Modeling and control methods for the reduction of traffic pollution and traffic stabilization

Transcription

1 Modeling and control methods for the reduction of traffic pollution and traffic stabilization Thesis by: Alfréd Csikós Supervisor: István Varga, PhD Co-supervisor: Katalin M. Hangos, DSc (MTA SZTAKI) Budapest University of Technology and Economics Department of Control for Transportation and Vehicle Systems Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy April 215

2 i Declaration Undersigned, Alfréd Csikós, hereby state that this PhD thesis is my own work wherein I have used only the sources listed in the Bibliography. All parts taken from other works, either in a word for word citation or rewritten keeping the original contents, have been unambiguously marked by a reference to the source. The reviews of this PhD thesis and the record of defense will be available later in the Dean Office of the Faculty of Transportation and Vehicle Engineering of the Budapest University of Technology and Economics. Nyilatkozat Alulírott Csikós Alfréd kijelentem, hogy ezt a doktori értekezést magam készítettem és abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem. A doktori értekezésről készült bírálatok és a jegyzőkönyv a későbbiekben a Budapesti Műszaki és Gazdaságtudományi Egyetem Közlekedés- és Járműmérnöki Karának Dékáni Hivatalában lesznek elérhetőek. Budapest, April 215. Csikós Alfréd

3 Acknowledgements First of all, I would like to express my gratitude to my supervisor, István Varga who introduced me to the scientific and engineering profession. I feel honored to be co-supervised by Prof. Katalin Hangos. I learned a lot from her both professionally and personally for which I will always be grateful. I would like to thank Prof. József Bokor for the opportunity to pursue my doctoral studies and researches in his scientific school. During the last five years it was an honor to work with Tamás Tettamanti at the Department of Control for Transportation and Vehicle Systems. His enthusiasm and support always inspired me. I am thankful to all my wonderful colleagues at the Systems and Control Laboratory. Especially to Tamás Péni, Gábor Rödönyi, Balázs Németh, Gabriella Varga, Bálint Vanek, Tamás Bartha, István Gőzse and Ádám Bakos, who all supported me both professionally and personally. I would also like to express my special appreciation for Prof. István Zobory at the Budapest University of Technology for the valuable lessons during both my MSc and PhD studies. I very appreciate the help of Balázs Kulcsár and Bálint Kiss who made valuable suggestions to improve my thesis. Finally, my greatest thank goes to my loving family for their persistent support. ii

4 Abstract In this work novel modeling and control problems are proposed in the area of road traffic modeling and control following the idea of sustainable mobility. This concept requires the involvement of traffic related environmental pollution to control design in addition to conventional traffic stabilizing and performance optimizing approaches. The proposed methods are developed both for motorways and urban networks, following different aims and approaches. The first part of the thesis focuses on involving the emission of motorway traffic into motorway control design. This problem is addressed in three steps. First, a macroscopic description of the spatiotemporal distribution of emissions is derived. The proposed macroscopic framework relies on the average-speed based vehicular emission modeling method and the available loop detector measurements of macroscopic traffic variables. The analytic form of emission distribution serves as a basis for a dynamic modeling of the pollutant concentrations at the built-in areas near motorways. A dynamic emission dispersion model is developed to describe the convection effect of the prevailing winds. The developed dispersion model is attached to the motorway traffic model METANET, and for the joint system a sensitivity analysis is carried out for an appropriate control problem formulation. In the third step, a model-based control is designed for the joint system: the conventional control purpose of traffic stabilizing and performance optimization is enhanced by the control objective of keeping pollutant concentrations below legislation limits. For this complex control problem, a hybrid controller is suggested. Two separate control modes are designed for the control tasks, among which the switching is realized by a finite automata. The behavior of the controller (switching stability, transient behaviour, and the performance in different modes) is analyzed in a complex case study. Simulation results indicate the stability of the switching controller, and an acceptable performance during different control tasks. The second part of the work focuses on the improvement of traffic performances in urban networks based on a network-level demand control approach, by using the concept of the network fundamental diagram (NFD). The urban gating problem is further improved by involving the performance of the exterior network to the control design, and a less greedy policy is offered for network gating. The application of predictive control is analyzed in a real life network. The suggested methods provide models and controllers with acceptable performance and low computational demands. Thus, the achieved research results serve as practical methods for the improvement of traffic control in both motorways and urban networks. iii

5 Contents Acknowledgements Contents List of Figures List of Tables ii iii vii ix 1 Introduction Background and motivation Overview of the thesis Preliminaries and literature review Modeling of motorway traffic Macroscopic model variables Dynamic models of motorway traffic Performances and control objectives Previous results of motorway traffic control Modeling of urban traffic Modeling of link dynamics and control of intersections Network level modeling and control Modeling of vehicular emissions Basic notions Emission models Emission models used in the thesis The use of emission models in ITS applications Models of emission dispersion Macroscopic static description of traffic emissions Problem formulation Derivation of the spatiotemporal distribution of emissions Macroscopic description of motorway network emissions Emission in the spatiotemporally discrete framework Concept verification Macroscopic emission modeling of urban networks Emission performance in urban networks Concept verification Analysis of the emission field function Conclusion and contributions iv

6 CONTENTS v 4 Dynamic model for the dispersion of motorway traffic emissions Problem formulation Modeling assumptions Derivation of model equations Conservation of pollution masses Normed dimensionless form Spatial and temporal discretization Initial and boundary conditions Determination of decay rate λ The state-space model of the joint traffic-emission dispersion system State dynamics System variables Model verification Model analysis Numerical analysis Sensitivity analysis Conclusion and contributions Hybrid control of motorway traffic flow Problem formulation Control objective statement Control system structure Controller mode no. 1 - ramp metering for concentration limitation Cost function Constraints Controller mode no. 2 - traffic stabilization Cost function Constraints Case studies Case study 1: different concentration limits during rush hour Case study 2: changing traffic loads for constant concentration limit Case study 3: rush hour shockwave Discussion and possible improvements Practical applicability on a realistic case study Measurements Control inputs Computational demand Computational parameters of a realistic network Conclusion and contributions Modeling and control of urban networks Problem formulation System model State-space model System variables Control design Cost function Constraints

7 CONTENTS vi 6.4 Case study network Network description Identification of network parameters Simulation Simulation environment and setup Simulation results Summary of results and possible improvements Conclusion and contributions Conclusions and future research 98 A Mathematical modeling of shockwaves 1 B Motorway traffic model 16 C Emission factor functions 18 D PID controller of the urban gating problem 11 E Analysis of horizon lengths in the gating control problem 111 F Acronyms 112 Bibliography 113

8 List of Figures 2.1 Approach 1: spatial measurement Approach 2: instantaneous measurement Spatial discretization of motorway networks Motorway network models The fundamental diagram of traffic Flow and conservation of vehicles in an analyzed area Link level modeling Network level modeling The network fundamental diagram Simulation no. 1.: traffic variables and emissions on the analyzed segment Simulation no. 2.: traffic variables and emissions on the analyzed segment Case study 1: gate inputs and number of vehicles in PN Case study 1: network emissions Case study 2: gate inputs and number of vehicles in PN Case study 2: network emissions Emission field as a function of traffic density and traffic mean speed Topographic layout Flow channel parameters Schematic representation of the Gaussian plume Flux decrease in the simplified plume model Decay rate as a function of wind speed and downwind distance Disturbances of verification scenario no Traffic density of verification scenario no CO concentration values of verification scenario no Disturbances of verification scenario no CO concentration values of verification scenario no Simulation of different ramp inputs - traffic variables Simulation of different ramp inputs - emissions and emerging concentration Simulation of different VSL inputs State transition diagram Macroscopic emission function ε CO [g/km h] Results of case study CO concentration limits and values - uncontrolled case Ramp control and ramp queue CO concentration and limits - controlled case Boundary conditions of case study CO concentration limits and values - uncontrolled case vii

9 LIST OF FIGURES viii 5.9 Ramp control and ramp queue CO concentration limits and values - controlled case The disturbances of case study State values for the uncontrolled case Ramp control and ramp queue - controller mode State values and prescribed limits - controller mode Ramp input and queue of the discrete VSL valued two-mode controller VSL inputs, decision density value and switching mode no. - two-mode control during the shock wave State values and prescribed limits - two mode control. Mode 1: concentration control; Mode 2: shockwave prevention control Scheme of the realistic network Network layout NFD of the protected network, i.e. the variation of TTD together with TTS Linear regression for parameter Γ, i.e. relationship between total outflow and TTD PN The closed-loop simulation environment Variation of disturbance- and controlled gate inflows - case study Variation of vehicle numbers inside and outside PN - case study Variation of TTD performances inside and outside PN - case study Phase plots of case study Case study 2: Traffic load at controlled and uncontrolled gates Variation of disturbance- and controlled gate flows - case study Variation of vehicle numbers inside and outside PN - case study Variation of TTD performances inside and outside PN - case study Phase plot of case study A.1 Illustration of propagation velocities A.2 Considerable density differences evocating a shockwave A.3 Traffic densities and flows around a shock front A.4 Shockwave on the fundamental diagram C.1 Emission factor function of CO for the Hungarian fleet composition of year 2119 C.2 Emission factor function of NO X for the Hungarian fleet composition of year

10 List of Tables 4.1 Parameter values of equation (4.35). Source: [14] Parameters of the optimization problems Optimization parameters of case study 3 (Section 5.6.3) Parameters of the affected inhabited areas Aggregated results of case study Aggregated results of case study Comparison: NMPC vs. greedy NMPC control B.1 Parameters of the motorway traffic model C.1 Coefficients of the emission factor functions C.2 Emission rates [g/h] of light-duty vehicles, up to 25 kg gross vehicle weight. Source: [15] E.1 Simulation results with different horizon lengths ix

11 Chapter 1 Introduction 1.1 Background and motivation Since the invention and wide spread of the automobile, engineers have been urged to analyze and manage the peculiar process of road traffic. The first model describing the flow conditions of public roads was presented by Greenshields in the 193 s [2]. A few decades later, a dynamic model was derived by Lighthill, Whithams and Richards [8], [9] based on the law of conservation to provide an analytic solution of complex traffic phenomena, such as the traffic shock waves. In the following decades, several further refinements were suggested for the sophistication of the model to describe motorway traffic. The modeling of urban networks appeared in the 196 s with the description of intersection dynamics in the work of Gazis, Herman and Rothery [36] and analyzing network level flow conditions (see the work of Godfrey [46]). Nevertheless, the control system applications of the traffic models only appeared in the last decades of the 2th century. These control strategies aimed at the reduction of congestions by changing traffic signal green times, or controlling motorways via ramp metering or variable speed limits. Efficient control applications were elaborated for both the urban (see [37] and [43]) and motorway topologies (for the milestone works, see [27], [28] and [29]). The challenge of modern traffic engineering lies in the design for sustainable mobility, which concept contains - among a number of aims - the reduction of air pollution in addition to the suppression of congestions. Thus, it is desired to involve this latter aspect to the control design as well, preferably by an integration of an appropriate emission model to the control system. The description of traffic-originated air pollution needs an analytic approach, involving the process characteristics. This presents a challenge in itself, because the existing vehicular emission models, developed since the 197 s are basically established for the analysis of the effect of automotive and fuel engineering technologies and therefore they describe the emission of an individual vehicle. To state the overall pollution emitted by traffic, a macroscopic level 1 description is needed. The problem can be further sophisticated by considering the emerging concentrations at the nearby built-in areas. There have been efforts for the incorporation of vehicular emission models into traffic control systems during the last decade (see e.g. [17], [18] or [11]). Nevertheless, in these works the overall emission of the process was not considered, and the optimization was done only for the 1 i.e. describing the process as a whole, using aggregated variables and neglecting the individual vehicles 1

12 Chapter 1 Introduction 2 emission of a single vehicle. This approach is reasonable in certain cases when one considers pollutants with global effect. However, for the ones causing local effects, a more reasoned approach is needed, that needs the modeling of the spatiotemporal distribution of emission and its dynamic dispersion. The modeling and control of roadside emission dispersion have been addressed, too, by Zegeye in [95] with a grid-based approach. In that work, however, the excitation of the dispersion dynamics is not declared appropriately. The main contribution of this work is a proposal for the integration of traffic emissions into the control designs of motorways, using the existing measurements of traffic systems. This goal is realized in three main steps. First, a static model function is stated for the spatiotemporal distribution of emission. This model function also facilitates a prior statement of control objectives for the different pollution types. In the second step, an emission dispersion model is developed for the description of emerging concentrations near motorways. The third step proposes a combined control approach, for the stabilization of traffic and reducing pollutant concentrations. A minor contribution of the thesis is a suggestion on the further improvement of the network level urban gating problem, by proposing a balanced loading of the controlled network and the exterior network. Although the static emission model is analyzed for urban networks as well, the model has not yet been integrated to the existing control design for that topology. 1.2 Overview of the thesis After the introduction, preliminary sections are developed in Chapter 2 by discussing the traffic models of motorways and urban networks, the modeling of vehicular emissions, as well as the models of emission dispersion. Chapter 3 presents a static description of macroscopic emissions, using the variables of the macroscopic modeling framework, and the emission factor function of the average-speed emission model method. The obtained model function is extended for the spatiotemporally discrete frameworks of motorways and urban networks. For both topologies, a simulation-based verification is given. Based on the analysis of the model function, control objectives are postulated for the different pollutant types. In Chapter 4, a simple dynamic model of emission dispersion is suggested to describe the emerging pollutant concentrations of built-in areas near motorways. The roadside dispersion model is based on the conservation of pollutant masses. The excitation of the process is modeled by the discrete version of the macroscopic emission model, introduced in Chapter 3. The absorption of pollution is modeled based on the Gaussian plume approach. After the discretization of the model, analysis is given in terms of computational stability and sensitivity to the control measures of motorway control systems. The contributions of Chapter 5 rely on the model given in Chapter 4. A hybrid controller is designed for the joint model of traffic dynamics and emission dispersion. Two control goals are aimed: on one hand, the stabilization of traffic flow by suppressing shock waves; on the other hand, keeping pollutant concentrations below legislation limits. The control tasks and the corresponding control modes are chosen according to the stability conditions of the traffic and are switched by a finite automata. In the control modes, the input signals of ramp metering and variable speed limits are optimized by a nonlinear model predictive algorithm. In Chapter 6 suggestions are given for the improvement of the NFD-based urban traffic model and gating problem. Additional state variables are allocated to describe the queue

13 Chapter 1 Introduction 3 dynamics at the network gates. The minimization of the queues are involved in the control problem, improving the traffic performance of both the interior and exterior networks. The gating signals are optimized by a nonlinear model predictive control algorithm. The designed controller is analyzed in a test network built in the traffic simulator VISSIM [88]. Finally, a conclusion is given in Chapter 7 and future research is stated. Additional material is attached in six appendix sections.

14 Chapter 2 Preliminaries and literature review In this chapter the most important results from the field of traffic and emission modeling are summarized, on which the contributions of this work are based. First, an overview of the macroscopic modeling of freeway and urban traffic is presented paying special attention to the formalization of system performances and control objectives. Then, the basic notions and the types of vehicular emission models are outlined. Finally, methods of emission dispersion modeling are reviewed. 2.1 Modeling of motorway traffic The uninterrupted and continuous flow of traffic on motorways (also called freeways) makes this network type the most versatile environment for the analysis of a variety of models. Traffic models can be classified based on several aspects from which the most important is featured here: the level-of-detail categorization. According to the level-of-detail, the following model classes can be distinguished: Microscopic traffic models provide a high-detail description of the dynamics of traffic particles, i.e. the vehicle individuals, based on their interactions. The most widespread microscopic approach is the car-following modeling. The basic linear car-following model developed by Gazis, Herman and Rothery [1] was first extended by Gipps [2] incorporating the nonlinear manner of the stimuli. Optimal velocity models such as [3], [4] or [5] extend the stimuli function by the flow characteristics concluding in a more sophisticated description of the velocity choice. Intelligent driver models [6] include the memory effect of the drivers of local historic traffic patterns. Another approach of microscopic modeling is represented by the cellular automaton models [7] describing the motorway as a network of connected cells. Each cell has a size of a vehicle, described by an integer value carrying the dynamical properties of the system. Although microscopic models give a high resolution modeling of traffic, which provides an opportunity to analyze vehicle-to-vehicle interactions, their use for a model-based control over a large network is not feasible because of the extreme computational demand and the lack of dynamic information of individual vehicles. Macroscopic traffic models give a low-detail representation of the process using aggregated variables (summarizing the information of multiple vehicles) neglecting the individual 4

15 Chapter 2 Preliminaries and literature review 5 vehicle dynamics. The clear advantage of these models is their lower computational complexity relative to microscopic models, making them applicable for simulation and control purposes of large networks. The basics of the modeling approach were established by Lighthill, Whitham and Richards in [8] and [9] offering a continuous description of traffic flow. The first-order dynamics of the traffic density of the Lighthill-Whitham-Richards (LWR) model were later extended with the mean speed dynamics by Payne [1] and Whitham [11] resulting in a second-order description. A higher-order model was developed for multi-lane traffic by Helbing [12]. Kerner [16] offers a model, focusing on the dynamic modeling of shockwaves, considering stable, unstable and metastable conditions of traffic. The discrete interpretation of the continuous LWR model was proposed by Daganzo in the Cell Transmission Model (CTM) [15]. The CTM model gives a numeric solution of the continuity equation of LWR following the Godunov scheme. The continuous models of [1] and [11] were reformalized in a spatiotemporally discrete manner by Messmer and Papageorgiou in [13] and [14], resulting in the METANET model. Although first-order models are capable of modeling complex traffic phenomena (i.e. shockwaves), second-order models provide a more realistic representation of the process due to the dynamic speed description. Therefore, in this work METANET is used as a modeling platform. In the following part of this section, the applied modeling framework of motorway networks is presented. First, the variables of the macroscopic description are introduced, which is followed by the review of the continuous motorway network models of Lighthill, Whitham and Richards, and its extension: the Payne-Whitham model is given. After introducing the basic modeling equations, the applied discrete model based on the METANET model is detailed. Then, the control objectives of motorway traffic control are summarized, as a corollary of the interpretation of system performances and the explanation of certain phenomena (i.e. the kinematic waves and shockwaves). Finally, the existing results of motorway traffic control are summarized Macroscopic model variables The macroscopic description of traffic is possible through aggregated variables, the meaning of which can be best interpreted through their measurements. According to the Eulerian view (i.e. in a fixed coordinate system, see [17]) two approaches can be followed: spatial and instantaneous measurements. x x +dx x dx T t t +T t Figure 2.1: Approach 1: spatial measurement Approach 1: spatial measurements. Let us analyze the traffic during a long period T over a short increment dx (thus in the infinitesimal spatiotemporal rectangle dx T ), see [18],

16 Chapter 2 Preliminaries and literature review 6 [19]. The definition of traffic flow in this case can be stated as: q dx T = n dx T T (2.1) where q dx T in unit [veh/h] denotes traffic flow in the infinitesimal rectangle dx T, and n dx T denotes the number of vehicles crossing dx during period T. In this case, the notion of time-mean speed can be defined as follows: v T dx T = ndx T i=1 v i ([x, x +dx], [t, t + T ]) n dx T (2.2) where v T dx T denotes the time-mean speed of traffic in dx T in unit [km/h], whereas v i([x, x + dx], [t, t + T ]) denotes the speed of vehicle i within dx T. By taking the limit dx, point-wise variables can be defined. The traffic flow and traffic mean speed, at the point x are as follows: q(x ) = n(x ) T (2.3) v T (x ) = n(x ) i=1 v i (x ) n(x ) (2.4) x x +L L x dt t t +dt t Figure 2.2: Approach 2: instantaneous measurement Approach 2: instantaneous measurements. Let us take measurements of a roadway section of length L during a short time interval dt [18], [19]. This approach provides an opportunity to directly measure traffic density in the spatiotemporal rectangle L dt: ρ L dt = n L dt L (2.5) where ρ L dt in unit [veh/km] denotes the traffic density over L dt, whereas n L dt denotes the number of vehicles in L dt. Space-mean speed can be defined using the measurements of vehicle speeds in L dt the following way: v S L dt = nl dt i=1 v i ([x, x + L], [t, t +dt]) n L dt (2.6) where v S L dt denotes the space-mean speed of traffic in L dt in unit [km/h], whereas v i([x, x + L], [t, t + dt]) denotes the speed of vehicle i within L dt.

