Perimeter and boundary flow control in multi-reservoir heterogeneous networks

TL;DR: In this article, the authors macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between networkwide space-mean flow and density of vehicles.

Abstract: In this paper, we macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling variations lead to two alternative optimal control methodologies for the design of perimeter and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as possible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system’s throughput is maximized. Based on the two control methodologies, perimeter and boundary control actions may be computed in real-time through a linear multivariable feedback regulator or a linear multivariable integral feedback regulator. Perimeter control occurs at the periphery of the network while boundary control occurs at the inter-transfers between neighborhood reservoirs. To this end, the heterogeneous network of San Francisco is partitioned into three homogeneous reservoirs and the proposed feedback regulators are compared with a pre-timed signal plan and a singlereservoir perimeter control strategy. Finally, the impact of the perimeter and boundary control actions is demonstrated via simulation by the use of the corresponding MFDs and other performance measures. A key advantage of the proposed approach is that it does not require high computational effort and future demand data if the current state of each reservoir can be observed with loop detector data.

Summary

1. Introduction

• Realistic modeling and efficient control of heterogeneous transportation networks remain a big challenge, due to the high unpredictability of choices of travelers (in terms of route, time of departure and mode of travel), the uncertainty in their reactions to the control, the spatiotemporal propagation of congestion, and the lack of coordinated actions coupled with the limited infrastructure available.
• A practicable work to address oversaturated traffic conditions was the recently developed feedback control strategy TUC (Diakaki et al., 2002; Diakaki et al., 2003; Aboudolas et al., 2009; Kouvelas et al., 2011).
• Given also the linear relationship between network outflow and circulating (or space-mean) flow (due to time-invariant regional trip length), MFD can also be expressed as space-mean flow vs. accumulation.
• Circulating flow can be directly measured by loop detectors while outflow requires a wide deployment of GPS.
• Perimeter flow control occurs at the periphery of the network while boundary flow control occurs at the inter-transfers between neighborhood reservoirs.

2. Dynamics for heterogeneous networks partitioned in N reservoirs

• While this simplification might not be always the case, the value of bii will be consistent with the physical properties of the system when the controller is active, i.e. for large input flow and/or congested conditions (see also (4) below).
• Without loss of generality, the authors assume that sji = 0, i.e., vehicles can immediately get access to the receiving reservoirs when exiting from the sending reservoirs of the network.
• This assumption can be readily removed if needed by introducing additional auxiliary variables (e.g. see Chapter 2 in Åström and Wittenmark (1996)).
• The linear system (8) approximates the original non-linear system (6) when the authors are near the equilibrium point about which the system was linearized.

3. Multivariable feedback regulators for perimeter and boundary flow control

• The linear control theory offers a number of methods and theoretical results for feedback regulator design in a systematic and efficient way.
• Multivariable feedback regulators have been applied in the transport area mainly for coordinated ramp metering (Papageorgiou et al., 1990) and traffic signal control (Diakaki et al., 2002; Diakaki et al., 2003; Aboudolas et al., 2009).
• In the sequel, the authors present two alternative optimal control methods for the design of feedback perimeter and boundary flow control strategies for multi-region and heterogeneously loaded networks.
• The first methodology is a multivariable feedback regulator derived through the formulation of the problem as a Linear-Quadratic (LQ) optimal control problem.
• The second methodology obtained through the formulation of the problem as a Linear-Quadratic-Integral (LQI) optimal control problem, which provides zero steady-state error under persistent disturbances and eliminates the need of set values b̂ji.

3.1. Perimeter and boundary flow control objectives

• In the case of a single-reservoir system (Fig. 1(a)) which exhibits an MFD, a suitable control objective is to minimize the total time that vehicles spend in the system including both time waiting to enter and time traveling in the network.
• It is known that the corresponding optimal policy is to allow as many vehicles to enter the network as possible without allowing the accumulation to reach states in the congested regime.
• 1Þ < ~n qmin else ð11Þ where qmin and qmax are the minimum and maximum entrance flow, respectively.
• In the case of a multi-reservoir system (Fig. 1(b)), however, a single-reservoir bang-bang policy (11) may induce uneven distribution of vehicles in the reservoirs, and, as a consequence, may invalidate the homogeneity assumption of traffic loads within the reservoirs and degrade the total network throughput and efficiency.
• The specification of set points n̂i (and corresponding b̂ji) for monocentric networks with well-defined destination attractions is easy, while heterogeneous networks with multiple regions of attraction would require a non-trivial choice of n̂i.

