

**2. Integration of ramp metering and route guidance for HDVs**

This section will focus on providing a review on the combined RM and RG control as two of the most common traffic control management techniques. To do so, first, a review on traffic flow models will be provided and then, the most common RM and RG strategies will be explained, respectively. At the end of this section, a review on the studies with the focus on the integration of RM and RG will

Traffic flow models can be categorized into first order and second order models.

The CTM was first developed by Daganzo [2] in 1992 and then, through out the following years, many other extensions of it were developed. The following version is the original version of the CTM with some minor modifications from [6] and

The most frequently used models are first order models, such as Lighthill-Whitham-Richards (LWR) model [1], which is a continuous model, and the celltransmission model (CTM) [2], which is a discretized version of the LWR model. The second-order traffic flow models, besides considering the dynamics of the traffic density, introduce a dynamic equation for the mean velocity. The most famous second order model is the Modèle d'Écoulement de Trafic sur Autoroute NETworks (METANET) model [3, 4]. In this section, a review on the CTM and METANET model, as the two most used discrete traffic flow models in the literature, will be provided. The notations adopted in this section are adopted from [5]. **Tables 1** and **2** describe the model variables and parameters of these two models

improving this overall process.

combination of these behaviors.

of HDVs.

be presented.

**120**

**2.1 Traffic flow models**

with their symbols, definitions, and units.

*2.1.1 The cell transmission model (CTM)*


*ri*ð Þ¼ *k*

*rC*

**Table 2.**

min *li*ð Þþ *<sup>k</sup> di*ð Þ*<sup>k</sup>* , *<sup>ρ</sup>max*

*DOI: http://dx.doi.org/10.5772/intechopen.94332*

min *li*ð Þþ *<sup>k</sup> di*ð Þ*<sup>k</sup>* , *<sup>ρ</sup>max*

order to agree with the other notations of this section.

ð Þþ *<sup>k</sup> <sup>T</sup> Lmλ<sup>m</sup>*

*<sup>T</sup> <sup>ρ</sup><sup>m</sup>*,*i*þ<sup>1</sup>ð Þ� *<sup>k</sup> <sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ*<sup>k</sup>* � � *<sup>τ</sup>Lm <sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þþ *<sup>k</sup> <sup>χ</sup>* � �

*qm*,*<sup>i</sup>*

*<sup>V</sup> <sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ*<sup>k</sup>* � � <sup>¼</sup> *<sup>ν</sup> <sup>f</sup>*

Metering rate variables *r*<sup>C</sup>

maximum possible value.

*during interval [kT,(k+1)T).*

*2.1.2 The METANET model*

**Freeway Links**

ð Þ¼ *k* þ 1 *ρ<sup>m</sup>*,*i*,*<sup>j</sup>*

*<sup>ν</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ν</sup><sup>m</sup>*,*<sup>i</sup>*ð Þþ *<sup>k</sup> <sup>T</sup>*

� *ν*0

*ρ<sup>m</sup>*,*i*,*<sup>j</sup>*

**123**

original CTM is.

*<sup>i</sup>* � *<sup>ρ</sup>i*ð Þ*<sup>k</sup>* � � Uncontrolled On � Ramps

*METANET model variables and parameters of mainline link m, section i, node n, origin link o, destination j*

mentioned in detail in Section 2.2. All variables are bounded between zero and their

Many extensions of the original CTM have been proposed in the literature in the last two decades. The CTM in a mixed-integer linear form [7], the CTM including capacity drop phenomena [8, 9], the CTM for a freeway network [10], the asymmetric CTM [6], the link-node CTM [11], and the variable-length CTM [12] are some of these extended versions. Although these models have been proposed in different years and are suitable for different networks and applications, the original CTM [2] is the underlying model in all of them and it proves how powerful the

The METANET model presented here is an improved version [4] of the original that was first presented in [3]. However, the notation has been adopted from [5] in

*γ<sup>m</sup>*,*i*�1,*<sup>j</sup>*

*<sup>γ</sup><sup>m</sup>*,*i*,*<sup>j</sup>*ð Þ¼ *<sup>k</sup> <sup>ρ</sup><sup>m</sup>*,*i*,*<sup>j</sup>*

*<sup>τ</sup> <sup>V</sup> <sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ*<sup>k</sup>* � � � *<sup>ν</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ*<sup>k</sup>* � � <sup>þ</sup>

*<sup>m</sup>* exp � <sup>1</sup>

*am*

*j*∈*Jm*

*ρ<sup>m</sup>*,*i*,*<sup>j</sup>*

ð Þ*k*

*<sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ¼ *<sup>k</sup>* <sup>X</sup>

*C*, *max i* � � Controlled On � Ramps (

*<sup>i</sup>* ð Þ*k* come from the RM control law which will be

ð Þ*<sup>k</sup> qm*,*i*�<sup>1</sup>ð Þ� *<sup>k</sup> <sup>γ</sup><sup>m</sup>*,*i*,*<sup>j</sup>*

*T Lm*

ð Þ¼ *k ρ<sup>m</sup>*,*<sup>i</sup>*ð Þ*k ν<sup>m</sup>*,*<sup>i</sup>*ð Þ*k λ<sup>m</sup>* (12)

� �*am* � � (13)

*ρ<sup>m</sup>*,*<sup>i</sup>*ð Þ*k ρcr m*

h i (8)

ð Þ*k qm*,*<sup>i</sup>*

ð Þ*k* (9)

*ν<sup>m</sup>*,*<sup>i</sup>*ð Þ*k* ½ � *ν<sup>m</sup>*,*i*�<sup>1</sup>ð Þ� *k ν<sup>m</sup>*,*<sup>i</sup>*ð Þ*k*