17 Chapter 2 Preliminaries and literature review 7 By taking the limit dt, the instantaneous traffic variables ρ(t ) and v S (t ) can be defined at t : ρ(t ) = n(t ) (2.7) L n(t ) v S i=1 (t ) = v i(t ) (2.8) n(t ) The traffic process is a distributed parameter system (DPS, [32]) i.e. its dynamic variables are bivariate functions of space and time. Their spatiotemporal distributions are called the traffic flow-, traffic density- and traffic mean speed fields denoted by q(x, t), ρ(x, t) and v(x, t), respectively. The continuous flow field q(x, t) can be obtained by taking an infinitesimal refinement of T (T ) for q(x ) in eq. (2.3). Similarly, the taking of the limit L for ρ(t ) in eq. (2.7) results in the continuous density field ρ(x, t). Applying the limits T and L, respectively, for the definitions of time mean speed v T (x ) (2.4) and space mean speed v S (t ) (2.8) results in the same continuous velocity field, denoted by v(x, t). Thus, from the definition of traffic measurements formalized for infinitesimal spatiotemporal rectangles, we concluded to distributions that are continuous in space and time. The modeling of the process is based on partial differential equations (PDEs), describing the dynamics of the macroscopic variable fields. In the following subsection, the continuous model of Lighthill, Whitham and Richards, and its extension: the Payne-Whitham model are summarized. The latter serves as the basis for the discrete-time motorway model METANET, which is used in the thesis Dynamic models of motorway traffic In this subsection the applied discrete model of the motorway traffic is presented. For a better understanding, it is introduced after giving insights to its continuous predecessors. As highlighted in the introduction of Section 2.1, the dynamic modeling of freeway traffic has gone through a significant evolution since the first works addressing the problems of the process. The basics of traffic modeling were established by the Lighthill-Whitham-Richards (LWR) model, providing a first-order description of the process through the dynamic modeling of traffic density. The continuous LWR model was extended to a second-order model by Payne and Whitham, providing the dynamic description of traffic mean speed. The model METANET suggests a further extension of the discrete Payne-Whitham model. In the thesis, a modified version of METANET is used. This subsection is organized as follows: first, the model equations of the LWR model are summarized. Then, the extension of the model is presented by the Payne-Whitham model. Finally, the applied form of the METANET model is outlined. The LWR model The LWR model, introduced in the papers [8] and [9] in 1956 was originally developed to explain the dynamic behavior of traffic through the modeling of complex traffic phenomena such as the shockwaves (see Appendix A). The spatiotemporally continuous model describes the dynamics of the density field ρ(x, t). The velocity field v(x, t) and the distribution of traffic flow q(x, t) are given as static functions of ρ(x, t). The model adopts the following assumptions:

18 Chapter 2 Preliminaries and literature review 8 Law of conservation for vehicles. Analyzing a stretch of motorway of finite length, the basic law of flow dynamics can be stated formalizing the conservation of the vehicles: the temporal change in the number of vehicles equals to the difference of the total inflow and outflow through the boundaries. By using the density and flow fields, the conservation of vehicles can be stated as follows: ρ(x, t) + q(x, t) = (2.9) t x Equilibrium speed-density equation. The LWR model highly relies on the hypothesis of a static relationship between traffic density and traffic mean speed, first presented by Greenshields [2]. In stationary homogeneous flow conditions [6], an equilibrium speed-density relationship can be stated: v(x, t) = V (ρ(x, t)) (2.1) where V =V (ρ) is a monotonously decreasing function. For the function V (ρ) several models are suggested. In the original LWR model, the equilibrium speed function according to Greenshields [2] was used: V (ρ) = v free ρ ρ jam (2.11) where v free denotes the maximal attainable speed on the road according to speed limits; ρ jam denotes the maximal traffic density. Fundamental equation. An important constitutional formula of the LWR model describes the relationship among the traffic variables. The fundamental (or also called flow) equation is given by: q(x, t) = ρ(x, t) v(x, t) (2.12) Substituting (2.1) and (2.12) to (2.9), the brief form of the LWR model can be presented, describing the dynamics of the density field through the following PDE (called the continuity equation [139]): ρ(x, t) t + (ρ(x, t) V (ρ(x, t)) x = (2.13) The solution of the PDE (2.13) for certain initial and boundary conditions leads to the description of kinematic waves and shockwaves, detailed in Appendix A. It is important to notice, that only eqs. (2.9) and (2.12) of the LWR model are satisfied under all circumstances. The equilibrium speed-density equation (2.1) implies a few simplifying presumptions [21]: The use of a static function supposes that drivers adopt their speed instantaneously according to the density observed at their location, neglecting the finite reaction time in driving. The anticipation of drivers is also neglected: in the reality, the speed of traffic also depends on the downstream density perceived by the drivers. Although the LWR model shows high fidelity under certain conditions, the above simplifications explain why it is not capable of describing the hysteretic behavior of the process during congesting and easing traffic. An improvement of the model was suggested by Payne and Whitham, resulting in a second-order dynamic model.

19 Chapter 2 Preliminaries and literature review 9 The Payne-Whitham model The need for further refinement of the LWR model was recognized by Payne [1] and Whitham [11] in the early 197 s, who further developed the model by involving the driving behavior. As a result of their work, a model extension was suggested by introducing the momentum equation describing the dynamics of the traffic mean speed. The main idea of the Payne-Whitham (PW) model can be summarized in two points: 1. The involvement of the reaction time of drivers (denoted by τ) is given in the left hand side of (2.1): v(x, t+τ) = V (ρ(x, t)) (2.14) 2. The anticipation of drivers is also modeled by introducing the spacing in the equilibrium speed-density relation: V (ρ(x, t)) V (ρ(x+s, t)) (2.15) where s denotes the spacing, approximated by the reciprocal of traffic density: s=ρ 1. As a result of the above suggestions, traffic speed is described by the following equation: v(x, t+τ) = V (ρ(x+s, t)) (2.16) Proceeding from the first-order Taylor expansion of (2.16), the PDE of the mean speed evolution can be obtained in the following form: v(x, t) t = 1 τ (V (ρ(x, t)) v(x, t)) v(x, t) v(x, t) x ν ρ(x, t) τρ(x, t) x (2.17) where the decrease rate of the equilibrium speed with increasing density is replaced by a constant value, ν = dv (ρ(x,t)) dρ(x,t). Eq. (2.17) is also called the momentum equation. Its right hand side contains three terms, addressing the different neglects of the LWR model: 1. The first term in the form 1 τ (V (ρ(x, t)) v(x, t)) is called the relaxation term, describing the desire of drivers to follow the static equilibrium speed function. 2. The second term in the form v(x, t) v(x,t) x, called convection term carries the effect of the speed change, caused by the in- and outflow of the vehicles. ρ(x,t) 3. The third term in the form of ν τρ(x,t) x, called the anticipation term models the foreseeing of the drivers, who adjust their speed considering downstream traffic densities. By virtue of the momentum equation, the second-order model of Payne and Whitham is capable of reproducing various phenomena, such as the hysteresis during congesting and easing traffic. An improvement of the discrete PW model was presented in the METANET model [13], which is reviewed in the following paragraph. The applied discrete model For the applicability in an intelligent transport system (ITS), a spatially and temporally discretized model of the process is needed. The reason for this on one hand is that measurements and intervention measures are not available continuously in space or time. On

20 Chapter 2 Preliminaries and literature review 1 the other hand, a discrete model has lower computational demands. Spatial discretization of the process model is realized through the division of the motorway network to segments of variable lengths (see Figure 2.3), thus, a lumping of the system is carried out [32]. As a result of the lumping, a set of ordinary differential equations is obtained for each segment i. The choice of segment length L i is determined by the topology of the inductive loop detectors (through which traffic variables are measured) and of the control measures. Measurements and the modification of control variables are possible in discrete time instances, thus a constant sampling of the process is needed with sampling time T. Sampling means a temporal discretization - its representation in the modeling is possible through the finite difference approximation [32], resulting in a set of difference equations 1. The choice of L i and T also effect the accuracy of the discrete model. An analysis on the effect of spatial and temporal interval sizes is shown in [34]. VSL 1 VSL i VSL Ns ρ 1, v 1 ρ i, v i ρ Ns, v Ns r 1 s 1 L 1 r i L i s i r Ns s Ns L Ns Figure 2.3: Spatial discretization of motorway networks In Figure 2.3 the two control variables for control of motorways are highlighted as well: 1. VSL i denotes the variable speed limit in units [km/h] on segment i, determining the maximal attainable speed on the segment. Hence, it provides a control tool for the main lane of the motorway. (Its effect is explained in Appendix A.) 2. r i denotes the manipulated (or also called metered) ramp flow feeding segment i. The unit of ramp metering is [veh/h]. The outlined discretization can be carried out for the PW model. A detailed overview of the discrete PW model is given in [21]. The model METANET [13] is based on the discrete PW model, extending it by two important contributions: 1. A sophisticated suggestion on the equilibrium speed-density formula (2.1). 2. An additional term in the momentum equation (2.17), describing the effect of ramp inflows. In this work, further modifications are proposed on the discrete METANET model [13] that are described together with the model equations. Before the statement of the applied discrete model, the motorway traffic models featured in this work are highlighted in Figure During the discretization of a DPS, attention has to be paid to the numerical stability of the obtained set of difference equations through the appropriate choice of L i and T. The necessary condition for the convergence of solving hyperbolic PDEs is given by the Courant-Friedrichs-Lewy condition [33]. According to the condition, max v(x, t) T L i < 1 should hold for every i to obtain a numerically stable solution of the continuity equation (2.13).

21 Chapter 2 Preliminaries and literature review 11 continuous LWR PW discrete discrete PW METANET modified METANET Figure 2.4: Motorway network models The model equations of the applied discrete model, regarding segment i during a discrete time step k of length T are as follows: The conservation of vehicles is given by the following equation: ρ i (k + 1) = ρ i (k) + T L i [q i 1 (k) q i (k) + r i (k) s i (k)] (2.18) where ρ i, v i, q i denote the traffic density, speed and flow of segment i; r i and s i denote the ramp inflow and outflow values of segment i, respectively. The basic equilibrium speed equation of the METANET model is as follows: ( V (ρ) = v free exp 1 ( ρ a )) (2.19) a where parameters v free, a, ρ cr are functions of the speed limit value. Here, a slight modification of the equilibrium speed equation of METANET is carried out. In order to obtain smooth input affine dynamics, the following assumption is used: by using variable speed limits VSL i on segment i, the free flow speed parameter v free is altered to VSL i. Thus, the equilibrium speed function in case of VSL control becomes a fraction of the uncontrolled equilibrium speed function (2.2)) 2. The same approach is applied for a real network case study in [23], leading to promising results. The following modified formula is suggested for the involvement of variable speed limits: ( V (ρ i (k)) = VSL i (k)exp 1 a where a, ρ cr are constant model parameters. ρ cr ( ) ρi (k) a ) ρ cr (2.2) Fundamental equation: q i (k) = ρ i (k)v i (k) (2.21) The momentum equation of the METANET model is in the following form: v i (k + 1) = v i (k)+ T τ (V (ρ i(k)) v i (k)) + T L i v i (k) (v i 1 (k) v i (k)) ηt ρ i+1 (k) ρ i (k) δt r i (k)v i (k) τl i ρ i (k) + κ τl i ρ i (k) + κ (2.22) 2 The result of speed limit is the reduced capacity of the road. Nevertheless, certain VSL models (i.e. [28] and [22]) also describe a slight increase in the critical density, thus the extension of the stable domain of the system, meaning an additional stabilizing effect of VSL control. However, this effect of VSL control has not yet been validated and is neglected in present work.

22 Chapter 2 Preliminaries and literature review 12 where τ, η, δ, κ are constant model parameters. The model METANET provides an additional term in the momentum equation of the PW model (2.17): δt r i (k)v i (k) τl i ρ i (k) + κ The term describes the slowing effect of the ramp inflow. Furthermore, an important element of the discrete motorway system is the dynamic modeling of the evolution of ramp queues: l i (k + 1) = l i (k) + T (d i (k) r i (k)) (2.23) where l i (k) denotes the queue length of vehicles on-ramp of segment i at time step k, d i (k) and r i (k) in units [veh/h] denote the ramp demand and the controlled flow allowed at the the ramp of segment i, respectively. As a result of this subsection, the engineering model of the motorway traffic is obtained. The basic METANET model is stated in eqs. (2.18), (2.19), (2.21), (2.22) and (2.23). The applied discrete model of this work uses the same model equations, with one difference: modeling the equilibrium speed-density function with (2.2) instead of (2.19). The mathematical model of the motorway traffic system used throughout the thesis is detailed in Appendix B Performances and control objectives In this subsection, traffic performances are formalized proceeding from the definitions of macroscopic variables of Subsection From the performance definitions, control objectives are concluded. The performances of the distributed parameter system of traffic are analyzed in the spacetime domain. First, examine the meaning of traffic flow in a finite spatiotemporal rectangle L T! By setting L dx, the case of spatial measurement method is followed. Multiply the definition of traffic flow (2.1) by dx/dx [18]: q dx T = n dx T dx dx T (2.24) Taking the limit dx, dx equals to the distance that each vehicle completes within the infinitesimal rectangle dx T. Thus, traffic flow can be interpreted as the total travel distance of traffic within dx T normalized by the size of the rectangle dx T : q dx T = TTD dx T dx T (2.25) Taking the limit T dt, the definition of total travel distance can be formalized using the flow distribution for a spatiotemporal rectangle of arbitrary size as a definite integral: TTD [x1,x 2 ] [t 1,t 2 ] = t 2 x 2 t 1 x 1 q(x, t) dx dt (2.26) To provide an approximation in the spatiotemporally discrete framework for a rectangle of arbitrary size (L T ), the traffic performance TTD can be stated as follows: TTD L T = q L T L T (2.27)

23 Chapter 2 Preliminaries and literature review 13 Now analyze the definition of traffic density in the spatiotemporal rectangle L dt. Multiplying (2.5) by dt/dt results in the following equation: ρ L dt = n L dt dt L dt (2.28) In case of taking the limit dt, dt equals to the time that each vehicle spends within the rectangle L dt. Thus, traffic density can be interpreted as the total time spent by traffic within L dt normalized by the size of the rectangle L dt: ρ L dt = TTS L dt L dt (2.29) Taking the limit L dx, the definition of total time spent can be formalized using the continuous traffic density field for a spatiotemporal rectangle of arbitrary size as a definite integral: t 2 x 2 TTS [x1,x 2 ] [t 1,t 2 ] = ρ(x, t) dx dt (2.3) x 1 To provide an approximation for TTS in the discrete framework for a rectangle of arbitrary size (L T ), the total time spent can be stated as follows: t 1 TTS L T = ρ L T L T (2.31) The above definitions of traffic performances TTD and TTS can be further used for control objective statements. In the followings, these objectives are stated and an ultimate control goal is chosen. A control problem may include a cost functional that is usually a function of state (and possibly control input and disturbance) variables. In case of optimal control, the input signal is calculated to minimize the cost functional. Basic cost functions for traffic control can be simply described using the above stated performances: A cost function for TTS can be stated for a finite spatiotemporal rectangle [x 1, x 2 ] [t 1, t 2 ]: J TTS (t 1 ) = TTS [x1,x 2 ] [t 1,t 2 ] min (2.32) As for TTD, the control goal is its maximization. Accordingly, a cost function involving TTD can be stated in the following form: J TTD (t 1 ) = TTD[x 1, x 2 ] [t 1, t 2 ] max (2.33) For a controller synthesis, cost functions need to be stated for the system states, i.e. for the traffic density, or possibly for traffic flow or speed. Using that TTS(x, t) is stated as a primitive function of the density field, a new cost function - equivalent to the one in (2.32) - can be given for the traffic density: J TTS,ρ = t 2 x 2 t 1 x 1 ρ(x, t) 2 2 dx dt min (2.34)

24 Chapter 2 Preliminaries and literature review 14 Similarly, TTD(x, t) is obtained as a primitive function of the traffic flow field, thus an equivalent cost function (2.33) can be formalized for the traffic flow: J TTD,q = t 2 x 2 t 1 x 1 q(x, t) 2 2 dx dt max (2.35) The obtained cost functions (2.34) and (2.35) both represent feasible and acceptable control goals: the maximization of traffic flow and the minimization of traffic density. However, for a comparability of these control goals, both functionals need to be interpreted as functions of the same variable: the traffic density, which can be considered the primary state variable 3 of the system. For this analysis the fundamental diagram of traffic is used. Assuming a stationary homogeneous traffic flow [6], an equilibrium flow-density relation can be stated, by substituting (2.1) to (2.12): Q(ρ) = ρ V (ρ) (2.36) where V (ρ) and Q(ρ) denote the equilibrium speed-density and flow-density functions, respectively. The latter function is illustrated on the so-called fundamental diagram depicted in Fig The shape of the fundamental diagram clearly shows, that a global maximum for Q Q cap ρ cr ρ Figure 2.5: The fundamental diagram of traffic traffic flow exists which is called the capacity flow (Q cap =Q(ρ cr )), i.e. the function value at the critical density. The significance of the critical density and the fundamental diagram is further explained in Appendix A. A cost function for TTD should be stated using the traffic density variable. However, as no inverse function exists for the fundamental relationship, the suggested functional is not equivalent to (2.34). Nevertheless, it can be accepted as an approximation around ρ cr and used for the understanding of control goals: J ρ = t 2 x 2 t 1 x 1 (ρ cr ρ(x, t)) 2 2 dx dt min (2.37) The control objectives for the minimization of TTS and the maximization of TTD are thus stated as regulation problems with the set-point values specified for traffic density (ρ ): For TTS=min, ρ = 3 The secondary state variable (the traffic speed) has dynamics around the equilibrium speed, which is a static function of traffic density.

