I Introduction
We study how to alleviate highway traffic congestions by encouraging plugin hybrid and electric vehicles to stop at charging stations around peak congestion time. In the first part of our work (Part I: Theory) [cenedese:2020:highway_control_pI], motivated by the rising number of PEV and inspired by the conventional ramp metering control, we have proposed a novel ATDM strategy based on soft measures to alleviate traffic congestion during rush hours. Specifically, we have designed a pricing policy to make the charging price dynamic and dependent on the (predicted) traffic congestion level, encouraging the PEV owners to stop for charging when the congestion level is (going to be) high. To achieve this goal, we have developed a novel framework that models how the proposed policy affects the drivers’ decisions as a MIGPG. From a technological point of view, we have introduced the concept of “roadtostation” r2s and “stationtoroad” s2r traffic flows, and shown that the selfish behavior of the drivers leads to decision strategies that are individually optimal in the sense of Nash.
In this paper (Part II: Case Study), the proposed ATDM strategy is validated in simulation on the A highway stretch between The Hague and Rotterdam, in The Netherlands. More precisely, the main contribution of this paper is to show by simulation that the adoption of a dynamic charging price policy depending on the traffic congestion can alleviate highway traffic congestion by encouraging plugin electric and hybrid vehicles to stop at one CS around peak congestion time. To the best of the authors’ knowledge, this is the first study that proposes the r2s and s2r technology to develop an ATDM. Furthermore, to support the reliability of our simulation results, we use real traffic data to identify the parameters of the corresponding CTM, which predicts the evolution of the traffic congestion [cenedese:2020:highway_control_pI, Sec. II]. Finally, to show the general validity of the proposed approach and to study its performance in different scenarios, we perform an extensive sensitivity analysis on the system parameters, which allows us to identify the “best” setup that maximizes the performance of the proposed strategy, based on empirical results.
Structure of the paper: In Section II, we recall the most important components of the ATDM strategy developed in the first part of this work (Part I: Theory) [cenedese:2020:highway_control_pI] and the associated notation. In Section III, we use real data to identify the parameters of the CTM and discuss the choice of the parameters adopted in the decision making process. The first simulation is analyzed in Section V, where the parameters reflect the current market scenario. For the most relevant decision parameters, a sensitivity analysis is carried out in Section VI. Finally, Sections VII and VIII, respectively, discuss the policies that emerge from the simulation results as the most effective and the conclusions that summarize the strengths of our approach together with promising future developments.
Ii The ATDM strategy proposed in Part I
In this section, we briefly review the ATDM strategy we have proposed in the first part of this work (Part I: Theory) [cenedese:2020:highway_control_pI], including the problem formulation, theoretical background and notations (see Table I).
The traffic evolution is modeled via the CTM that is a simple, yet sufficently accurate, first order model for traffic evolution on a highway [rinaldi:ferrara:2012:CTM_identification]. We consider a highway stretch in which there are no entering or exiting ramps, divided in cells of length where . Each cell has an entering flow of vehicles and exiting one . We assume that the PEV leaving cell can choose to stop at the unique CS, that we assume to be located between cells and (see [cenedese:2020:highway_control_pI, Fig. 1]). The flows of PEV entering and exiting the CS are r2s and s2r respectively. The relation between a cell’s flow and its density describes the traffic evolution as defined in [cenedese:2020:highway_control_pI, Eq.1–4].
We consider that the HO regulates the price of the electricity purchased by the PEV at the CS. This price is dynamic and subject to a discount that is proportional to the level of the actual (or predicted) traffic congestion:
(1) 
where and is the cell’s extra travel time due to the traffic congestion.
CTM cell  
flow entering the cell  
r2s  flow entering the CS  
s2r  flow exiting the CS  
traffic density  
extra travel time due to congestion  
predicted future extra travel time  
due to congestion  
cell length  
freeflow velocity  
congestion wave speed  
maximum cell capacity  
maximum jam density  
length of the time interval  
Decision making process of agent  
–  battery SoC of PEV  
energy purchased by PEV  
–  binary var. iff PEV is charging  
total energy purchased by PEV  
€  price for at CS  
€  predicted future price at CS  
interval in which PEV enters cell  
–  rectangular function centered in  
(over)estimation of the extra travel time 

