Vehicular Demand Management in Traffic Networks Comprised by Macroscopically Homogeneous Regions under Inter-boundary Constraints

Vehicula Demand Managemen in T a ic Ne wo ks Comp ised by
Mac oscopically Homogeneous Regions unde In e -bounda y
Cons ain s
Michalis Rampa, And eas Kasisa, Cha alambos Menelaouaand S elios Timo heoua
Abs ac — Vehicula demand managemen s a egies eme ge
as a po en ial solu ion o a ic conges ion. These s a e-
gies aim o egula e he in low o ehicles ac oss speci ic
egions o a ehicula anspo a ion ne wo k. This wo k
in es iga es he s abili y and op imali y o ehicula demand
managemen schemes in egional a ic ne wo ks, conside ing
in e -bounda y low cons ain s and iangula mac oscopic
undamen al diag am ela ionships be ween densi ies and lows.
We i s o mula e an op imiza ion p oblem ha aims o
maximize he o al ehicula h oughpu a s eady-s a e. Due o
he iangula mac oscopic undamen al diag am ela ionships
and in e -bounda y low cons ain s, he op imiza ion p oblem
is noncon ex. We ackle his challenge by e o mula ing he
p oblem as a Mixed In ege Linea P og am ha can be
sol ed wi h s anda d ma hema ical p og amming sol e s o
de e mine he op imal ope a ing se -poin s. None heless, i has
been demons a ed ha ope a ing a maximum h oughpu se -
poin s, pa icula ly nea local c i ical densi y poin s, may lead
o ins abili y and g idlock. To add ess his issue, we p opose
a decen alized p opo ional ehicula demand managemen
con olle , accompanied by p ope local design condi ions, such
ha s abili y is gua an eed. The e ec i eness and p ac icali y
o he p oposed app oach a e demons a ed h ough nume ical
simula ions in a six- egion a ic ne wo k sys em, ha showcase
he impac , in e ms o maximum h oughpu , o inco po a ing
in e -bounda y cons ain s.
I. INTRODUCTION
T a ic conges ion emains one o he mos p essing chal-
lenges o mode n ehicula anspo a ion ne wo ks (VTN).
Conges ion a ises om high a ic demand, wi h many
ehicles using he same oad, especially du ing peak hou s.
When demand su passes he in as uc u e’s capaci y, a ic
low de e io a es, leading o conges ion [1]. In ex eme cases,
his can escala e o g idlock, a s a e whe e ehicles a e a a
comple e s ands ill, and he VTN is unable o admi any mo e
a ic, causing a o al sys em b eakdown.
To educe conges ion, a ious a ic and demand manage-
men s a egies ha e been p oposed. Gi en he la ge scale o
VTN, mos con empo a y app oaches ely on mac oscopic
This wo k is suppo ed by he Eu opean Union (i. ERC, URANUS, No.
101088124, and ii. Ho izon 2020 Teaming, KIOS CoE, No. 739551), and
he Go e nmen o he Republic o Cyp us h ough he Depu y Minis y o
Resea ch, Inno a ion, and Digi al S a egy. Views and opinions exp essed
a e howe e hose o he au ho (s) only and do no necessa ily e lec
hose o he Eu opean Union o he Eu opean Resea ch Council Execu i e
Agency. Nei he he Eu opean Union no he g an ing au ho i y can be held
esponsible o hem.
aAll he au ho s a e wi h KIOS Resea ch and Inno a ion Cen e o Ex-
cellence, 1 Panepis imiou A enue, 2109 Aglan zia, Nicosia, Cyp us. Emails:
{ amp.michalis,kasis.and eas,menelaou.cha alampos,
imo heou.s elios}@ucy.ac.cy
a ic dynamics, which o e a simpli ied, agg ega ed iew
o he sys em by di iding he ne wo k in o homogeneous e-
gions [2]. Wi hin each egion, a ic dynamics a e modelled
using key pa ame e s such as low, densi y, and speed [3].
Mac oscopic app oaches educe compu a ional complexi y
by ocusing on he collec i e beha iou o a ic a he han
acking indi idual ehicles, which would be compu a ion-
ally in easible in la ge-scale VTNs. The p ima y goal o
bo h a ic and demand managemen s a egies is o sus ain
ne wo k ope a ion a maximum capaci y, which is achie ed
nea he c i ical densi y poin —a h eshold ha sepa a es he
ee- low egime om he conges ed egime [3].
T a ic managemen s a egies ocus on op imizing he
low o ehicles ac oss he ne wo k using me hods such
as ou e guidance, ga ing s a egies, and pe ime e con ol
[4]–[11]. These app oaches aim o edis ibu e a ic o
es ic access o c i ical a eas, such as ci y cen e s, o
a oid conges ion. Howe e , hey o en lead o conges ion in
unp o ec ed egions, as a ic is me ely shi ed a he han
educed [12].
In con as , ehicula demand managemen (VDM) s a e-
gies ha e he po en ial o comple ely elimina e conges ion
by egula ing he a e a which ehicles en e a VTN. Unlike
a ic managemen , VDM con ols bo h he spa ial and
empo al dis ibu ion o a ic low, making i possible o
op imize o e all sys em pe o mance. VDM s a egies can
queue ehicles ou side he VTN, schedule depa u e imes,
o encou age he use o al e na i e anspo a ion modes
o a oid o e loading he VTN [13]. By egula ing ehicle
in low, VDM ensu es ha he VTN ope a es nea i s c i ical
densi y, he eby p e en ing excessi e ehicle accumula ion
and main aining smoo h a ic low [14].
