Del Uni e si y o Technology
Del Cen e o Sys ems and Con ol
Technical epo 14-015
Hyb id model p edic i e con ol o
eeway a ic using disc e e speed limi
signals∗
J.R.D. F ejo, A. N´
u˜
nez, B. De Schu e , and E.F. Camacho
I you wan o ci e his epo , please use he ollowing e e ence ins ead:
J.R.D. F ejo, A. N´
u˜
nez, B. De Schu e , and E.F. Camacho, “Hyb id model p edic-
i e con ol o eeway a ic using disc e e speed limi signals,” T anspo a ion
Resea ch Pa C, ol. 46, pp. 309–325, Sep . 2014.
Del Cen e o Sys ems and Con ol
Del Uni e si y o Technology
Mekelweg 2, 2628 CD Del
The Ne he lands
phone: +31-15-278.51.19 (sec e a y)
ax: +31-15-278.66.79
URL: h p://www.dcsc. udel .nl
∗This epo can also be downloaded ia h p://pub.deschu e .in o/abs/14_015.h ml
Hyb id model p edic i e con ol o eeway a ic
using disc e e speed limi signals
Jos´
e Ram´
on D. F ejo1,Al edo N´
u˜
nez,2, Ba De Schu e ,3and
Edua do F. Camacho1
Abs ac
In his pape , wo hyb id Model P edic i e Con ol (MPC) app oaches o eeway a ic
con ol a e p oposed conside ing a iable speed limi s (VSL) as disc e e a iables as in cu en
eal wo ld implemen a ions. These disc e e cha ac e is ics o he speed limi s alues and some
necessa y cons ain s o he ac ual ope a ion o VSL a e usually unde es ima ed in he li e a u e,
so we p opose a way o include hem using a mac oscopic a ic model wi hin an MPC amewo k.
Fo ob aining disc e e signals, he MPC con olle has o sol e a highly non-linea op imiza ion
p oblem, including mixed-in ege a iables. Since sol ing such a p oblem is complex and di icul
o execu e in eal- ime, we p opose some me hods o ob ain easonable con ol ac ions in a limi ed
compu a ion ime. The i s wo me hods (θ-exhaus i e and θ-gene ic disc e iza ion) consis o i s
elaxing he disc e e cons ain s o he VSL inpu s; and hen, based on his con inuous solu ion
and using a gene ic o an exhaus i e algo i hm, o ind disc e e solu ions wi hin a dis ance θo
he con inuous solu ion ha p o ide a good pe o mance. The second class o me hods spli he
p oblem in a con inuous op imiza ion o he amp me e ing signals and in a disc e e op imiza ion
o speed limi s. The speed limi s op imiza ion, which is much mo e ime-consuming han he amp
me e ing one, is sol ed by a gene ic o an exhaus i e algo i hm in communica ion wi h a non-linea
sol e o he amp me e ing. The p oposed me hods a e es ed by simula ion, showing no only a
good pe o mance, bu also keeping he compu a ion ime educed.
*The esea ch leading o hese esul s has ecei ed unding om he Eu opean Union Se en h F amewo k P og amme
[FP7/2007-2013] unde g an ag eemen 257462 HYCON2 Ne wo k o excellence.
1J.R.D. F ejo and E.F. Camacho a e wi h he Dep . de Ingenie ´
ıa de Sis emas y Au om´
a ica, Escuela Supe io de
Ingenie os, Uni e si y o Se ille, Spain. {jdominguez3,edua do}@us.es
2A. N´
u˜
nez is wi h he Sec ion o Road and Railway Enginee ing, Del Uni e si y o Technology, The Ne he lands.
a.a.nunez icencio udel .nl
3B. De Schu e is wi h he Del Cen e o Sys ems and Con ol, Del Uni e si y o Technology, The Ne he lands.
b.deschu e udel .nl
I. INTRODUCTION
T a ic conges ion on eeways is a c i ical p oblem due o i s nega i e impac on he
en i onmen and many o he impo an consequences like highe delays, was e o uel, a
highe acciden isk p obabili y, e c. In he las decades, a lo o esea ch has been ocused on
making a be e use o he a ailable a ic in as uc u e, since solu ions like he cons uc ion
o new eeways a e no always iable o implemen in he sho - e m due o echnical,
poli ical, legal, o economic easons. I has been epo ed in he li e a u e unde di e en
condi ions ha dynamic a ic con ol is a good solu ion o dec ease conges ion [1]–[3]. In
gene al, dynamic a ic con ol uses measu emen s o he a ic condi ions o e ime and
compu es dynamic con ol signals o in luence he beha io o he d i e s and o gene a e a
esponse in such a way ha he pe o mance o he ne wo k is imp o ed, by educing o
example he delays, emissions, e c.