3.2. Multivariable feedback regulator

• A first approach towards feedback perimeter and boundary control based on the dynamics for a network partitioned in N reservoirs in (10) and the control objective mentioned in the previous section is derived as follows.
• The authors consider the following quadratic cost criterion that expresses the control objectives in mathematical terms: LðbÞ ¼ 1 2 X1 k¼0 kDnðkÞk2Q þ kDbðkÞk 2 R ð12Þ where Q and R are diagonal weighting matrices that are positive semi-definite and positive definite, respectively.
• The first term in (12) is responsible for minimization and balancing of the relative accumulation of vehicles ni/ni,max in each reservoir i (objective (I)).
• To this end, the diagonal elements of Q are set equal to the inverses of the maximum accumulation of the corresponding reservoirs (see Diakaki et al., 2002; Aboudolas et al., 2009 for details).
• The second term in (12) is responsible for objective (II) in Section 3.1 and the choice of the weighting matrix R = rI can influence the magnitude of the control actions.

3.3. Multivariable integral feedback regulator

• The basic approach in integral feedback control is to create a state within the controller that computes the integral of the error signal, which is then used as a feedback term to provide zero steady-state error.
• The augmented discrete-time system (10), (14) can be written in compact form as D~nðkþ 1Þ ¼ eAD~nðkÞ þ eBDbðkÞ þ eHDdðkÞ ð15Þ where ~nðkÞ ¼ ½nðkÞ zðkÞ s is the augmented state vector, and eA; eB; eH are the augmented state, control, and demand matrices, respectively .
• Similarly to the LQ cost criterion (12), the first term in (16) is responsible for objective (I) in Section 3.1, i.e. minimization and balancing of the relative accumulation of vehicles ni/ni,max in each reservoir i.
• The second term is responsible for objective (II), while the third term corresponds to the magnitude of the error signal.
• Minimization of the performance criterion (16) subject to (15) (assuming Dd(k) = 0) leads to the LQI multivariable feedback regulator DbðkÞ ¼ eK DnðkÞ zðkÞ ð17Þ where eK is the steady-state solution of the corresponding Riccati equation.

3.4. Constraints and implementation issues

• The authors conclude this section with some remarks pertaining to the control and state constraints of a multi-reservoir system and to the implementation of the multivariable feedback perimeter and boundary flow control in real-time.
• Alternatively, one can solve the same problem as a Model-Predictive perimeter Control (MPC) problem including all constraints by using the current state (current estimates of the accumulation in each reservoir) of the traffic system as the initial state n(0) as well as predicted demand flows d(k) over the a finite-time horizon (Geroliminis et al., 2013).
• The loop detectors may be placed anywhere within the link, but the estimation is most accurate for detector locations around the upstream middle of the link.
• After the application of the feedback regulators (13) or (18), if the ordered value b(k) violates the operational constraints (3), it should be adjusted to become feasible, i.e. truncated to [bmin, bmax].

4.1. Network description and simulation setup

• The test site is a 2.5 square mile area of Downtown San Francisco (Financial District and South of Market Area), including about 100 intersections and 400 links with lengths varying from 400 to 1300 feet (Fig. 2(a)).
• Traffic signals are all multiphase fixed-time operating on a common cycle length of 90 s for the west boundary of the area (The Embarcadero) and 60 s for the rest.
• The simulation step for the microscopic simulation model of the test site, was set to 0.5 s. Initially, to derive and investigate the shape of the MFDs of the three reservoirs, simulations are performed with a fieldapplied, fixed-time signal control plan.
• Nevertheless, in their experiments the authors only control the traffic signal at the perimeter of the network and the boundaries of the three reservoirs and they observe that the critical accumulations do not differ noticeably in the no control and perimeter flow control cases.

4.2. Macroscopic fundamental diagrams and heterogeneity

• Fig. 3(a) displays the MFD resulting for the considered demand scenario and ten replications (R1 to R10), each with different seed.
• It can be seen that all three reservoirs experience MFD with quite moderate scatter across different replications.
• This establishes their heterogeneity presumption stated in Section 3.1.
• Definitely, this strategy will be suboptimal as each reservoir should be treated differently.

4.3. Design of the perimeter flow control strategies

• The authors now perform the implementation and comparison of the proposed strategies FPC-LQ, FPC-LQI with the BBC strategy corresponding to the design and application of the feedback regulators (13), (18) and the bang-bang controller (11).
• The set accumulation n̂i for each reservoir i is selected within the optimal range of the corresponding MFD for maximum output, given the analysis in Section 4.2.
• All state matrices are developed for the particular network on the basis of the selected set point n̂; b̂, the matrix Y = I3 (only for (18)), and the linearization according to (8) and (10).

5. Results and insights

• In the sequel, the authors present simulation results for the proposed perimeter control strategies that are obtained by applying a non-adaptive demand scenario (Scenario 1, based on pre-specified turning movements at intersections) and two OD demand profiles (Scenarios 2 and 3).
• In the OD scenarios the DTA module is activated in the microsimulator and (some of) the drivers choose their routes adaptively in response to traffic conditions.
• Results include the most important traffic performance indices.

5.1. Non-adaptive demand scenarios

• The simulation results for non-adaptive demand are summarized in (i) Fig. 4 that graphically describes in details the evolution of congestion for each reservoir under no control and FPC-LQ control and (ii) Table 1 that presents different performance indices (average of all replications).
• Clearly, a high virtual waiting queue in perimeter control (compared to no control) is the price to pay for this particular improvement.