(11)

*<sup>ρ</sup><sup>m</sup>*,*<sup>i</sup>*ð Þ*<sup>k</sup>* (10)

ð Þ*k*

(7)

*<sup>i</sup>* � *ρi*ð Þ*k* ,*r*

*qo*ð Þ*k* Total traffic volume leaving origin link *o* ½ � *veh=h*

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected…*

*<sup>o</sup>* ð Þ*k* Ramp metering control variable ½ � *veh=h Qn*,*<sup>j</sup>*ð Þ*k* Flow entering node *n* ½ � *veh=h β<sup>m</sup>*,*n*,*<sup>j</sup>*ð Þ*k* Split ratio ∈½ � 0, 1 *Models and Methods for Intelligent Highway Routing of Human-Driven and Connected… DOI: http://dx.doi.org/10.5772/intechopen.94332*


**Table 2.**

**Symbol Description Unit/Range**

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

*k* Time index *k* ¼ f g 0, … , *K* � 1

*m* Mainline link index *m* ¼ f g 1, … , *M*

*i* Section index *i* ¼ f g 1, … , *Nm*

*o* Origin link index *i* ¼ f g 1, … , *O*

*T* Sampling time ½ � *h K* Time horizon int

*M* Number of mainline links int

*Nm* Number of sections of mainline link *m* int

*O* Number of origin links int

*Lm* Length of each mainline link *m* ½ � *km λ<sup>m</sup>* Lane numbers of each mainline link *m* int *On* Set of exiting mainline links from node *n* - *In* Set of entering mainline links to node *n* - *In* Set of entering origin links to node *n* - *Jm* Set of destinations reachable from mainline link *m* - *Jo* Set of destinations reachable from origin link *o* - ̿*Jn* Set of destinations reachable from node *n* -

*<sup>m</sup>*,*<sup>i</sup>* Free-flow speed in section *<sup>i</sup>* of link *<sup>m</sup>* ½ � *km=<sup>h</sup>*

*<sup>m</sup>* Critical density ½ � *veh=km*

*<sup>m</sup>* Jam density ½ � *veh=km*

*<sup>o</sup>* Capacity of origin link *o* ½ � *veh=h τ* Model parameter *η* Model parameter *χ* Model parameter *ϕ* Model parameter *am* Model parameter *δon* Model parameter -

*ρ<sup>m</sup>*,*i*,*<sup>j</sup>*ð Þ*k* Partial density ½ � *veh=km ρ<sup>m</sup>*,*<sup>i</sup>*ð Þ *k* Total density ½ � *veh=km ν<sup>m</sup>*,*<sup>i</sup>*ð Þ*k* Mean traffic speed ½ � *km=h*

ð Þ *k* Traffic flow leaving section *i* of link *m* ½ � *veh=h γ<sup>m</sup>*,*i*,*<sup>j</sup>*ð Þ *k* Composition rate ∈½ � 0, 1 *do*,*<sup>j</sup>*ð Þ*k* Partial origin demand at origin link *o* ½ � *veh=h do*ð Þ*k* Total origin demand at origin link *o* ½ � *veh=h lo*,*<sup>j</sup>*ð Þ *k* Partial queue length at origin link *o* ½ � *veh lo*ð Þ *k* Total queue length at origin link *o* ½ � *veh γ<sup>o</sup>*,*<sup>j</sup>*ð Þ*k* Composition rate ∈½ � 0, 1 *θ<sup>o</sup>*,*<sup>j</sup>*ð Þ*k* Portion of demand originating in origin link *o* ∈½ � 0, 1

*ν f*

*ρcr*

*ρmax*

*qmax*

*qm*,*<sup>i</sup>*

**122**

*METANET model variables and parameters of mainline link m, section i, node n, origin link o, destination j during interval [kT,(k+1)T).*

$$r\_i(k) = \begin{cases} \min\left\{l\_i(k) + d\_i(k), \rho\_i^{\max} - \rho\_i(k)\right\} & \text{Uncontrolled On} - \text{Ramps} \\ \min\left\{l\_i(k) + d\_i(k), \rho\_i^{\max} - \rho\_i(k), r\_i^{C,\max}\right\} & \text{ControlledOn} - \text{Ramps} \end{cases} \tag{7}$$

Metering rate variables *r*<sup>C</sup> *<sup>i</sup>* ð Þ*k* come from the RM control law which will be mentioned in detail in Section 2.2. All variables are bounded between zero and their maximum possible value.

Many extensions of the original CTM have been proposed in the literature in the last two decades. The CTM in a mixed-integer linear form [7], the CTM including capacity drop phenomena [8, 9], the CTM for a freeway network [10], the asymmetric CTM [6], the link-node CTM [11], and the variable-length CTM [12] are some of these extended versions. Although these models have been proposed in different years and are suitable for different networks and applications, the original CTM [2] is the underlying model in all of them and it proves how powerful the original CTM is.

#### *2.1.2 The METANET model*

The METANET model presented here is an improved version [4] of the original that was first presented in [3]. However, the notation has been adopted from [5] in order to agree with the other notations of this section.