25 Chapter 2 Preliminaries and literature review 15 For TTD=max, ρ = ρ cr From the two possibilities an ultimate control criterion needs to be chosen. Considering a non-zero traffic demand, the objective is to maximize the capacity utilization of the road. As a result of that, the maximization of TTD is aimed, and a regulator problem for the set-point ρ = ρ cr is prescribed. Furthermore, the stabilization of traffic by the suppression of shockwaves requires the regulation for the same setpoint. For a detailed explanation, see Appendix A. In the spatiotemporally discrete framework, the cost function (2.37) takes the following form: K N s J ρ (k) = (ρ cr ρ i (k + l)) 2 2 min (2.38) l=1 i=1 where K denotes the control horizon, and N s determines the size of the network, on which the functional needs to be minimized. In the literature of the motorway control problem, different approaches have been elaborated. Certain works concentrate on the minimization of TTS (e.g. [28]). This leads, however, to an under-exploitation of road capacity. In this work, the functional in (2.38) serves as the basis of the cost function for control design Previous results of motorway traffic control In the literature of model-based motorway traffic control, a plethora of works can be found. Here the milestone papers of this field are highlighted. The first freeway control was formulated as a fix time strategy in the middle of the 196 s [24]. The feedback structure and the linear design techniques (i.e. the linear quadratic control) were first adopted for traffic regulation problems only in 1983 by Papageorgiou [25]. Feedback appeared first in a freeway traffic context with the so-called ALINEA ramp metering control in 199, (see [26] and [27]), using a PID controller. A process control method, the so-called Model Predictive Control (MPC) framework had a significant impact on traffic control in the 2 s as it is capable of handling various state and input constraints, which mean serious inconveniences in control design. As a result of the MPC control, a coordinated control was suggested for variable speed limit and ramp metering control. The first contribution of this control design concept was presented by Hegyi in [28]. In the early 21 s, an important milestone in motorway traffic modeling and control was presented by Luspay, who introduced the Linear Parameter Varying (LPV) description of the second-order motorway model in [29]. Based on the LPV model a H controller for the ramp metering problem was presented in [3]. Current work does not concentrate on developing novel control design methodologies. Instead, an application of an elaborated control method, the nonlinear model predictive control (NMPC, see [31]) is examined for a complex control problem. The developed controller is introduced in Chapter 5.

26 Chapter 2 Preliminaries and literature review Modeling of urban traffic While continuum models of motorway traffic basically describe the flow conditions along arterials, an urban network means a web of short links, where the flow is interrupted by intersections. This fundamentally different topology means that freeway traffic models cannot be adapted to urban networks. Although microscopic traffic models are capable of describing urban traffic as well, their high computational demands and the lack of data of vehicle dynamics make them unsuitable for control design. However, they are widely used for simulations, e.g. in the VISSIM traffic simulator the microscopic model of Wiedemann is used [45]. Urban models mainly focus on the conservation of vehicles rather than the flow conditions. This principle is satisfied for an analyzed area of arbitrary size (see Figure 2.6). In the discrete framework, a general form of the conservation law (2.9) can be stated as follows: N(k+1) = N(k) + T (q in (k) q out (k)) (2.39) where N(k) denotes the number of vehicles in the analyzed area, q in (k) and q out (k) denote the in- and outflow in units [veh/h], respectively, within the time period [kt, (k+1)t ] with discrete time step index k and sampling time T (see Figure 2.6). The equation states, that the change in the number of vehicles in a time period equals to the balance of inflows and outflows during the examined time period. q in (k) N(k) q out (k) Figure 2.6: Flow and conservation of vehicles in an analyzed area The size and topology of the analyzed area gives a fundamental distinction in modeling: Choosing links as the analyzed area gives the opportunity to describe dynamics around signalized intersections and to design their control. If a an entire network is chosen as the analyzed area, the aggregated flow performance of the network can be modeled and used for a high level network control. Both modeling approaches have an extensive literature. In the followings, a brief summary is given on the above defined model categories and the corresponding control problems Modeling of link dynamics and control of intersections Model description The modeling basics of traffic link dynamics were established by Gazis and Potts in the 196 s with the store-and-forward model [36]. Each analyzed link is defined between two intersections: for link j, A j is the upstream intersection and B j denotes the downstream intersection (see Figure 2.7). The inflow to link j is a sum of flows from intersection A j, whereas its outflow is determined by a signal control at intersection B j. The law of conservation (2.39) in this case is formalized as follows: N j (k+1) = N j (k) + T C (q in,j (k)) q out,j (k)) (2.4)

27 Chapter 2 Preliminaries and literature review 17 A j q in,j q out,j N j B j Figure 2.7: Link level modeling where N j is the number of vehicles in link j, q in,j denotes the total inflow to link j from intersection A j. Sampling time T C is chosen as the cycle time of the intersection signal controller. The outflow from link j is calculated by the following relationship: q out,j (k) = u j (k)s j (2.41) where S j denotes the saturation flow of link j, u j denotes the manipulated input: the green time of link j. The saturation flow of a road segment depends on a number of parameters: e.g. the width of the lane, traffic composition, etc. For a thorough analysis on saturation flow, the reader is referred to [141]. It is worth noticing that the store-and-forward approach is basically a modified queuing model, where the exit flow of the queue is determined by the green time of the lane, and the inflow to the lane is fed by the links attached to the upstream intersection. The inflow to each link, however, comes from controlled links: this fact is used in an extension of the model, suggested by Diakaki in the late 199 s. The model TUC (traffic-responsive urban control method, see [37]) is created for a control-oriented model description for a web of links. In the TUC model, the network is described by graphs. The conservation dynamics of the store-and-forward model is applied for each edge of the graph. At the graph nodes, a further notion: the turning rate is used to describe the division of traffic at nodes. Based on the turning rates, the exact knowledge of link inflows is provided. Thus, a network of multiple intersections can be modeled and controlled. The basic linear model was further improved by Lin, suggesting the S-model [38], which aims to describe flow conditions in links as a function of several variables: the intersection capacity, the number of waiting and arriving vehicles and the available space in the downstream links. The nonlinearity of the S-model results however in a high computation complexity for largescale traffic networks, especially in case of real-time MPC. Control objectives and solutions The basic problem of intersection control can be stated as follows: minimize the queues for links j, j = 1,..., n Z at intersection Z and suppress the deviation from the nominal values in green times. The control problem is given with the following cost function: K min J(k) = x Z (k + l) 2 Q + u Z (k + l) 2 R [ u Z ] l=1 (2.42) subject to: u Z,min u Z u Z,max where Q and R are positive definite weighting matrices. In eq. (2.42), x Z denotes the state vector, built from the link queues: x Z = [N 1... N nz ] T. u Z, in the form u Z = [u 1 u N 1... u n Z u N n Z ] T denotes the deviation of commanded green times [u 1... u nz ] T from the

28 Chapter 2 Preliminaries and literature review 18 nominal green times [u N 1... un n Z ] T. Lower and upper bounds of the control signal are given in (2.42) with u Z,min = [ u 1,min... u nz,min] and u Z,max = [ u 1,max... u nz,max]. The elements of the lower and upper bounds are given so that the commanded green times should be between the minimal and maximal green times. The length of the control horizon K can be determined by the network size [152] and the control method. In the TUC control, a linear quadratic (LQ) regulator is used, applying for the control horizon. This approach designs green times as a linear function of the state vector x Z. However, the LQ technique by itself is not capable of handling certain constraints on the input, i.e. the elimination of giving green light for concurrent directions. This leads to an additional optimization problem, which can be formalized as follows: min [u Z ] subject to: u Z ũ Z 2 2 u Z 1 T C T (2.43) loss u Z,min ũ Z u Z,max where T loss denotes the loss time during a cycle (summing the simultaneous red intervals when no flow is realized). Solving (2.43), the green times are found that are both optimal in terms of the original problem (2.42) and satisfy the condition on concurrent green lights. A more sophisticated solution of the link level control is solved by the model predictive control (MPC) approach, which is capable of handling input constraints. This algorithm optimizes the cost function on a finite control horizon, usually determined by the network extent. By following this line of research, several papers were published investigating the applicability of MPC for urban control problems, e.g. [39], [4] and [41]. Beside the efficiency of the MPC-based control, the computational burdens must be mentioned as well. In large-scale urban networks, the centralized MPC might not guarantee the real-time feasibility in all cases. Therefore, a distributed control scheme is proposed for signal split optimization by Tettamanti in [42]. Tettamanti also suggests a method for the robust control of intersections in [43], by considering the unmeasured disturbances of inner link sources and sinks as additive uncertainties in the states. A potential solution to increase the real-time feasibility of the S-model based control problem is provided by Lin by reformulating the online optimization into a mixed-integer linear programming problem [44] Network level modeling and control The intersection control methods perform very efficiently as traffic-responsive strategies. However, they are not able to deal with extreme traffic conditions when demands extremely overpass the network capacity for a long period of time. A plausible solution for this problem is to design a high-level control, optimizing traffic conditions on a network level through the manipulation of the entering flows. In this case, actually the demand of the network is controlled, however, at the expense of the outside network where the eliminated traffic may cause congestion. The concept of the protected network (PN) has been highlighted recently as an efficient solution to prevent traffic jams in certain networks. PNs are usually located in a city center or a dense urban area that needs protection against insatiable demands during rush hours. The topology of the network level model is demonstrated in Figure 2.8. The network is basically modeled through the overall number of vehicles in network, for which the law of conservation is applied. The flow conditions and network performance are characterized by an aggregated variable, that can be expressed as a function of the number of vehicles. The

29 Chapter 2 Preliminaries and literature review 19 Controlled gates Non-controlled gates Exit gates Figure 2.8: Network level modeling relationship between these variables is described by the theory of the urban fundamental diagram which was first proposed by [46]. The approach is also called the macroscopic fundamental diagram (MFD) or network fundamental diagram (NFD). This concept has been widely investigated during the past decades, see e.g. [47], [49], [48]. The task here is to control traffic flow through gates so that the network performance is maximized, by manipulating the green times of the controlled gates. Green times are designed based on the knowledge of the presence of non-controlled gates (which means a disturbance for the system) and the exit flow from the network, which is modeled as a function of the overall network performance. In the followings, the network level modeling is reviewed. The applied framework is based on the work of Keyvan-Ekbatani [5], as it is the most thorough model description in the literature. Model description formalized as follows: The conservation of vehicles (2.39) for a protected network can be N PN (k+1)=n PN (k)+t C [Q in (k)+q d (k) Q out (k)] (2.44) where N PN (k) denotes the number of vehicles within the protected network, and T C denotes the cycle time of the signal control. In the literature, N PN (k) is also referred as the accumulation in the network. It is given by the formula: N PN (k) = n link j=1 ρ j (k)l j (2.45) where ρ j (k) denotes the traffic density on link j, and n link is the number of links in the network. In the practice, ρ j can be obtained by appropriate estimation of link detector measurements, see e.g. [51]. The inflow to the network is served by two types of sources: from controlled and uncontrolled gates. Q in (k)= n in j=1 q in,j(k) in units of [veh/h] is the sum of the controlled vehicle inflows to the protected network with controlled gate flow q in,j of gate j, and n in denotes the number of controlled gates. Q d (k)= n d j=1 q d,j(k) is the sum of uncontrolled inflow with n d denoting the number of uncontrolled gates.

30 Chapter 2 Preliminaries and literature review 2 Q out (k)= n out j=1 q out,j(k) denotes the total outflow of the network with n out denoting the number of exit gates. The outflow is the most interesting aspect of the model, as it reflects the overall flow conditions of the network. Q out can be modeled as a linear function of the inner flow in the following form: Q out (k) = Γ L Q PN (k) (2.46) where Γ is a model parameter, L denotes the average length of network links in the PN; Q PN (k) denotes the sum of inner traffic flows in the network: Q PN (k) = n link j=1 q j (k) (2.47) where q j (k) is the traffic flow on link j. For Q PN the term production of the network is also used in the literature. A basic concept in network modeling is the so-called network fundamental diagram (NFD), suggested by Godfrey in [46] and Daganzo and Geroliminis in [49]. The NFD depicts the static relationship between the accumulation and the production of the analyzed network: Q PN (k) = F (N PN (k)) (2.48) In the literature, the accumulation and production of the network are often substituted by performances, which can be obtained by following the reasoning of Section From the accumulation in the network a normalized performance notion can be derived: the total time spent in the network, denoted by TTS PN, given in units of [veh h]. It is given with the following formula: n link TTS PN (k) = T C j=1 N j (k) (2.49) From the production of the network the total travel distance in the network can be derived, using the following formula: n link TTD PN (k) = T C j=1 q j (k)l j (2.5) where TTD PN is given in units of [veh km]. As a corollary of the above definitions, NFD can be also stated as the relationship between the times spent and travel distances: TTD PN (k) = F (TTS PN (k)) (2.51) Control problems and results for the PN concept The function F ( ), similarly to the fundamental diagram of the LWR model, describes the capacity of the network as a function of the number of vehicles. The illustration of the NFD (see Figure 2.9) helps the understanding the classic objective of the network control problem. The highest production of the network is reached at the critical number of vehicles N PN,cr : then, Q PN reaches the production capacity of the network. In this case, the exiting traffic of the network Q PN,out is also maximized due to the linear relationship in (2.47). As a corollary of the shape of the NFD, the objective function for the gating control problem can be formalized as follows: J(k)= K N PN (k + l) N PN,cr 2 2 min (2.52) l=1

31 Chapter 2 Preliminaries and literature review 21 Q PN Q PN,cap N PN,cr N PN Figure 2.9: The network fundamental diagram In the literature, a classification is made for the network control problems. The gating problem focuses on optimizing the traffic performance of one protected network only, based on the cost function (2.52). In case of a topologically inhomogeneous network, it is divided to several homogeneous subnetworks, and the dynamics of each are represented by individual NFDs. In the perimeter control problem the control task is to optimize the traffic flows among the subnetworks, so that the cost function (2.52) for each network is minimized. For both problems, several results are presented. For the basic gating problem, Daganzo introduced a control rule based on time-dependent switching conditions [52]. The work [53] analyzes the effect of different signal strategies within the network on the shape of the NFD, and proposes a control system, separating the control along the links at the boundary, and inside the network. Keyvan-Ekbatani [54] presented a linear feedback regulator (PID control) for the gating problem. In [55], a perimeter controller is suggested, by designing a flow-transmission control for a set of subnetworks using an LQ optimal controller. In [56] the optimal control problem is formulated as a mixed integer linear optimization problem, with two types of controllers: perimeter controllers, and a switching controller of fix-time signal plans. The most recent results are presented in [57] and [58], investigating the NFD-based perimeter control strategy with the use of model predictive control. 2.3 Modeling of vehicular emissions The estimation of traffic emissions has gained significant attention in the past decades in the discussion of air quality and climate change problems, due to the continued growth of motorization. Exhaust gases, such as CO, HC (unburned hydrocarbons), CO 2 and NO X (nitrous oxides) play a significant role in air pollution. Vehicular emission models are originally established for two main reasons: first, to assess the effects of automotive and petroleum industry technologies on fuel consumption and exhaust emissions; and second, to support the estimations of overall national emissions. Their presence only motivated traffic engineers in the past decade to use them as a performance criterion for the analysis and synthesis of intelligent transportation system (ITS) measures. In this section, models describing vehicular emissions are reviewed. First, the basic notions of this field are given. Then, a classification of the recognized models are carried out. Finally, an overview is presented on the use of emission models in ITS solutions.

32 Chapter 2 Preliminaries and literature review Basic notions The basic knowledge in the modeling of vehicular emissions is expressed in characteristic describing quantities, and the relationships among them. Emission factor Emission factor (denoted by ef ) is the usual output of emission models. It gives the intensity of the emission, expressing it by pollutant quantity per energy consumed, pollutant mass per fuel used, or pollutant mass per distance driven, in units of [g/kwh], [g/kg] or [g/km]. In a narrower sense, emission factors of vehicular emission models are the distance specific emissions, given in units [g/km]. Throughout the thesis, the notion emission factor is used in this narrower sense. Emission rate Apart from the distance-specific function, a useful description of vehicular pollution is the emission rate (denoted by e), which is the time-specific intensity of the outflow of pollution, given in the units of [g/h]. Its importance is straightforward considering that distance-specific emission cannot be defined for idling vehicles with zero speed. Between the emission rate and the emission factor of a moving vehicle the following relationship can be stated ([61]): e p (t) = ef p (t) v(t) (2.53) where e p (t), ef p (t) and v(t) denote the instantaneous emission rate and emission factor for pollutant p and the speed of a vehicle, respectively. Emission inventory An important aim of macroscopic emission models is to provide a standardized method for the estimation of national emissions of traffic related pollutants via emission inventories [59], listing the amount of air pollutants discharged by a certain network for a certain time period. Emission inventories contribute to the definition of air quality standards and emission ceilings. The inventory of pollutant p on network ν over time period τ is calculated as follows: E p ν,τ = VKT ν,τ ef p ν,τ (2.54) where traffic activity value VKT ν,τ denotes the total vehicle kilometers travelled (in units [vehkm]) on network ν over time period τ, and is obtained from the average mileage (daily, monthly, annual mileage) of vehicles (e.g. in [62, 63]) or traffic stream level (e.g. in [64]). ef p ν,τ is the average emission factor of pollutant p of the analyzed network ν over time period τ, obtained in the form ef p ν,τ = ef p (v ν,τ ) where v ν,τ is usually approximated by the speed limit over network ν. Relationship between fuel consumption and CO 2 emission Elaborating the reaction stochiometrics of internal combustion engines a linear connection between fuel consumption and CO 2 emission of a vehicle can be stated, see [6], [61]. ef CO 2c = κ fc c (2.55) where fc c in units [l/1km] is the fuel consumption of vehicle type c. Parameter κ is a constant coefficient, in case of Diesel fuel κ = 26.29, for gasoline κ = Thus the criterion of fuel consumption minimization is equivalent to the criterion of CO 2 minimization.

