Multicriteria ramp metering control considering environmental and traffic aspects

15th IFAC Workshop on Control Applications of Optimization The International Federation of Automatic Control September 13-16, 2012. Rimini, Italy

Multicriteria ramp metering control considering environmental and traffic aspects Alfr´ ed Csik´ os ∗ Istv´ an Varga ∗∗ ∗

Department of Control and Transport Automation, Budapest University of Technology and Economics, Hungary Bertalan L. u. 2., H-1111 Budapest, Hungary ∗∗ Systems and Control Laboratory, Computer and Automation Research Institute, Hungarian Academy of Sciences Kende u. 13-17, H-1111 Budapest, Hungary Abstract: In this paper a multicriteria ramp metering control is designed considering three criteria: traffic performance and the local– and global effects of traffic emission. For realtime emission modeling a macroscopic modeling framework is proposed using an average-speed emission model. Based on the proposed model function cost functions regarding the global and local effects of traffic emission are stated. Traffic control is realized via ramp metering and the control input is optimized by using the NMPC technique. The closed-loop simulation of the designed control is carried out in VISSIM microscopic simulator. Keywords: freeway traffic; real-time traffic emission modeling; ramp metering; multicriteria control; nonlinear MPC 1. INTRODUCTION Following modern policies, engineers are prompted to optimize the process of road traffic not only in terms of travel times but also considering environmental impacts. A variety of control strategies have been used to moderate traffic congestions but the idea to create an environmentally optimal traffic is a new policy for traffic control engineering. The main goal is to maintain a process that is optimal both in terms of social costs and environmental aspects through a multicriteria control design. In this paper a multicriteria optimal traffic control design is detailed for motorway networks. The multicriteria control design is based on macroscopic traffic and emission models. While traffic models and their applications for model-based control are elaborated in Payne [1971], Hegyi et al. [2005], Papamichail et al. [2008], range of emission models for control purposes is not wide yet. As measurement and intervention is executed on macroscopic level, macroscopic emission modeling is needed. Macroscopic models (EMFAC, Copert) see CARB [1991], Ntziachristos et al. [2000] use aggregated traffic variables (such as V KT - vehicle kilometers traveled and emission factors as functions of the average speed input) to estimate network-wide fuel consumption and emissions of long time periods. This modeling approach was motivated by the need of emission inventories Smit et al. [2010] and Yao et al. [2011], control possibilities of road traffic were not considered. However, in the traffic control framework real-time data are available thus a modeling real-time emergence of traffic emission is possible. In Xia and Shao [2005], Zegeye et al. [2009] emission models of different levels were used for control purposes, and although 978-3-902823-14-4/12/$20.00 © 2012 IFAC

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these model functions rest on theoretic considerations, their analyses are not profound. For emission modeling the macroscopic emission modeling framework is utilized introduced in Csikos and Varga [2011] which is based on an average speed emission model. Using this model, traffic emission can be formalized as control criterion concerning both its local and global effects. The proposed control approach is designed for freeway environment using ramp metering. Models describing freeway traffic are nonlinear, therefore a nonlinear model predictive control is designed, based on the work of Gr¨ une and Pannek [2011]. The designed control is applied on a stretch model of the Hungarian freeway M5, and the closed loop control is realized in a VISSIM-MatLab environment. The control design is presented in four sections. After the introductory section, preliminaries regarding emission modeling and macroscopic traffic description are summarized. In Section 4 the control framework is outlined and cost functions are formalized. Then, in Section 5 closedloop simulations are presented. Finally, in Section 6 conclusions and further research directions are stated. 2. MACROSCOPIC TRAFFIC MODELING Freeway traffic is most commonly described by macroscopic traffic models. This approach handles traffic as a compressible fluid neglecting individual vehicle dynamics and describing it by aggregated variables. Road traffic variables are bivariate functions of space (x) and time (t): traffic flow (denoted by q(x, t) = [veh/h]), traffic density (denoted by ρ(x, t) = [veh/km] and space mean speed of traffic (denoted by v(x, t) = [km/h]). These continuous variables are approximated by discrete values in time and 10.3182/20120913-4-IT-4027.00011

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space by loop detector measurements (for measurement resolutions see Cremer and Papageorgiou [1981]).