#### **Freeway Links**

$$\rho\_{m,i,j}(k+1) = \rho\_{m,i,j}(k) + \frac{T}{L\_m \lambda\_m} \left[ \chi\_{m,i-1,j}(k) q\_{m,i-1}(k) - \chi\_{m,i,j}(k) q\_{m,i}(k) \right] \tag{8}$$

$$\rho\_{m,i}(k) = \sum\_{j \in J\_m} \rho\_{m,i,j}(k) \tag{9}$$

$$\gamma\_{m,i,j}(k) = \frac{\rho\_{m,i,j}(k)}{\rho\_{m,i}(k)} \tag{10}$$

$$\begin{split} \nu\_{m,i}(k+1) &= \nu\_{m,i}(k) + \frac{T}{\tau} \left[ V(\rho\_{m,i}(k)) - \nu\_{m,i}(k) \right] + \frac{T}{L\_m} \nu\_{m,i}(k) \left[ \nu\_{m,i-1}(k) - \nu\_{m,i}(k) \right] \\ &- \frac{\nu' T \left[ \rho\_{m,i+1}(k) - \rho\_{m,i}(k) \right]}{\tau L\_m \left[ \rho\_{m,i}(k) + \chi \right]} \end{split}$$

(11)

$$q\_{m,i}(k) = \rho\_{m,i}(k)\nu\_{m,i}(k)\lambda\_m \tag{12}$$

$$V(\rho\_{m,i}(k)) = \nu\_m^f \exp\left[-\frac{1}{a\_m} \left(\frac{\rho\_{m,i}(k)}{\rho\_m^{cr}}\right)^{a\_m}\right] \tag{13}$$

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

The speed reduction caused by merging phenomena near on-ramps (possible additional term to Eq. (11)):

$$-\delta\_{on}T\frac{\nu\_{m,1}(k)q\_o(k)}{L\_m\lambda\_m\left[\rho\_{m,1}(k)+\chi\right]}\tag{14}$$

*qm*,0ð Þ¼ *<sup>k</sup>* <sup>X</sup>

*γm*,0,*<sup>j</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.94332*

they lead to more complex optimization problems.

traffic congestion from one location to another.

used to determine the control variables.

**125**

Section 2.3.

recent years.

**2.2 Ramp metering**

*j*∈*Jm*

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected…*

ð Þ¼ *<sup>k</sup> <sup>β</sup>m*,*n*,*<sup>j</sup>*

are calculated based on the RG control law. It will be mentioned in detail in

In presence of RG control, the splitting rates become the control variables and

Few research studies have developed different versions of the METANET model due to the complexity and non-linearity of the second-order models. However, the extensions of the METANET for a freeway network [13], and the multi-class METANET both for a freeway stretch [14] and for a freeway network [15] are examples of the extensions of this second-order traffic model developed in the

Both the first-order and second-order models are capable of developing the evolution of traffic flow in both urban and non-urban network. However, to highlight their difference, it is necessary to emphasize that first order models focus on the evolution of the density while the second-order traffic flow models, besides considering the dynamics of the traffic density, explicitly introduce a dynamic equation for the mean speed. Second order models have the distinct advantage over first order models that they can reproduce the capacity drop, which is the observed difference between the freeway capacity and the queue discharge rate. First order models, because they do not capture this phenomenon, are incapable of exploiting the benefits of increasing bottleneck flow. They can only reduce travel time by increasing off-ramp flow. The obvious disadvantage to second order models is that

The focus of the rest of this section will be on ramp metering and route guidance

Ramp metering is achieved by placing traffic signals at on-ramps to control the

Ramp metering control strategies can be classified in the following categories [16]: (1) local system where the control is applied to a single on-ramp, (2) coordinated system where the control is applied to a group of on-ramps, considering the traffic conditions of the whole network, (3) integrated system where a combination of ramp metering, signal timing, and route guidance is applied as the control system. Also, from another point of view, there are two types of RM control schemes [16]: (1) pre-timed or isolated where metering rates are fixed and predefined, (2) traffic-responsive control where real time freeway measurements are

control schemes as two of the most famous traffic management techniques.

flow rate at which vehicles enter the freeway. The ramp metering controller computes the metering rate to be applied. Ramp metering has various goals [16]: to improve or remove congestion, to alleviate freeway flow, traffic safety and air quality, to reduce total travel time and the number of peak-period accidents, to regulate the input demand of the freeway system so that a truly operationally balanced corridor system is achieved. Although the ramp metering provides many advantages, at the same time, it can have disadvantages too. The following are two of the most plausible ones [5]: (1) drivers may use parallel routes to avoid ramp meters which may lead to increased travel time and distance, (2) it can shift the

*βm*,*n*,*<sup>j</sup>*

ð Þ*k Qn*,*<sup>j</sup>*

ð Þ*k*

ð Þ*k Qn*,*<sup>j</sup>*

ð Þ*k* (25)

*qm*,0ð Þ*<sup>k</sup>* (26)

The speed reduction due to weaving phenomena in case of lane reductions in the mainstream (possible additional term to Eq. (11)):

$$-\phi T \Delta \lambda \frac{\nu\_{m,N\_m}(k)^2 \rho\_{m,N\_m}(k)}{L\_m \lambda\_m \rho\_m^{cr}} \tag{15}$$

The virtual downstream density at the end of the link (for node *n* at the end of link *m* with more than one outgoing link):

$$\rho\_{m,N\_m+1}(k) = \frac{\sum\_{\mu \in O\_\pi} \rho\_{\mu,1}(k)^2}{\sum\_{\mu \in O\_\pi} \rho\_{\mu,1}(k)}\tag{16}$$

The virtual upstream speed at the beginning of the link (for node *n* at the beginning of link *m* with more than one entering freeway link):

$$\nu\_{m,0}(k) = \frac{\sum\_{\mu \in I\_u} \nu\_{\mu, N\_\mu}(k) q\_{\mu, N\_\mu}(k)}{\sum\_{\mu \in I\_u} q\_{\mu, N\_\mu}(k)} \tag{17}$$