33 Chapter 2 Preliminaries and literature review Emission models The output of vehicular emission models is the emission factor in every case. A classification of the models can be carried out according to the scale of input variables. This approach highlights the level of incorporation of the driving behavior into the modeling, thus - similarly to traffic models - a level-of-detail categorization can be presented. However, one should remember that the use of average traffic variables as input variables does not mean a macroscopic emission model, as always the emission of a vehicle individual - considered an average member of traffic - is modeled. In the order of complexity, the following model categories can be distinguished [66]: Area-wide models (e.g. [67]) use data on total VKT and a single emission factor value to compute total area emissions. This low level model is used to compute emission and fuel consumption inventories. Average speed models provide emission factors as functions of speeds, obtained as average travelling speeds of driving cycles 4 [68]. Examples for this modeling class are: COPERT [69], MOBILE [7] and EMFAC [71]. Traffic situation models (e.g. HBEFA [72] or ARTEMIS [73] define classes of driving conditions (e.g. stop/go-, idling- and free-flow traffic) and for these classes different average speed model functions and thus emission relationships are stated. Cycle-variable models (e.g. MEASURE [74], VERSIT+ [75], VT-Micro [76]). In this case emission factors are functions of various driving cycle variables (e.g. instantaneous speeds, idle time, acceleration, road slopes). These models require detailed information on vehicle movements, which can only be acquired from microscopic traffic models or, from GPS measurements. Modal models produce emisson factors from engine operating conditons. Examples for this modeling class: CMEM [77], MOVES [78] or PHEM [79]. The validation and development of emission models can be carried out by several methods (see [65] and [66]). In this work, these methods are classified to direct and indirect methods. Direct methods, such as laboratory measurements and the PEMS (portable emission measurement system) method provide emission factors of specific vehicle types, obtained for specified driving cycles [68]. Laboratory measurements mean the use of chassis- or engine dynamometers (see [8], [81] and [82]). In this case, driving cycles are reproduced for a number of vehicles which are run on rollers, simulating the resistive power of drag, road slopes and the rolling resistance. The exhaust gases are then collected in sampling bags and exhaust rates are exactly monitored. This method provides an opportunity to analyze a number of vehicles under the same conditions, developing emission models of entire vehicle classes. The PEMS method means that a complete set of emission measurement instrument is carried onboard [83] or by a chasing vehicle [84]. Indirect methods of emission model development mean real-world measurements, that calculate average emission factors from measured concentrations of average traffic fleets. These methods involve the remote sensing method and tunnel studies. In case of the remote sensing method [85] IR and UV lights of specific wavelengths are used. The beam passes through the exhaust plume to a detector, and pollutant concentrations are obtained from 4 Driving cycles are standard route cycles during which a scale of operation modes of a vehicle is represented.

34 Chapter 2 Preliminaries and literature review 24 the absorption of lights with specific wavelengths. In tunnel studies [86], [87] the fluxes of pollutants are measured at the entrances/exits of tunnels, and given the wind conditions within the tunnel, average levels of emissions can be obtained Emission models used in the thesis Two emission models are used in this work. The first one is the average speed emission model COPERT [69], which provides an open-source platform for the modeling of emissions of EU countries through the specification of vehicle compositions. The model is integrated with the traffic model METANET through the macroscopic static emission function stated in Chapter 4. Throughout the thesis, the vehicle composition of Hungary for year 21 is used for this model. The coefficients of the emission factor functions are given in Appendix C. For the proposed macroscopic approach of Chapter 3, a concept verification is given on a microscopic level. For this end, the a cycle-variable model VERSIT+ [75] is used with the same vehicle composition. In the case studies, CO and NO X emissions are modeled The use of emission models in ITS applications The effect of traffic management systems on vehicular emissions have been investigated both in motorways and urban networks by several authors in the past decades. The exploitation of vehicular emission models in traffic engineering research can be realized on different levels. The first level is the use of ITS data for offline or online emission modeling. Offline emission modeling efforts mean an improved approach for making emission inventories. In [64], the effect of mean speed distributions on traffic emission inventories is examined. A similar analysis is carried out in [89]. In [62], the correlation of traffic patterns and emission data are analyzed, aiming to understand the correlation between flow levels and emissions. The work [9] compares a number of emission models in terms of the incorporation of traffic conditions and their effect on the fidelity of emission inventories. Online emission modeling may serve as to establish control strategies including emission performances as control criteria. The paper [91] analyzes the substitution of traffic variables to emission models resulting in average emission factors, normalized for vehicle unit. An interesting analysis is presented in [92], examining the effect of microscopic traffic model parameters on emissions. In [93], a bottom-up vehicle emission model is proposed to estimate real-time CO 2 emissions using real-time data. The proposed method uses loop detectors and floating car data to express average emission factors of local fleet compositions. The work [94] proposes a method to use probe vehicle data (or floating car data) to express average emissions emerging at link units. In [95] Zegeye suggests the substitution of macroscopic traffic variables in a cycle-variable model VT-Micro [76] to express average emission factors of vehicles. This approach, however, neglects the low fidelity of macroscopic acceleration [96] to individual accelerations in the classic spatial discretization scale and that the effect of acceleration on emissions at high speeds (i.e. on motorways) is negligible to the effect of speeds. On the middle-level emission models are used for the analysis of the environmental impact of ITS tools. In [97] and [98] microscopic simulation platforms are developed for the analysis of traffic control measures. Analyses are carried out regarding the influence of traffic intensity, signal coordination schemes and signal parameters on the gaseous emissions in urban networks in [99], [1] and [11]. The effects of speed limit control is examined in urban networks [12]

35 Chapter 2 Preliminaries and literature review 25 and on freeways ([13] and [14]). Furthermore, the effect of speed limit control is compared to the effect of road pricing in [15] in terms of traffic performance and emission. The incorporation of emission models to the control design is the highest level of use of emission models in traffic engineering. This was first done by Zegeye in [17] using the model presented in [95]. The same approach is presented in [18]. A variable speed limit control scheme is designed to reduce emission factors on freeways [19]. The data transferred by vehicle-to-vehicle (V2V) technologies can be best exploited for cycle variable or higher level emission models, an interesting control approach in this field is presented in [11]. It has to be pointed out, that although several approaches have been suggested to use emission models in traffic control analysis and synthesis, none of these approaches define a macroscopic description for the spatiotemporal distribution of emissions. This issue is discussed in Chapter Models of emission dispersion The understanding of the pollution dispersion in the atmosphere has been of paramount importance for decades. Emission dispersion models are primarily developed to describe the spreading of industrial pollution from point sources (e.g. [111]) and to examine the effect of buildings and rural settlements on the dispersion. Also, these models help to understand certain physical phenomena of the atmosphere (e.g. eddies) 5 as concentrations of different gases can serve as markers [112]. In these models the accurate modeling of the spatial distribution of atmospheric parameters, such as the temperature and pressure of the air, the direction and the speed of the wind play a key role, while the exact knowledge of the excitation (i.e. the emission rate of the pollution source) is supposed to be known. The modeling of roadside dispersion of gases gained attention in the seventies by the introduction of line-source modeling [113], [114]. In these models area-wide emission models are used with constant emission factors which can be considered as a macroscopic-level approach. Efforts have been made to handle the problem on vehicular level, as well, by modeling the turbulent dispersion of exhaust gases around vehicles, as a function of vehicle speed [115], [116]. For urban networks, a special method is developed, called the street-canyon modeling approach (see e.g. [117], [118]). This approach models the mixture of pollution by skimming flows [119] and the stack effect [12] in narrow streets surrounded by high buildings. Examples of street canyon models: Aeiolus [121], CPRM [122] and CAR [123]. In this work, the focus is on the emission dispersion of motorways. For motorways, roadside emission dispersion models can be applied. There are no clear-cut distinctions among modeling categories, however, a classification can be made according to the used physical principles (as suggested by Vardoulakis in [124]). Thus, dispersion models can be grouped into the following categories: CFD (Computational Fluid Dynamics) modeling CFD refers to a branch of models using numerical methods for the analysis of fluid dynamics[112]. CFD provides a high-resolution description including advanced turbulence treatment schemes, making them suitable for small-scale dispersion modeling. The 5 Eddy is the swirling flow of a fluid or gas caused by obstacles in the general flow line.

36 Chapter 2 Preliminaries and literature review 26 governing fluid flow and dispersion equations of the CFD models are derived from the basic conservation and transport principles that are the laws of mass and momentum conservation and pollutant transport principles [124]. The direct numerical simulation (DNS) for the solution of these equations requires a fine grid to capture all the relevant scales of turbulence [125]. Thus, DNS demands high computational performance the reduction of which is addressed by several techniques. A possible solution for this is the so-called eddy simulation, which means the roughening of the grid to a stable solution of the smallest eddies [125]. Another approach is the use of Reynolds averaged Navier-Stokes equations (RANS). The RANS equations are averaged in time over all turbulent scales, to directly yield the statistically steady-state solution of the mean and turbulent flows [125], [126]. According to the analyses of Zegeye [129] and He [13], the computation time of the reduced CFD models remains still so high that they are not suitable for real-time applications in spite of the efforts on reducing their computational demand. Nevertheless, their high modeling resolution makes them applicable for complex topologies, thus they are extensively used for street canyon modeling (see e.g. [125], [127], [128]). Gaussian dispersion models This modeling method dates back to the 193 s based on the work of Bosanquet and Pearson [111]. The approach is based on the observation of chimney smokes which are spreading in a plume shape. In the plume, the diffusion of gases are described perpendicular to the wind direction. The horizontal and vertical distribution of pollutant concentrations are characterized by Gaussian distributions which are parametrized by functions of atmospheric stability and wind speed. The spreading pollution is supposed to be reflected from ground surface as suggested by Sutton in [131]. Gaussian plume models are basically developed for point sources with continuous release of emissions. An extension to line sources [114], [132] makes this concept applicable for roadside dispersion modeling. However, the accuracy of line-source Gaussian dispersion models declines by decreasing the slope of the wind and the line source. A correcting term is suggested in [137]. Line-source Gaussian models thus serve as a good basis for roadside emission dispersion modeling, see e.g. the models CALINE 4 [133], CAR-FMI [134], DISPERSION21 [135] and ADMS [136]. The existing dispersion models in their current form are not applicable for control purposes. As highlighted, CFD models have an extreme computational demand; line-source Gaussian plume models reach high complexities when describing the mixing of pollution in case of nonperpendicular wind slope. Addressing the above difficulties, Zegeye [95] suggested a pointsource model, later used for model-based control. This work, however can be further improved to meet the requirements of practical applicability: to design a controller of a complex traffic system, both the traffic flow and the dispersion process have to be represented by dynamic models, and the control has to be designed for the state-space model of the joint dynamics. In this work, a simple dynamic model is proposed for roadside dispersion modeling in Chapter 4. The suggested method is a modified Gaussian dispersion model resting on the principle of mass conservation, with a simple consideration on pollution dissolution. Special topological assumptions are made to establish a realistic yet a compact model with low computational demands.

37 Chapter 3 Macroscopic static description of traffic emissions In this chapter a macroscopic description of traffic emissions is presented. The spatiotemporal distribution of emission is derived as a function of traffic variables. The continuous distribution is extended to the spatiotemporally discrete framework of motorways and urban networks. For these topologies, the discrete forms of the derived model function is verified by simulations. Based on the analysis of the suggested model function the total emission of traffic is formalized as a possible control performance. As a corollary of the performance statement, control objectives are postulated for the different pollutant types. 3.1 Problem formulation As shown in the Preliminaries (see Section 2.3.4), the existing control strategies involving emission as a performance criterion use a simplified modeling approach: average speed measurements of loop detectors and probe vehicle data are adopted as model inputs. By this approach an average emission factor of the traffic is modeled, rather than the total emitted pollution. Nevertheless, the latter quantity is more appropriate for formalizing control objectives. As it can be suspected, the emergence of pollution depends on the intensity of the traffic as well, not only on its average speed. Thus, the need arises to provide a closed form representing the spatiotemporal distribution of emission, possibly as a function of the macroscopic variables, Also, for control purposes, the total emission of traffic should be formalized as a performance, and based on an analysis, a control design problem needs to be specified. In this chapter the above described problems are addressed. The spatiotemporal distribution of emission is first derived in continuous manner as a function of the macroscopic traffic variables (and an average-speed emission model). Then, the suggested model function is analyzed in the spatiotemporally discrete framework of motorway networks, where simulations are shown to verify the modeling approach. The framework is also extended to urban networks using the notions of NFD modeling, and a simulation-based analysis is carried out on the applicability of network aggregated variables for emission modeling. The analysis of the model function leads to the statement of separate control objectives for different pollutant types, distinguished by their spatial scale of effect. 27

38 Chapter 3 Macroscopic static description of traffic emissions Derivation of the spatiotemporal distribution of emissions The purpose of this section is to provide a closed form for the spatiotemporal distribution of traffic emissions, given as a function of the macroscopic traffic variables. The variable is sought so that the total emitted pollution is obtained as its spatiotemporal aggregation. The derivation is proceeded from the review of different performance definitions. In the continuous space-time domain, traffic performances TTS (2.3) and TTD (2.26) in an arbitrary spatiotemporal rectangle [x 1, x 2 ] [t 1, t 2 ] are given as follows: TTS [x1,x 2 ] [t 1,t 2 ] = t 2 x 2 t 1 x 1 ρ(x, t) dx dt (3.1) TTD [x1,x 2 ] [t 1,t 2 ] = t 2 x 2 t 1 x 1 q(x, t) dx dt (3.2) Eqs. (3.1) and (3.2) give the performances as spatiotemporal aggregations of the density and flow fields. The modeling framework of vehicular emissions also give an opportunity to express an aggregated performance of an individual vehicle: the total emission of vehicle i for a space domain [x 1, x 2 ] can be expressed based on the emission factor function as follows: x 2 E[x i 1,x 2 ] = where v i (x) denotes the speed of vehicle i. or, during time interval [t 1, t 2 ], with the emission rate function: t 2 E[t i 1,t 2 ] = where v i (t) denotes the speed of vehicle i. x 1 ef (v i (x)) dx (3.3) t 1 e(v i (t)) dt (3.4) The definite integrals of eqs. (3.3) and (3.4) give the total emission of an individual vehicle only. The object is to calculate the total emission generated by the traffic, as an aggregated performance, similarly to the definitions in (3.1) and (3.2). The total emission of pollutant p is sought as a space- and time-parametered variable in the following form: t 2 E p [x 1,x 2 ] [t 1,t 2 ] = t 1 x 2 x 1 ε p (x, t) dx dt (3.5) where ε p (x, t) denotes the spatiotemporal distribution of emission for pollutant p, in the units of [g/km h], E p [x 1,x 2 ] [t 1,t 2 ] denotes the total emission of traffic for pollutant p in [x 1, x 2 ] [t 1, t 2 ] in units of [veh g]. Thus, the total emission of traffic as a performance is obtained in mass dimension.

39 Chapter 3 Macroscopic static description of traffic emissions 29 The problem is thus to explicitly formalize the distribution ε p (x, t) which has to satisfy the following requirements: 1. ε p (x, t) should be expressed as a function of macroscopic variable fields (q(x, t), ρ(x, t) and v(x, t)) 2. The stated variable ε p (x, t) should provide information on the traffic emission via the emission factor function of an average-speed emission model. 3. ε p (x, t) should possess information on the vehicle composition. The sought ε p (x, t) is called the emission field of pollutant p, and the objective is to provide an analytic form of ε p = f(q(x, t), ρ(x, t), v(x, t)) by using the definitions of TTS and TTD, and the emission factor function. During the derivation, it is formalized generally for any pollutant p and a homogeneous traffic composition. Hence, first requirements no. 1 and 2 are addressed. Using the velocity field v(x, t), define the function e(x, t) in the following form: e(x, t) = e(v(x, t)) (3.6) where e(v(x, t)) is given by the emission rate function (2.53) in units [g/h]. In case of the COPERT average speed emission modeling it is a rational fractional function of v(x, t). In the followings, the emission of traffic is formalized for an arbitrary [x, x + x] [t, t + t] spatiotemporal rectangle. Also, the continuity properties of traffic variables (2.7) and (2.8) are used: The traffic density is considered continuous in t and approximated by ρ([x, x + x], t) For the mean speed of traffic the time-mean speed is used which is continuous in x: v T (x, [t, t + t]). By applying the definition of e(x, t) (3.6) for the time-mean speed v T (x, t), the obtained function is continuous in x and is approximated as follows: e = e(x, [t, t + t]) Using the above definitions, the emission of traffic in [x, x + x] [t, t + t] can be formalized as the product of the average emission rate field and the total time spent in [x, x + x] [t, t + t]: E [x,x + x] [t,t + t] = ē [x,x + x] [t,t + t] TTS [x,x + x] [t,t + t] (3.7) where ē [x,x + x] [t,t + t] denotes the average emission rate field in [x, x + x] [t, t + t], calculated from the time-mean speed as follows: ē [x,x + x] [t,t + t] = e(v T [x,x + x] [t,t + t] ) By using (3.1), the expression of (3.7) can be stated as follows: E [x,x + x] [t,t + t] = 1 x t In a simple form: x + x x e(x, t ) dx t t + t t ρ(x, t) dt x (3.8) E [x,x + x] [t,t + t] = x + x x e(x, t )dx t + t t ρ(x, t)dt (3.9)

40 Chapter 3 Macroscopic static description of traffic emissions 3 Applying Clairaut s theorem [138] on the equality of mixed partials on eq. (3.9): E [x,x + x] [t,t + t] = t + t t x + x x e(x, t ) ρ(x, t) dx dt (3.1) Taking the limits x and t the indefinite integral form of (3.1) independent of x and t : E(x, t) = e(x, t) ρ(x, t) dx dt (3.11) Taking the partial derivatives of E(x, t), the sought function ε(x, t) is obtained (3.5): ε(x, t) = E(x, t) = e(x, t) ρ(x, t) (3.12) t x As a result of the above, the emission field is expressed as a function of macroscopic variables and the emission factor function ef (v): ε(x, t) = ef (v(x, t)) v(x, t) ρ(x, t) = f(v(x, t), ρ(x, t)) (3.13) The obtained function satisfies requirements no. 1 and 2. Requirement no. 3 can be addressed by the involvement of the vehicle composition in the emission field function. Traffic density can be formalized as a sum of densities of the vehicle classes: N c ρ(x, t) = ρ c (x, t) (3.14) c=1 where ρ c (x, t) denotes the density field of vehicle class c, N c denotes the number of vehicle classes. By using (3.14), the emission field of an inhomogeneous traffic can be modeled as follows: N c ε(x, t) = ef c (v c (x, t)) v c (x, t) ρ c (x, t) (3.15) c=1 where v c (x, t) denotes the velocity field of vehicle class c, ef c (v c ) denotes the emission factor function of vehicle class c. The total emission of traffic in units [g] in the spatiotemporal rectangle [x 1, x 2 ] [t 1, t 2 ] can be calculated from the following definite integral: E [x1,x 2 ] [t 1,t 2 ] = t 2 x 2 t 1 x 1 ε(x, t) dx dt = t 2 x 2 t 1 x 1 e(v(x, t)) ρ(x, t) dx dt (3.16) In the following sections, the proposed model function is applied for motorways and urban networks. Then, an analysis of the emission field function is presented in Section 3.5.