5.2. Simulation results for adaptive drivers and hysteresis loops

• These scenarios also include an offset of congestion to highlight the additional improvements of FPC.
• This underlines that appropriate designed perimeter control strategies for multi-reservoir systems might prove beneficial in ameliorating deficiencies associated with single-reservoir systems (e.g. propagation of congestion).
• These happens because reservoir 2 accumulation is retained around 2300 veh for FPC (compared to 2700 veh for BBC) without significant capacity loss, while the other 2 reservoirs obtain higher accumulations that result to higher flows (compare Fig. 5(c) with Fig. 5(e)).
• Thus, even if vehicles are restricted in the perimeter of the network, they are able to reach their destinations faster than in the no control case (‘‘slower is faster’’ effect, see Helbing and Mazloumian, 2009).

6. Discussion

• The authors addressed the problem of perimeter control for congested networks partitioned in reservoirs.
• This can be of great importance towards the development of generic, elegant, and efficient perimeter control strategies that appropriately account for the spatial and temporal heterogeneity of congestion between the reservoirs.
• A key advantage of their approach is that it does not require high computational effort and future demand data if the state can be observed.
• The proposed strategy was demonstrated to preserve high network performance and equity in the heterogeneous test network and significantly reduce the hysteresis loops in the MFD.
• These findings are of great importance for the traffic engineering community because the concept of an MFD (a) can be applied for heterogeneously loaded large-scale networks with multiple centers of congestion, if these networks can be partitioned into a small number of homogeneous regions, and (b) can be used towards the development of efficient perimeter and boundary flow control strategies.

Figures

Table 1 PIs of the FPC strategy vs. NC for the non-adaptive demand scenario.

Fig. 4. Simulation results for one replication of the non-adaptive demand Scenario 1: (a) MFDs of the three reservoirs with NC; (b) MFDs of the three reservoirs with FPC-LQ. The continuous curves in (a) and (b) represent the best fit of data to a third-degree polynomial; (c) Reservoir flow over time with NC; (d) Reservoir flow over time with FPC-LQ; (e) Total network flow over time; (f) Vehicles waiting out of the network over time.

Fig. 1. A network modeled as (a) single-reservoir system and (b) multi-reservoir system. In (a), the macroscopic fundamental diagram O(n) defines a unimodal, low-scatter relation between network vehicle accumulations n (veh) and network outflow or output (veh/h) for all road sections. The maximum outflow in the network may be observed over a range of accumulation-values that is close to a critical accumulation ~n. In (b), each reservoir i exhibits a macroscopic fundamental diagram Oi(ni) where ni is the regional accumulation; each destination reservoir i is reachable (from the perimeter or boundary) from a number of origin reservoirs, which defines the set Si , e.g. Si ¼ fi; j; k;p; qg (Si includes i since the destination reservoir i is reachable from the perimeter) and Sj ¼ fi; k; l; o;p; qg.

Table 2 PIs of the FPC strategy vs. NC for the adaptive OD Scenario 2.

Table 3 PIs of the FPC strategy vs. BBC for the adaptive OD Scenario 3.

Fig. 6. Comparison of the FPC-LQI with BBC for one replication.

Fig. 2. The test site of Downtown San Francisco: (a) real network; (b) simulation model; (c) partitioning of the network into 3 reservoirs.

Fig. 5. Simulation results for one replication of the adaptive OD scenario: (a), (c), (e) MFDs of the three reservoirs under NC, BBC, and FPC-LQI, respectively; (b), (d), (f) MFD of the whole network under NC, BBC, and FPC-LQI, respectively.

Fig. 3. Macroscopic Fundamental Diagrams for the: (a) whole network; (b) three reservoirs.