3. EMISSION MODELING

First order models characterize flow speed as a static function of traffic density (3), whereas second-order modeling (see Payne [1971]) engages a speed momentum equation (4) in addition to the equilibrium relation (3) of traffic mean speed and density. Equations of the second-order model regarding segment i of length L during a discrete time step k of length T are as follows:

In the macroscopic traffic control problem, the available driving pattern information is quite small: only average traffic speed data are available from loop detector measurements. In Csikos et al. [2011] it is shown that macroscopic approximation of vehicle accelerations at high speeds are of low fidelity, so the use of an average speed model is reasonable. Thus, an average-speed model is embedded into the macroscopic modeling framework.

- conservation equation: T ρi (k + 1) = ρi (k) + [qi−1 (k) − qi (k) + ri (k) − si (k)] (1) L

3.1 Average-speed emission modeling

- fundamental equation: qi (k) = ρi (k)vi (k)

(2)

- equilibrium speed equation – formula approximating the traffic dynamics of the software VISSIM:

ρ (k)−b2 2 ρ (k)−b1 2 − i c − i c 1 2 + a2 · e (3) V (ρi (k)) = a1 · e - momentum equation: T vi (k + 1) = vi (k) + (V (ρi (k) − vi (k))) τ T + vi (k) (vi−1 (k) − vi (k)) L ηT ρi+1 (k) − ρi (k) δT ri (k)vi (k) − − τL ρi (k) + κ τ L ρi (k) + κ

In case of average speed emission modeling, the sole model input variable is average speed as registered average speeds of the function are using different driving patterns Smit et al. [2008a]. Model outputs are the emission factors of different pollutants [g/km] which are m-order convex polynomial functions of the average speed devised by certain driving profiles. The emission factor functions are specific for different vehicle classes, fuel types, Euro norms and engine capacities. For a vehicle type c and pollutant p, emission factor function is usually (or can be approximated) as m-order polynomials of average speed: p,c p,c m ef p,c = αm v + αm−1 v m−1 + ... + α0p,c = [g/km]

(4)

where qi , ρi and vi denote respectively the flow, traffic density and mean speed of segment i, ri denotes the flow of the on-ramps on segment i, si the flow of the off-ramps of segment i. In eq. (3) and (4) a1 , a2 , b1 , b2 , c1 , c2 , τ , η, δ, κ are constant model parameters, for their estimation see Section 4.1. In the following, a special interpretation of traffic variables is highlighted. Derivation of the definitions is detailed in Ashton [1966]. Consider a homogeneous traffic moving along a motorway and analyze the traffic variables in the short motorway segment [l0 ; l0 + L] and a short period of time [t0 ; t0 + T ] (analysis in a spatiotemporal window of size L × T ). Average traffic density in a spatiotemporal window L × T is equal to the Total Time Spent (T T S) in that window: T T SL×T (5) L·T where ρL×T denotes the traffic density measured in L × T . Average traffic flow in the spatiotemporal window is equal to the Total Travel Distance (T T D) in that window: ρ[l0 ;l0 +L]×[t0 ;t0 +T ] = ρL×T =

T T DL×T (6) L·T where qL×T denotes the traffic flow measured in L × T . By using (5) and (6) traffic performance in a spatiotemporal window can be expressed by the size of the window and the average traffic variables within. In small windows, from loop detector measurements using relationships (5) and (6), real-time traffic performance can be expressed. In Section 4.2 ’emission performance’ is formalized for small rectangles based on the emission inventory approach. q[l0 ;l0 +L]×[t0 ;t0 +T ] = qL×T =

195

(7)

For heterogeneous traffic, an average emission factor can be formalized as: Nc X γc ef p,c = [g/km] (8) ef p = c=1

Emission can be analyzed by its temporal rate (emission rate function e) or throughout a journey, by its spatial rate (emission factor function ef ). The relationship between emission rate ep (t) and emission factor functions ef p (t) for pollutant p is as follows Tiwary and Colls [2010]: epj = efjp vj (9) 3.2 Emission inventories and formulation of the real-time emission model framework The original aim of macroscopic emission modeling is to provide a standardized method for the estimation of national emissions of traffic related pollutants via emission inventories. Emission inventories list the amount of air pollutants discharged by a certain network, for a certain time period. Emission of pollutant p on network ν over time period τ is calculated as follows: p Eν,τ = ef p (vν,τ )V KTν,τ (10) where ef p (vν,τ ) is the average emission factor of pollutant p of the analyzed period obtained from the average-speed emission model (using average speed level vν,τ of the network ν over time τ ). The ’traffic activity’ variable V KTν,τ denotes vehicle kilometers travelled [vehkm] on network ν over time period τ , and is obtained from the average (daily, monthly or annual) mileage of vehicles Liu et al. [2011], Yao et al. [2011] or traffic stream level Smit et al. [2008b]. Applying the method of emission inventories on small spatiotemporal rectangles, real-time modeling of traffic emissions can be formalized with the following contexture:

CAO 2012 September 13-16, 2012. Rimini, Italy

Calculate emission factors using traffic mean speed obtained from real-time measurements of the spatiotemporal window L × T : ef p (vν,τ ) = ef p (vL×T ) (11) Substitute T T D to V KT , using (6). Thus, emission inventory in a small spatiotemporal segment T × L using real-time macroscopic variables: p EL×T = ef p (vL×T ) · T T DL×T = ef p (vL×T ) · qL×T · L · T (12) Substitute the fundamental relationship (2): p EL×T = ef p (vL×T ) · vL×T · ρL×T · L · T (13) Use the macroscopic meaning of (9): p EL×T = ep (vL×T )·T T SL×T = ep (vL×T )·ρL×T ·L·T (14) Using the notations introduced in Section 2, real-time macroscopic emission of pollutant p in the spatiotemporal window L × T – in section i, during time step k: TS p m+1 p m Eip (k) = Li ρi (k) αm vi (k) + αm vi (k) + ... 3600 p +α0 vi (k)) = [vehg/samplestep · segment] (15) This notation is according to the formula derived and validated in Csikos and Varga [2011]. In case of inhomogeneous traffic composition, the formula is as in (16). As single-variable polynomials of order m (in this case the emission factors) constitute an m-dimensional linear space, the real-time macroscopic emission function turns out as a linear combination of the functions of traffic fractions containing different vehicle types: Nc X γc (k)Eip,c (k) = [vehg/samplestep · segment] Eip (k) = c=1

(16) where c denotes vehicle type, Nc the number of vehicle types, γc the proportion of vehicle class c of whole traffic. 4. CONTROL DESIGN

In our work, a model predictive control is designed for the nonlinear system model. Model Predictive Control (MPC) is a control method that calculates optimal input of a dynamic system throughout a certain control horizon N , using predictions on future system dynamics (in our case using (1)–(4)) and future disturbances. Based on the predicted state-, disturbance- and input values, predefined objective functions are calculated, and optimized over the prediction horizon. The optimal control input is obtained from the optimal input sequence that minimizes the objective function. MPC is used in a rolling horizon manner, i.e. optimization is carried out and optimal control sequence is obtained in each sample step k, for a horizon of size N , but only the first element of the control sequence is applied on the system, and the optimization is repeated for the same horizon length N in each step. Calculation of the optimal input is possible analytically (turning cost functions to linear programming) or using on-line optimization tools (e.g. fmincon in the MatLab environment). The major advantage of MPC is that constraints on control input (upper and lower bound) and the nonlinear nature of the system can be handled. In this work the nonlinear MPC (NMPC) technique of Gr¨ une and Pannek [2011] is applied on the nonlinear system model of eqs. (1)-(4). 196

MPC is frequently used for traffic control purposes (see Tettamanti et al. [2011], Papamichail et al. [2008], Hegyi et al. [2005]). For emission dispersion modeling, Zegeye et al. [2011] created an MPC framework as well, but closed loop simulations were not applied to real traffic systems or simulation tools. In our work, the designed control is applied in a VISSIM environment, on a model stretch of the Hungarian freeway M5. The layout of the controlled system is illustrated on Fig. 1. Ramp metering is used thus control input is the flow of the ramp. The outputs of the controlled system are the density and speed variables of the modeled segments. Disturbances of the system are the speed and flow variables of the antecedent segment, and the density of the subsequent segment (see eq. (1)- (4)).

VISSIM System Control input

(M5 motorway stretch)

Measurements (states, disturbances)