**Origin links**

$$l\_{oj}(k+1) = l\_{oj}(k) + T \left[ d\_{oj}(k) - \gamma\_{oj}(k) q\_o(k) \right] \tag{18}$$

$$l\_o(k) = \sum\_{j \in \mathcal{J}\_o} l\_{oj}(k) \tag{19}$$

$$\gamma\_{oj}(k) = \frac{l\_{oj}(k)}{l\_o(k)}\tag{20}$$

$$d\_{o,j}(k) = \theta\_{o,j}(k)d\_o(k)\tag{21}$$

For uncontrolled on-ramps:

$$q\_o(k) = \min\left\{ d\_o(k) + \frac{l\_o(k)}{T}, q\_o^{\max}, q\_o^{\max} \frac{\rho\_m^{\max} - \rho\_{m,1}(k)}{\rho\_m^{\max} - \rho\_m^{cr}} \right\} \tag{22}$$

For controlled on-ramps:

$$q\_o(k) = \min\left\{ d\_o(k) + \frac{l\_o(k)}{T}, q\_o^{\max}, r\_o^{\mathcal{C}}(k), q\_o^{\max} \frac{\rho\_m^{\max} - \rho\_{m,1}(k)}{\rho\_m^{\max} - \rho\_m^{cr}} \right\} \tag{23}$$

where *r<sup>C</sup> <sup>o</sup>* ð Þ*k* come from the RM control law which will be mentioned in detail in Section 2.2.

**Nodes**

$$Q\_{n,j}(k) = \sum\_{\mu \in I\_n} q\_{\mu, N\_{\mu}}(k) \chi\_{\mu, N\_{\mu j}}(k) + \sum\_{o \in \overline{I}\_n} q\_o(k) \chi\_{oj}(k) \tag{24}$$

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected… DOI: http://dx.doi.org/10.5772/intechopen.94332*

$$q\_{m,0}(k) = \sum\_{j \in J\_m} \beta\_{m,n,j}(k) Q\_{n,j}(k) \tag{25}$$

$$\gamma\_{m,0,j}(k) = \frac{\beta\_{m,n,j}(k)Q\_{n,j}(k)}{q\_{m,0}(k)}\tag{26}$$

In presence of RG control, the splitting rates become the control variables and are calculated based on the RG control law. It will be mentioned in detail in Section 2.3.

Few research studies have developed different versions of the METANET model due to the complexity and non-linearity of the second-order models. However, the extensions of the METANET for a freeway network [13], and the multi-class METANET both for a freeway stretch [14] and for a freeway network [15] are examples of the extensions of this second-order traffic model developed in the recent years.

Both the first-order and second-order models are capable of developing the evolution of traffic flow in both urban and non-urban network. However, to highlight their difference, it is necessary to emphasize that first order models focus on the evolution of the density while the second-order traffic flow models, besides considering the dynamics of the traffic density, explicitly introduce a dynamic equation for the mean speed. Second order models have the distinct advantage over first order models that they can reproduce the capacity drop, which is the observed difference between the freeway capacity and the queue discharge rate. First order models, because they do not capture this phenomenon, are incapable of exploiting the benefits of increasing bottleneck flow. They can only reduce travel time by increasing off-ramp flow. The obvious disadvantage to second order models is that they lead to more complex optimization problems.

The focus of the rest of this section will be on ramp metering and route guidance control schemes as two of the most famous traffic management techniques.

#### **2.2 Ramp metering**

The speed reduction caused by merging phenomena near on-ramps (possible

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

�*δonT <sup>ν</sup>m*,1ð Þ*<sup>k</sup> qo*ð Þ*<sup>k</sup>*

The speed reduction due to weaving phenomena in case of lane reductions in the

*Lmλmρcr m*

*ρm*,*Nm* ð Þ*k*

*ρμ*,1ð Þ*<sup>k</sup>* <sup>2</sup>

ð Þ*k*

h i

*νμ*,*N<sup>μ</sup>* ð Þ*k qμ*,*N<sup>μ</sup>*

*<sup>ν</sup>m*,*Nm* ð Þ*<sup>k</sup>* <sup>2</sup>

The virtual downstream density at the end of the link (for node *n* at the end of

P *μ* ∈ *On*

P *μ* ∈ *On*

The virtual upstream speed at the beginning of the link (for node *n* at the

P *μ*∈*In qμ*,*N<sup>μ</sup>*

*lo*,*<sup>j</sup>*ð Þ¼ *k* þ 1 *lo*,*<sup>j</sup>*ð Þþ *k T do*,*<sup>j</sup>*ð Þ� *k γ<sup>o</sup>*,*<sup>j</sup>*ð Þ*k qo*ð Þ*k*

*j* ∈*Jo*

ð Þ¼ *<sup>k</sup> lo*,*<sup>j</sup>*ð Þ*<sup>k</sup>*

*<sup>T</sup>* , *<sup>q</sup>max*

� �

*<sup>o</sup>* ð Þ*k* come from the RM control law which will be mentioned in detail in

ð Þþ *<sup>k</sup>* <sup>X</sup> *o* ∈*In*

*<sup>o</sup>* , *qmax o*

� �

*lo*ð Þ¼ *<sup>k</sup>* <sup>X</sup>

*γo*,*j*

*<sup>T</sup>* , *qmax <sup>o</sup>* ,*r C <sup>o</sup>* ð Þ*<sup>k</sup>* , *<sup>q</sup>max o*