41 Chapter 3 Macroscopic static description of traffic emissions Macroscopic description of motorway network emissions In this section, the application of the emission field function is analyzed for motorway networks. The notion emission field is interpreted in the spatiotemporally discrete framework, and the performance of traffic emissions is also formalized using the loop detector measurements. A simulation-based verification is carried out to examine the consistency of the suggested macroscopic framework relative to a microscopic level approach Emission in the spatiotemporally discrete framework The discrete description of traffic emissions is proceeded from the definition of the average emission over [l; l+l i ] [kt ; (k+1)t ]. Discrete emission values can be given as an approximation of the emission field function (3.13): ε p [l;l+l i ] [kt ;(k+1)t ] = ef p (v [l;l+li ] [kt ;(k+1)t ]) v [l;l+li ] [kt ;(k+1)t ] ρ [l;l+li ] [kt ;(k+1)t ] (3.17) The space-time window [l; l+l i ] [kt ; (k+1)t ] is specified based on the spatiotemporal grid of the motorway model, for segment length L i and sampling time T. In the discrete framework, traffic states are approximated by loop detector measurements, which can be further used for emission approximation, applying them for the formula in (3.17): ε p i (k) = ef p (v i (k)) v i (k) ρ i (k) (3.18) where ε p i (k) denotes the macroscopic emission of pollutant p in segment i in discrete step k. For an inhomogeneous traffic, its macroscopic emission can be calculated by using (3.15): ε p N c i (k) = ef p c(v i,c (k)) v i,c (k) ρ i,c (k) (3.19) c=1 where N c denotes the number of vehicle types. v i,c (k) and ρ i,c (k) denote the average speed and density measurements on segment i in sample step k. The total emission of pollutant p over [l; l+l i ] [kt ; (k+1)t ] can be expressed in the same manner, giving an approximation for the integral (3.16) by using (3.17): E p [l;l+l i ] [kt ;(k+1)t ] =εp [l;l+l i ] [kt ;(k+1)t ] L i T =ef p (v [l;l+li ] [kt ;(k+1)t ]) v [l;l+li ] [kt ;(k+1)t ] ρ [l;l+li ] [kt ;(k+1)t ] L i T (3.2) Similar to (3.19), the total emission of a homogeneous traffic can be described as a function of loop detector variables: E p i (k) = ef p (v i (k)) v i (k) ρ i (k) L i T (3.21) For an inhomogeneous traffic, the total emission traffic is obtained as follows: E p N c i (k) = ef p c(v i,c (k)) v i,c (k) ρ i,c (k) L i T (3.22) c=1

42 Chapter 3 Macroscopic static description of traffic emissions Concept verification In this section, the proposed macroscopic modeling framework (based on an average speed emission model) is compared to a modal emission model in two case studies. This comparison of the emission calculations does not provide a validation for the suggested macroscopic static emission function, as the level-of-detail of the analyzed models are different. Also, different models may lead to biased results as their emission factors are validated by different data sets. The aim of the concept verification is to analyze the behavior of the macroscopic modeling approach, i.e. the fidelity to the expected changes of the emission values in different traffic conditions. The simulations are realized in the traffic simulation tool VISSIM [88]. The program is widely used for traffic signal planning and considered as a reliable, state-of-the-art modeling software. The traffic is run on a 1 km long, three-lane motorway stretch modeled as a single segment network, the measurements of which are provided by loop detector measurements. Two scenarios are examined: in the first one, an accident and a resulting bottleneck are featured. The second case study simulates the effect of variable speed limits. Emission values are expected to be lower at high speeds and low densities, and higher at low speeds and high densities (according to the static emission function (3.21)). In both cases, the macroscopic static emission function is compared to a high level reference model. Macroscopic calculation means that the total emission of traffic is calculated using loop detector measurements which are substituted to the formula (3.21) to obtain the total emission of traffic. The emission factor (obtained from the model COPERT) in the macroscopic model function is stated in Appendix C. The calculation of the high level reference model, to which the macroscopic emission calculation is compared is provided by the EnViVer module of VISSIM which is capable of calculating traffic emissions offline in a microscopic approach using the modal emission model VERSIT+ (see Section 2.3.2). The comparison is carried out for the NO X and CO pollution. MacroscopicmodelingbasedonCOPERT Microscopicmodeling(VERSIT+) Time[1s] Figure 3.1: Simulation no. 1.: traffic variables and emissions on the analyzed segment In the first simulation a complex traffic situation is modeled (see Figure 3.1), with a congestion caused by a bottleneck, which disappears after 13s. Thus, in the scenario free flow

43 Chapter 3 Macroscopic static description of traffic emissions 33 conditions (with low traffic flow volumes and high traffic speeds) are succeeded by a short period with saturated traffic (between 4s to 6s). After 6s, traffic speed decreases considerably and a congestion is formed which lasts until 14s. Finally, after the removal of the bottleneck, the congestion is dissolved. The simulation results highlight that the emission of macroscopic calculation is similar to the microscopic reference values, and shows a behavior as expected: during congestion higher emission is modeled relative to free-flow and saturating conditions. Surprisingly, the values of the calculations overlap in free flow conditions, which may be a result of the clean constantspeed traffic flow. The modeling errors are significant in transitional conditions (e.g. during 4-6s and 1-16s reaching 3%), which is a result of the neglected accelerations of average speed modeling in the macroscopic calculation. The second simulation (3.2) helps the further examination of model behavior at transitional traffic conditions: the changing of variable speed limits is modeled. Three speed limit values are operated: 11 km/h, 9 km/h and 7 km/h. The 2 km/h steps provide an opportunity to analyze the model behavior during transitions of considerable deceleration/acceleration. Traffic speed [km/h] Traffic density [veh/km] CO emission [g/s.step x km] NO X emission [g/s.step x km] Macroscopic modeling based on COPERT Microscopic modeling (VERSIT+) Time [1s] Figure 3.2: Simulation no. 2.: traffic variables and emissions on the analyzed segment Analyzing the simulation results, the macroscopic approach shows again a behavior as expected: the total emission is lower at low traffic densities and high speeds, and becomes high at low speeds and high densities. During transitions, especially during accelerations from low speeds, high modeling errors are present (e.g. at 275s), the accelerating traffic causes a significant increase in emission, which is not modeled by the macroscopic approach. A constant bias is present for constant speeds, which is as expected: the models COPERT and VERSIT+ are not necessarily fitted for the same measurement data. Nevertheless, the purpose of the simulation is not the validation of the method, only a concept verification for the modeling approach which is considered fulfilled.

44 Chapter 3 Macroscopic static description of traffic emissions 34 The simulations show, that the proposed modeling approach can be used for the modeling of emissions of motorway traffic, with acceptable behavior. The model error is fairly high at transitional states of traffic, but this is the result of the neglect of acceleration in average speed modeling. Considering the available level of vehicular data (i.e. the macroscopic average speeds), the modeling approach can be accepted for emission modeling. In the following section, the applicability of the suggested static emission function is analyzed in the NFD modeling framework (introduced in Section 2.2.2). 3.4 Macroscopic emission modeling of urban networks Akin to motorway networks, the modeling of the traffic emissions of an urban network can be based on the loop detector measurements. However, the calculation of emissions for each link of a network may lead to extreme computation demands. In case of large networks, the idea of using the NFD modeling framework (introduced in Section 2.2) arises to reduce computation times. In this section, the applicability of network aggregated parameters of urban NFD models for the suggested macroscopic emission modeling framework is analyzed. First, the static emission function is stated for the NFD framework. Then, a simulation-based analysis is carried out, similar to the concept verification shown in Section In the analysis, special attention is payed to the differences between the aggregated emission modeling and the linkwise emission modeling Emission performance in urban networks In this section, the emission of an urban network is formalized following two separate approaches: in the first one, emission is calculated for each link from loop detector measurements and the total emission of the network is an aggregation of link emissions. The second approach uses only aggregated network parameters for emission calculation. Approach 1: link-wise emission calculation This approach assumes that speed and traffic flow measurements are available for each link i in the network. In the followings, the notation L PN represents the set of network links, (characterized by their length) within the network: where L i denotes the length of link i. L PN = {L i }, i = 1,..., n link (3.23) The total emission in the network is the sum of the emissions of the network links: E LPN T C = n link i=1 E Li T C (3.24) where T C denotes the sample time. T C also equals to the signaling cycle time, during which a uniform flow distribution is supposed. Thus, the emission of link i can be given as follows: E Li T C = ef (v Li T s ) q Li T C L i T C (3.25)

45 Chapter 3 Macroscopic static description of traffic emissions 35 In the spatiotemporally discrete framework, emission of link i in sample step k: E i (k) = ef (v i (k)) q i (k) L i T C (3.26) Approach 2: emission calculation based on aggregated network parameters In case of NFD network modeling, from link measurements the aggregated variables TTS PN and TTD PN are known (using eqs and 2.5). However, average speeds are not necessarily measured for each link i. Nevertheless, the average cruising speed in unit [km/h] of the network can be expressed using the fundamental relationship among the traffic variables ((2.12): Applying performance definitions (2.49) and (2.5) to (3.27): v LPN T C = q L PN T C ρ LPN T C (3.27) v LPN T C = TTD L PN T C TTS LPN T C (3.28) The average cruising speed v PN is supposed to represent the speed conditions of the protected network. The emission emerging in the protected network during time T C can be stated as follows, using (3.28) and (3.2): E LPN T C = ef (v LPN T C ) TTD LPN T C (3.29) where v LPN T C is calculated as in (3.28). In spatiotemporally discrete form: E PN (k) = ef (v PN (k)) TTD PN (k) (3.3) where v PN (k) = TTD PN (k) TTS PN (k) (3.31) By using the above formulae, the emission of the protected network can be stated using aggregated network parameters v PN (k) and TTD PN (k). However, it needs to be analyzed, whether the network-level description of emissions is accurate enough for emission modeling. For this end, simulations are run in which the microscopic emission of the vehicles, and linkwise emissions of the traffic (3.24) are simulated alongside with the network-level emissions. The network-level values (using eqs. (3.3)) are compared to the link-wise calculations and the reference microscopic emissions Concept verification The objective of this section is twofold: first, to verify the suggested macroscopic emission modeling framework based on the aggregated network parameters (Approach 2), and second: to analyze the differences between the link-wise emission modeling and the use of aggregated variables (by stating the modeling errors between Approach 1 and 2). The behavior of the developed model framework is analyzed through simulations. The suggested model framework, i.e. the modeling of the emission of the network using aggregated traffic variables, is used alongside with the COPERT average speed model. The aggregated modeling of emission is compared to two levels of emission modeling:

46 Chapter 3 Macroscopic static description of traffic emissions 36 Emission is calculated for each link, and summed for the entire network (link-wise emission modeling). Emission factors are provided by the COPERT average speed model. As a reference, the highly accurate microscopic description is used. The emission of the entire network is calculated on a microscopic level by the VERSIT+ emission model (see [75]) via the EnViVer add-on module of VISSIM. The simulations are run with the emission factors see Appendix C. Two scenarios are used for the comparison. The first scenario represents a rush hour situation with changing traffic loads and a fixed time signal control. The second scenario realizes the same traffic load, (i.e. a rush hour traffic with changing loads), but uses a PID controller tuned to work with oscillations (see Appendix D). The oscillations provide an opportunity to analyze the model accuracy for different state values. For the simulations, the traffic simulator VISSIM is used again alongside its offline microscopic emission calculation add-on, EnViVer. The case study applies the same network as used and identified in Section 6.4. For the simulations, the following parameters are set. The sampling time is chosen as the signal controller cycle time: T C = 9s. The simulations are run for 72s. The applied PID control parameters are detailed in Appendix D. Simulation results - case study no. 1 The accuracy of the model can be best examined, if a wide range of the state domain is used within a simulation. For this end, two scenarios are modeled. First, a congested situation is presented by scenario 1. The total flow entering the network, via controlled and uncontrolled Total flow into network [veh/cycle] Number of vehicles in network TTD in network [vehkm/h] Network average speed [km/h] Time [s] Figure 3.3: Case study 1: gate inputs and number of vehicles in PN gates of scenario no. 1 are plotted in Fig. 3.3, whereas the network performances (TTD,

47 Chapter 3 Macroscopic static description of traffic emissions 37 Total CO emission in network [g/sample step] Total NO X emission in network [g/sample step] Microscopic emission, using Versit+ Link wise emission, using Copert Emission based on network average speed, using Copert Time [s] Figure 3.4: Case study 1: network emissions network average speed) are presented in Fig This basic scenario simulates congesting conditions, thus mainly low speeds are present. The model accuracy is highlighted in Fig The emission using aggregated variables (and network average speed) is very similar to the link-wise emission calculation. However, both provide low estimations, compared to the reference emissions calculated by VERSIT+. This high relative error (in average 21.3%) can be a result of the neglected effects of accelerations in average-speed emission modeling. The aim of the concept verification is not the analysis of model accuracy but the behavior of both macroscopic model approaches are acceptable. Nevertheless, the difference between the two macroscopic approaches is represented by an average of 1.9 % relative error, which can be considered negligible. Simulation results - case study no. 2 Scenario no. 2 features a simple PID controller (for the details of the controller see Appendix D), which is capable of preventing the congestion, however, causes significant oscillations. This gives an opportunity to analyze the model during transients and covering a wide range of the state domain. The gate control and disturbance signals of scenario no. 2 are plotted in Fig. 3.5, whereas the network performances (TTD, network average speed) are presented in Fig In this scenario both high and low traffic accumulation is present, thus the accuracy can be analyzed through both high and low speeds. Fig. 3.6 highlights the model accuracy. In this case, the emission using aggregated variables shows less resemblance to the link-wise emission calculation. The average model error of macroscopic model approaches, again, is considerable at almost 3% in average. This is a direct result of the speed oscillations of this scenario. Nevertheless, the difference between the two macroscopic approaches is represented again by an average

48 Chapter 3 Macroscopic static description of traffic emissions 38 Total flow into network [veh/cycle] Number of vehicles in network TTD in network [veh km/h] Network average speed [km/h] Time [s] Figure 3.5: Case study 2: gate inputs and number of vehicles in PN 7 Total CO emission in network [g/sample step] Total NO X emission in network [g/sample step] Microscopic emission, using Versit+ Link wise emission, using Copert Emission based on network average speed, using Copert Time [s] Figure 3.6: Case study 2: network emissions of 3.4% relative error, which is a meaningful result: for given link measurements networkaverage emission model shows no significant modeling errors relative to the link-wise emission modeling approach, thus it can be accepted for emission modeling.

49 Chapter 3 Macroscopic static description of traffic emissions 39 According to the verification results, the suggested macroscopic framework (based on the macroscopic static function of emission) provides data that is consistent with the microscopic model and thus can be accepted for modeling purposes both in motorways and urban networks. In the following section, a model analysis is carried out, based on which control objectives are postulated. 3.5 Analysis of the emission field function CO emission [veh g/kmxh] The surface of the emission field function (3.13) for CO is demonstrated in Figure 3.7 for the emission factor function given in Appendix C. To highlight the range of the function, function values of measured traffic states are plotted as dots. The shape of the macroscopic emission Traffic speed [km/h] Traffic density [veh/km] Figure 3.7: Emission field as a function of traffic density and traffic mean speed (or emission field) function is clearly determined by its linear dependence on traffic density. At high speeds and low traffic densities, the slope of the linear dependence is higher (because of the higher emission of single vehicles). At low speeds the slope is lower, as the emission rate of a single vehicle stays low; nevertheless, the high density of the congested traffic leads to high macroscopic emission values. This trait helps to understand the specification of control objectives for different types of pollutants. In the followings, control objectives are postulated for different pollutant types. The exact statement of control objective for the motorway framework is detailed in the control design Section 5.4. Hereunder a preliminary proposal is given, distinguishing the following pollutant types: pollutants causing global effects are responsible for the greenhouse effect, the main contributor of traffic emission is the CO2 pollution. When considering the global environmental effects of traffic, the exhaustion of fossil energy reserves also arises as a concern. However, as shown in Section 2.3.1, the control objective formalized for the CO2 emission involves the handling of fuel consumption as well.

50 Chapter 3 Macroscopic static description of traffic emissions 4 certain pollutants (e.g. NO X, CO and HC) exert their effects locally, (leading to e.g. health problems, acid rain). To ease these effects of pollution, concentration of polluting gases needs to be kept low. The control objective for global pollutants is the minimization of the total emission over an infinite horizon in space and time: J global = ε(x, t) dx dt = ef (v(x, t)) v(x, t) ρ(x, t) dx dt min (3.32) For a non-zero ρ(x, t), the improper integral is minimal, if the emission factor is minimized for all x and t. ef (v(x, t)) min, x, t (3.33) Thus, the control objective can be stated as a regulator problem for the traffic speed: J global = where the setpoint v opt is defined as follows: v opt v(x, t) 2 2 dx dt min (3.34) v opt = arg min ef (v) (3.35) Remark: the existing control policies address the minimization of traffic-averaged emission factors. Actually, for the handling of global pollution, this is the proper control objective. As the control design for this aim is considered elaborated in the literature, in this work the focus is on the control of pollutants with local effects. In the case of pollutants with local effects, the obvious aim is to minimize their concentrations. As seen in emission dispersion models, the excitation of dispersion is provided by the pollutant sources. The control objective then is to minimize the instantaneous excitation, in our case: the traffic emissions. This requirement in the spatiotemporally discrete framework means the minimization of the total emission produced in finite spatiotemporal rectangles, i.e. for any [x 1, x 2 ] [t 1, t 2 ]: t 2 x 2 J local = ε(x, t) dx dt min (3.36) x 1 t 1 By using the condition for any [x 1, x 2 ] [t 1, t 2 ], the above objective is equivalent to the minimization of the emission field function: ε(x, t) dx dt min, x, t (3.37) As ε(ρ, v) is a non-convex function, (3.37) leads to a non-convex optimization problem, the handling of which is not in the scope of this work. However, a non-equivalent cost function of (3.37) for the state variables can be used for control objective statement. The primary state variable of traffic systems is the traffic density. Exploiting that ε(ρ, v) is a monotonously increasing function of ρ, a simplified cost function can be given by J local = t 2 x 2 t 1 x 1 ρ(x, t) 2 2 dx dt min (3.38)

51 Chapter 3 Macroscopic static description of traffic emissions 41 which means a regulation for ρ=, ( x, t). This aim, however, is not always a feasible objective in traffic control, even though its satisfaction would minimize the concentrations around road networks. Nevertheless, an acceptable control problem is to keep pollutant concentrations below prescribed limits. The statement of such an objective demands the dynamic modeling of pollutant concentrations near road networks. This approach is followed for motorway networks, for which a roadside emission dispersion model is proposed in Chapter 4. The emission dispersion model is further used for motorway control presented in Chapter Conclusion and contributions A static model function for traffic emissions is suggested in this chapter. To describe the spatiotemporal distribution of emission, a variable called the emission field is derived as a function of the macroscopic variables. The main requirement towards the emission variable is that the total emission of traffic in mass dimension is obtained as its definite integral over the space-time domain. The modeling of vehicular emissions is incorporated to the model via the average-speed emission modeling method. The model function of the emission field is derived in the continuous space-time domain, and extended to the spatiotemporally discrete frameworks of motorways and urban networks for a real-time use. In the latter case, the model function is extended to the scale of aggregated model variables of NFD modeling. For both network types, a simulation-based analysis is carried out to analyze the applicability of the suggested model framework. Finally, a model analysis is carried out, concluding in preliminary suggestions on the control objectives for different pollutant types. Thesis 1 A modeling approach is proposed for the macroscopic description of traffic emissions. The spatiotemporal distribution of emission is derived as a function of macroscopic traffic variables. The advantage of the proposed approach is that it requires only the existing measurement data of traffic networks. The modeling of vehicular emissions is incorporated into the derived model function via the average-speed modeling method. The obtained continuous space-time distribution variable is extended to the discrete frameworks of motorways and the NFD-based modeling of urban networks. The suggested macroscopic modeling framework is compared to microscopic level emission modeling in terms of consistency with acceptable simulation results. As a corollary of the analysis of the suggested model function, control objectives are postulated for the different pollution types. Related publications: [CsA211], [CsA212a], [CsA212b], [CsA215c].