Despi e he success o VDM s a egies in op imizing
a ic lows, a c ucial aspec o a ic managemen is o en
o e looked: s abili y. S abili y is i al, pa icula ly when
ope a ing nea he c i ical densi y poin , whe e small dis u -
bances, such as sligh inc eases in demand o i egula i ies in
low, can lead o apid des abiliza ion o he VTN, se e ely
deg ading pe o mance and po en ially leading o g idlock
[15]. Al hough many exis ing a ic and demand manage-
men s a egies e ec i ely mi iga e conges ion, ew explici ly
accoun o s abili y, despi e i s impo ance in ensu ing he
long- e m success o conges ion managemen .
An addi ional ac o ha con ibu es o conges ion is he
p esence o inhe en in e -bounda y low cons ain s (IBCs)
be ween egions ha a e inna e cha ac e is ics o VTNs.
IBCs limi he amoun o ehicles a egion can admi om
neighbou ing a eas based on i s cu en occupancy, becoming
igh e as local ehicle numbe s inc ease. This beha iou
signi ican ly complica es he ask o conges ion mi iga ion
as i in oduces addi ional non-linea i ies and challenges in
he VTN op imiza ion and con ol design, making i essen ial
o accoun o . Mo eo e , o he au ho ’s bes knowledge, he
s abili y and op imali y p ope ies o VDM schemes in VTNs
in he p esence o IBCs ha e no been explo ed.
Con ibu ion: This wo k add esses his c i ical gap, by
ocusing on he s abili y p ope ies o VDM schemes and
in es iga ing op imali y condi ions ha achie e maximum
h oughpu esponse in VTNs, while explici ly accoun ing
o egional IBCs. These cons ain s in oduce non-linea i ies
in o a ic ne wo k dynamics and addi ional challenges o he
op imiza ion and con ol design. Speci ically, we conside
a class o VTNs wi h densi y- low ela ionships desc ibed
by iangula mac oscopic undamen al diag ams (MFD) and
in es iga e how mac oscopic VDM schemes can be designed
o main ain ne wo k s abili y. Fi s , an op imiza ion p oblem
aiming o maximize ehicle h oughpu wi h espec o a
VTN unde s eady-s a e condi ions is o mula ed. Regional
IBCs ende he o mula ed op imiza ion p oblem nonlinea
and non-con ex, making s anda d sol ing ools no di ec ly
applicable. We esol e his by ans o ming i o a Mixed
In ege Linea P og am (MILP). Howe e , as demons a ed
in [15], maximum- h oughpu solu ions, which o en co-
incide wi h local c i ical densi y alues, migh esul o
uns able beha iou . To add ess his, a decen alized p o-
po ional eedback VDM scheme is de eloped and supple-
men ed wi h local condi ions on i s pa ame e s o analy ically
gua an ee VTN s abili y. The de eloped s abili y condi ions
explici ly accoun o IBCs. The de eloped analy ic esul s
a e alida ed h ough nume ical simula ions on a 6- egion
VTN, which demons a e he e ec i eness o he p oposed
s a egy, and highligh he impo ance o IBCs, showing a
signi ican imp o emen in h oughpu when hose a e aken
in o accoun . By simul aneously enabling op imal h oughpu
and s abili y gua an ees, his wo k p o ides a obus and
p ac ical solu ion o conges ion in mode n VTNs.
The main con ibu ions o his wo k a e summa ized
below.
(i) We o mula e an op imiza ion p oblem aimed a max-
imizing ehicle h oughpu in la ge-scale ne wo ks,
explici ly conside ing egional IBCs. The op imiza ion
p oblem is ans o med in o a MILP o accoun o he
non-linea i ies p esen in he sys em, and enable he use
o s anda d sol e s.
(ii) We p opose a decen alized p opo ional VDM con-
olle , and p o ide locally e i iable condi ions ha
ake in o accoun egional IBCs, such ha VTN s abili y
is analy ically gua an eed.
Pape s uc u e: The pape is o ganized as ollows. The
p oblem o mula ion is gi en in Sec ion II. I en ails he
dynamics o he VTN, he o mula ion o an op imiza ion
p oblem ha aims o maximize ehicula h oughpu and
he p oblem s a emen . Ou app oach in sol ing he maxi-
mum ehicula h oughpu op imiza ion p oblem is gi en in
Sec ion III. Sec ion IV p esen s a decen alized p opo ional
VDM scheme wi h enhanced s abili y p ope ies and he
main s abili y esul s o his wo k. The e ec i eness o he
de eloped analy ic esul s is showcased in Sec ion V h ough
simula ions on a 6- egion VTN. Sec ion VI con ains he
conclusions o his wo k.
No a ion: We deno e eal numbe s by R. The se o
eal non-nega i e numbe s is deno ed by R+. Small bold
le e s deno e ec o s wi h he i h componen o a ec o
xdeno ed by xi. We use Rn o deno e he se o n-
dimensional ec o s wi h eal en ies wi h he non-nega i e
o han o Rngi en by Rn
+. The min-max ope a o is gi en
by [x]b
a= max(min(x, b), a), whe e a, b ∈Rand a≤b.