due to congestion  
estimation of the extra travel time  
due to PEV exiting the CS  
–  min. SoC of PEV before leaving the CS  
–  percentage of PEV in the market  
length of the time interval  
–  number of intervals composing the horizon  
battery capacity  
scaling associated to battery efficiency  
–  interest of PEV in saving money  
–  number of charging spots at the CS  
CS max. energy providable in an interval  
upper(lower) bound on purchasable energy  
base energy demand in the grid  
€  price scaling factors  
€  price scaling factor  
€  fixed avg. price for not at CS  
€  estimated price scaling factor  
€  estimated price scaling factor  
–  half width of  
scaling used to compute in  
timevarying scaling of the travel time  
–  weight over the time spent at the CS 
The PEV exiting cell face a choice, namely, whether or not it is convenient to stop at the CS given their SoC , the available plugs and the current/predicted traffic situation. Sharing the facility and the dependency of in (1) on make the choice of the optimal charging strategy a challenging problem that has to be solved locally by each PEV. Each driver aims at finding the best tradeoff between the money saved by exploiting the electricity price discount and the extra travel time due to the stop at the CS. This is modeled via the following twoterm cost function:
(2) 
which each PEV (or agent) , with being the set of all the agents involved in the game, desires to minimize. The coefficient
is the realization of a Gaussian random variable, with mean
and variance
. It represents the interest of agent in saving money rather than time. The first term in (2) describes the total savings, i.e.,(3) 
where is the prediction horizon. The predicted price that the HO applies during is computed by the following approximation:
(4) 
To characterize the interval during which agent enters cell , we introduced in [cenedese:2020:highway_control_pI, Sec. III.A.3] the rectangular function that takes nonzero values during the intervals preceding and succeeding . The second term in (2) models the extra travel time that PEV experiences for stopping at the CS and/or for the traffic situation, i.e.,
(5) 
If agent enters cell during , describes the estimated remaining travel time to complete the transit through cells . The value of is computed by exploiting the CTM as an oracle to predict the evolution of the traffic congestion (see [cenedese:2020:highway_control_pI, Sec. III.A.3] for further details).
The resources available at the CS are finite and must be shared among the PEV. This creates a coupling between the agents’ decisions, both in the cost (5) and in the constraints. Next, we qualitatively list the most important constraints that PEV has to meet during the choice of the optimal feasible charging schedule. For clarity we refer the reader to the corresponding formal definitions in [cenedese:2020:highway_control_pI, Sec. III.B]:

Charging only if is at the CS, [cenedese:2020:highway_control_pI, Eq. 16],

exit the CS if stops charging, [cenedese:2020:highway_control_pI, Eq. 17],

if stops, it has to remain at the CS for at least intervals, [cenedese:2020:highway_control_pI, Eq. 19],

exit the CS only if , [cenedese:2020:highway_control_pI, Eq. 20],

total energy purchased by the PEV has to be lower than , [cenedese:2020:highway_control_pI, Eq. 21],