Full text

Perimeter and boundary flow control in multi-reservoir
heterogeneous networks
Konstantinos Aboudolas
⇑
, Nikolas Geroliminis
Urban Transport Systems Laboratory, School of Architecture, Civil & Environmental Engineering, École Polytechnique Fédérale de Lausanne (EPFL),
CH-1015 Lausanne, Switzerland
article info
Article history:
Received 28 March 2013
Received in revised form 30 June 2013
Accepted 2 July 2013
Keywords:
Macroscopic fundamental diagram
Heterogeneous networks
Perimeter and boundary flow control
Multivariable feedback regulators
abstract
In this paper, we macroscopically describe the traffic dynamics in heterogeneous transpor-
tation urban networks by utilizing the Macroscopic Fundamental Diagram (MFD), a widely
observed relation between network-wide space-mean flow and density of vehicles. A gen-
eric mathematical model for multi-reservoir networks with well-defined MFDs for each
reservoir is presented first. Then, two modeling variations lead to two alternative optimal
control methodologies for the design of perimeter and boundary flow control strategies
that aim at distributing the accumulation in each reservoir as homogeneously as possible,
and maintaining the rate of vehicles that are allowed to enter each reservoir around a
desired point, while the system’s throughput is maximized. Based on the two control
methodologies, perimeter and boundary control actions may be computed in real-time
through a linear multivariable feedback regulator or a linear multivariable integral feed-
back regulator. Perimeter control occurs at the periphery of the network while boundary
control occurs at the inter-transfers between neighborhood reservoirs. To this end, the het-
erogeneous network of San Francisco is partitioned into three homogeneous reservoirs and
the proposed feedback regulators are compared with a pre-timed signal plan and a single-
reservoir perimeter control strategy. Finally, the impact of the perimeter and boundary
control actions is demonstrated via simulation by the use of the corresponding MFDs
and other performance measures. A key advantage of the proposed approach is that it does
not require high computational effort and future demand data if the current state of each
reservoir can be observed with loop detector data.
 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Realistic modeling and efficient control of heterogeneous transportation networks remain a big challenge, due to the high
unpredictability of choices of travelers (in terms of route, time of departure and mode of travel), the uncertainty in their
reactions to the control, the spatiotemporal propagation of congestion, and the lack of coordinated actions coupled with
the limited infrastructure available. While there is a vast literature of congestion dynamics, control and spreading in one-
dimensional traffic systems with a single mode of traffic, most of the analysis at the network level is based on simplistic
models or simulation, which require a large number of input parameters (sometimes unobservable with existing data)
and cannot be solved in real-time. Still congestion governance in large-scale systems is currently fragmented and
uncoordinated with respect to optimizing the goals of travel efficiency and equity for multiple entities. Understanding these
0191-2615/$ - see front matter  2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.trb.2013.07.003
⇑
Corresponding author. Tel.: +41 (0)21 69 32481.
E-mail addresses: konstantinos.ampountolas@epfl.ch (K. Aboudolas), nikolas.geroliminis@epfl.ch (N. Geroliminis).
Transportation Research Part B 55 (2013) 265–281
Contents lists available at SciVerse ScienceDirect
Transportation Research Part B
journal homepage: www.elsevier.com/locate/trb

interactions for complex and congested cities is a big challenge, which will allow revisiting, redesigning and integrating
smarter traffic management approaches to generate more sustainable cities.
With respect to traffic signal control, many methodologies have been developed, but still a major challenge is the deploy-
ment of advanced and efficient traffic control strategies in heterogeneous large-scale networks, with particular focus on
addressing traffic congestion and propagation phenomena. Widely used strategies like SCOOT (Hunt et al., 1982) and SCATS
(Lowrie, 1982), although applicable to large-scale networks, are less efficient under oversaturated traffic conditions with
long queues and spillbacks. However, recently ad hoc gating schemes (engineering solutions) have been incorporated in
these systems to resolve local spill-over situations (Bretherton et al., 2003; Luk and Green, 2010). Other advanced traffic-
responsive strategies (Gartner et al., 2001, 2005) use complex optimization algorithms, which do not permit a real-time net-
work-wide application. A practicable work to address oversaturated traffic conditions was the recently developed feedback
control strategy TUC (Diakaki et al., 2002; Diakaki et al., 2003; Aboudolas et al., 2009; Kouvelas et al., 2011). TUC attempts to
minimize the risk of oversaturation and spillback of link queues by minimizing and balancing the links’ relative occupancies.
Furthermore, TUC also includes a local gating feature to protect downstream links from overload in the sense of limiting the
entrance in a link when close to overload. However, these policies might be suboptimal or delayed reactive for heteroge-
neous networks with multiple centers of congestion and heavily directional demand flows.
An alternative avenue to real-time network-wide traffic signal control for urban networks is a hierarchical two-level ap-
proach, where at the first level perimeter and boundary flow control between different regions of the network advances the
aggregated performance, while at the second level a more detailed control can be applied to smooth traffic movements with-
in these regions (e.g. TUC). The physical tool to advance in a systematic way this research is the Macroscopic Fundamental
Diagram (MFD) of urban traffic, which provides for network regions under specific regularity conditions (mainly homogene-
ity in the spatial distribution of congestion and the network topology), a unimodal, low-scatter relationship between net-
work vehicle accumulations n (veh) and network outflow (veh/h), as shown in Fig. 1(a). The idea of an MFD with an
optimum (critical) accumulation
~
n at which capacity is reached (maximum circulating flow or throughput) belongs to God-
frey (1969), but the empirical verification of its existence with dynamic features is recent (Geroliminis and Daganzo, 2008).
Given also the linear relationship between network outflow and circulating (or space-mean) flow (due to time-invariant re-
gional trip length), MFD can also be expressed as space-mean flow vs. accumulation. Both expressions are utilized in this
paper due to their similarity. Circulating flow can be directly measured by loop detectors while outflow requires a wide
deployment of GPS. This property is important for modeling purposes as details in individual links are not needed to describe
the congestion level of cities and its dynamics. It can also be utilized to introduce simple perimeter flow control policies to
improve mobility in homogeneous networks (Daganzo, 2007; Keyvan-Ekbatani et al., 2012; Geroliminis et al., 2013). The
general idea of a perimeter flow control policy is to ‘‘meter’’ the input flow to the system and to hold vehicles outside the
controlled area if necessary. A key advantage of this approach is that it does not require high computational effort if proxies
for
~
n are available (e.g. critical accumulation, critical average occupancy or critical density) and the current state of the net-
work n can be observed with loop detector data in real-time (see Fig. 1(a)). A drawback of this approach is that it creates
queues blocking the urban roads outside the controlled area. Alternatively, route choice or dynamic pricing models can
be directly incorporated in the perimeter flow control problem to avoid long queues and delays at the perimeter of the con-
trolled area (see e.g. Haddad et al., 2013; Knoop et al., 2012; Geroliminis and Levinson, 2009).
(a)
(b)
Fig. 1. A network modeled as (a) single-reservoir system and (b) multi-reservoir system. In (a), the macroscopic fundamental diagram O(n) defines a
unimodal, low-scatter relation between network vehicle accumulations n (veh) and network outflow or output (veh/h) for all road sections. The maximum
outflow in the network may be observed over a range of accumulation-values that is close to a critical accumulation
~
n. In (b), each reservoir i exhibits a
macroscopic fundamental diagram O
i
(n
i
) where n
i
is the regional accumulation; each destination reservoir i is reachable (from the perimeter or boundary)
from a number of origin reservoirs, which defines the set S
i
, e.g. S
i
¼fi; j; k; p; qg (S
i
includes i since the destination reservoir i is reachable from the
perimeter) and S
j
¼fi; k; l; o; p; qg.
266 K. Aboudolas, N. Geroliminis / Transportation Research Part B 55 (2013) 265–281