52 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions The aim of the chapter is to introduce a novel modeling method for the dispersion of motorway traffic emissions. After the introductory part, the derivation of the model follows. During the derivation special attention is paid to the modeling of pollution decay. The model is formalized in a spatiotemporally discrete form, and the obtained dynamic system is attached to the second-order freeway traffic model, introduced in Section 2.1. The verification of the model is followed by an analysis, in which the numerical stability issues and the sensitivity to the motorway control measures are examined. 4.1 Problem formulation The dispersion of vehicular emissions is a significant environmental problem. Exhaust gases with local effects (i.e. CO, HC or NO X ) can cause serious health issues. Therefore, the dynamic modeling of the emerging concentrations and their reduction in built-in areas needs to be addressed by traffic engineering design. As highlighted in Section 2.4, a number of high resolution models exist for the description of dispersion. However, these models have high computational complexity making them unsuitable for real-time modeling supporting an online control algorithm. The control system should be based on a model that meets rigorous requirements: on one hand, it has to be accurate, yet numerically simple enough to be used for online control purposes. On the other hand, during its development the model needs to be built up so that the notions of controller design (i.e. the control problem, the system variables and their constraints) can be clearly stated and the model can be attached to the existing motorway traffic system through appropriately formalized boundary conditions. To meet the above specified requirements, the development of a novel model is suggested to describe the dispersion of traffic emissions between motorways and local built-in areas. The research results presented in this chapter serve as a basis of a controller to keep pollutant concentrations emerging at built-in areas in the proximity of a motorway stretch under legislation limits. The topographic layout of the addressed modeling problem is illustrated in Fig The space between the motorway and the built-in area is divided to constant cross-section channels of equal width, parallel to the wind direction. In the suggested approach, the dispersion 42

53 X j Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 43 is described as a distributed parameter system (DPS) [32], incorporating the spatial and temporal variation of the process. The dispersion dynamics are modeled for each flow channel separately, which are treated as balance volumes satisfying the law of mass conservation. The excitation of the system comes from the motorway traffic emissions, described by the model function suggested in Chapter 3 and the output of the model dynamics are the pollutant concentrations at the boundary of the inhabited area. i=1... wind N s j=1... Inhabited area N b Figure 4.1: Topographic layout The modeling assumptions are detailed and justified in the following section. 4.2 Modeling assumptions In this section the modeling assumptions are outlined which will lead to a mathematical model in Section In the flow channels (also called balance volumes) the conservation law is satisfied. 2. For the boundary of the channels, the concentration is calculated from the spatiotemporal emission model proposed in Chapter Plug flow is assumed within the balance volumes. Using this assumption, the wind speed is considered the same in different elevations within the flow channels, and the effect of wind can be described by a single coordinate in wind direction. This simplification can be justified by the acceptance of moderate or higher wind speeds. 4. The pollution is ideally mixed over the cross section of the flow channels. Only axial dispersion is present through the channels. In case of modeling the dispersion of pollution in a moderate or higher wind speed and constant wind direction, the diffusion is considered to be negligible compared to the convection as diffusion does not depend on wind direction, while convection does. Using this assumption only one spatial coordinate is needed in wind direction. 5. Constant wind direction is supposed. The analysis is carried out for different dedicated wind directions that are most characteristic for the area. The model aims to describe the effect of prevailing winds, which basically have constant wind directions.

54 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions The flow channels are parallel to the wind direction and are of equal width. The reason for this is that the whole length of the built-in area is considered by separated balance volumes of equal significance. 7. The dissolution of gases is modeled by a simplified version of the Gaussian plume model. The suggested approach considers the vertical diffusion of high temperature gases. The decay rate coefficient is developed in Section Remark: The number of the flow channels depends on the wind direction. Two important points have to be satisfied: first, the flow channels are parallel to the wind direction. Second, their width is equal, according to assumption 6. However, the motorway is not necessarily straight, thus its curvature implies that a balance volume may involve a different number of motorway segments. This is highlighted in Fig. 4.1: there are balance volumes containing multiple motorway segments, while others may contain only one segment. As a result of this trait, the indices of the balance volumes (j) do not necessarily check up with those of the road segments (i). In most cases, N s N b, i.e. the number of motorway segments and that of the flow channels differ. When grouping the balance volumes, further aspects have to be considered as well: on the one hand, the extra dimensions of the system (which is the number of the balance volumes) coming from the emission dispersion modeling should be kept low because of computational reasons. On the other hand, the practical choice of flow channel positioning follows a rule that segments of coherent control decisions (e.g VSL signs) should be grouped in the same flow channel. 4.3 Derivation of model equations In this section the modeling assumptions are used to develop a mathematical model. The law of mass conservation results in a partial differential equation which is then discretized both in time and space. A key feature of the model is the description of the decay of pollution for which a simplified approach is suggested in Section Conservation of pollution masses The outlined process system can be modeled through the conservation of pollutant masses within the balance volumes. Each flow channel is considered as an autonomous balance volume with individual dynamics. In the following, the conservation is formalized for an arbitrary flow channel j. The mass balance for pollutant p in flow channel j is described through the variable m p j (x, t) (measured in the units of [g]) as a bivariate function of time ( t) and space ( x j X j ), where X j denotes the length of flow channel j. The mass conservation equation of pollutant p for balance volume j is the following [32]: m p j (x j, t) = φ p j,in t (x j, t) φ p j,out (x j, t) ψ j,dis (x j, t) (4.1) where φ p j,in (x j, t), φ p j,out (x j, t) and ψ j,dis (x j, t) in the units of [g/s] are the inflow, outflow and dissolution rate of the pollutant, respectively.

55 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 45 The inflow of pollutants at the border of the balance volume comes from the emission of the motorway segments involved in the balance volumes: N j,i φ p j,in (, t) = ε p j,i (t) L j,i (4.2) i=1 where the notation j, i refers to the segments feeding balance volume j. N j,i denotes the number of segments involved in balance volume j; ε p j,i, measured in units of [g/km h] denotes the macroscopic ( emission of motorway segment j, i (see eq. (3.17)); L j,i denotes the length of Nj,i segment j, i i=1 L j,i=l j ). The outflow of pollutants is the direct effect of the wind: where w(t) (in units of [m/s]) denotes the wind speed. φ p j,out (x j, t) = w(t) mp j (x j, t) x j (4.3) The dissolution of pollutants is described by the following formula: ψ p j,dis (x j, t) = λ p (x j, t)m p j (x j, t) (4.4) where λ p (x j, t) (in units of [s 1 ]) denotes the decay rate of the pollutant p. For the analytic approximation of λ p (x j, t) see Section In the following, the conservation of the pollutant is formalized by means of concentration in a balance volume increment V j. This volume increment represents the flow within the balance volume using the assumption of plug flow and considering a constant height and cross-section of the flow channel. The volume of the flow channel is calculated using the surface of the inhabited zone and the length of the channel. The size of volume increment V j = X j L j H j can be obtained similarly. For the parameters of a flow channel see Fig The relationship between the mass of pollutant p and its concentration in an infinitesimal X j motorway W j L j H j Figure 4.2: Flow channel parameters segment V j of balance volume j is described as: m p j (x j+ x j, t) = c p j (x j+ x j, t) V j = c p j (x j+ x j, t) H j L j x j (4.5) where c p j = cp j (x j+ x j, t) denotes the concentration of pollutant p in the balance volume increment V j, measured in units of [kg/m 3 ].

56 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 46 The outflow of the pollutant can be reformalized using (4.5): φ p j,out (x j+ x j, t) = w(t) H jl j x j c p j (x j+ x j, t) x j = w(t) c p j (x j+ x j, t)h j L j (4.6) The dissolution increment can also be easily calculated as ψ p j,dis (x j+ x j, t) = λ p (x j + x j, t) c p j (x j+ x j, t)h j L j x j (4.7) The inflow of the pollutant is described differently for the origin of the balance volume and an arbitrary volume increment V j within the balance volume. At the origin of the balance volume, the inflow of pollutant p is an external excitation, formalized as the rate of traffic emission, emerging uniformly distributed above the motorway in the volume H j L j W j. φ p j,in (+ x j, t) = Nj,i i=1 εp j,i (t) L j,i H j L j W j (4.8) For an arbitrary volume increment V j within the flow channel, the inflow to V j equals to the outflow of the previous increment: φ p j,in (x j+ x j, t) = φ p j,out (x j, t) = w(t)c p j (x j, t)h j L j (4.9) By virtue of the above formulae, the conservation for the balance volume increment V j can be formalized also for two cases. At the origin of the balance volume, substituting (4.2), (4.5), (4.6), (4.7), and (4.8) to the conservation equation (4.1) results in the following equation: H j L j x j c p j (x j, t) t N j,i = i=1 ε p j,i (t)l j,i w(t)c p j (+ x j)h j L j (4.1) λ p (+ x j )c p j (+ x j)h j L j x j By further arrangement: c p j (X j, t) t = Nj,i i=1 εp j,i (t)l j,i w(t)c p j (+ x j)h j L j H j L j x j λ p (+ x j )c p j (+ x j) (4.11) Within the balance volume, substituting (4.2), (4.5), (4.6), (4.7), and (4.9) to the conservation equation (4.1): H j L j x j c p j (x j, t) t Further arrangement results in the following form: = w(t)c p j (x j, t)h j L j w(t)c p j (x j+ x j, t)h j L j λ p (x j + x j, t)c p j (x j+ x j, t)h j L j x j (4.12) c p j (x j, t) t = w(t) cp j (x j+ x j, t) c p j (x j, t) x j λ p (x j + x j, t)c p j (x j+ x j, t) (4.13)

57 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 47 Formalizing (4.11) and (4.13) in one continuous partial differential equation by taking the limit x j : c p j (x j, t) = w(t) cp j (x j, t) λ p (x j, t)c p j t x (x j, t) (4.14) j The boundary condition of the partial differential equation (PDE) in (4.14) is obtained from (4.11) w(t) cp j (x j, t) Nj,i i=1 = εp j,i (t) L j,i (4.15) x j H j L j W j xj = Thus, the dispersion dynamics is now stated as a partial differential equation (4.14) with boundary conditions as given in (4.15) Normed dimensionless form In the following, a normed dimensionless form of the dynamic equation (4.14) is formalized. The reasons for this step are the following: The distances X j are not equal. For the pollutants, the maximal allowed concentrations are different. In the dimensionless form, ratios to the concentration regulations are used. Thus the equal weighting of the pollutants in the control design is simplified. The following normalized terms are introduced: Thus, in differential form: For the concentrations: ˆx j = x j X j, ˆx j [, 1] j (4.16) x j = X j ˆx j (4.17) ĉ p j (x j, t) = cp j (x j, t) c p, ĉ p j (x j, t) [, 1] j (4.18) max where c p max denotes the maximal allowed concentration of pollutant p. Relative concentrations in differential form: c p j (x j, t) = c p max ĉ p j (x j, t) (4.19) By substituting the dimensionless groups, the PDE (4.14) can be reformalized as: ĉ p j (x j, t) t = w(t) ĉp j (x j, t) X j ˆx λ p(x j, t)ĉ p j (x j, t) (4.2) Eq. (4.2) describes the dynamics of the relative concentration balance of pollutant p in balance volume j.

58 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 48 The normalized form of the boundary conditions can be stated similarly: w(t) ĉp j (x j, t) x j = xj = Nj,i i=1 εp j,i (t) L j,i H j L j W j c p max (4.21) Spatial and temporal discretization Spatial discretization In the following, the continuous PDE (4.2) is converted to a set of ordinary difference equations. First, by lumping the system, a set of ordinary differential equations is obtained. The lumped model of the DPS is a finite approximation in the spatial variable with the temporal variable remaining the only independent variable in the lumped system. For the sake of simplicity, each balance volume is considered as a single lump. The number of lumps in a flow channel is determined by using Assumptions 4, 5 and 6 in Subsection 4.2. The aim of lumping is to turn the PDE-s of the model to ordinary differential equations. In case of plug flow (Assumption 6), there is a linear relationship between the spatial and temporal variable where the ratio between the space and time coordinate is the speed of the medium. In our case, the wind speed does not change along in wind direction as a result of plug flow modeling. By using that the cross-section of the balance volume is also constant as a function of the downwind distance (Assumption 5), the same result is obtained for the pollutant concentration at the end of the balance volume, regardless the choice of the number of lumps. As the number of dynamic equations in the model increases by the number of modeled lumps, a minimal extension of system dynamics can be achieved by choosing only one lump per balance volume. The numeric solution of the PDE (4.2) needs to be consistent with its analytic solution. The PDE in eq. (4.2) is an inhomogeneous advection equation, the solution of which is constant along a characteristic that is equal to the transport velocity [139]. The transport velocity in our case is the wind speed, which is considered always positive in the spatial direction. In numerical solutions, the forward propagating information of the positive characteristics can be represented via the backward difference method. By using the first order approximation of backward difference we obtain: f (x l j) = f(xl j ) x j j ) f(xl j ) f(xl 1 x l j, xl 1 j where the spatial points x l j and xl 1 j are the boundary points of the lth lump. Then the system dynamics of the lumped system in lump l of balance volume j are as follows: dĉ p,l j (t) dt w(t)ĉp,l j = w(t)ĉp,l 1 j = (t) ĉp,l 1(t) j X j (ˆx l j ˆxl 1 j ) (t) ĉ p,l (t) X j j λ l p(t)ĉ p,l j (t) λ l p(t)ĉ p,l j (t) (4.22) In our case, the number of lumps l=1, thus x l j xl 1 j =X j, and ĉ p,l 1 j (t)=ĉ p, j (t) is the relative concentration at the origin of balance volume j - the boundary condition of the PDE. Finally,

59 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 49 we obtain the following model equation for the single lump: dĉ p j (t) dt w(t)ĉp, j (t) ĉ p j = (t) λ p (t)ĉ p j (t) (4.23) X j Temporal discretization The ordinary differential equation (4.23) describes the dynamics of the relative concentration at the boundary of the inhabited area. However, the model structure of the motorway system is spatiotemporally discrete. To embed the emission dispersion dynamics into the existing traffic system (see Appendix A), the differential equations of the balance volumes need to be turned to difference equations. This is carried out by using finite difference approximation in the time domain. Differences are calculated based on the temporal increment of the original traffic system by using the sample time T. The difference equation of the conservation in balance volume j at discrete time step k: ĉ p j (k + 1) ĉp j (k) T w(k)ĉp, j (k) ĉ p j = (k) λ p (k)ĉ p j (k) (4.24) X j From the difference, the discrete dynamics of relative concentration in balance volume j for time step k is in the form: ( w(k)ĉp, ĉ p j (k + 1) = ĉp j (k) + T j (k) ĉ p j (k) ) λ p (k)ĉ p j (k) X j (4.25) Initial and boundary conditions In order to get a well-posed initial value problem, boundary conditions ((4.26)) and initial values for the PDE (4.14) should be transformed to conditions for its spatiotemporal, discretized version: The condition at the boundary ĉ p, j (k) is obtained from (4.15), and formalized as: T ĉ p, j (k) = Nj,i i=1 εp j,i (k) L j,i H j L j W j c p max, (4.26) i.e. the concentration appearing at sample step k is the temporal integral of the macroscopic emission rate within the volume H j L j W j during time step k. Initial condition: c p j (x j, ) =, x j [, X j ] (4.27) Thus, with eqs. (4.26) and (4.27) the discrete dynamics of relative concentrations at the boundary of the inhabited area are formalized. Accompanying eq. (4.25), the dynamics of the dispersion process is stated in spatiotemporally discrete form. In the following section, the modeling of decay rate coefficient λ p is elaborated.

60 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions Determination of decay rate λ According to the modeling assumptions, the dissolution of the pollution within balance volumes is modeled as a linear function of the concentration with the coefficient decay rate λ p, see (4.4). It is of key importance to provide an accurate estimation for λ p for the exact modeling of pollution dispersion. In this section an analytic approximation is proposed. For the approximation, the Gaussian plume dispersion model [111] is used. The model assumes a point source of emission from which the pollution is dispersed by diffusion in plume shape (see Fig. 4.3) and there are no chemical or other removal processes taking place. The change in concentration distribution is the result of the extension of high temperature gases. The Gaussian plume model gives a distribution function of the concentration within the plume. The dispersion is modeled by the following equation: [ ] c p (x, t)= Qp (t) 1 [ ] exp y2 1 w(t) 2πσ y (x) 2σy(x) 2 dy exp z2 2πσ z (x) 2σz(x) 2 dz (4.28) where the concentration c p (x, t) (in units of [g/m 3 ]) is obtained as a function of the pollutant emission rate Q p (t) (measured in [g/s]) and the wind speed w(t) (in [m/s]). Parameters σ y (x) and σ z (x) denote the crosswind- and vertical direction standard deviations of the concentration distribution at downwind distance x, respectively, and H plume denotes the height of the plume centerline. The Gaussian plume model is adopted under the following 3σ y (x) 3σ z (x) z w x H plume y Figure 4.3: Schematic representation of the Gaussian plume assumptions: 1. For all pollutants, λ p = λ is assumed equal. 2. A simplified model is considered: the pollution disperses only in vertical (z) direction (see Fig. 4.4). (This condition eliminates the cross-effects among the flow channels / preserves the independence of concentration dynamics within the flow channels.) 3. Mass flux decrease is described via the pollution getting outside balance volume boundaries. 4. The origin of the plume is at half the height of the flow channel. This assumption checks up with the condition of ideal mixing over the cross section of the flow channels.