Fig. 1. VSL implemen a ion on A13. Del , The Ne he lands.
Va iable speed limi s, amp me e ing, and ou e guidance a e some o he mos o en
used examples o eeway a ic signals ha can be used o dynamically con ol a ic.
These measu es ha e been al eady success ully implemen ed in USA, Ge many, Spain, he
Ne he lands, and o he coun ies [3]–[5]. When selec ing he con ol signals, among he
a ailable op ions desc ibed in he li e a u e, he me hods based on he use o ad anced
con ol echniques like Model P edic i e Con ol (MPC) [6] ha e by simula ion p o ed
o subs an ially imp o e he pe o mance o he con olled a ic sys em [1], [2], [7], [8].
In he cu en pape , we ocus on con ol using a iable speed limi s (VSL) oge he wi h
amp me e ing. In mos o he wo ks abou VSL compu ed wi h MPC, he VSL signals a e
assumed o ha e con inuous alues, meaning ha he eal VSL panels implemen ed in he
ne wo k should display o he d i e hose alues [1], [2], [7], [9], [10]. Howe e , in he eal
implemen a ions o VSL panels, he displayed signals a e jus allowed o ake a limi ed se
o disc e e alues. Fo example, in he Du ch eeway A12 he signals o he panels a e jus
allowed o ake alues in he se {60,80,100}km/h [3]. Mo eo e , o sa e y easons, some
ex a cons ain s should be conside ed like a limi ed a ia ion o e ime ( o each panel) and
a limi ed a ia ion o e space (consecu i e panels) so o a oid d as ic changes in speed [11],
[12].
The ollowing wo ks ha e p oposed ways o deal explici ly wi h disc e e VSL. In [13], i
is p oposed a disc e e VSL con olle based on shock wa e heo y. In [14], a a ic model
wi h a iable leng h segmen s is used o compu e a simple bes -e o con olle ha educes
conges ion conside ing VSL signals ha can only be dec eased o inc eased by s eps o
10 km/h. Bo h con olle s, [13] and [14], use simple con ol laws ha a e no explici ly
designed o op imize a pe o mance index o he ne wo k. In [11] and [12] he VSL a e
disc e ized (by ounding, ceiling, o loo ing) a e compu ing hem in a con inuous way.
These pape s conclude ha he pe o mance o he disc e ized speed limi s was compa able
wi h he con inuous case. Howe e , hose esul s depend on he ne wo k con igu a ion and he
demand condi ions, as in ou case s udy we ound some impo an loss o pe o mance due o
he disc e iza ion. So, in his pape , we conside explici ly he e ec s o using disc e e signals
o he VSL panels in an MPC amewo k. Subsequen ly, we p opose new e icien algo i hms
o he compu a ion o disc e e VSL signals oge he wi h con inuous amp me e ing a es.
Sec ion II b ie ly in oduces he gene al concep s o a ic model METANET, o Model
P edic i e Con ol, and he sa e y and ope a ional cons ain s ha appea in a eal VSL
implemen a ion. Sec ion III in oduces he common cha ac e is ics o he p oposed me hods
o ob ain disc e e signals using he con inuous solu ion o he VSL panels. In Sec ion IV,
an exhaus i e and a gene ic p ocedu e a e used o ob ain easible disc e e solu ions wi hin
aθdis ance o he con inuous solu ion p o ided by he MPC con olle . Sec ion V p oposes
me hods o ob ain he MPC solu ion by sol ing i e a i ely a con inuous op imiza ion o he
amp me e ing and a disc e e op imiza ion o he VSL wi hou using he con inuous MPC
solu ion. Again, an exhaus i e algo i hm and a gene ic algo i hm a e p oposed. The scena io
and he nume ical esul s a e p esen ed and discussed in Sec ion VI. Finally, Sec ion VII
shows a summa y o he esul s and he conclusions.
II. FREEWAY TRAFFIC CONTROL USING MODEL PREDICTIVE CONTROL
A. T a ic Model METANET
In his pape , we ha e selec ed he a ic model METANET [15]. Howe e , i is impo an
o no e ha he me hods we p opose a e independen o he a ic model used, so hey can be
equi alen ly applied using o he mac oscopic a ic models, i hose a e capable o including
he e ec o VSL in hei o mula ion (like some e sions o he Cell T ansmission Model
CTM [16]).