the number of PEV simultaneously charging must be lower than , [cenedese:2020:highway_control_pI, Eq. 21].
All the constraints are cast in an affine form via auxiliary variables and described in a compact form by means of the matrices and . Organizing all the decision variables associated to each PEV
in a vector
(and ) allows us to cast the charging problem above as an exact MIGPG, defined via the following set of interdependent optimization problems(6) 
where is the timedependent set of feasible decisions of each agent , [cenedese:2020:highway_control_pI, Eq. 24]. Algorithm in [cenedese:2020:highway_control_pI] is based on the sequential mixedinteger best response. It ensures convergence to an MINE, since (6) is an exact potential game [cenedese:2020:highway_control_pI, Th. 1] . The remainder of the paper is devoted to the implementation of the ATDM strategy proposed in (Part I: Theory) [cenedese:2020:highway_control_pI], and to show via numerical simulations its beneficial effects on traffic alleviation. Moreover, we analyze how different configurations of the parameters in Table I affect the travel time.
Iii CTM identification
Let us introduce the examined case study and identify the parameters of the associated CTM starting from realworld data and following the schematic procedure depicted in Figure 2. We consider a stretch of the A highway in The Netherlands connecting the cities of The Hague and Rotterdam. The Dutch government embraces the policy of making traffic data accessible via the NDW [ndw:portal]. In particular, the NDWDexter portal [ndw:dexter] allows us to access the sensors placed along the highway and extract the traffic data related to a particular date and time. Here, we select sensors, denoted in the following by , on the roadway that from The Hague enters Rotterdam (Figure 1).
The sensors are placed approximately at from each other and measure the number of vehicles and their average speed during a period of minute. We consider the data of the from : to :, since there is no traffic congestion during the remaining hours.
Remark 1 (Minimum sample frequency)
The frequency of the data samples has to be high enough to allow the CTM to evolve correctly. In fact, if for some the inequality
is not satisfied, then the associated cell’s density can take negative values. This is a consequence of the density definition in [cenedese:2020:highway_control_pI, Eq. 1]
and the lower bound for the sampling frequency is obtained from the worstcase scenario. One can solve this issue by considering longer cells or by interpolating the data obtained from the sensors to decrease the sampling time. For the problem at hand, we show that the free flow speed is approximately
, therefore we interpolated the data to obtain a sampling time of , so .We construct a CTM composed of cells, in which the flows between cells are obtained directly from the raw data provided by the sensors, i.e., for all
The the average speed of the vehicles is also measured by each sensor, thus for each cell the density is
The relation between the flow and the density is represented for cell in the fundamental diagram depicted in Figure (a)a, where we used a threshold on the average vehicles’ velocity to distinguish between the data associated with the free flow and those measured during a traffic congestion. As in [ferrara2018freeway] and more in details in [li:2014:traffic_parameters, Sec. 3.2]
, we identified the parameters of each cell of the CTM by using linear regression. For a cell
, the free flow speed is the slope of the linear regression applied to the data measured during the free flow (the yellow line in Figure (a)a). Similarly, the congestion wave speedis the slope of the quantile regression applied to the leftover data. The result of the regression is illustrated via the blue line in Figure
(a)a, and the density at which it intersects the horizontal axis represents the (theoretical) cell’s maximum density . Finally, the intersection of the two regression lines (i.e., the blue and the yellow) defines the maximum cell capacity . Therefore, by applying these steps, sketched in Figure 2 for all the cells, we identify the parameters that, together with and , fully describe the CTM. Their final values are reported in Table II and the code used is available at [cenedese:github_ctm].Parameters  Identified values of the CTM  

–  
Iv Decision process’ parameters selection
In this section, we focus on the parameters describing the decision making process (or game) in which the PEV are involved during every time interval (see Table I for the description of each parameter). The chosen values are reported in Table III and characterize what we call the base case in the remaining of the paper. Now, we list the motivations behind the choice of the values of the most significant parameters in the game.

: the current market share of PEV in the Netherlands is around . However, PEV are expected to constitute more than of total vehicles in the market by [iea:2020:ev_report]. For this reason, we choose .

: it is natural to consider the interval during which each player does not vary its strategy (namely if/when and for how long to stop at the CS) longer than . Then, we choose .

: in [verhoef:2010:incentive_traffic_managment] a group of commuters traveling via the A in the Netherlands participated in a test, whose results indicate that of the involved commuters could start their work within later than usual. Therefore, we considered , implying that the maximum time PEV can spend at the CS (i.e., the length of the prediction horizon) is equal to .