*qo*ð Þ¼ *<sup>k</sup>* min *do*ð Þþ *<sup>k</sup> lo*ð Þ*<sup>k</sup>*

P *μ* ∈*In*

*Lmλ<sup>m</sup> <sup>ρ</sup>m*,1ð Þþ *<sup>k</sup> <sup>χ</sup>* � � (14)

*ρμ*,1ð Þ*<sup>k</sup>* (16)

ð Þ*<sup>k</sup>* (17)

*lo*,*<sup>j</sup>*ð Þ*k* (19)

*lo*ð Þ*<sup>k</sup>* (20)

*<sup>m</sup>* � *ρ<sup>m</sup>*,1ð Þ*k ρmax <sup>m</sup>* � *ρcr m*

*<sup>m</sup>* � *ρ<sup>m</sup>*,1ð Þ*k ρmax <sup>m</sup>* � *ρcr m*

*qo*ð Þ*k γ<sup>o</sup>*,*<sup>j</sup>*

*do*,*<sup>j</sup>*ð Þ¼ *k θ<sup>o</sup>*,*<sup>j</sup>*ð Þ*k do*ð Þ*k* (21)

*ρmax*

*ρmax*

(15)

(18)

(22)

(23)

ð Þ*k* (24)

additional term to Eq. (11)):

**Origin links**

For uncontrolled on-ramps:

For controlled on-ramps:

where *r<sup>C</sup>*

Section 2.2. **Nodes**

**124**

*qo*ð Þ¼ *<sup>k</sup>* min *do*ð Þþ *<sup>k</sup> lo*ð Þ*<sup>k</sup>*

*Qn*,*<sup>j</sup>*ð Þ¼ *<sup>k</sup>* <sup>X</sup>

*μ* ∈*In*

*q<sup>μ</sup>*,*N<sup>μ</sup>*

ð Þ*k γμ*,*Nμ*,*<sup>j</sup>*

mainstream (possible additional term to Eq. (11)):

link *m* with more than one outgoing link):

�*ϕT*Δ*λ*

*ρ<sup>m</sup>*,*Nm*þ<sup>1</sup>ð Þ¼ *k*

beginning of link *m* with more than one entering freeway link):

*ν<sup>m</sup>*,0ð Þ¼ *k*

Ramp metering is achieved by placing traffic signals at on-ramps to control the flow rate at which vehicles enter the freeway. The ramp metering controller computes the metering rate to be applied. Ramp metering has various goals [16]: to improve or remove congestion, to alleviate freeway flow, traffic safety and air quality, to reduce total travel time and the number of peak-period accidents, to regulate the input demand of the freeway system so that a truly operationally balanced corridor system is achieved. Although the ramp metering provides many advantages, at the same time, it can have disadvantages too. The following are two of the most plausible ones [5]: (1) drivers may use parallel routes to avoid ramp meters which may lead to increased travel time and distance, (2) it can shift the traffic congestion from one location to another.

Ramp metering control strategies can be classified in the following categories [16]: (1) local system where the control is applied to a single on-ramp, (2) coordinated system where the control is applied to a group of on-ramps, considering the traffic conditions of the whole network, (3) integrated system where a combination of ramp metering, signal timing, and route guidance is applied as the control system. Also, from another point of view, there are two types of RM control schemes [16]: (1) pre-timed or isolated where metering rates are fixed and predefined, (2) traffic-responsive control where real time freeway measurements are used to determine the control variables.

where *ρdown*

*2.2.2 PI-ALINEA*

C

*<sup>i</sup>* ð Þ� *<sup>k</sup>* � <sup>1</sup> *KP <sup>ρ</sup>down*

*DOI: http://dx.doi.org/10.5772/intechopen.94332*

where *KP* is another regulator parameter.

the cost is higher that on the utilized routes.

may be different from the predicted travel time [5].

, where it is easily computed: *β*<sup>C</sup>

*β*C *m*,*n*,*j*

destination *j* is computed as

tions. In particular, the control variable is the splitting rate *β*<sup>C</sup>

choose link *m* to reach destination *j*. The other control variable is *β*<sup>C</sup>

ð Þ¼ *<sup>k</sup> <sup>β</sup><sup>N</sup>*

*r* C *<sup>i</sup>* ð Þ¼ *k r*

ALINEA.

link *m*<sup>0</sup>

**127**

**2.3 Route guidance**

*<sup>i</sup>* ð Þ*<sup>k</sup>* is the density measured downstream the on-ramp, *<sup>ρ</sup>* <sup>∗</sup>

A very famous extension of ALINEA is the PI-ALINEA, in which a proportional

*<sup>i</sup>* ð Þ *<sup>k</sup>* � <sup>1</sup> <sup>þ</sup> *KR <sup>ρ</sup>* <sup>∗</sup>

Based on the stability analysis of the closed-loop ramp metering system provided in [20], it can be stated that PI-ALINEA is able to show a better performance than

Route guidance (RG) is an efficient technique to distribute the traffic demand over the network, by providing information about alternative paths to drivers. The variable message signs (VMSs) are one of the main actuators which can provide route information to drivers in the RG control scheme. In the RG control, the concepts of equilibrium play an important role. Wardrop has offered the two following principles [21]: (1) The *system optimum (SO)* is achieved when the vehicles are guided such that the total costs of all drivers (typically the TTS) is minimized, (2) The traffic network is in *user equilibrium (UE)* when the costs on each utilized alternative route is equal and minimal, and on routes that are not utilized,