Despite these findings for the existence of an MFD with low scatter, these curves should not be a universal law. Recent
works (Mazloumian et al., 2010; Geroliminis and Sun, 2011; Daganzo et al., 2011; Knoop et al., 2012; Saberi and
Mahmassani, 2012) have identified the spatial distribution of vehicle density in the network as one of the key components
that affect the scatter of an MFD and its shape. They observed that the average network flow is consistently higher when link
density variance is low for the same network density, but higher densities can create points below an MFD when they are
heterogeneously distributed. Other investigations of empirical and simulated studies for network level traffic patterns can be
found in Buisson and Ladier (2009), Aboudolas et al. (2010), Ji et al. (2010), Gayah and Daganzo (2011a), Wu et al. (2011),
Mahmassani et al. (2013a), Mahmassani et al. (2013), Zhang et al. (2013) and elsewhere.
These results are of great importance because the concept of an MFD can be applied for heterogeneously loaded networks
with multiple centers of congestion, if these networks can be partitioned into a small number of homogeneous reservoirs
(regions). The objectives of partitioning are to obtain (i) small variance of link densities within a reservoir, which increases
the network flow for the same average density and (ii) spatial compactness of each reservoir which makes feasible the appli-
cation of perimeter and boundary flow control (Ji and Geroliminis, 2012). The objective is to partition a heterogeneous net-
work into homogeneous reservoirs with small variance of link densities and well-defined MFDs, as shown in Fig. 1(b). On the
other hand, single-reservoir perimeter flow control (Daganzo, 2007; Keyvan-Ekbatani et al., 2012) may enhance an uneven
distribution of vehicles in different parts of the network (for example due to asymmetric route choices and origin–destina-
tion matrices), and, as a consequence, may invalidate the homogeneity assumption of traffic loads and degrade the total net-
work throughput. Thus, in this work we put some effort to deal with the important issues of efficiency, heterogeneity and
equity in perimeter flow control. In particular, for a given partition of a heterogeneous network into some homogeneous re-
gions and corresponding MFDs (see Fig. 1(b)) with a critical (sweet spot) accumulation that maximizes the regional circu-
lating flow (outflow or trip completion rate), we develop perimeter and boundary flow control strategies to improve
mobility in heterogeneous networks. In this approach, perimeter flow control occurs at the periphery of the network while
boundary flow control occurs at the inter-transfers between neighborhood reservoirs.
More specifically, a generic mathematical model of an N-reservoir network with well-defined MFDs for each reservoir is
presented first. Two modeling variations lead to two alternative optimal control methodologies for the design of perimeter
and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as pos-
sible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system’s
throughput is maximized. Based on the two control methodologies, perimeter and boundary control actions may be com-
puted in real-time through a linear multivariable feedback regulator or a linear multivariable integral feedback regulator.
To this end, the heterogeneous network of the Downtown of San Francisco is partitioned into three homogeneous regions
that exhibit well-defined MFDs. These MFDs are then used to design and compare the two feedback regulators with a
pre-timed signal control plan and a single-reservoir perimeter control strategy. Finally, the impact of the perimeter and
boundary control actions to the three reservoirs and the whole network is demonstrated via simulation by the use of the
corresponding MFDs and other performance measures, under a number of different demand scenarios.
2. Dynamics for heterogeneous networks partitioned in N reservoirs
Consider a heterogeneous network partitioned in N reservoirs (Fig. 1(b)). Denote by i =1,..., N a reservoir in the system,
and let n
i
(t) be the accumulation of vehicles in reservoir i at time t; n
i,max
be the maximum accumulation of vehicles in res-
ervoir i. We assume that for each reservoir i =1,..., N there exists an MFD, O
i
(n
i
(t)), between accumulation n
i
and output O
i
(number of trips exiting reservoir i per unit time either because they finished their trip or because they move to another res-
ervoir), which describes the behavior of the system when it evolves slowly with time t.
Let q
i,in
(t) and q
i,out
(t) be the inflow and outflow in reservoir i at time t, respectively; S
i
be the set of origin reservoirs
whose outflow will go to destination reservoir i (including reservoir i in case that reservoir i is reachable from the perimeter,
see Fig. 1(b)). Also, let d
i
(t) be the uncontrolled traffic demand (disturbances) in reservoir i at time t. Note that d
i
(t) includes
both internal (off-street parking for taxis and pockets for private vehicles) and external non-controlled inflows. The conser-
vation equation for each reservoir i =1,..., N reads:
dn
i
ðtÞ
dt
¼ q
i;in
ðtÞq
i;out
ðtÞþd
i
ðtÞ ð1Þ
Since the system of each reservoir evolves slowly with time t, we may assume that the outflow q
i,out
(t) is given by the output
O
i
(n
i
(t)) (the MFD), which is a function of the accumulation n
i
(t), where output O
i
(n
i
(t)) is the sum of the exit flows from res-
ervoir i to reservoir j, plus the internal output (internal trip completion rates at i). If i and j are two reservoirs sharing a com-
mon boundary, we denote by b
ji
(j – i) the fraction of the flow rate in reservoir j that are allowed to enter reservoir i and by b
ii
the fraction of the flow rate in the perimeter of the network allowed to enter reservoir i (see Fig. 1(b)). The inflow to reservoir
i is given by
q
i;in
ðtÞ¼
X
j2S
i
b
ji
ðt 
s
ji
ÞO
j
ðn
j
ðtÞÞ ð2Þ
where b
ji
(t 
s
ji
) are the input variables from reservoir j to reservoir i at time t, to be calculated by the perimeter and bound-
ary controller, and
s
ji
is the travel time needed for vehicles to approach reservoir i from origin reservoir j. Given that we
K. Aboudolas, N. Geroliminis / Transportation Research Part B 55 (2013) 265–281
267