Capi al callig aphic le e s deno e se s. The n×1 ec o wi h
all elemen s equal o 0 is deno ed by 0n∈Rn. A unc ion
:Rn→Ris posi i e (nega i e) de ini e i (0n) = 0
and (x)>0( (x)<0) o e e y x=0n. We w i e
N {i} o deno e he se Nexcluding he elemen i. The igh
supe sc ip x∗deno es he equilib ium alue o xand he
ha designa ion, ˆx, deno es ha xis an op imal elemen wi h
espec o a gi en op imiza ion p oblem. Uni s o conside ed
a iables a e gi en only a he i s ins ance o compac ness.
II. PROBLEM FORMULATION
A. Model desc ip ion
The a ic a ea unde in e es is modelled as a VTN o n
nodes (n≥2) connec ed by edges be ween nodes. Each node
co esponds o a speci ic egion o he VTN and each edge
o a speci ic oad segmen connec ing wo nodes. A di ec ed
g aph G={N,E} desc ibes he VTN s uc u e wi h he
se s o egions1and oads deno ed by N={1,2, ..., n},
and E ⊆ N × N , espec i ely. We use ϵi,j = (i, j)∈ E o
deno e he oad allowing egion j o ecei e ehicles om
egion i. P edecesso egions ha can di ec ly send ehicles
o a egion ibelong o he se Pi={j∈ N :ϵj,i ∈ E}.
Successo egions ha can di ec ly ecei e ehicles om a
egion ibelong o he se Si={l∈ N :ϵi,l ∈ E}. Vehicles
en e he mul i- egion VTN ia o igin egions and e e y
ehicle ha en e s he ne wo k, does so o each a des ina ion
egion ha belongs o he VTN. In his wo k, all egions a e
conside ed o be bo h o igin and des ina ion egions.
Since egions a e ea ed as homogeneous, he collec-
i e in e - egional a ic low beha iou o a egion i,
gi(ρi( )) [ eh/h], whe e ρi( ) [ eh/km]is he egion’s
ehicle densi y, is desc ibed by a iangula (piecewise-
linea ) MFD [3], [16], [17], see Fig. 1, i.e. a iangula
unc ion o he egion’s ehicle densi y, gi en by
gi(ρi) = i i(ρi), i ∈ N,(1a)
i(ρi) = min(
iρi, bC
i− C
iρi), i ∈ N,(1b)
i=Lil−1
i, i ∈ N,(1c)
1In his wo k he e ms node and egion and edge and oad a e used
in e changeably.
whe e, i∈R+is he egion i ip comple ion a io, Li[km]
is he leng h o egion i,li[km]is he a e age ip leng h o
a ehicle in he same egion, and
i[km/h]is he ee- low
speed o egion igi en by
i=qC
i(ρC
i)−1, i ∈ N,(2)
wi h qC
i[ eh/h]being he capaci y low2o egion i, and
ρC
i[ eh/km], being he c i ical densi y h eshold o egion
i. The backwa d conges ion p opaga ion speed, C
i[km/h],
and he cons an bC
i[ eh/h], o egion iin (1b) a e gi en
by
C
i=qC
i(ρJ
i−ρC
i)−1, i ∈ N,(3a)
bC
i=ρJ
iqC
i(ρJ
i−ρC
i)−1, i ∈ N,(3b)
whe e ρJ
i[ eh/km]is he jam densi y h eshold o egion i.
Two a ic modes o ope a ion a e iden i ied; a ee- low
mode, [0, ρC
i), and a conges ed mode, [ρC
i, ρJ
i], see Fig. 1.
The MFDs o al in e - egional low, gi(ρi( )), is needed in
he calcula ion o he lows ha exi he ne wo k h ough
des ina ion egions and he ans e lows be ween egions.
T ans e lows be ween egions a e limi ed by successo
egions in e -bounda y capaci ies ia in e -bounda y egion
i o egion jcapaci y unc ions, cij :R+→R+, gi en by
cij(ρj) = min(cmax
ij , cmax
ij
ρJ
j−ρj
ρJ
j−¯ρij
), i ∈ N, j ∈ Si,(4)
whe e cmax
ij ∈R+is he c i ical in e -bounda y capaci y low
h eshold be ween egion iand egion j, and ¯ρij ∈(0, ρJ
j)is
he c i ical in e -bounda y densi y h eshold be ween egion
iand egion j. Equa ions (1a) and (4) yield he ans e low
om egion i o egion j, gi en by
gij(ρi, ρj) = min (wij gi(ρi), cij (ρj)) ,(i, j)∈ E,(5)
whe e wij ∈R+a e he egion iou low spli cons an s
sa is ying
wii +X
j∈Si
wij = 1, i ∈ N.(6)
The e m wii ∈R+in (6) is he a e ha ehicles end hei
ip in egion i. The VTN densi y s a e dynamics o each
egion i,ρi( ), is gi en by
˙ρi( ) = 1
Li−X
l∈Si
gil(ρi( ), ρl( )) −wiigi(ρi( ))
+ui( ) + X
j∈Pi
gji(ρj( ), ρi( )),∀i∈ N.(7)
In he igh hand side o (7), he i s e m is he low owa ds
successo nodes, he second e m is he low exi ing he VTN
h ough des ina ion egion i, he hi d e m, ui( ) [ eh/h], is
he se iced demand admi ed o he ne wo k h ough he
o igin egion i(conside ed as a con ol a iable), and he
las e m is he in low om p edecesso nodes. Fo ease o
2Capaci y low is he maximum low ha can be suppo ed by egion i;
i is yielded by he iangula unc ion i(ρi)a he c i ical densi y poin ,
ρC
i.