Optimization

Cost function

Constraints

Nonlinear system model

MatLab Fig. 1. Controlled system loop 4.1 System model in VISSIM The first task of the control design is to build the macroscopic traffic model of VISSIM on which the control design can be applied. VISSIM is a microscopic simulation tool but the control framework is macroscopic, hence the corresponding model on macroscopic level is needed. VISSIM uses the microscopic model of Wiedemann [1974], which is a non-continuous function of speed and spacing, thus the analytic derivation of a matching continuous equilibrium speed model (3) (by using the method shown in Gazis [2002] is not possible, parameter estimation is needed for the parameters in equations (3) and (4)). For parameters L and T the following choice is made based on network characteristics: T = 10 s, L = 500 m. These values satisfy the Courant-Friedrichs-Lewy condition (see in Courant et al. [1928]). The estimation of the remaining parameters is carried out in batch mode. First, equilibrium speed function was fitted to the available simulation data according to the formula (3). Then, parameters of the momentum equation are estimated in a least squares manner. The estimated parameter values of the analyzed functions are summarized in the Appendix in Table 1. 4.2 Objective functions of optimal control Multicriteria optimization in this work is carried out by the minimization of an objective function that is formalized as a weighted sum of different performance criteria. Three performance criteria are invoked to the design: traffic

CAO 2012 September 13-16, 2012. Rimini, Italy

performance, minimization of emission of pollutants with local effects in each sample step, and the minimization of emission of pollutants with global effects throughout the simulation period. The evaluation of traffic performance is possible using different cost functions. In Hegyi et al. [2005] and Papamichail et al. [2008] total time spent (TTS) is minimized throughout the simulation period. However, it is shown in Section 2 (Eq. (5)) that TTS is minimized if traffic density on the main lane is minimized, thus free-flow conditions are maintained, but in this case capacity of the freeway is not fully exploited, traffic flow is minimized as well. Thus, the vehicles on the main lane travel with high speeds, but also with high spacing values. This can be maintained by keeping traffic at the origins idling. Although in the works Hegyi et al. [2005] and Papamichail et al. [2008] idling traffic at the ramps are considered, a more simple and more suggestive performance function is needed. In Section 2 the relationship between traffic flow and the other traffic performance: total travel distance (TTD) is shown. TTD is maximized, if capacity of the road is exploited, thus traffic flow is maximal. This is fulfilled if traffic density is at the critical density (ρcrit ). In the performance function, only TTD on the main lane is involved, travel distances on the ramps are neglected, as they are travel distances outside the freeway. Traffic performance function is formalized as follows: Ni X K X V (ρcrit ) · ρcrit → min (17) Jtr = qi (k) i=1

modeling which is not in the scope of this paper; however, the model function total emission rate can be utilized also for immission modeling as it provides exact data of emission distribution in space and time. Thus, cost function of the emission of local pollutant p is as follows: Nc K X n X X E p,c (v(k), ρ(k)) p → min (19) γc Jlocal = p,c Emax c=1 i=1 k=1

Cost function Jlocal is minimized if the bivariate total emission rate function (16) is minimal for each control time step in the network. This is not equal to minimizing the sum of emission throughout the simulation period thus the overall amount of pollutants (which was the aim for optimizing the emission of pollutants with global effects). The minimum of Jlocal is at [v; ρ] = [vf ree ; 0] regardless of pollutants and traffic composition. Thus the control task is to minimize traffic density for the controlled network. Cost function (19) provides the weighting function for the control design. The cost function of the multicriteria control is formalized as follows: Np Np X X p p J = αtr Jtr + αglobal Jglobal + αlocal Jlocal → min p=1

p=1

(20) where αi denotes the weighting parameters of the performance criteria i. 5. SIMULATION

k=1

where Ni denotes the number of segments in the controlled network, K denotes the simulation horizon.

When considering global effects caused by traffic, two main points arise: the exhaustion of fossil energy reserves because of high fuel consumption and the greenhouse effect. The most important exhaust gases responsible for global warming are CO2 and NOX , but using GW P (Global Warming Potential Elrod [1999]) factors the effect of each exhaust gas can be characterized, thus all of them can be considered and put into the overall weighting function of emission optimization. Emission leading to global effects is optimal if each vehicle is travelling at its optimal emission speed, i.e. the global optimum of (8). Hence, the cost function of the emission of global pollutant p can be formalized as: p Jglobal =

Nc K X n X X i=1 k=1 c=1

γc GW Pp

ef p,c (vi (k)) → min (18) p,c efmax

where n denotes the number of controlled freeway segp,c ments; K denotes control horizon; efmax = maxv (ef p,c (v)); GW Pp denotes the global warming potential factor of pollutant p. This cost function is minimized, if ef p,c (k) is minimized for all k, thus traffic mean speeds for segments i = 1, ..., n equal vopt = arg min ef (v). Apart from CO2 , all of the exhaust pollutants cause local harms (health problems, acid rain, etc). To ease these effects of pollution, local concentration of polluting gases needs to be kept low. This is aimed by keeping the total emission rate function (15) of local pollutants low in each spatiotemporal window over the control horizon. Concentration of pollutants is a matter of immission 197