assume no knowledge of generated demand from areas outside of the entire network (external perimeter), the input flow
from the perimeter to reservoir i is considered proportional to the outflow of region i, O
i
(n
i
). While this simplification might
not be always the case, the value of b
ii
will be consistent with the physical properties of the system when the controller is
active, i.e. for large input flow and/or congested conditions (see also (4) below). Without loss of generality, we assume that
s
ji
= 0, i.e., vehicles can immediately get access to the receiving reservoirs when exiting from the sending reservoirs of the
network. This assumption can be readily removed if needed by introducing additional auxiliary variables (e.g. see Chapter
2inÅström and Wittenmark (1996)).
Additionally, b
ji
(t) is constrained as follows
b
ji;min
6 b
ji
ðtÞ 6 b
ji;max
ð3Þ
where b
ji,min
, b
ji,max
are the minimum and maximum permissible entrance rate of vehicles, respectively, and b
ji,min
>0 to
avoid long queues and delays at the perimeter of the network and the boundary of neighborhood reservoirs. Moreover,
the following constraints are introduced to prevent overflow phenomena within the reservoirs
X
N
i¼1
ðb
ji
ðtÞþ
e
i
Þ 6 1;
8
j ¼ 1; ...; N ð4Þ
where
e
i
> 0 is a portion of uncontrolled flow that enters reservoir i. Finally the accumulation n
i
(t) cannot be higher than the
maximum accumulation n
i,max
for each reservoir i
0 6 n
i
ðtÞ 6 n
i;max
;
8
i ¼ 1; ...; N ð5Þ
Introducing (2) in (1) we obtain the following non-linear state equation
dn
i
ðtÞ
dt
¼
X
j2S
i
b
ji
ðtÞO
j
ðn
j
ðtÞÞ  O
i
ðn
i
ðtÞÞ þ d
i
ðtÞ ð6Þ
Given the existence of MFDs O
i
(n
i
(t)) with an optimum (critical) accumulation
~
n
i
at which capacity is reached for each res-
ervoir i =1,..., N (see Fig. 1(b)), the non-linear model (6) may be linearized around some set point
^
n
i
;
^
b
ji
, and
^
d
i
that satisfies
the steady state version of (6), given by
0 ¼
X
j2S
i
^
b
ji
ðtÞO
j
ð
^
n
j
ðtÞÞ  O
i
ð
^
n
i
ðtÞÞ þ
^
d
i
ðtÞ ð7Þ
Denoting
D
x ¼ x 
^
x analogously for all variables and assuming first-order Taylor approximation, the linearization yields
D
_
n
i
ðtÞ¼
X
j2S
i
D
b
ji
ðtÞO
j
ð
^
n
j
ðtÞÞ þ
X
j2S
i
^
b
ji
ðtÞ
D
n
j
ðtÞO
0
j
ð
^
n
j
ðtÞÞ 
D
n
i
ðtÞO
0
i
ð
^
n
i
ðtÞÞ þ
D
d
i
ðtÞð8Þ
The linear system (8) approximates the original non-linear system (6) when we are near the equilibrium point about which
the system was linearized. In our case, this equilibrium point should be close to the critical accumulation
~
n
i
for each reservoir
i =1,..., N, where the individual reservoirs’ output is maximized.
Applying (8) to a network partitioned in N reservoirs the following state equation (in vector form) describes the evolution
of the system in time
D
_
nðtÞ¼F
D
nðtÞþG
D
bðtÞþH
D
dðtÞ ð9Þ
where
D
n 2 R
N
is the state deviations vector of
D
n
i
¼ n
i