ρi
gi
gC
i
ρC
iρJ
i
gi(ρi)
cji(ρi)
¯ρji
Fig. 1. Region i o al in e - egional a ic low gi(ρi( )), app oxima ed
by a iangula mac oscopic undamen al diag am and egion iin e - egional
cons ain unc ion cji(ρi( )).
e e encing he dynamics a e de ined in a compac o m as
ollows:
T a ic Model 1: Equa ions (1), (4), (5) and (7) cap u e
he e olu ion o he a ic s a es o he conside ed n- egion
connec ed VTN.
B. Op imiza ion p oblem
The design o e icien VDM schemes equi es he o -
mula ion o an op imiza ion p oblem sa is ying key me ics
o demand alloca ion. He e, by explici ly aking in o accoun
egional IBCs, an op imiza ion p oblem yielding h oughpu -
op imal equilib ia unde s eady-s a e condi ions is o mu-
la ed, i.e. a s a e o maximum ehicle h oughpu ela i e o
he VTN bounda ies.
Op imiza ion P oblem 1: Fo a VTN wi h dynamics de-
sc ibed by T a ic Model 1, sol e:
max
(ρ,u)X
i∈N
ui(8a)
s. . : 0 ≤ρi≤ρC
i, i ∈ N,(8b)
umin
i≤ui≤umax
i, i ∈ N,(8c)
X
j∈Pi
gji(ρj, ρi) + ui
−wiigi(ρi)−X
l∈Si
gil(ρi, ρl)=0, i ∈ N,(8d)
whe e umin
i∈R+, umax
i∈R+a e cons an s associa ed wi h
he abili y o each egion o accommoda e demand.
Rema k 1: Cons ain (8b) s ems om he selec ion o an
equilib ium poin o be cha ac e ized by ee- low condi ions.
In (8c), cons ain umin
iis employed o ensu e ha he
solu ion o Op imiza ion P oblem 1 simul aneously se es
he ehicle demand goals and ensu es su icien con ol
au ho i y since no nega i e con ol alues a e allowed, i.e.
he con olle canno ake ehicles ou om a egion o
he ne wo k. An admi ed demand uppe bound cons ain ,
umax
i, is also placed since in p ac ice he a ic ne wo k
pa ame e s and he ne wo k s uc u e cons ain he ange o
alues o ui,∀i∈ N. Also, cons ain (8d) ollows om (7)
a equilib ium. Finally, i is assumed ha a easible solu ion
o his p oblem exis s.
C. P oblem S a emen
Since ehicula conges ion con inues o be a owe ing
issue o mode n VTNs, his wo k in ends o add ess he
ollowing p oblem:
P oblem 1: Fo he T a ic Model 1:
(i) ob ain an op imal VDM solu ion, by sol ing Op imiza-
ion P oblem 1,
(ii) de elop a decen alized VDM con ol scheme ha is
based on locally a ailable in o ma ion and condi ions
ha enable con e gence gua an ees and a e applicable
o a bi a y connec ed VTN con igu a ions.
The i s objec i e aims o he VDM solu ion o enable
maximum h oughpu . The second objec i e aims o de elop
a decen alized VDM con olle , accompanied by sui able
s abili y gua an ees, ha s abilizes he sys em s a es o a
e e ence equilib ium and is applicable o a bi a y connec ed
VTN con igu a ions enabling scalable designs.
III. OPTIMAL VEHICULAR DEMAND MANAGEMENT
SOLUTION
Op imiza ion P oblem 1 does no ollow any s anda d
o m, as i is nonlinea due o he p esence o he nonlinea
unc ions gij and he e ms cij nes ed in gij, wi hin cons ain
(8d). The cij and gij unc ions employ min ope a o s (see (4),
(5)) which signi ican ly complica es sol ing (8). To acili a e
ob aining a solu ion, Op imiza ion P oblem 1 is ans o med
in o a MILP by ollowing he me hodology desc ibed in [18,
Sec. 2.2]. Poin edly in [0, ρC
i], i ∈ N , he unc ions gij in
(8d) sa is y
gij(ρi, ρj) = min wij i
iρi, cmax
ij , cmax
ij
ρJ
j−ρj
ρJ
j−¯ρij , j ∈ Si.
(9)
Acco ding o [18, Sec. 2.2], in any op imiza ion p oblem,
o con inuous a iables x1, x2, x3, he e m min{x1, x2, x3}
can be eplaced in he op imiza ion p oblem wi h a new a i-
able y, by in oducing he ollowing addi ional cons ain s
xmin
i≤xi≤xmax
i,∀i∈ {1,2,3},(10a)
y≤xi,∀i∈ {1,2,3},(10b)
y≥xi−(xmax
i−min{xmin
1, xmin
2
, xmin
3})(1 −di),∀i∈ {1,2,3},(10c)
3
X
i=1
di=1,(10d)
di=(1,i ,min{x1, x2, x3}=xi,
0,o he wise, ∀i∈ {1,2,3}.
(10e)
This subs i u ion gene a es a s anda d MILP o m, p esen ed
below, ha can be sol ed by con en ional sol e s. Thus
he e ms gil(ρi, ρl)in (8d) a e eplaced wi h in e media e
a iables yil, and app op ia e MILP cons ain s by using
bina y a iables, dilk ∈ {0,1},(i, l)∈ E,k∈ {1,2,3}.