: in The Netherlands, the government is putting a lot of effort in increasing the number of available charging points, doubling it between and [rvo_chargign_spots]. Thus, we choose to model the foreseeable increment in charging spots.

: the battery capacity of each PEV is randomly chosen from a pool of the most common PEV models. Specifically, , where represents the capacity of a Mitsubishi Outlander and the one of a Tesla Model S.

: for the realization of this Gaussian variable we choose mean and variance equal to and , respectively. This choice reflects the assumption that agents are more keen to save time rather than money. The small variance mimics an homogeneous interest among the PEV owners. The case of a more heterogeneous population is analyzed in Section VI.

: the charging spots are able to fast charge the PEV and the maximum energy that can be delivered by each spot in an hour is . Thus, during a single time interval we have .

: the CS is assumed to be able to deliver the maximum power to all the charging spots simultaneously.

: this energy price is chosen equal to €/kWh and represents the average between the price for fast charging (see [tesla_charger_price, fastnet]) and the one for charging at home (see [iea:2020:energy_price_nl]).

: the base demand profile of the grid is obtained from [eia_beta] and scaled to match the average demand of a regional grid in the Netherlands, so .

: these values must be chosen simultaneously to make the average value of equal to and normalize the three components in the price. Furthermore, is particularly important, since it defines the discount due to the traffic congestion. We introduce here what we refer to as incentive in the remainder of the paper. We say that we apply an incentive of if the term in the price function directly associated to the traffic congestion makes decrease of during the intervals of peak traffic congestion.

: this normalizing factor takes different values along the prediction horizon. Specifically, for the first time intervals in and for the others. This is due to the definition of and constraints on (see [cenedese:2020:highway_control_pI] for more details).
Base case: Game parameters  