If the goal of a control strategy is defined as the travel time, it is typically defined as the *predicted travel time* or as the *instantaneous travel time*. The predicted travel time is the time that the driver will experience when he drives along the given route, while the instantaneous travel time is the travel time determined based on the current speeds on the route. In a dynamic setting, the instantaneous travel time

In route guidance control strategies, the control variable is the splitting rate at a given node. Considering the simple case of only two alternative paths [22], originating from node *n*, let us denote with *m* and *m*<sup>0</sup> the two links exiting node *n*, corresponding respectively to the primary and secondary path. The primary path is the one characterised by the shortest travel time, in case of regular traffic condi-

representing the portion of flow present in node *n* at time instant *kT* which should

,*n*,*<sup>j</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>C</sup>

*m*,*n*,*j*

*m*0

regulators of P-type or PI-type are the most used strategies for route guidance systems in the literature [5, 23]. According to a proportional control law, the portion of flow present in node *n* at time instant *kT* which should choose link *m* to reach

*m*,*n*,*j*

point value for the downstream density, and *KR* is the integral gain. Note that, in case the main objective of the traffic controller is to reduce congestion and to

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected…*

maximise the throughput, a good choice for the set-point is *ρ* <sup>∗</sup>

term is added to result in a PI regulator. The metering rate is given by

*<sup>i</sup>* ð Þ� *<sup>k</sup> <sup>ρ</sup>down*

*<sup>i</sup>* is a set-

*<sup>i</sup>* <sup>¼</sup> *<sup>ρ</sup>cr i* .

*<sup>i</sup>* � *<sup>ρ</sup>down*

*<sup>m</sup>*,*n*,*<sup>j</sup>* ∈ ½ � 0, 1 ,

*m*0 ,*n*,*j*

. The following feedback

ð Þþ *k KP*Δ*τ<sup>n</sup>*,*<sup>j</sup>*ð Þ*k* (29)

, referred to

*<sup>i</sup>* ð Þ*<sup>k</sup>* (28)

**Figure 1.** *Ramp metering algorithms classification.*

A classification of ramp metering algorithms based on a study by Papageorgiou and Kotsialos [17] is presented in **Figure 1**. Fixed time metering is the simplest strategy which is usually adjusted based on historical data and applied during particular times of day. Reactive ramp metering techniques are based on real time traffic metrics. Local ramp metering uses traffic measures collected form the ramp vicinity. Demand-capacity, and occupancy-based strategies allow as much traffic inflow as possible to reach the freeway capacity. ALINEA and PI-AlINEA offer a more complex and more responsive strategy that, unlike capacity and occupancy strategies, generates smoother responses towards changes in metrics. Multi-variable regulator strategies perform the same as local strategies, but more comprehensively and independently on a set of ramps and usually outperform local strategies. METALINE can be viewed as a more general and extended form of ALINEA. Nonlinear optimal control strategy considers local traffic parameters and metrics as well as nonlinear traffic flow dynamics, incidents, and demand predictions in a freeway network and outputs a consistent control strategy. Knowledge-based control systems are developed based on historical data and human expertise. Integrated freeway network traffic control is a more general approach to nonlinear control that extends application of optimal control strategies to all forms of freeway traffic control. In case of knowledge based systems, inability to learn and adapt to temporal evolution of the system being controlled can be an issue, so knowledge based systems need to be periodically updated to remain efficient. Artificial intelligence and machine learning approaches like reinforcement learning (RL) and artificial neural networks (ANN) are new techniques being implemented recently for RM control [18].

Two of the most common ramp metering strategies are described in the following. Here, the flow that can enter section *i* of a freeway from the on-ramp during time interval ½ Þ *kT*,ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* is shown by *<sup>r</sup>*<sup>C</sup> *<sup>i</sup>* ð Þ*k* where *k* is the time index and *T* is the sampling time.

#### *2.2.1 ALINEA*

ALINEA [19] is an I-type controller in which the metering rate is given by

$$r\_i^{\mathbb{C}}(k) = r\_i^{\mathbb{C}}(k-1) - K\_R \left[ \rho\_i^\* - \rho\_i^{down}(k) \right] \tag{27}$$

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected… DOI: http://dx.doi.org/10.5772/intechopen.94332*

where *ρdown <sup>i</sup>* ð Þ*<sup>k</sup>* is the density measured downstream the on-ramp, *<sup>ρ</sup>* <sup>∗</sup> *<sup>i</sup>* is a setpoint value for the downstream density, and *KR* is the integral gain. Note that, in case the main objective of the traffic controller is to reduce congestion and to maximise the throughput, a good choice for the set-point is *ρ* <sup>∗</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>ρ</sup>cr i* .

### *2.2.2 PI-ALINEA*

A very famous extension of ALINEA is the PI-ALINEA, in which a proportional term is added to result in a PI regulator. The metering rate is given by

$$r\_i^{\mathbb{C}}(k) = r\_i^{\mathbb{C}}(k-1) - K\_P \left[ \rho\_i^{down}(k) - \rho\_i^{down}(k-1) \right] + K\_R \left[ \rho\_i^{\*} - \rho\_i^{down}(k) \right] \tag{28}$$

where *KP* is another regulator parameter.

Based on the stability analysis of the closed-loop ramp metering system provided in [20], it can be stated that PI-ALINEA is able to show a better performance than ALINEA.