Fo compac ness we de ine he in e -bounda y low ec o ,
y∈R|E|
+comp ised by all he elemen s yil such ha
(i, l)∈ E. We also de ine he bina y ec o s d1∈ B|E|,d2∈
B|E|,d3∈ B|E|, whe e B={0,1}, comp ised espec i ely
by all he elemen s dil1,dil2, and dil3, such ha (i, l)∈ E.
These ac ions esul in he ollowing MILP o mula ion.
Op imiza ion P oblem 2: Fo a VTN wi h a ic dynam-
ics desc ibed by T a ic Model 1, sol e:
max
(ρ,u,y,
d1,d2,d3)X
i∈N
ui(11a)
s. . :Cons ain s (8b),(8c),(11b)
X
j∈Pi
yji +ui−wii i
iρi−X
k∈Si
yik = 0, i ∈ N,
(11c)
yνl −wνl ν
νρν≤0,(ν, l)∈ E,(11d)
yνl −cmax
νl ≤0,(ν, l)∈ E,(11e)
yνl +cmax
νl ρl
ρJ
l−¯ρνl
≤cmax
νl ρJ
l
ρJ
l−¯ρνl
,(ν, l)∈ E,(11 )
wνl ν
ν(ρν+ρC
νdνl1)
−yνl ≤wνl ν
νρC
ν,(ν, l)∈ E,(11g)
cmax
νl dνl2−yνl ≤0,(ν, l)∈ E,(11h)
cmax
νl (ρJ
ldνl3−ρl)
ρJ
l−¯ρνl
−yνl ≤0,(ν, l)∈ E,(11i)
dνl1+dνl2+dνl3= 1,(ν, l)∈ E,(11j)
whe e o he bina y a iables, dνlk,(ν, l)∈ E,k∈ {1,2,3},
he ollowing hold
dνl1=(1,i gνl(ρν, ρl) = wνl ν
νρν
0,o he wise ,(12a)
dνl2=(1,i gνl(ρν, ρl) = cmax
νl
0,o he wise ,(12b)
dνl3=(1,i gνl(ρν, ρl) = cmax
νl
ρJ
l−ρl
ρJ
l−¯ρνl
0,o he wise .(12c)
I is signi ied ha Op imiza ion P oblem 2 is equi alen o
Op imiza ion P oblem 1 and as al eady men ioned Op imiza-
ion P oblem 2 has mixed-in ege linea cons ain s and can
be sol ed ia s anda d comme cial sol e s o ob ain op imal
ne wo k s a es/equilib ia, (ˆ
ρ,ˆ
u), cha ac e ized by maximum
se iced ehicle demand and ee- low ope a ing condi ions.
This se es he VTN demand alloca ion ask as de ined in
P oblem 1, I em i.
I should be no ed ha op imal maximum- h oughpu
solu ions o en en ail local equilib ium densi ies a c i ical
alues. Ope a ion a hese alues migh esul o ins abili y
and g idlock [15]. Below, we de elop a VDM s abilizing
con olle ha allows s abili y gua an ees o be deduced in
he p esence o IBCs, e en when equilib ium poin s coincide
wi h c i ical densi ies. Such scheme enables s abili y and
maximum h oughpu o be simul aneously a ained, while
explici ly accoun ing o IBCs.
IV. TRAFFIC NETWORK STABILITY
By explici ly aking in o accoun egional IBCs, his
sec ion in es iga es he s abili y p ope ies o a VTN unde
a de eloped p opo ional eedback VDM scheme. A o mal
de ini ion o he equilib ium poin s o T a ic Model 1, o be
e e enced du ing he analysis, is gi en nex .
De ini ion 1: An equilib ium poin , ρ∗∈Rn
+, o T a ic
Model 1, sa is ies
−wiigi(ρ∗
i)−X
l∈Si
gil(ρ∗
i, ρ∗
l)
+X
j∈Pi
gji(ρ∗
j, ρ∗
i) + u∗
i= 0,∀i∈ N.(13)
whe e u∗
iis he s eady-s a e alue o ui.
An in es iga ion on he esponse o T a ic Model 1 when
he equilib ium poin s a e wi hin an open se o he domain,
i.e. ∀i∈ N, ρ∗
i∈(0, ρJ
i), is p o ided nex . In addi ion,
su icien condi ions o s abili y a e de i ed, in he p esence
o IBCs, ia Lyapuno analysis.
The inhe en comp omise be ween op imali y and s abili y
iden i ied in [15] (ins abili y-p one beha iou ) d i es he
de elopmen o a s abilizing con olle ha accoun s o IBCs
and ensu es sys em s a e con e gence gua an ees. I is gi en
by
ui( ) = [¯ui−kpiρi( )]umax
i
0, kpi∈R+ {0}, i ∈ N,(14)
whe e he admi ed low cons an , ¯ui, sa is ies by design he
ollowing condi ion
0<¯ui−kpiρ∗
i<¯umax
i.(15)
and can be selec ed by aking in o accoun op imali y goals.
Knowledge o he local equilib ium densi y, ρ∗
i, i ∈ N, is
equi ed o p ope selec ion o he cons an s ¯uiand kpi. This
alue can be ex ac ed by his o ical da a, o when easible,
by sol ing Op imiza ion P oblem 2. I he local equilib ium
is no a ailable, ¯uiand kpican be selec ed empi ically o
sa is y (15), hus yielding a ange o local equilib ium alues.