The METANET model is a mac oscopic second-o de a ic model ha p o ides a good
ade-o be ween simula ion speed and accu acy o online con ol pu poses. The METANET
model is de e minis ic and can be adap ed o eeway ne wo ks o a bi a y opology and
cha ac e is ics, including eeway s e ches, bi u ca ions, on- amps, and o - amps, and i
akes in o accoun he e ec s o con ol ac ions such as amp me e ing, ou e guidance, and
VSL. This model disc e izes he eeway in consecu i e links wi h segmen s o leng h Liand
uses densi y ρi(km)and speed i(km)as s a e a iables whe e kmis he model ime s ep.
Fo simplici y, in his pape he au ho s do no di e en ia e be ween links and segmen s in
con as wi h he o iginal METANET model. The model is o mula ed as ollows:
-Densi y equa ion:
ρi(km+ 1) = ρi(km) + Tm
λiLi
(qi−1(km)−qi(km) + q ,i(km)−βi(km)qi−1(km)) (1)
whe e λiis he numbe o lanes, βi(km)is he spli a io o o - amp in be ween segmen
iand segmen i+ 1 (βi(km) = 0 i he e is no an o - amp), Tmis he model sample ime,
qi(km)is he low lea ing segmen i, and q ,i(km)is low en e ing by an on- amp a he s a
o segmen i(q ,i(km) = 0 o a segmen wi hou an on- amp).
-Speed equa ion:
i(km+ 1) = i(km) + Tm
τi
(V(ρi(km)) − i(km)) + Tm
Li
i(km)( i−1(km)− i(km)) (2)
−µiTm
τiLi
ρi(km+ 1) −ρi(km)
ρi(km) + Ki
−δiTmq ,i(km) i(km)
Liλi(ρi(km) + Ki)
whe e Ki,τi,δi, and µia e model pa ame e s and V(ρi(km)) is he speed desi ed o he
d i e s. As p oposed in [1], he model can ake di e en alues o µi, depending on whe he
he downs eam densi y is highe o lowe han he densi y in he co esponding segmen .
Howe e , he algo i hms we p opose a e independen o his choice and could be applied in
bo h cases.
-Flow equa ion:
qi(km) = λiρi(km) i(km)
-Desi ed speed equa ion:
V(ρi(km)) = min ,i exp −1
aiρi(km)
ρc,i ai,(1 + αi)Vc,i(km)(3)
whe e ai,αia e model pa ame e s, ρc,i is he c i ical densi y, ,i is he ee low speed and
Vc,i(km)is he a iable speed limi applied o segmen i.
In his pape , he VSL a e included by a minimum e m in he desi ed speed equa ion
(3) as p oposed in [1]. Howe e , in [15], VSL a e included in he model h ough he h ee
pa ame e s o he undamen al diag am ρc,i, ,i and ai. In [17] he e ec o VSL on agg ega e
a ic low beha io is s udied on he basis o a ic da a compa ing he equa ion (3) used
in his pape wi h o he op ions. The me hods p oposed in his pape a e independen om
he op ion selec ed.
I he desi ed low Q(ρi(k)) = V(ρi(k)) ·ρi(k)is ep esen ed g aphically wi hou con-
side ing VSL, he undamen al diag am (which can be seen in Fig.2.) o a ic sys ems is
ob ained. The undamen al diag am gi es us he s a ic cha ac e is ic o he sys em.
Fig. 2. Fundamen al diag am o a ic
-Ramp low equa ion:
q ,i(km) = min i(km)C ,i, Di(km) + wi(km)
Tm
, C ,i
ρm,i −ρi(km)
ρm,i −ρc,i (4)
whe e C ,i is he amp capaci y, Di(km)is he amp demand, wi(km)is he amp queue
leng h, ρm,i is he maximum densi y, and i(km)is he amp me e ing a e.
-Queue leng h equa ion:
wi(km+ 1) = wi(km) + Tm·(Di(km)−q ,i(km)) (5)
Fo he sake o simplici y, me ge and join nodes, and o he ex ensions a e no conside ed
in his pape (See [15], [18] o u he de ails).
B. Model P edic i e Con ol
The main concep s behind a model-based p edic i e con ol (MPC) [6] s a egy a e:
1) The use o a p edic ion model o ob ain he ajec o ies o ele an a iables o he sys em.
2) The op imiza ion o a objec i e unc ion o de e mine he bes sequence o con ol ac ions
o he sys em.