The closed-loop simulations of the designed controller are simulated in VISSIM and with emission module EnViVer. No. 1 M5 No. 11 Ramp from M0

No. 14 No. 19

Fig. 2. The simulated network layout In VISSIM, an existing motorway stretch at the border of Budapest is modeled: the motorway M5 from waymark 18 km (segment no. 1) to 27 km (segment no. 19), in northsouth direction. For the network layout see Fig. 2. An hourlong period is simulated and the control input calculated by the NMPC controller is applied on the system via the COM interface introduced in Tettamanti et al. [2011]. The control input is the ramp control of segment no. 14. For the simulation, the Hungarian vehicle composition of the year 2010 (detailed in Csikos and Varga [2011] and the emission factor functions of the corresponding vehicle classes of COPERT IV were used. During the simulation a bottleneck is modeled which is triggered by an accident. For the first 5 minutes (30 steps) free flow traffic is present with no congestion. An accident occurs at 300 s on segment no. 14 which leads to a solid increase in traffic density until 1200 sec, when the congestion is formed. The congestion starts to dissolve at the 220th step (2200 s), and a shock-wave propagates from segment no. 14 to segment no. 8 in the non-controlled case.

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For the spatiotemporal dynamics of traffic density see Fig. 3.

Fig. 5. PM10 emission without control Fig. 3. Traffic density without control

Fig. 6. PM10 emission using NMPC ramp control

Fig. 4. Traffic density using NMPC ramp control Multi-criteria performance of traffic control is analyzed regarding three aspects: traffic performance (TTD) and congestion management; real-time emission of pollutants causing global effects; overall emission of pollutants causing local effects. The comparison is carried out in the main lane only as appropriate model for idling vehicle emission model for all the analyzed pollutants (CO2 , NOX , PM10 ) [is not available. In the following, results of the compromised control design are detailed. Traffic performance is represented by TTD values. In the uncontrolled case TTD of the analyzed stretch throughout the simulation run results 20625.3 vehkm, using the control a TTD value of 22473.9 vehkm is obtained which is 8.96% improvement. The shockwave is also shorter, propagating until segment no. 10 in space, and at 1800 s (180th step) the density starts to decrease (see Fig. 3 and 4).

Fig. 7. N OX emission without control

Emission of pollutants causing global effects is restricted to CO2 analysis. The control intervention reduces CO2 emission from 8.39 · 105 g to 7.49 · 105 g. This is 10.7% reduction of the overall amount of CO2 emission. Emission performance of pollutants causing local effects (PM10 and NOX is illustrated on Fig. 5 to 8. The aim of the control is to reduce emissions in each spatiotemporal rectangle (in each segment, during each sample step). Reduction of emission levels is mainly caused by the shorter shock wave, but reduction is observable on the whole network throughout the simulation period. 198

Fig. 8. N OX emission using NMPC ramp control

CAO 2012 September 13-16, 2012. Rimini, Italy

6. SUMMARY In this work a multicriteria ramp metering control is outlined. The control is designed to optimize three control criteria: traffic performance and the emissions of pollutants causing global and local effects. First, the real-time macroscopic emission model is reviewed, based on which the emission cost functions of local and global pollutants are derived. Accompanied by the cost function for traffic performance, the objective function of the control system is optimized using the NMPC technique. After the estimation of the system parameters of VISSIM, the designed control is tested in a closed-loop VISSIM/MatLab environment. The compromised control leads to the amelioration of all three control criteria. A possible future research direction is a sophisticated synthesis of control weightings. ACKNOWLEDGEMENTS Special thanks to Tam´ as Tettamanti for providing the MatLab/VISSIM COM interface framework. The work is connected to the scientific program of the ”Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project. This project is supported by the New Sz´echenyi Plan (Project ID: ´ TAMOP-4.2.1/B-09/1/KMR-2010-0002), the Hungarian National Scientific Research Fund (OTKA CNK 78168) and by J´anos B´ olyai Research Scholarship of the Hungarian Academy of Sciences which are gratefully acknowledged. APPENDIX Table 1. Estimated parameters of the secondorder model a1 b1 c1 a2 b2 c2

5.97 · 1013 -1952 40.39 15.01 375.1 16.24

L T τ κ η δ

0.5 0.0028 0.00355 39.58 100.85 0.21

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