Finally, i is no ed ha wi hin he heo em we will make use
o he ollowing Lipschi z cons an s cha ac e izing (1b)3and
(4)4
L
i= max(
i, C
i), i ∈ N,(16a)
L
ij =cmax
ij /(ρJ
j−¯ρij),(i, j)∈ E.(16b)
Theo em 1: Conside T a ic Model 1 unde he ac ion
o (14), (15), and conside an equilib ium poin ρ∗
i∈
(0, ρJ
i),∀i∈ N. Then, i he e exis γil ∈R,(i, l)∈ E,
such ha
kpi>max{X
l∈Si
αil,X
l∈Pi
L
li
2γli
+X
l∈Si
γil L
il
2}
+ max{X
j∈Pi
L
ji,X
j∈Si
αij
2γij
+X
j∈Pi
γjiαji
2}+wii i L
i,
(17a)
αji =wji j L
j, j ∈ Pi,(17b)
3Equa ion (1b) implies ha i:R+→R+sa is ies he Lipschi z
condi ion globally wi h he Lipschi z cons an L
igi en by (16a).
4Simila ly, (4) implies ha each IBC unc ion, cij :R+→R+also
sa is ies he Lipschi z condi ion globally, wi h Lipschi z cons an L
ij gi en
by (16b).
o all i∈ N hen, he solu ions o he T a ic Model 1
locally con e ge o he equilib ium poin ρ∗. Addi ionally,
i u∗=ˆ
u, whe e ˆ
uis an elemen o he solu ion o
Op imiza ion P oblem 1, hen he equilib ium poin globally
sol es Op imiza ion P oblem 1.
P oo : Onwa d, ime dependence is d opped o com-
pac ness. We use e= [ρ1−ρ∗
1, .., ρn−ρ∗
n]T∈Rn o deno e
he ehicula densi y e o s a e. A Lyapuno candida e
V:Rn→Ris selec ed as
V(e) = X
i∈N
Li
2e2
i.(18)
The conside ed solu ions o T a ic Model 1 lie wi hin a
connec ed local subse K ⊂ Rn, wi h Kgi en by
K=e:ρ∗
i∈(0, ρJ
i), i ∈ N, V (e)≤ε.(19)
Due o (15), a p ope choice o ε∈R+ {0}, esul s in a
VDM con olle esponse ha does no iola e i s bounds.
Via (7), (14), he de i a i e o Vis
˙
V=X
i∈N
ei−wiigi(ρi)−X
l∈Si
gil(ρi, ρl)
+X
j∈Pi
gji(ρj, ρi) + ¯ui−kpiρi.(20)
Sub ac ing (13) om (20), and no ing ha due o (15) he e
exis s εsuch ha u∗
i= ¯ui−kpiρ∗
i, yields
˙
V=X
i∈N
ei−kpiei−wii(gi(ρi)−gi(ρ∗
i))
−X
l∈Si
(gil(ρi, ρl)−gil(ρ∗
i, ρ∗
l))
+X
j∈Pi
(gji(ρj, ρi( )) −gji(ρ∗
j, ρ∗
i)).(21)
Employing (16a), (16b), (17b), in (21) esul s in
˙
V≤X
i∈N (wii i L
i−kpi)e2
i+X
l∈Si
max{αile2
i, L
il|el||ei|}
+X
j∈Pi
max{αji|ej||ei|, L
jie2
i}.(22)
Using he ollowing pe ec squa es iden i y
βxy =β
2γx2+γβ
2y2− sβ
2γx− γβ
2y!2
(23)
on (22) leads o
˙
V≤X
i∈N (wii i L
i−kpi)e2
i+ max{X
l∈Pi
L
li
2γli
e2
i
+X
l∈Si
γil L
il
2e2
i−s L
il
2γil
el− γil L
il
2ei2,X
l∈Si
αile2
i}
+ max{X
j∈Si
αij
2γij
e2
i+X
j∈Pi
γjiαji
2e2
i
− αji
2γji
ej− γjiαji
2ei2,X
j∈Pi
L
jie2
i}.(24)

F om (24) and (17a), i ollows ha
˙
V(e)≤ − X
i∈N
ξie2
i<0,e∈ K {0},(25a)
ξi=kpi−wii i L
i−max{X
l∈Si
αil,X
l∈Pi
L
li
2γli
+X
l∈Si
γil L
il
2} − max{X
j∈Pi
L
ji,X
j∈Si
αij
2γij
+X
j∈Pi
γjiαji
2},(25b)
˙
V(0) =0,(25c)
and ia [19, Theo em 4.1], asymp o ic s abili y o e=0
ollows sui ; i.e. lim →∞ ρ( ) = ρ∗. I is signi ied ha , i
u∗
i= ˆui,∀i∈ N hen by (13), and he analysis abo e,
he VTN dynamics con e ge o ˆ
ρ, hus globally sol ing
Op imiza ion P oblem 1.
Theo em 1 p o ides s abili y gua an ees o he VDM
con olle (14) and also yields a su icien condi ion o
gain selec ion, i.e. (17a), ha ensu es VTN s abili y. I
is no ed ha , when ¯ui= ˆui+kpiˆρi, hen he ob ained
equilib ium poin sol es Op imiza ion P oblem 1, enabling
globally op imal h oughpu beha iou . I is also no ed ha
he ob ained esul s apply o any connec ed VTN con igu a-
ion. Concluding, all objec i es o P oblem 1 a e achie ed.