3) The applica ion o he olling ho izon p ocedu e: om he bes sequence o con ol ac ions
only he i s componen is applied o he sys em and in he nex con ol s ep he ini ial
condi ions a e upda ed and he p ocedu e is epea ed again.
k−1 k+1 k+2
k
u(k+Nu)
...
...
k+Np
Np
Nu
y(k)
y(k+1)
y(k+Np)
u(k+1)
u(k)
Fig. 3. Receding ho izon s a egy. The se o u u e con ol signals a e compu ed conside ing he p edic ed ou pu s du ing
he p edic ion ho izon, bu jus he i s con ol u(k)is applied
Since MPC is based on an op imiza ion p ocedu e, in he case o sys ems wi h disc e e
con ol ac ions we can include ha cha ac e is ic as a cons ain , and ob ain sui able con ol
laws by using a p ope mixed-in ege op imiza ion sol e .
To o malize all hese concep s, conside he disc e e- ime non-linea sys em whose dy-
namic e olu ion is desc ibed by he ollowing s a e-space model:
x(km+ 1) = (x(km), u(km), d(km)) (6)
wi h x(km) he s a e, u(km) = [(ud(km))T,(uc(km))T]T he disc e e and con inuous inpu
ec o , and d(km) he non-con ollable inpu ec o , usually demand p o iles and o he ex-
ogenous a iables. In some ci cums ances, he con olle sample ime may be di e en om
he model sample ime. In a ic we will use a con olle sample ime Tlonge han he
simula ion sample ime Tm(i.e. =km·Tm=k·T). The con ol inpu s will be conside ed
cons an du ing one con olle sample esul ing in he ollowing s a e-space model:
x(k+ 1) = (x(k), u(k), d(k)) (7)
In an MPC con olle , he co e is he op imiza ion o a cos unc ion J(x (k), u (k), d (k)),
which is used o measu e he pe o mance o he sys em whe e he ec o s x (k) = [x(k+
1), ..., x(k+Np)]Tand d (k) = [d(k), ..., d(k+Np−1)]Ta e he s a e and non-con ollable
inpu p edic ions along he p edic ion ho izon Np; and u (k) = [u(k), ..., u(k+Nu−1)]Tis
he con ol sequence along he con ol ho izon Nu. The con ol inpu s a e kep cons an a e
he con ol ho izon Nu(i.e. u(k+Nu−1) = u(k+Nu) = ... =u(k+Np−1)).
As he o mula ion is based on he solu ion o an op imiza ion p oblem, i is possible o
explici ly include cons ain s. Assume ha he easible alues o he s a es and he inpu s a e
gi en by he ollowing gene ic cons ain s x(k)∈X,uc(k)∈U, and ud(k)∈S o all k,
ep esen ing explici ly physical o ope a ional cons ain s o he sys em. The e o e, he MPC
p oblem can be o mula ed as he ollowing mixed-in ege non-linea op imiza ion p oblem:
min
u (k)J(u (k), x (k), d (k)) (8)
subjec o:
x(k+ℓ+ 1) = (x(k+ℓ), u(k+ℓ), d(k+ℓ)), x(k) = xk
x(k+ℓ+ 1) ∈X, uc(k+ℓ)∈U, ud(k+ℓ)∈S,
h(u (k), x (k), d (k)) ∈D,
o ℓ= 0,1, ..., Np−1,
wi h S he se o possible con ol alues S={u1, u2, ..., uM} o he co esponding easible
disc e e inpu componen s (in ou case, he VSL). The e m h(u (k), x (k), d (k)) ∈D
co esponds o he es o he cons ain s and xkis las measu ed s a e a ime s ep k. Using
he olling ho izon p ocedu e, only he i s con ol ac ion u(k)o he op imal sequence is
applied o he sys em, and in he nex ime s ep he ini ial condi ions a e upda ed and he
p ocedu e is epea ed.
To p ope ly conside p ocesses wi h disc e e and con inuous a iables (hyb id sys ems) in
an MPC o mula ion, hyb id p edic i e con ol echniques ha e been de eloped [19]–[21]. Fo
his o mula ion, he main di icul y is he compu a ion ime needed o sol e he op imiza ion
p oblem because we a e dealing wi h mixed in ege nonlinea p og amming (MINLP). In
cases when he p oblem can be ecas in o a MILP, well-known op imiza ion me hods a e
e icien and many so wa e/ oolboxes a e a ailable o sol e hem [22]. Howe e , in he
case o a ic sys ems, he p oblem is in insically non-linea and any simpli ica ion o he
model may lead o non-accep able p edic ions, which a e incapable o inco po a ing he eal
beha io o a ic. As a way o deal wi h complexi y, and o include he disc e e cha ac e is ic
explici ly in he solu ion, we p opose a se o me hods based on gene ic algo i hms.