^
n
i
for each reservoir i =1,..., N;
D
b 2 R
M
is the control deviations
vector of
D
b
ji
¼ b
ji

^
b
ji
, "i =1,..., N, j 2S
i
;
D
d 2 R
N
is the demand deviations vector of
D
d
i
¼ d
i

^
d
i
for each reservoir i =1,
..., N; and F, G, and H are the state, control, and demand matrices, respectively. In particular, F 2 R
NN
is a square matrix with
diagonal elements F
ii
¼ð1 
^
b
ii
ðtÞÞO
0
i
ð
^
n
i
ðtÞÞ if i 2S
i
, and F
ii
¼O
0
i
ð
^
n
i
ðtÞÞ otherwise, and off-diagonal elements
F
ji
¼
^
b
ji
ðtÞO
0
j
ð
^
n
j
ðtÞÞ if j 2S
i
, and F
ji
= 0 otherwise; G 2 R
NM
is a rectangular matrix, where M 6 N
2
(depends on the network
partition and the set S
i
; i ¼ 1; ...; N) with elements G
ji
¼ O
j
ð
^
n
j
ðtÞÞ if the origin reservoir j is reachable from the destination
reservoir i, and G
ji
= 0 otherwise; H is an identity square matrix of dimension N (see Appendix A.1 for more details). It should
be noted that each reservoir i =1,..., N is equipped with (at least) one boundary controller b
ij
, j 2S
i
and it might be equipped
with one perimeter controller b
ii
(depends on the network partition and the set S
i
, i =1,..., N) thus the number of control
variables M is greater than the number of state variables N for any network partition and the linear system (9) of a multi-
reservoir network is completely controllable.
The continuous-time linear state system (9) of the multi-reservoir network may be directly translated in discrete-time,
using Euler first-order time discretization with sample time T, as follows
D
nðk þ 1Þ¼A
D
nðkÞþB
D
bðkÞþ
D
dðkÞ ð10Þ
where k is the discrete time index, and A ¼ e
FT
 I þ
1
2
AT

I 
1
2
AT

1
, B = F
1
(A  I)G (if F is non-singular) are the state and
control matrices of the corresponding discrete-time system. This discrete-time linear model (10) will be used as a basis for
feedback control design in the subsequent sections (see Appendix A).
268 K. Aboudolas, N. Geroliminis / Transportation Research Part B 55 (2013) 265–281

3. Multivariable feedback regulators for perimeter and boundary flow control
The linear control theory offers a number of methods and theoretical results for feedback regulator design in a systematic
and efficient way. Multivariable feedback regulators have been applied in the transport area mainly for coordinated ramp
metering (Papageorgiou et al., 1990) and traffic signal control (Diakaki et al., 2002; Diakaki et al., 2003; Aboudolas et al.,
2009). In the sequel, we present two alternative optimal control methods for the design of feedback perimeter and boundary
flow control strategies for multi-region and heterogeneously loaded networks. The first methodology is a multivariable feed-
back regulator derived through the formulation of the problem as a Linear-Quadratic (LQ) optimal control problem. The sec-
ond methodology obtained through the formulation of the problem as a Linear-Quadratic-Integral (LQI) optimal control
problem, which provides zero steady-state error under persistent disturbances and eliminates the need of set values
^
b
ji
.
3.1. Perimeter and boundary flow control objectives
In the case of a single-reservoir system (Fig. 1(a)) which exhibits an MFD, a suitable control objective is to minimize the
total time that vehicles spend in the system including both time waiting to enter and time traveling in the network. It is
known that the corresponding optimal policy is to allow as many vehicles to enter the network as possible without allowing
the accumulation to reach states in the congested regime. This policy can be formalized as follows Daganzo (2007): when the
network operating in the uncongested regime (n <
~
n), vehicles are allowed to enter the perimeter of the network as quickly
as they arrive with respect to the critical accumulation
~
n; once accumulation reaches
~
n (i.e. n P
~
n) entrance to the network is
limited to the minimum entrance flow. It is well-known that this policy corresponds to the so-called ‘‘bang-bang control’’
(BBC) given by
q
in
ðkÞ¼
q
max
if nðkÞ <
~
n and nðk þ 1Þ <
~
n
q
min
else