61 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions The pollution that attempts to move outside the balance volume in surface direction is reflected from the ground and remains in the balance volume. (The same assumption is suggested in [131].) 6. Wind speed is only the function of time, and is constant along the balance volume, i.e. w(x j, t) = w(t). 7. Decay rate is calculated for different atmospheric stability classes (each corresponds to an approximately constant wind speed). By using Assumption 2. and 6. for eq. (4.28), the concentration in the simplified plume can be given as follows: [ ] c p (x, t) = Qp (t) 1 exp z2 w(t) 2πσ z (x) 2σz(x) 2 dz (4.29) It is important to notice that within the plume the conservation law is satisfied as the improper integral of the Gaussian bell function equals to 1 regardless the value of σ z. The loss of pollution is modeled by the decrease of mass within the intersection of the plume and the balance volume (Assumption 3). The decay rate λ p is derived based on the downwind change of the mass flux of pollution. Mass flux is calculated using the concentrations of the intersection of the balance volume and the plume (highlighted as red in Fig. 4.4). The mass flux at point x j of balance volume j is φ p (x j, t) = w(t)c p (x j, t) (4.3) w z H j H j /2 φ p 1 φ p 2 3σ z1 3σ z2 x 1 x 2 x Figure 4.4: Flux decrease in the simplified plume model Based on the Gaussian plume approach, by using eq. (4.29), the relative change in mass flux between point 1 and 2 along balance volume j can be written as follows: φ p (x 1, t) φ p (x 2, t) = φ(x 1, t) 1 σ z (x 1, t)2π 1 = σ z (x 2, t)2π exp z 2 2σ 2 z(x 1, t) dz 1 Hj /2 σ z (x 2, t)2π 1 σ z (x 1, t)2π exp z 2 2σz(x 2 1, t) dz exp z 2 2σ 2 z(x 2, t) dz Note, that eq. (4.31) incorporates Assumptions 4 and 5. H j /2 exp z 2 2σ 2 z(x 2, t) dz (4.31)

62 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 52 Using the assumption of small variations in wind speed, the propagation time from point 1 to point 2 can be calculated based on the distance of the points (x 2 x 1 ): T wind12 (t) = x 2 x 1 w(t) (4.32) The decay rate λ can be stated as the relative change in mass flux during the propagation time (using (4.31) and (4.32)): λ j (t) = (φp (x 1, t) φ p (x 2, t))/φ p (x 1, t) T wind12 (t) = 1 σ z (x 2, t)2π z 2 2σz(x 2 2, t) dz w(t) (4.33) x 2 x 1 Hj /2 exp For a particular flow channel j, x 1 = and x 2 =X j is substituted to (4.33). Then the decay rate of flow channel j is a function of its length and the wind speed: λ j (t) = w(t) Hj /2 z 2 exp X j σ z (X j, t)2π 2σz(X 2 dz (4.34) j, t) Parameter σ z (x j, t) is a function of the centerline distance from the source (x) in the form: σ z (x j, t) = a(w(t))x b(w(t)) j (4.35) Based on the wind speed, different atmospheric stability classes are specified and for them, the parameter values a and b are shown in Table 4.1 (Assumption 7). Stability class Wind speed [m/s] a b A B C D Table 4.1: Parameter values of equation (4.35). Source: [14] Parameter σ z (X j, t) and thus λ(t) is a bivariate function of wind speed w(t) and downwind distance X j, piecewise constant in w(t) and continuous in X j. By using the formula of σ z (X j, t) (4.35), λ(t) can be computed as follows: λ(t) = w(t) X j a(w(t))x b(w(t)) j 2π Hj /2 exp z 2 2a(w(t))X 2b(w(t)) j dz (4.36) Thus, an explicit form of the decay rate coefficient λ(t) is obtained. For the bivariate function plot of λ see Fig The state-space model of the joint traffic-emission dispersion system In this section, the developed dispersion model is attached to the existing dynamic traffic model (given in Appendix B) and recast in a state-space form.

63 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 53 Figure 4.5: Decay rate as a function of wind speed and downwind distance State dynamics The emission dispersion system is in the form of a DAE system (differential-algebraic equation system). Based on eqs. (4.27), (4.26), (4.25) and (4.36), the initial value problem describing the dynamics of the emission dispersion model is as follows: ( w(k)ĉp, ĉ p j (k + 1) = ĉp j (k) + T j (k) ĉ p j (k) ) λ(x j, w(k))ĉ p j (k) where λ(k) is obtained from the following formula: X j (4.37) λ(k) = w(k) X j a(w(k))x b(w(k)) j 2π Hj /2 exp z 2 2a(w(k))X 2b(w(k)) j dz (4.38) The problem is well-posed if both the initial values and conditions at the boundary are given. Initial values are stated in the following form: c p j (x j, ) =, x j [, X j ] (4.39) The condition at the boundary, ĉ p, j (k) is obtained from the following formula: T ĉ p, j (k) = Nj,i i=1 εp j,i (k) L j,i H j L j W j c p max (4.4) The proposed process model is attached to the dynamic traffic model (see Appendix (B)). The original state space form of traffic dynamics is extended by the above emission dispersion model carrying a certain degree of freedom (N b ). The dimension size of concentration dynamics N b depends on the wind direction and topographic characteristics. The two dynamics are coupled by the boundary conditions of the concentration dynamics as it is formalized as the function of the traffic variables. The state equations of the overall

64 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 54 traffic-emission dispersion system are in the following form: ρ 1 (k+1) v 1 (k+1) l 1 (k+1)... ρ i (k+1) v i (k+1) l i (k+1)... ρ Ns (k+1) = v Ns (k+1) l Ns (k+1) ĉ p 1 (k+1)... ĉ p N b (k+1)... + ( T τ VSL 1(k) exp ( T τ VSL i(k) exp ( T τ VSL 1 N s (k) exp a ρ 1 (k) + T L 1 [ ρ 1 (k)v 1 (k)] v 1 (k) T τ v 1(k) T L v 1(k)v 1 (k) ηt ρ 2(k) ρ 1(k) τl 1 ρ 1(k)+κ l 1 (k)... ρ i (k) + T L i [ρ i 1 (k)v i 1 (k) ρ i (k)v i (k)] v i (k) T τ v i(k) + T L i v i (k) (v i 1 (k) v i (k)) ηt ρ i+1(k) ρ i(k) τl i ρ i(k)+κ l i (k)... ρ Ns (k) + T L Ns [ρ Ns 1(k)v Ns 1(k) ρ Ns (k)v Ns (k)] v Ns (k) T τ v N s (k)+ T L Ns v Ns (k) (v Ns 1(k) v Ns (k)) ηt ρ Ns+1 (k) ρ Ns (k) τl Ns ρ Ns (k)+κ ( l Ns (k) ) ĉ p 1 (k) + T w(k) ĉp, 1 (ρ1(k),v1(k),...,ρ Ns (k),v Ns (k)) ĉp 1 (k) X 1 λ 1 (k)ĉ p 1 (k)... ) ĉ p N b (k)+t (w(k) ĉp, (ρ N b 1(k),v 1(k),...,ρ Ns (k),v Ns (k)) ĉ p (k) N b X Nb λ Nb (k)ĉ p N b (k) 1 a T ( ρ1(k) L 1 r 1 (k) ) a ) ρ cr T r 1 (k)... T L i r i (k) ( ) a ) 1 ρi(k) a ρ cr T r i (k)... T L Ns r Ns (k) ( ) a ) ρns (k) ρ cr T r Ns (k) δt r 1(k)v 1(k) τl 1 ρ 1(k)+κ δt r i(k)v i(k) τl i ρ i(k)+κ... δt r Ns (k)v Ns (k) τl Ns ρ Ns (k)+κ + T L 1 q (k) T L 1 v 1 (k)v (k) T d 1 (k)... T d i (k)... ηt ρ Ns+1 (k) τl Ns ρ Ns (k)+κ T d Ns (k) (4.41) The notation... implies that the last block below the dashed line containing the concentration dynamics has to be repeated for all the pollutants (p = {CO, HC, NO X }) System variables As state variables, the traffic density ρ i, the traffic mean speed v i of each road segment i, and the relative concentrations ĉ p j of each flow channel j are considered: [ ] x(k)= ρ 1 (k), v 1 (k), l 1 (k),..., ρ Ns (k), v Ns (k), l Ns (k), ĉ p 1 (k),..., ĉp N b (k),... R 3Ns+3N b The disturbances are collected in the following vector: d(k)= [q (k), v (k), ρ Ns+1(k), w(k), d 1 (k),..., d Ns (k)] T R Ns+4 where q and v denote the traffic flow and speed of the upstream segment of the motorway stretch, ρ Ns+1 denotes the traffic density of the downstream segment of the network. These variables at the boundary of the network are considered as disturbances.

65 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 55 The input vector is in the form: u(k)= [r 1 (k),..., r Ns (k), VSL 1 (k),..., VSL Ns (k)] T R 2Ns where r i and VSL i denote the on-ramp and the dynamic speed limit of segment i. It is important to note that all control inputs are not necessarily present. The highest dimension of the control input (2N s ) is stated. Measured outputs are in the output vector y(k): y(k)= [ρ 1 (k), v 1 (k), l 1 (k),..., ρ Ns (k), v Ns (k), l Ns (k)] T R 3Ns 4.5 Model verification The aim of this section is to determine if the suggested joint model is behaving as expected in different situations. The traffic model itself has been verified and analyzed in several works (see e.g. [26]), thus the emphasis is on the emission dispersion model. For its verification, two scenarios are analyzed: in the first one, the effect of traffic variables (i.e. the conditions at the boundary of the dispersion dynamics) are examined. The second scenario highlights the effects of changes in wind speed. The first scenario features a motorway rush hour with changing loads, also with a shockwave occurence. The simulation network is a 1 km long, four-lane motorway stretch split to ten segments of equal length. No control is applied on the network. Wind direction is perpendicular to the motorway, and for the sake of simplicity, flow channels are connected to the motorway segments their numbering is chosen accordingly. The parameters of the flow channels: X j =1m; H j =3m; L j =1m, j. The disturbances of the traffic system 22 q [veh/h] v [km/h] ρ 11 [veh/km] Upstream disturbance traffic flow Upstream disturbance traffic speed Downstream disturbance traffic density Time [s] Figure 4.6: Disturbances of verification scenario no. 1 during the simulation are plotted in Fig A constant w=4m/s value is used for wind speed. The steadiness of the rush hour traffic is changing at two points: first at 1s, upstream traffic load drops for a period of 1s. Then, at 36s, high density occurs downstram, and

66 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 56 leads to a backwards propagating shockwave. The simulation period is chosen for 72 s, thus the transients in concentration dynamics can be clearly observed. Simulation results Traffic density [veh/km] Segment no Time [s] Figure 4.7: Traffic density of verification scenario no. 1.2 CO concentration [g/m 3 ] Balance volume no Time [s] Figure 4.8: CO concentration values of verification scenario no. 1 are as expected: the density profile gets lower during the drop of upstream traffic load (see Fig. 4.7). The shockwave appears around 36s, and propagates through the network, at the last segment causing a constant traffic jam. The concentration profile (see Fig. 4.8) of the simulation meets the expectations: during lower traffic load, lower densities and lower concentrations appear. The shockwave leads to higher densities, thus high concentration values are present in the according flow channels. In the second scenario the effect of sudden squalls is analyzed. The traffic load is constant and represents a rush hour demand, but without a shockwave. The disturbances of the system during the simulation are plotted in Fig. 4.9 involving the wind speed, in which a squall is present from 12s to 24s. The analysis of simulation results is narrowed for the concentration values, as a constant traffic density profile was obtained for the simulation. In

67 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 57 v [km/h] ρ 11 [veh/km] w [m/s] Upstream disturbance traffic speed q [veh/h]215 8 Upstream disturbance traffic flow Downstream disturbance traffic density Wind disturbance Time [s] Figure 4.9: Disturbances of verification scenario no. 2 the concentration profile, the effect of the squall is observable with a transient. The length of the transient depends on the wind speed: the first change, with high wind speed leads to a fast transient, whereas the second change results in slow wind speed, and a slow transient can be observed. The simulation results of wind speed changes thus meets the engineering expectations. 8 x CO concentration [g/m 3 ] Balance volume no Time [s] Figure 4.1: CO concentration values of verification scenario no Model analysis The aim of this section is twofold: first, a numerical stability analysis is carried out to determine the adequacy of the spatial and temporal discretization. Second, the sensitivity analysis highlights the effects of the control measures (i.e. the variable speed limits (VSL)

68 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 58 and the ramp metering (RM)) on the concentration values supporting the control design, that will be presented in Chapter Numerical analysis As the proposed process model describes the DPS with finite difference approximations both in space and time, the satisfaction of the Courant-Friedrichs-Lewy (CFL) condition [33] needs to be analyzed. Considering, that a maximum of w max =8[m/s]=29[km/h] wind speed is assumed (Pasquill-Gifford stability class D - during daytime, with moderate solar radiation), the CFL condition: T w max 1, j X j Thus, the shortest distance for X j can be calculated as follows: w max T X j In our case, X j 8.56m - this is the minimal distance of inhabited areas from the motorway for which the lumping and sampling assumptions respect the CFL condition Sensitivity analysis In this section, the effects of the motorway traffic control inputs (i.e. dynamic speed limits and ramp metering) are analyzed on the pollutant concentrations. The case study is utilized to obtain preliminary information on the controller structure design, i.e. to test the efficiency of the control inputs. Low concentration values are expected to be present in case of low main lane densities, which can be provided by low ramp inputs or high dynamic speed limits in case of no congestion. The sensitivity analysis also serves to quantify the effect of control measures on the states. The analysis is carried out for a single, 1 km long four-lane motorway stretch with all four lanes heading to the same direction, and the CO concentration at the border of a builtin area 1 km far from the road is investigated for a 1.5 hour long period. This rather simple yet realistic network is designed so that the simulation results can serve for a clear-cut quantitative analysis. The parameters of the flow channel: X j =1m; H j =3m; L j =1m. A rush hour situation is modeled with no congestion. The reason for this is that pollutant concentrations are necessarily exceeded in case of traffic jams as seen in section 4.5, and the presence of congestion may shadow the effect of control measures on the concentrations. On the other hand, in case of a congestion, the importance of traffic stabilization is superior to pollution reduction and the controller needs to be developed accordingly. The analysis of the effect of control measures on pollutant concentrations are thus analyzed in stable conditions. Traffic variables without control are chosen to correspond to an average flow on the main lane and the ramp: q=18veh/h/lane and r=4veh/h/lane. An average traffic density with ρ=25veh/km/lane is used with mean speed v=9km/h. Throughout the simulation, constant wind direction and speed (w=4m/s) is present, perpendicular to the road. Constant traffic composition is supposed, and the concentration of CO is modeled (for the emission factor function, see Appendix C.

69 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 59 r [veh/h] ρ [veh/km] 4 2 Ramp control Traffic density v [km/h] l [veh] 1 Traffic speed Ramp queue Time [s] Figure 4.11: Simulation of different ramp inputs - traffic variables E CO, main lane [g/s. step x segment] E CO, ramp [g/s. step x segment] c CO [g/m 3 ] CO emission of main lane CO emission of ramp queue x 1 3 CO concentration Time [s] Figure 4.12: Simulation of different ramp inputs - emissions and emerging concentration For the analysis of ramp metering, the control input is changed in 1veh/h steps. Its effect on the traffic variables is illustrated in Fig During the simulations, the length of the ramp queue and its emission are also modeled. Fig shows that emission of the ramp queue has a lower order of magnitude than that of the main lane. However, when keeping ramp flows low for long periods, the emerging concentrations start to increase due to the increasing emission of the ramp queue. Considering, that in spite of that the case study features an extreme limitation of ramp inputs resulting in a queue length of 3 vehicles after 48 s, the effect of ramp queues on the emerging concentrations is negligible. The simulation

70 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions 6 shows, that the lower ramp input is set, the lower concentration can be maintained. Lower boundary conditions of low main lane densities explain this phenomenon. It is clear, that for the minimization of concentrations, the minimal ramp input is required. By the total withdrawal of the ramp traffic (4veh/h, which is 1/4 of the total traffic demand), almost 25% decrease of the concentration can be achieved. Thus, the reduction of ramp input can result in proportionally lower concentration values, however, with transients. The length of the transients (appearing at input switches around 1s, 2s, etc) depends on the wind speed and needs further analysis. The recommendation for ramp metering control is thus to design the input value so that the state constraints of concentrations are satisfied while ramp queues are minimized. VSL [km/h] 15 1 Dynamic speed limit ρ [veh/km] v [km/h] E CO [g/s.step x segm.] Traffic density Traffic speed CO emission c CO [g/m 3 ].1 CO concentration Time [s] Figure 4.13: Simulation of different VSL inputs The result of variable speed limit analysis (shown in Fig. 4.13) is similar to that of the ramp metering: the lowest concentration levels can be achieved by keeping the main lane density as low as possible. However, this needs an opposite regulation to that of the ramp metering by keeping speed limits high: as the decreasing of speed limits increases traffic density, the no speed limit case provides the best concentration values. The sensitivity analysis agrees with the engineering expectations: while ramp metering reduces concentrations as well, the stabilizing effect of variable speed limits entails an increase in concentration. Thus, the recommendation for VSL control is to keep speed limit values as high as possible to maintain low concentrations. The sensitivity analysis shows that it is hard to handle pollutant concentration as a control objective because traffic stabilizing interventions do not necessarily improve concentration levels. This observation suggests not to incorporate the minimization of the pollution concentrations into the control goal, but instead keeping them under constraints, specified by the international legislation limits. In case of a threat of congestion, a separate controller should be used to stabilize traffic. In this case, concentration limits are necessarily neglected. For the controller design, see Chapter 5.

71 Chapter 4 Dynamic model for the dispersion of motorway traffic emissions Conclusion and contributions A simple dynamic dispersion model is introduced for motorway traffic emissions in this chapter. The process model is formalized for the distributed parameter system, and is derived based on the mass conservation laws within balance volumes, specified by the wind direction. In the balance volumes plug flow is considered with changing wind speed. For the boundary conditions of the model, the output of the macroscopic static emission model, introduced in Chapter 3 is used. The developed partial differential equation is converted to a set of ordinary differential equations by lumping the system. These differential equations are then reformalized by finite difference approximation resulting in a model being discrete both in space and time. For modeling the decay rate, an analytical approximation is suggested based on a simplified version of the Gaussian plume model. The discretization is analyzed in terms of computational stability following the CFL condition. On the joint traffic-emission dispersion model a sensitivity analysis is performed which justifies the idea of considering the states of concentration dynamics to be kept under constraints in the controller design. Thesis 2 A dynamic model is developed for the description of emission dispersion of vehicular pollutants of motorway traffic. The process dynamics is formalized based on the law of mass conservation within the balance volumes defined between the motorway and the border of the built-in area. The excitation of the system is described by the static model function suggested in Thesis 1. Supposing a constant wind direction with changing wind speed, plug flow is considered within the balance volumes. The dissolution of pollution is stated as a linear function of the concentration, the decay rate coefficient is derived using a modified version of the Gaussian line source plume model. The mathematical formalization of the modelling assumptions leads to a hyperbolic PDE, the numeric solution of which is given based on process characteristics and topological considerations. The proposed model is analyzed in terms of computational stability and sensitivity to the control measures of motorway control systems, leading to preliminary considerations on control design. Related publications: [CsA213a], [CsA215b].