Nex , he esul s p esen ed in his wo k a e e i ied wi h
nume ical simula ions.
V. NUMERICAL RESULTS
A ep esen a i e simula ion o a 6- egion VTN is con-
duc ed o alida e he analy ical esul s o his s udy. The
simula ions aim o demons a e he s abili y p ope ies o
he de eloped con olle in he p esence o dis u bances,
he op imali y p ope ies o he conside ed se up, and also
highligh he impo ance o including IBCs, by showcasing
he impac , in e ms o h oughpu , when IBCs a e neglec ed.
The VTN pa ame e s a e gi en in Table I. The VTN opology
can be in e ed om he spli - a io ma ix wgi en in Table I.
Two simula ions a e conduc ed, o be e e ed o as Cases
1 and 2, as desc ibed below.
(i) Case 1 (black line) employs he de eloped VDM con-
ol law (14) wi h ¯ui= ˆui+kpiˆρi, i ∈ N (see Table II
o ˆui,ˆρi) whe e ˆui,ˆρiwe e ob ained by sol ing
Op imiza ion P oblem 1 wi hou aking in o accoun
he exis ence o he IBCs5.
(ii) Case 2 (blue line) also uses he de eloped VDM con ol
law (14) wi h ¯ui= ˆui+kpiˆρi, i ∈ N bu he
op imal solu ion ˆui,ˆρi akes in o accoun he IBCs as
i is ob ained by sol ing Op imiza ion P oblem 2 (see
Table III o ˆui,ˆρi).
The gain alues kpi o bo h con olle s a e he same and
a e gi en in Table I; hey we e selec ed o sa is y he gain
selec ion condi ion (17) o Theo em 1.
5When he IBCs a e dis ega ded, Op imiza ion P oblem 1 becomes linea
and can be easily sol ed wi h a s anda d sol e .
TABLE I
6-REGION NETWORK PARAMETERS AND CONTROL GAIN VALUES
Pa ame e & Value
L= [1.2,1,0.85,0.9,1.02,0.88]
l= [0.6,0.45,0.35,0.4,0.48,0.34]
= [30,35,32,34,35,31]
ρJ= [118,125,98,115,120,106]
ρC= [26.3,28.2,24.4,25.3,23.8,21.9]
w=







0.25 0.25 0 0 0.25 0.25
0.15 0.35 0.3 0 0 0.2
0 0.1 0.3 0.4 0 0.2
0 0 0.24 0.16 0.3 0.3
0.05 0.1 0 0.25 0.3 0.3
0.32 0.03 0.23 0.17 0.1 0.15







cmax =







0 117.6 0 0 106.1 124.3
151.3 0 140.9 0 0 155.5
0 116.4 0 134.5 0 123
0 0 122.8 0 115.6 135.5
127.7 124.2 0 143.4 0 131.2
104.1 101.2 97 116.9 91.2 0







¯
ρ=







0 19.2 0 0 13.2 21.1
16.9 0 21.2 0 0 22.6
0 17.8 0 17.1 0 19.5
0 0 19 0 12.7 20.2
14.3 17.4 0 16.7 0 19.1
13.1 16 16.4 15.3 11 0







kp= [103.1,115.8,135.1,140.4,123.5,163.9]
γil = 1,(i, l)∈ E
qC= [789,987,780.8,860.2,833,678.9]
umin = 0.01qC
umax =qC
TABLE II
THROUGHPUT-OPTIMAL SOLUTION (WITHOUT IBCS)
Pa ame e & Value
ˆ
ρ= [24.2,21.7,24.4,17,12.6,21.9]
ˆ
u= [589.4,987,674.3,8.6,8.3,6.8]
A1[h] simula ion scena io is ca ied ou whe e, by
u ilizing he co esponding op imal solu ions o Cases 1 and
2, he VDM con olle s aim o achie e maximum h oughpu
ope a ion. Fo bo h cases, a = 40 [min] a dis u bance is
in oduced in egion 2 o a du a ion o 1[min], by inc easing
u2by 0.5ˆu2.
The simula ion esul s a e discussed nex . A key poin
o no e is he signi ican di e ence in h oughpu esponse
shown in Fig. 2, be ween he op imal solu ions (ˆui,ˆρi)
ob ained o Case-1 (see Table II) and Case-2 (see Table III).
Fo his example, when IBCs a e aken in o accoun in he
op imiza ion p oblem, he h oughpu is imp o ed by 24.4%.