–  
–  
–  
–  
–  
€  
€  
€  
€  
€  
–  
– 
Finally, let us suppose that the percentage of the PEV users that are traveling with an insufficient SoC is very low, since we focus on drivers commuting during the rush hours so that the optimization problem is always feasible. From a theoretical point of view, one can prevent feasibility problems by increasing the length of the prediction horizon or by translating the hard constraint on the minimum SoC into a soft one. In a real scenario, this would not be an issue, since these users will simply extend their staying at the CS.
V Numerical results
In this section, we present and discuss the simulation results for the base case introduced in the previous section (see Table III), focusing on the benefit of the ATDM strategy in terms of the average total travel time for all the drivers.
Va Performance index
The goal of the proposed policy is to alleviate traffic congestion during the rush hours for all the drivers on the highway, not only for the PEV owners. To quantify the benefit (and performance) of the proposed policy, we first define the quantity , which denotes the sum of the extra travel times of the drivers in the absence of the CS. Furthermore, we define as the extra travel time in the presence of the CS and supposing that PEV subscribe the proposed policy. We then introduce the following performance index:
(7) 
The larger , the greater the benefit emerging from the behavior of PEV owners. The index (7) is a good yardstick to evaluate the performance of the policy over the whole day and whether or not its adoption achieves the desired traffic congestion alleviation.
VB Simulation base case
We are now ready to discuss the effects of the ATDM strategy proposed in (Part I: Theory) [cenedese:2020:highway_control_pI]. Figure 6 highlights that there are two time intervals during which a traffic congestion occurs and they are centered around : and :, respectively, and reflect the real data in Figure (b)b, where the velocity drops exactly in those time intervals. The traffic jam in the morning lasts fro almost one hour while the one in the afternoon for three hours, with a particularly high peak (in terms of travel time) at :. Remarkably, despite the percentage of PEV on the highway is modest, our algorithm manages to decrease the afternoon peak of more than in terms of . If there is no congestion all the cells are traveled by a vehicle in . During peak traffic congestion this time becomes , the proposed ATDM strategy reduces it to .
The overall achievement in terms of travel time is .
In Figure 9, we highlight the difference between the travel time with vs without our ATDM strategy. The ATDM manages to decrease the peaks and redistribute the vehicles in the “valleys”, i.e., the parts in which the curve takes lower values with respect to the closest peaks. This emerging behavior matches the expectation: in fact, it relates to the “valley filling” phenomenon, well known in the electricity market, where the energy price depends on the energy demand, see [J_Hiskens_2013, cenedese:2019:PEV_MIG]. The redistribution of the vehicles creates a global reduction of the travel time. This effect of the ATDM on the PEV is confirmed by the flows shown in Figure 12, where r2s increases before the two peaks of , while s2r peaks immediately after the reduction of after the rush hours.
The dynamic energy price designed by the HO is reported in Figure 13. It shows a slow variation due to the base energy demand and a fast one associated with the traffic congestion on the highway stretch.
Interestingly, these two quantities act in opposite directions since the major traffic congestions happen during the rush hours, which are notoriously those during which the demand of energy is at its highest. The parameters and are chosen such that the incentive provided is the , i.e., the maximum discount is equal to €/kWh with respect to the average price €/kWh.
Finally, in Figures 14 and 15, we illustrate the number of PEV simultaneously connected to the grid, i.e., , and the amount of energy purchased . These two quantities are closely related and for this reason their profiles are almost identical. In both cases, the strategies implemented satisfy the upper bound constraints. As expected, during the period of high traffic congestion all the charging spots at the CS are busy. This highlights a key difference between longer and shorter traffic congestion; in the latter, there are only brief periods during which no charging spot is available, i.e., . Therefore, if a PEV decides to enter the CS, then it is almost sure to connect immediately. On the other hand, a traffic congestion that lasts for a substantial amount of time creates a queue of PEV at the CS waiting for charging. This idle waiting time creates an extra cost for the PEV stopping. Moreover, this phenomenon generates detrimental effects in the concurrence of high traffic congestion peaks surrounded by an overall elevated traffic congestion level. In fact, if we consider PEV exiting cell at :, then it should be beneficial for them to stop and charge for a short period, waiting for the traffic congestion to disappear. Unfortunately, all the charging spots are occupied, so the PEV cannot stop, hence contributing to increase the traffic congestion. In fact, Figure 12 shows that in concurrence with the highest congestion peak, the r2s flow decreases.
This first example highlighted the overall benefit of adopting a dynamic energy price to enforce an ATDM. The most important macroscopic effect is the redistribution of the vehicles that creates a valley filling like phenomenon in the travel time.
Vi Sensitivity analysis
Next, we investigate the effects that a single parameter has on . They are difficult to extrapolate directly from the base case due to the complexity of the decision making process and the multitude of variables involved. In this section, we propose a sensitivity analysis that focuses on the most compelling features of the proposed ATDM. Specifically, we start from the parameter configuration in Table III and vary one parameter (or at most two) to highlight how it affects the travel time. Note that the values used for this variation may deviate from what one can reasonably expect in a realworld scenario, e.g. a CS with , but they are meant to investigate the performance in extreme configurations or approximations of possible future configurations, e.g. one CS that approximates the charging capacity of two or more nearby CS.
Via Number of available charging spots
Intuitively, a larger number of charging spots mitigates the effects of prolonged traffic congestion; in fact, the PEV take more time to saturate the available spots at the CS, and, as a consequence, the congestion alleviation during the rush hours can be more effective.
The variation of with respect to the increment of charging spots is presented in Figure 16 for different PEV shares, i.e., . The curve profile for shows two plateaus, one for small values of and the other for values greater than . The transit between the two happens between and . It represents the setting in which an investment to increase the number of charging spots is the most profitable in terms of performance. In fact, going from to triples the value of , which rises from to .
Assuming a higher number of PEV on the highway does not avoid the initial plateau, but, surprisingly, it prevents the occurrence of an analogous plateau for greater values of . This implies that investing in more charging spots creates a high return in terms of performance, after a certain “critical” value, i.e., the value of after which the initial plateau ends. Interestingly, even if varies, the critical value of remains between and . In Figures 6 and 9, we note that the beneficial effect of an increment of the PEV number. In fact, the more PEV subscribe the higher the traffic alleviation is. Figure 9, clearly shows the valley filling effect, namely a higher peak reduction implies a subsequent increment of travel time during the lesscongested period.