ð11Þ
where q
min
and q
max
are the minimum and maximum entrance flow, respectively. Bang-bang control works well when the
system under consideration has relatively slow dynamics, but tends to oscillate between the extremes q
min
and q
max
.
In the case of a multi-reservoir system (Fig. 1(b)), however, a single-reservoir bang-bang policy (11) may induce uneven
distribution of vehicles in the reservoirs, and, as a consequence, may invalidate the homogeneity assumption of traffic loads
within the reservoirs and degrade the total network throughput and efficiency. As it is demonstrated later in the paper, the
critical accumulation
~
n and the maximum output Oð
~
nÞ of a network modeled as a single-reservoir system can be different
from the critical accumulation
~
n
i
; i ¼ 1; ...; N and the maximum output O
i
ð
~
n
i
Þ; i ¼ 1; ...; N of the same network partitioned
in N reservoirs, i.e.
~
n is not necessarily equal to
P
N
i¼1
~
n
i
, as the different regions might not reach the critical values simulta-
neously. Moreover, the time each of the reservoirs reaches the congested regime is very different.
With these observations at hand, a suitable control objective for a multi-reservoir system aims at: (I) distributing the
accumulation of vehicles n
i
in each reservoir i as homogeneously as possible over time and the network reservoirs, and
(II) maintaining the rate of vehicles b
ji
that are allowed to enter each reservoir around a set (desired) point
^
b
ji
while the sys-
tem’s throughput is maximized. A possible way to act in the sense of point (I) is to equalize the distribution of the relative
accumulation of vehicles n
i
/n
i,max
despite inhomogeneous time and space distribution of arrival flows. Requirement (II) is
taken by setting the desired point
^
b
ji
be equal to the rate of vehicles correspond to output O
j
ð
^
n
j
Þ; i ¼ 1; ...; N; j 2S
i
.
The specification of set points
^
n
i
(and corresponding
^
b
ji
) for monocentric networks with well-defined destination attrac-
tions is easy, while heterogeneous networks with multiple regions of attraction would require a non-trivial choice of
^
n
i
.
Physically speaking, if a control approach can keep all regions below or close to the critical accumulation of each MFD,
~
n
i
that maximizes the regional outflow, then the problem is well resolved. A challenge, which will be investigated in the future,
is the dynamic partitioning of a network and the dynamic choice of
^
n
i
as a functions of the level of congestion in each region,
n
i
(k) and the distribution of destinations across the network. For example if heavily directional flows from the periphery of a
network pass through a small region to enter the center, the set point for the small region should be smaller than the set
point of the periphery. In case it is not possible to keep n
i
ðkÞ <
~
n
i
, "i =1,..., N , a controller can be designed with
^
n
i
deviating
by little (e.g. 10–20%) from the critical accumulation
~
n
i
, in such a way to prevent congestion from the reservoir with the high-
est density of destinations.
3.2. Multivariable feedback regulator
A first approach towards feedback perimeter and boundary control based on the dynamics for a network partitioned in N
reservoirs in (10) and the control objective mentioned in the previous section is derived as follows. We consider the follow-
ing quadratic cost criterion that expresses the control objectives in mathematical terms:
LðbÞ¼
1
2
X
1
k¼0
k
D
nðkÞk
2
Q
þk
D
bðkÞk
2
R

ð12Þ
where Q and R are diagonal weighting matrices that are positive semi-definite and positive definite, respectively. The first
term in (12) is responsible for minimization and balancing of the relative accumulation of vehicles n
i
/n
i,max
in each reservoir
K. Aboudolas, N. Geroliminis / Transportation Research Part B 55 (2013) 265–281
269

Frequently asked questions

Q1. What are the contributions in "Perimeter and boundary flow control in multi-reservoir heterogeneous networks" ?

In this paper, the authors macroscopically describe the traffic dynamics in heterogeneous transportation urban networks by utilizing the Macroscopic Fundamental Diagram ( MFD ), a widely observed relation between network-wide space-mean flow and density of vehicles. A generic mathematical model for multi-reservoir networks with well-defined MFDs for each reservoir is presented first. Then, two modeling variations lead to two alternative optimal control methodologies for the design of perimeter and boundary flow control strategies that aim at distributing the accumulation in each reservoir as homogeneously as possible, and maintaining the rate of vehicles that are allowed to enter each reservoir around a desired point, while the system ’ s throughput is maximized. To this end, the heterogeneous network of San Francisco is partitioned into three homogeneous reservoirs and the proposed feedback regulators are compared with a pre-timed signal plan and a singlereservoir perimeter control strategy.