Mo eo e , o Case-1, a s eady s a e sys em-wide equi-
lib ium is achie ed. Howe e , i) he sys em s abilizes a a
sub-op imal ope a ion poin (see Fig. 2), and ii) i is unable
o con e ge o he op imal solu ion gi en in Table II (see
Fig. 3). This is no due o he inabili y o he con olle
o s ee he sys em o a desi ed equilib ium bu due o he
ac ha Case-1 doesn’ accoun o IBCs despi e hei ac i e
p esence in he ac ual ne wo k. Thus he ob ained ”op imal”
solu ion when IBCs a e excluded yields a subop imal sys em-
wide equilib ium ha also co esponds o egion 3 being
conges ed (see Fig. 5). The ac ha egion 3 is conges ed is
in e ed by obse ing ha he egion low eloci y in Fig. 5
TABLE III
THROUGHPUT-OPTIMAL SOLUTION (WITH IBCS)
Pa ame e & Value
ˆ
ρ= [26.3,28.2,24.4,25.3,23.8,21.9]
ˆ
u= [403.8,818.2,579.9,301.3,709.4,22.7]
0 10 20 30 40 50 60
0
1000
2000
3000
4000 24.4% imp o emen in h oughpu o Case-2
Fig. 2. To al ne wo k ou low [ eh/h] o Case-1 (black line) and Case-2
(blue line). Due o he ac ha he Case-1 op imal solu ion dis ega ds IBCs
he ne wo k ope a es sub-op imally wi h 24.4% educed h oughpu . Case-2
ope a es op imally wi h maximum h oughpu esponse.
is lowe han he ee- low alue.
In con as , o Case-2, a sys em-wide equilib ium is
apidly achie ed as he solu ions a e unhinde ed o con e ge
o he desi ed op imal solu ion gi en in Table III (see Fig. 4).
The con olle easily s ee s he sys em o he desi ed equilib-
ium since Case-2 solu ion accoun s o IBCs. Mo eo e all
egions ope a e in ee- low mode as i is e iden by he low
speed (see Fig. 5) and a maximum h oughpu (see Fig. 2).
Despi e he demand a ia ion a = 40 [min] he de i ed
con olle eco e s he s a es o an equilib ium e y as o
bo h cases hus demons a ing i s e ec i eness. Addi ionally,
he abili y o he de eloped con olle o s abilize he sys em
e en om conges ion is highligh ed om he ac ha i
eco e s he sys em o equilib ium o Case-2. Poin edly,
he op imal solu ion o egion 2 coincides wi h he c i ical
densi y (see ˆρ2= 28.2in Table III and ρC
2= 28.2in
Table I) and due o he dis u bance, egion 2 ge s conges ed
momen a ily. This is e iden by he speed d op om he ee-
low nominal alue obse ed in Fig. 5 (blue line). Despi e
his, he con olle is able o eco e and s ee he solu ions
back o he desi ed ope a ing equilib ium, ensu ing as
eco e y o maximum h oughpu ope a ion. The simula ion
scena ios clea ly alida e he de i ed heo e ic esul s and
highligh he impo ance o aking IBCs in o conside a ion
in he design o sys em ope a ion.
VI. CONCLUSIONS
This wo k conside ed he s abili y and op imali y p op-
e ies o ehicula demand managemen schemes in he
p esence o in e -bounda y egion o egion low cons ain s.
An op imiza ion p oblem ha aimed o maximize ehicle
h oughpu was o mula ed, aking in o accoun ehicula
0 50
0
20
0 50
0
20
0 50
0
20
0 50
0
20
0 50
0
10
20
0 50
0
10
20
Fig. 3. Case-1 ne wo k densi y [ eh/km]. S eady s a e is achie ed in bo h
cases. Howe e , he sys em s abilizes a a sub-op imal ope a ion poin due o
he ac ion o IBCs. The solu ions a e unable o a ain he op imal solu ion
yielded by Op imiza ion P oblem 1 (IBCs no aken in o accoun in his
solu ion). Also egion 3 is conges ed since he low eloci y in egion 3
(see Fig. 5) is lowe han he ee- low alue. Despi e he demand a ia ion
a = 40 [min] he de i ed con olle eco e s he s a es o an equilib ium
e y as .
0 50
0
20
0 50
0
20
0 50
0
20
0 50
0
20
0 50
0
10
20
0 50
0
10
20
Fig. 4. Case-2 ne wo k densi y [ eh/km]. S eady s a e is apidly achie ed
and a he in ended op imal ope a ion poin as he solu ions con e ge o he
op imal solu ion yielded by Op imiza ion P oblem 2 (IBCs a e aken in o
accoun in his solu ion). All egions ope a e in ee- low mode. Despi e he
demand a ia ion a = 40 [min] he de i ed con olle eco e s he s a es
o an equilib ium e y as .
a ic ne wo k dynamics and in e -bounda y cons ain s. The
p oblem was non-con ex and non-linea and was ans o med
o a Mixed In ege Linea P og am o enable he use o s an-
da d sol e s. A decen alized p opo ional ehicula demand
managemen con olle was de eloped, accompanied by local
design condi ions ha explici ly accoun o in e -bounda y
cons ain s, such ha s abili y is analy ically gua an eed.
The e ec i eness and p ac icali y o he de i ed analy ic
esul s was showcased wi h nume ical simula ions in a 6-
egion ehicula a ic ne wo k sys em, which highligh he
impo ance o in e -bounda y cons ain s, demons a ing a
signi ican imp o emen in ehicle h oughpu when hose
a e aken in o accoun .
0 50
0
20
0 50
0
20
40
0 50
0
20
0 50
0
20
40
0 50
0
20
40
0 50
0
20
Fig. 5. Ne wo k speed o Case-1 (black line) and Case-2 (blue-line)
[km/h]. Case-1 ope a es subop imally since he low eloci y in egion 3
is lowe han he ee- low alue. Hence egion 3 is conges ed. In con as ,
Case-2 ope a es op imally and e en eco e s om conges ion ( e y as ) a
= 40 [min] when he dis u bance pushes he solu ions o egion 2 in o
conges ion.
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