72 Chapter 5 Hybrid control of traffic flow stabilization and pollution reduction of motorways Previously, in Chapters 3 and 4 the modeling framework of pollutant concentrations have been established. In the sequel, these results are implemented in a control system framework. The conventional traffic stabilizing control problem (detailed in Section 2.1.3) is extended by the task of limiting pollutant concentrations. For the solution of this complex control problem, a hybrid controller is designed, presented in the chapter. 5.1 Problem formulation The complexity of traffic engineering can be best observed in the evolution of the problems in motorway traffic control. The basic control problem for traffic stabilization and capacity maximization was first addressed in [25], and a number of methods have been proposed for improving controller performance in the same problem. In the short past, attention has been drawn to the concept of sustainable development, the most significant goal of which in road traffic is the reduction of fuel consumption and pollutant emissions. By adding these requirements to the basic problem, a multicriteria control problem can be formalized. Following this approach, remarkable results have been shown, e.g. in [17] and [18]. However, these control strategies focus on the minimizing of normalized emission factor functions of the traffic, which is - as shown in Section only usable for pollutants with global effects. The control of emerging concentrations has only been considered once as a control problem in [95]. Here, a multicriteria approach was followed as well. However, the minimization of concentrations is not necessary by all means, a satisfactory goal is keeping them under the prescribed limits. In this work, the complex control problem is separated to the basic objectives: in case of congestion, the control goal is to stabilize traffic; whereas in stable conditions, the limitation of pollutant concentrations is aimed together with the maximizing of the traffic flow. This separation can be conveniently realized, as the domain of the primary state variable (i.e. the traffic density) can be divided to stable and unstable regimes. However, attention has to be payed to the switching rule between the control modes. In this chapter the above described control approach is realized. The controller is based on the joint traffic-emission dispersion system, presented in Chapter 4. After the statement 62

73 Chapter 5 Hybrid control of motorway traffic flow 63 of control objectives, the control system structure is reviewed, with special attention to the switching rule between the modes. The control modes are specified through the utilized control measures, the cost functions and the constraints. The proposed controller is evaluated in case studies. 5.2 Control objective statement As mentioned in the previous section, the contemplated controller has two tasks. It is basically used for keeping pollutant concentrations under legislation limits, and is also capable of suppressing shockwave formulation in case of high downstream disturbances. The sensitivity analysis of the emission dispersion model, presented in Chapter 4, showed that it is hard to handle concentration regulation as a control objective in a control system structure as traffic stabilizing interventions do not necessarily improve concentration levels. Therefore, the task of pollution reduction needs to be separated from the traffic stabilization task, and different controllers should be designed featuring different control inputs. For the outlined complex control problem a two-mode controller is suggested. In both modes, control input is designed by the nonlinear model predictive control (NMPC) method (see [31]), however, with different cost functions and constraints. The following two control modes are suggested: In case of stable traffic conditions, (under the critical density) the controller aims to keep the concentrations below legislation limits using the ramp metering only. State constraints are defined as follows: c p j (k) cp limit The cost function of the control is specified so that the ramp queues are minimized. Thus, its main term is postulated in the following form for a motorway network of N s segments: K N s 1 2 J stable (k)= l i (k) min (5.1) l k=1 i=1 i,max 2 where l i,max denotes the maximal queue length of ramp i and K is the control horizon. In case of unstable traffic conditions (above the critical density), concentration constraints are neglected and a traffic stabilizing controller is used. This controller uses both ramp metering and variable speed limits. As shown in Section 2.1.3, traffic stabilization is possible by regulation to the critical density. Thus, the key element of the cost function for this case is as follows: K N s 1 2 J unstable (k)= (ρ i (k) ρ cr ) min (5.2) ρ jam ρ cr k=1 i=1 where ρ jam denotes the jam density (i.e. the maximal attainable traffic density). The proposed two-mode controller has to handle both the uncongested and congested situations, aiming for the appropriate control objective. The key feature of the controller is the switching rule set that is able to recognize the neccessity of switching in both directions based on the system states. The controllers are embedded in a hybrid automata model framework that also involves the switching logic, outlined in section 5.3. The controller modes are detailed in Sections 5.4 and

74 Chapter 5 Hybrid control of motorway traffic flow Control system structure In this section the operation of the two-mode controller is outlined by the formal description of the control system structure. A system is considered a hybrid system if it combines subsystems that are continuous in behavior (having a continuous set of states) with discrete event subsystems (having discrete states only). In our case, the subsystem with continuous behavior is the motorway traffic system, and the discrete event system is the subsystem that assigns the control mode (control for concentration/congestion). The switching rule is thus given through the discrete event system specification. The discrete event system is represented by a finite automata model, which is designed based on both analytical considerations and heuristic design. Following the conventional formal description [32], a hybrid system is described by the elements: HA=(C HA, D HA ) (5.3) where HA denotes the hybrid automata model, which is composed of the discrete event system D HA and the continuous system C HA. The continuous state system C HA of the structure is the controlled system. The nonlinear state dynamics are described by eq. (4.41). The closed loop systems of the control modes are structurally consistent, however, the applied control inputs differ. This is described as equality constraints in the optimal control problems. For the controllers, see Sections 5.4 and 5.5. The discrete event system is described by a finite automata model in the form of D HA =(Q HA, Σ HA, δ HA ) (5.4) where Q HA is a set of states that correspond to different operational regions of the continuous state system. In our case, Q HA ={ congested traffic, free flow } (5.5) The operating modes of the controller are determined accordingly: Controller mode 1 works in case of free flow. Controller mode 2 works in case of congested traffic. Σ HA is the set of input elements of the finite automata that consists of autonomous plant events: Σ HA ={ forming congestion, dissolving congestion, saturation } (5.6) δ HA is the state transition function that describes the possible operational regions that occur as a result of the state-event pairs: δ HA : Q HA Σ HA Q HA (5.7)

75 Chapter 5 Hybrid control of motorway traffic flow 65 In our case, δ HA ( free flow, forming congestion )= congested traffic δ HA ( congested traffic, forming congestion )= congested traffic δ HA ( congested traffic, dissolving congestion )= free flow δ HA ( free flow, dissolving congestion )= free flow δ HA ( congested traffic, saturation )= congested traffic δ HA ( free flow, saturation )= free flow (5.8) The state transition diagram of the discrete event system D HA is shown in Figure 5.1. Figure 5.1: State transition diagram The events that form the set Σ HA in (5.6) are determined as follows. For the traffic process system values around the critical density are chosen to exploit the hysteresis characteristics of the system. The decision values are chosen based on manual tuning following a set of test runs. forming congestion = max ρ n ρ sat,high <n N s dissolving congestion = max ρ n ρ sat,low (5.9) <n N s saturation = ρ sat,low max ρ n ρ sat,high <n N s For the case study system, the decision values are chosen based on manual tuning following a set of test runs for the case study system (see Appendix B). As a result of the tuning, the choice of ρ sat,low =23[veh/km] and ρ sat,high =28[veh/km] is made. Note, that the decision values ρ sat,low and ρ sat,high highly depend on both the critical density parameter ρ cr and the shape of the fundamental diagram. 5.4 Controller mode no. 1 - ramp metering for concentration limitation In this subsection, the first mode of the controller - working under stable traffic conditions - is presented through the definition of its cost function and constraints. Certain specifications of the controller modes are common, these are detailed in Section Cost function The cost function of the controller handles the operation of the ramp and the resulting ramp queue. Thus it formalizes the objective to allow as much of the ramp demand as possible to the main lane so that the concentration constraint is fulfilled while the ramp queue is

76 Chapter 5 Hybrid control of motorway traffic flow 66 minimized. K N s J(k)= 1 2 l i (k+l) l l=1 i=1 i,max 2 1 +ω 1 S (d 2 ) (5.1) i(k+l) r i (k+l)) 1 + ω 2 2 S (r 2 i(k+l 1) r i (k+l)) min 2 where S denotes the saturation flow (S=18 [veh/h)], the highest possible flow of a starting traffic 1. The constant parameters ω i work as weighting parameters and are chosen so that the magnitude of the cost function elements are the same order. (By this choice, ω 1 =1, ω 2 =.1 for the case study system, see Appendix B). There are three terms in the cost function: The first term, in the form of l i (k+l) 2 2 aims to minimize the ramp queue. The second term, in the form of d i (k+l) r i (k+l) 2 2 instantaneous high demands on the ramp. aims to involve the effect of The third term, in the form of r i (k+l 1) r i (k+l) 2 2 of the ramp input. aims to minimize the variation Constraints The following constraints are set for the system variables (states, inputs and disturbances). State constraints The specification of state constraints is a key element of the design of the first controller mode as they represent the prescribed concentration limits: c p j (k + l) cp limit (5.11) for each l=1,..., K sample step and each j=1,..., N b balance volume. c p limit is determined by the contribution of the motorway to the local pollution of pollutant p, the value is usually a fraction of the legislation limit. However, the constraint (5.11) can not be directly prescribed for the concentrations, since the optimization problem is ill-conditioned for these states. In Chapter 4, a low sensitivity of the concentration dynamics was obtained for both control inputs r i and VSL i. A solution for the problem is to apply state constraints on the traffic variables. This can be realized through the boundary concentration values of the emission dispersion model as they can also be considered as external excitations of the system (i.e. the emission inputs). By solving the discrete concentration dynamic equation (4.25) for steady-state conditions, the maximal emission can be expressed for a specified c p limit concentration constraint as an external excitation: ε limit (k) = c p limit H w(k) + λ j (k)x j jl j W j (5.12) w(k) 1 The saturation flow of a road segment depends on a number of parameters: e.g. the width of the lane, traffic composition, etc. Here, an average value is used. For a thorough analysis on saturation flow, the reader is referred to [141].

77 Chapter 5 Hybrid control of motorway traffic flow 67 According to eq. (3.13), the macroscopic emission ε is a function of traffic states and by further analysis, the state constraints can be chosen by using measurement data of the system and applying them for eqs. (3.13) and (5.12). By illustrating the macroscopic emission function, useful observations can be taken (see Fig. 5.2). In Fig. 5.2, a special view of the function (3.13) is plotted: the value of emissions are represented in color scale for the corresponding traffic state (speed and density) pairs. Also, measured values of the traffic states are highlighted with dots. The primary state variable of the system is the traffic density, with the speed as a secondary variable, expressing its dynamics around the equilibrium speed function. Below the critical density (i.e. free flow conditions), the static dependence of momentum on traffic density is present in a dominant way. As a conclusion, below the critical density (25 veh/km) which is the operation domain of this controller mode - traffic speed, and also, macroscopic emission (which is a function of traffic density and speed) can be accurately approximated as a function of the density variable only. Hence, the maximal macroscopic emission can also be represented as a function of the traffic density, and correspondingly, the state constraints should be specified for that variable Traffic speed [km/h] Traffic density [veh/km] 1 12 Figure 5.2: Macroscopic emission function εco [g/km h] The state constraint is thus given so that the highest emission level is a function of traffic density. For this function, the envelope of the highest emission values for pollutant p can be considered by substituting traffic measurement data to the emission function (3.13): εpmax = max εpi (ρi, vi ) vi (5.13) For each pollutant p, the maximal traffic density can be obtained as follows: p ρpi = arg(emax (ρ)) (5.14) The ultimate density constraint for segment i is the lowest of the density bounds, calculated in (5.14): ρi (k + `) min ρpi (k) p (5.15)

78 Chapter 5 Hybrid control of motorway traffic flow 68 for each k sample step and each l = 1,..., K control horizon step. Disturbances Throughout the control horizon, constant disturbance values are considered: q (k + l)=q (k) v (k + l)=v (k) ρ Ns+1(k + l)=ρ Ns+1(k) w(k + l)=w(k) (5.16) for each k sample step and each l=1,..., K control horizon step. Input constraints Ramp metering is constrained between zero and the possible admissible traffic flow: r i (k + l) min {S, 36 l i T } (5.17) for each k sample step and each l = 1,..., K control horizon step. The formula 36 l i T the highest admissible flow from the existing queue i in units [veh/h]. gives In this mode the VSL control inputs are fixed to: VSL i (k + l)=13, i=1,..., N s, l=1,..., K (5.18) 5.5 Controller mode no. 2 - traffic stabilization In this section, the detailed description of controller mode 2 is given. The mode is active if any segment of the controlled network gets to the unstable domain Cost function The cost function represents the goal of this control mode: the stabilization of the main lane states. It is formalized as follows: K N s J(k)= 1 2 N s (ρ i (k+l) ρ cr ) + w ρ l=1 i=1 jam ρ cr 1 1 S (d 2 i(k+l) r i (k+l)) 2 i=1 2 N s +ω (VSL i (k+l) VSL i (k+l 1)) (5.19) VSL i=1 max VSL min 2 N s 1 ) +ω (VSL i (k+l) VSL i+1 (k+l)) VSL i=1 max VSL min min 2 where VSL max and VSL min denote the maximal and minimal speed limit, respectively (in our case, VSL max =13 km/h, VSL min =6 km/h). Shockwaves appear in case of significant density differences with a ρ>ρ cr downstream density. Cost function (5.19) eliminates shockwaves through the reduction of downstream densities to the critical value. For the review of shockwave suppression approaches, see Appendix A. The required behavior of the controller is implied through different terms of the cost function:

79 Chapter 5 Hybrid control of motorway traffic flow 69 The most important element of the cost function is the first term ρ i (k+l) ρ cr 2 2, which realizes the regulator problem for traffic stabilization. The length of ramp queues are not involved explicitly. Nevertheless, the second term, in the form of d i (k+l) r i (k+l) 2 2 implicitly minimizes queue lengths in case of extreme d i loads. The last two terms (stated as VSL i (k+l) VSL i (k+l 1) 2 2 and VSL i (k+l) VSL i+1 (k+l) 2 2 ) are responsible for the reduction of spatial and temporal oscillation in VSL signals. The post-switching transient stability of both control modes are ensured with the appropriate weighting of the control signals. While oscillation of ramp metering does not lead to main lane instabilities, the VSL oscillation weighting is of key importance, as extreme spatial and temporal differences in VSL values may lead to shockwaves. The values of the tuned weighting parameters for the case study traffic system are as follows: w 1 =.2, w 2 =.2, w 3 = Constraints A positivity constraint is set on the states: x(k + l), l=1,..., K (5.2) Similarly to controller mode 1, throughout the control horizon, constant disturbance values are considered (see (5.16)). Input constraints For the ramp control: the same conditions are used as in controller mode no. 1, see (5.17). For the VSL control input a discrete set of control input is defined: VSL i (k + l) {6, 7, 8, 9, 1, 11, 12, 13}, i=1,..., N s, l=1,..., K (5.21) The design of the optimal speed limits is carried out in a continuous manner, the applied control is chosen by rounding the designed input to the possible discrete values. Nevertheless, for the elimination of VSL input oscillation a two-step optimization is carried out. In the first step, optimal input is calculated, considering continuous set for VSL input. Then, input variables for VSL control are rounded to the elements of the discrete set. After setting the VSL values, another optimization is run for the ramp control considering the fixed VSL values as input constraints. 5.6 Case studies In this section the developed controller is analyzed in three case studies. In these scenarios, the controller is evaluated in terms of the following important aspects: first, controller mode 1 is analyzed by two scenarios. In the first one, the tracking of the concentration limits and the satisfying of overlapping constraint are analyzed; then, the fulfilling of constant concentration limits in case of considerably changing traffic loads. Finally, the switching rule and controller mode 2 is tested in a complex traffic situation, featuring a shockwave.

80 Chapter 5 Hybrid control of motorway traffic flow 7 The simulation network is a 1 km long, four-lane motorway stretch, with a ramp at the first segment. Wind direction is perpendicular to the motorway, and the parameters of the flow channel are equally X j =1m; H j =3m; L j =1m. A constant w=4m/s value is used for wind speed. The case study network serves as a basis for quantitative analyses of the proposed control system. The modeled topology is a simplified representation of realistic networks with one ramp only. This topologic simplification, however, does not degrade the case study problem in terms of required control performance. Since realistic networks have multiple ramps on which ramp disturbances appear distributed, a higher number of manipulable inputs is available for their control. An important yet delicate point of the case studies is the specification of the concentration limits, as motorway traffic cannot be considered the sole source of air pollution. In our approach, we follow the figures of [142]. Road transport was responsible for 31.7% of the air pollution of the EU in 212. Considering that 5% of traffic emissions come from motorway networks, the contribution of this road type can be estimated as 15.85% to overall pollution. The legislation limit of CO for 1-hour daytime period is.3 g/m 3, thus the fraction that motorway traffic can reach is.47 g/m 3. Hence, the concentration limits are prescribed around this value. The optimization of the nonlinear model predictive control is carried out by the active set algorithm (via fmincon in MATLAB, as suggested in [31]). For prediction horizon length, the value K=5 is chosen, any further expansion of the prediction horizon provides no significant improvement of control performance, however leads to high computational requirements and thus long simulation periods. In the first case only ramp metering, and in the second case both the ramp metering and the variable speed limits (VSL) are used as manipulated input variables Case study 1: different concentration limits during rush hour The first case study features a rush hour scenario with constant traffic loads, and a prescription of changing concentration limits. The disturbances of the traffic system during the simulation are plotted in Fig. 5.3a. The concentration limits for the rush hour scenario are illustrated in Fig. 5.3b. q [veh/h] v [km/h] ρ 11 [veh/km] d 1 [veh/km] Upstream disturbance traffic flow Upstream disturbance traffic speed Downstream disturbance traffic density Ramp demand Time [1s] (a) Boundary conditions of case study 1 (b) Concentration limits of case study 1 Figure 5.3: Results of case study 1 The simulation period is chosen for 3 hours, thus the transients in concentration dynamics can be clearly observed. The concentration profile results of the uncontrolled scenario are plotted in Fig During the first 36 s, concentration limits are obeyed thus no control

81 Chapter 5 Hybrid control of motorway traffic flow 71 intervention is expected. Later on smaller concentration levels are specified, and the lowest possibly compliable limit is searched during the controlled simulations. Figure 5.4: CO concentration limits and values - uncontrolled case Fig. 5.5 shows the ramp input of the controlled case. Input values change at the shifting points of the concentration constraint. In case of the lowest concentration value the ramp is closed. CO concentrations for the controlled case are plotted in Fig The results show that the Ramp flow [veh/h] Ramp demand Ramp control Ramp queue [veh] Time [1s] Figure 5.5: Ramp control and ramp queue controller accurately meets the admissible state constraint requirements down to.4 g/m 3 values. However, arbitrary state constraints are not reachable because of the positive input constraint of the ramp controller. Yet, satisfying concentration limit prescriptions may lead to long ramp queues (almost 8 vehicles in 3 hours, see Fig. 5.5).