#### **2.3 Route guidance**

A classification of ramp metering algorithms based on a study by Papageorgiou and Kotsialos [17] is presented in **Figure 1**. Fixed time metering is the simplest strategy which is usually adjusted based on historical data and applied during particular times of day. Reactive ramp metering techniques are based on real time traffic metrics. Local ramp metering uses traffic measures collected form the ramp vicinity. Demand-capacity, and occupancy-based strategies allow as much traffic inflow as possible to reach the freeway capacity. ALINEA and PI-AlINEA offer a more complex and more responsive strategy that, unlike capacity and occupancy strategies, generates smoother responses towards changes in metrics. Multi-variable regulator strategies perform the same as local strategies, but more comprehensively and independently on a set of ramps and usually outperform local strategies. METALINE can be viewed as a more general and extended form of ALINEA. Nonlinear optimal control strategy considers local traffic parameters and metrics as well as nonlinear traffic flow dynamics, incidents, and demand predictions in a freeway network and outputs a consistent control strategy. Knowledge-based control systems are developed based on historical data and human expertise. Integrated freeway network traffic control is a more general approach to nonlinear control that extends application of optimal control strategies to all forms of freeway traffic control. In case of knowledge based systems, inability to learn and adapt to temporal evolution of the system being controlled can be an issue, so knowledge based systems need to be periodically updated to remain efficient. Artificial intelligence and machine learning approaches like reinforcement learning (RL) and artificial neural networks (ANN) are new techniques being implemented recently for RM control [18]. Two of the most common ramp metering strategies are described in the following. Here, the flow that can enter section *i* of a freeway from the on-ramp

*Models and Technologies for Smart, Sustainable and Safe Transportation Systems*

*<sup>i</sup>* ð Þ*k* where *k* is the time index

*<sup>i</sup>* ð Þ*<sup>k</sup>* (27)

during time interval ½ Þ *kT*,ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* is shown by *<sup>r</sup>*<sup>C</sup>

*r* C *<sup>i</sup>* ð Þ¼ *k r*

ALINEA [19] is an I-type controller in which the metering rate is given by

*<sup>i</sup>* ð Þ� *<sup>k</sup>* � <sup>1</sup> *KR <sup>ρ</sup>* <sup>∗</sup>

*<sup>i</sup>* � *<sup>ρ</sup>down*

C

and *T* is the sampling time.

*2.2.1 ALINEA*

**126**

**Figure 1.**

*Ramp metering algorithms classification.*

Route guidance (RG) is an efficient technique to distribute the traffic demand over the network, by providing information about alternative paths to drivers. The variable message signs (VMSs) are one of the main actuators which can provide route information to drivers in the RG control scheme. In the RG control, the concepts of equilibrium play an important role. Wardrop has offered the two following principles [21]: (1) The *system optimum (SO)* is achieved when the vehicles are guided such that the total costs of all drivers (typically the TTS) is minimized, (2) The traffic network is in *user equilibrium (UE)* when the costs on each utilized alternative route is equal and minimal, and on routes that are not utilized, the cost is higher that on the utilized routes.

If the goal of a control strategy is defined as the travel time, it is typically defined as the *predicted travel time* or as the *instantaneous travel time*. The predicted travel time is the time that the driver will experience when he drives along the given route, while the instantaneous travel time is the travel time determined based on the current speeds on the route. In a dynamic setting, the instantaneous travel time may be different from the predicted travel time [5].

In route guidance control strategies, the control variable is the splitting rate at a given node. Considering the simple case of only two alternative paths [22], originating from node *n*, let us denote with *m* and *m*<sup>0</sup> the two links exiting node *n*, corresponding respectively to the primary and secondary path. The primary path is the one characterised by the shortest travel time, in case of regular traffic conditions. In particular, the control variable is the splitting rate *β*<sup>C</sup> *<sup>m</sup>*,*n*,*<sup>j</sup>* ∈ ½ � 0, 1 , representing the portion of flow present in node *n* at time instant *kT* which should choose link *m* to reach destination *j*. The other control variable is *β*<sup>C</sup> *m*0 ,*n*,*j* , referred to link *m*<sup>0</sup> , where it is easily computed: *β*<sup>C</sup> *m*0 ,*n*,*<sup>j</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>C</sup> *m*,*n*,*j* . The following feedback regulators of P-type or PI-type are the most used strategies for route guidance systems in the literature [5, 23]. According to a proportional control law, the portion of flow present in node *n* at time instant *kT* which should choose link *m* to reach destination *j* is computed as

$$
\boldsymbol{\beta}^{\mathbb{C}}\_{m,n,j}(\boldsymbol{k}) = \boldsymbol{\beta}^{N}\_{m,n,j}(\boldsymbol{k}) + K\_P \Delta \boldsymbol{\tau}\_{n,j}(\boldsymbol{k}) \tag{29}
$$

where *β<sup>N</sup> m*,*n*,*j* ð Þ*k* is the nominal splitting rate, *KP* is a gain, Δ*τn*,*j*ð Þ*k* is the instantaneous travel time difference between the secondary and primary direction from *n* to *j*. In proportional-integral regulators, the splitting rate is

$$\boldsymbol{\beta}\_{m,n,j}^{\mathbb{C}}(k) = \boldsymbol{\beta}\_{m,n,j}^{\mathbb{C}}(k-1) + K\_P \left[ \Delta \tau\_{n,j}(k) - \Delta \tau\_{n,j}(k-1) \right] + K\_I \Delta \tau\_{n,j}(k) \tag{30}$$

system measures of the network would be maximized, while the drivers would choose the path provided by the RG control. They discussed their mathematical

*Models and Methods for Intelligent Highway Routing of Human-Driven and Connected…*