ViB Charging spots and incentive
Starting from the parameters in Table III, we analyze the performance obtained by varying simultaneously the number of available charging spots, i.e., , and the incentive on the energy price.
Figure 17 depicts the different curves resulting by increasing the number of available charging spots, i.e., , when the incentive ranges between and . An increment in the number of plugs generates, in general, an higher performance increment compared to a greater incentive. Surprisingly, the value of at which the initial plateau of performance stops is not affected by the incentive. In fact, the consequent steep increment in settles between and ; this can be considered the critical value for this configuration. On the other hand, an increment in the incentive creates a smaller effect on the performance yet it is more predictable especially if the number of plugs is high. This result suggests that an investment in increasing leads to a high return only if the additional charging spots allow to surpass the critical level. If this is not possible, then increasing the incentive is a more profitable strategy.
ViC Heterogeneous population of Pev
In the simulations performed for the base case in Section VB
, we have considered a fairly homogeneous population of PEV owners, i.e., drivers that value similarly their interest of saving time or money. The probability distribution, from which all the
are drawn, has variance . Next, we consider a collection of heterogeneous PEV, in which some are more interested in maintaining the travel time at its minimum and others in saving money.In Figure 18, we show the performance increment attained by assuming a population of agents with increasingly diverse interests (all the other parameters are set as in Table III). Interestingly, the value of increases from , when , all the way to , if . The trend of the increment is almost linear and does not seem to slow down for higher values of . From [verhoef:2010:incentive_traffic_managment], it seems reasonable to assume that the commuters involved in the decision making process form a heterogeneous set; nevertheless, the fact that we focus only on PEV owners may reduce this variability. The value of that describes the best a realworld scenario is for these reasons an open problem that requires further investigation. The increment in the performance can be explained by looking at Figure 15. In fact, a more heterogeneous population leads to fewer PEV stopping at the CS during the long afternoon congestion peak. This alleviates the problem of overcrowding the CS where all the charging spots are generally busy during the largest traffic congestion peaks.
ViD Incentive
Based on our numerical study, we have chosen the incentive used in the base case heuristically. In this subsection, we study the effect of different incentives if
varies.In Figure 19, we report the performance computed for all the combinations of the incentive and . If the value of is almost constant with fluctuations of . In the other cases, an interesting trend arises: the optimal incentive that maximizes the performance is not the highest one. In fact, if the optimal incentive is while for and it is . This relates to the phenomena described at the end of Section VB. Inflating the incentive makes the number of PEV willing to stop grow. This has a beneficial effect on the morning peak, but a detrimental one on the afternoon rush hour, since it exacerbates the problem of the overly crowded CS and the lack of available charging spots.
To mitigate this issue, we propose in the next simulation a discount that varies over time and, in particular, takes a higher value during the morning traffic congestion peak and a lower value during the afternoon one. Specifically, if we say we apply an incentive of , then it means that we apply a incentive from : to :, and for the remainder of the day. Implementing this smarter type of incentive leads to striking results. In Figure 20, for the travel time significantly decreases with the increment of the incentive, i.e., goes from (see Figure 19) to . This trend holds also for higher values of , increasing the overall maximum performance from to . This suggests that the problem of over incentivizing the PEV to stop during the afternoon should be solvable with incentives decreased for the afternoon peak only. Finally, the fact that the maximum value of always coincides with the maximum incentive suggests that there is still room for improving the performance by increasing the incentive even further. Nevertheless, one should be careful in implementing a very large incentive, because it might be overly costly for the HO that has to take care of the economical burden associated with the energy price discount.
Vii Policy recommendation
Let us now extrapolate the key messages based on the numerical analysis and results of the previous sections, and then elaborate on possible policies that can improve the performance of the ATDM.
As illustrated in the previous section, tuning the parameters involved in the decision making process translates into infrastructural or sociological actions. The former are related to tangible features of the infrastructure, e.g., the number of available charging spots or the maximum power that fast charging can deliver ; the latter instead are intrinsically linked to the behaviors and decisions of the drivers, e.g., the interests of each agent or the incentive applied to steer the drivers’ behavior. As expected, the infrastructural improvements create greater outcomes in terms of performance, as shown in Figure 17 where the performance improvement by increasing the number of available charging spots overcomes that due to a higher monetary discount. This type of interventions is expensive and produces inelastic changes with respect to the demand. As an illustrative example, let us focus on : increasing it leads to a CS that has a low percentage of charging spots occupied throughout most of the day, and thus the return of the investment is limited. This economic point of view and the existence of a critical value of (see Section VIA) suggest that the HO should carefully decide whether it is convenient to invest in this direction. Infrastructural actions should be rare and performed when a substantial leap forward can be achieved, while small incremental improvement should be avoided.
Enforcing a change in the drivers’ behavior is not as straightforward as changing some physical features. Nevertheless, our formal analysis and modeling of the decision making process help us to infer some possible policies to achieve an improvement in the traffic congestion alleviation. The energy price discount generates remarkable performance, if it is designed by the HO based on the severity of the traffic congestions, see Figures 19 and 20. This policy does not require massive investment to be implemented and it is elastic with respect to the current traffic situation. Thus, from an economic perspective, a dynamic energy price currently seems the safest and the most remunerative option for the HO. The growing trend of incentivizing smart working and flexible working hours has a beneficial effect on the proposed policy for ATDM. In fact, it will increase the variety of interests among the PEV owners and consequently the performance should also increase, see Figure 18. Moreover, flexible schedules may shift the overall interest of the drivers towards saving money rather than time. This variation will allow the HO to attain the same performance even with lower incentives.
To summarize, the theoretical framework developed and the numerical simulations carried out in this work have convinced us that a solid policy to maximize the efficacy of the ATDM over time should be composed of short term and long term actions following two broad guidelines:

Short term: a dynamic energy price should be implemented where the incentive varies with respect to the intensity and (estimated) duration of the traffic congestion.

Long term: consider a major infrastructural investment to increase the CS capabilities, e.g., increase the number of charging spots, and consequently achieve a performance boost. In the A case study analyzed in this paper, the long term actions should be introduced, in addition to the short term ones, when the HO can increase the charging plugs above . This ensures a substantial improvement in performance that motivates the monetary investment.
Viii Conclusion and outlook
The ATDM strategy proposed in (Part I: Theory) [cenedese:2020:highway_control_pI] has been validated by simulation in a highly congested highway stretch, achieving a reduction of the peak traffic congestion of about in terms of travel time. This result is achieved with the current value of PEV market shares that is around the . This method creates a redistribution of the vehicles throughout the day, achieving a travel time that is on average lower for all the vehicles (not only PEV) traveling on the highway. The foreseeable increment of the PEV and the strengthening of the charging infrastructure, together with the increasing popularity of flexible working schedules, will enhance the benefits of applying the proposed policy, making it even more effective over the upcoming years. This paper is a selfcontained analysis that discovers interesting results. Nevertheless, it can act as a stepping stone to develop further research based on a similar concept. Perhaps, the most valuable extension can be a field study in which this approach is validated with humans in the loop. The framework of potential games well suits the introduction of partially rational agents (see for instance [zino:2017:imitation, cenedese:2019:RBR]) that may introduce new algorithmic dynamics in the framework. In the literature, other ATDM strategies have been proposed, e.g. [metropia:2019:bart2]. It can be of great value to carry out a comprehensive study that analyzes if and how well they work in combination to improve the traffic congestion alleviation.
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