In 1999 and then with some modifications in 2002, Apostolos Kotsialos et al. in [28, 29], considered the design of an integrated traffic control system for motorway networks with the use of ramp metering, motorway-to-motorway control, and route guidance. They offered a generic problem formulation in the format of a discrete time optimal control problem. They assumed that both RM control measures and RG are available. The METANET model was used for the description of traffic flow. A hypothetical test network was considered to evaluate the performance of the proposed control system. The control measure considered was the minimisation of the travel time spent (TTS). The results showed the high efficiency

In 2004, Karimi et al. [30] considered the integration of dynamic RG and RM based on MPC. They used the dynamic route guidance panels (DRIPs) as both a control tool and an information provider to the drivers, and ramp metering as a control tool to spread the congestion over the network. This resulted in a control strategy that reduced the total time spent by optimally re-routing traffic over the available alternative routes in the network, and also kept the difference between the travel times shown on the DRIPs and the travel times actually realized by the drivers as small as possible. The simulations done for the case study showed that rerouting of traffic and on-ramp metering using MPC has lead to a significant

In 2015, Yu Han et al. [31] proposed an extended version of the CTM first proposed in [2] with the ability to reproduce the capacity drop at both the on-ramp bottleneck and the lane drop bottleneck. Based on this model, a linear quadratic model predictive control strategy for the integration of dynamic RG and RM was offered with the objective of minimizing the TTS of a traffic network. In this paper, a RG model based on the perceived travel time of each route by drivers was also offered. If the instantaneous travel time of each route is provided to travelers, the perceived travel time is assumed to be the same as the instantaneous travel time. If not, the perceived travel time is assumed to be the free flow travel time. The splitting rates at a bifurcation are determined by the well-known Logit model [26, 30]. A test case network containing both on-ramp bottlenecks and lane drop bottlenecks was used to investigate the effectiveness of the proposed framework and the results showed the improvement the proposed control strategy brought for

In 2017, Cecilia Pasquale et al. [22] offered a multi-class control scheme for freeway traffic networks with the integration of RM and RG in order to reduce the TTS and the total emissions in a balanced way. Their two controllers were feedback predictive controllers and it was shown how this choice for their controllers can benefit the performance of the controllers. They applied the multi-class METANET model and the multi-class macroscopic VERSIT+ model for prediction of the traffic dynamics in the network. In addition, they designed a controller gain selector to compute the gains of the RM and RG controllers. The simulation results showed significant improvements of the freeway network performance, in terms of

In 2018, Hirsh Majid et al. [32] designed integrated traffic control strategies for highway networks with the use of RG and RM. The highway network was simulated using the LWR model. A control algorithm was designed to solve the proposed problem, based on the inverse control technique and variable structure control (super twisting sliding mode). Three case studies were tested in the presence of an on-ramp at each alternate route and where there was a capacity constraint in the

formulation in detail and provided the solution finding algorithm.

of the proposed control system.

*DOI: http://dx.doi.org/10.5772/intechopen.94332*

improvement in performance.

the network performance.

**129**

reduction of the TTS and the total emissions.

where *KP* and *KI* are other controller gains.

Another possible class of RG strategies is *iterative strategies*, where the splitting rate is computed by iteratively running different simulations in real time with different RG, in order to achieve conditions of either user equilibrium or system optimum [24, 25]. Iterative strategies are very beneficial, however, their high computational effort is a major drawback for this category of RG control techniques.

It is also interesting to describe how drivers react to travel time information and how they adapt their route choice. A well-known behavior model used for this purpose is the logit model [26], which is used to model all kinds of consumer behavior based on the cost of several alternatives. The lower the cost of an alternative, the more consumers will choose that alternative. In the case of traffic management, consumers are the drivers, and the cost is the comfort, safety, or travel time of the alternative routes to reach the desired destination. The logit model calculates the probability that a driver chooses one of more alternatives based on the difference in travel time between the alternatives. Assume that we have two possible choices *m*<sup>1</sup> and *m*<sup>2</sup> at node *n* to get to destination *j*. For the calculation of the split rates out of the travel time difference between two alternatives, the logit model results in:

$$\beta\_{m,n,j}(k) = \frac{\exp\left(\sigma \theta\_{n,m,j}(k)\right)}{\exp\left(\sigma \theta\_{n,m,j}(k)\right) + \exp\left(\sigma \theta\_{n,m,j}(k)\right)}\tag{31}$$

for *m* ¼ *m*<sup>1</sup> or *m* ¼ *m*2, where *θ<sup>n</sup>*,*m*1,*<sup>j</sup>*ð Þ*k* is the travel time shown on the DRIP at node *n* to travel to destination *j* via link *m*. The parameter *σ* describes how drivers react on a travel time difference between two alternatives. The higher *σ*, the less travel time difference is needed to convince drivers to choose the fastest alternative route.

#### **2.4 Integration of ramp metering and route guidance**

The RM and RG controllers are both feedback and predictive controllers as they apply not only on the real-time measurements of the system to calculate the control actions, but they also use information about the prediction of the system evolution. The combination of RM and RG controllers has shown promising results in network performance from the point of view of different performance measures. The focus of this section is on providing a review on the related studies on the combination of these two main traffic management techniques.

In 1990, a study performed by Iida et al. [27] considered the development of an improved on-ramp traffic control technique of urban expressway. They extended the conventional LP control method to consider the multiple paths between onramps and off-ramps of the test case network and also the route choice behavior of drivers. They assumed that in the future, the drivers would have the travel information offered by the route guidance system. Their formulation combined the user equilibrium with the available LP traffic control formulation at the time. In their problem statement, the goal was to determine the optimal metering rate so that the