C. Pa icula i ies o he MPC con olle s used in his pape :
This subsec ion explains he main pa icula i ies o he MPC con olle s used in his pape :
–The MPC con olle uses an objec i e unc ion (9) con aining one e m o he To al Time
Spen (TTS) and wo e ms ha penalize ab up a ia ions in he amp me e ing and VSL:
J(k) =
Np
X
ℓ=1
T[X
i∈O
wi(k+ℓ) + X
li∈I
(ρi(k+ℓ)Liλi) ] (9)
+
Nu−1
X
ℓ=0
ǫ c kVc,j(k+ℓ)−Vc,j(k+ℓ−1) k2(10)
+
Nu−1
X
ℓ=0
ǫ k c,j(k+ℓ)− c,j(k+ℓ−1) k2
whe e ǫ c,ǫ a e weigh ing pa ame e s, Ois he se o all he segmen s wi h an on- amp,
and I he se o all he segmen s.
–The con olled sys em is subjec o cons ain s on he maximum and minimum alues o
densi ies, speeds, queues, amp me e ing a es, and VSL. The cons ain s on speed, densi y,
and queue a e made so by including hem as penaliza ion e ms in he o e all cos unc ion:
¯
J(k) = J(k) +
Np
X
ℓ=1
Ncons
X
i=1
Ωi(k+ℓ)(11)
whe e Ωi(k+ℓ)is a penaliza ion e m ha is di e en o ze o i he co esponding so
cons ain is iola ed and Ncons is he o al numbe o so cons ain s.
–The op imiza ion is compu ed using SQP op imiza ion echniques by he Ma lab unc ion
mincon wi h a con ol and p edic ion ho izon o Nu= 4 and Np= 6, espec i ely. The
unc ion uses he in e io poin me hod wi h a maximum numbe o 20000 cos unc ion
e alua ions.
–In o de o y o a oid ha he algo i hm ends up in a local minimum, he algo i hm uns
whe e {m1, m2, ..., mN }is he se o segmen s wi h a VSL as explained p e iously.
In GA he idea is o ind he i es indi iduals (solu ions wi h bes objec i e unc ion alues)
wi hin a gene a ion, o apply gene ic ope a o s o he ecombina ion o hose indi iduals,
and o gene a e a good o sp ing [24]. In his pape , o he selec ion, a oule e me hod is
applied, gi ing he bes indi iduals mo e chances o be selec ed o ecombina ion. Fo he e-
combina ion, wo undamen al ope a o s a e used: c osso e and mu a ion. Fo he c osso e ,
po ions o he ch omosomes o wo indi iduals a e exchanged wi h a gi en p obabili y pc;
and he mu a ion ope a o modi ies each gene andomly wi h a gi en p obabili y pm. This is
jus an speci ic implemen a ion and al e na i e gene ic me hods could be used.
New Indi idual 1 New Indi idual 2
Indi idual 1a Indi idual 2b
Indi idual 2b
40
40 50
50
Indi idual 2a
50 60 50
50 50
50 50 50
50
40
Indi idual 1b
40
40
30 30
60
60
40
40
40
40
50
50
60
60
30
20
30 20
20
30
Indi idual 1a Indi idual 1b
Indi idual 2a
50 60
C osso e poin C osso e poin
Mu a ion Mu a ion
Fig. 6. The basic ope a o s in a GA-based con ol s a egy o VSL panels.
Fig. 6 shows an example o he ecombina ion s eps o wo indi iduals assuming ha
we ha e wo VSLs and he con ol ho izon is 3. Fi s wo indi iduals a e selec ed o
ecombina ion wi h a highe p obabili y i hei objec i e unc ion is lowe . Then, a andom
c osso e poin is selec ed, and wo new indi iduals a e gene a ed. Then he mu a ion ope a o
selec s andomly a con ol o modi y and i s alues a e changed. Finally wo new indi iduals
o a new gene a ion a e ob ained.
We ha e o poin ou ha due o he limi ed ime eaching he global op imum is no
gua an eed. Since o a ic con ol he op imiza ion is a complex mixed-in ege and nonlinea
p oblem, using he GA op imiza ion is jus i ied. Many di e en app oaches and adap a ions
o gene ic algo i hms ha e been p oposed in he li e a u e in o de o deal wi h many issues
like cons ain handling, di e si y o he solu ions, combina ion wi h classical op imiza ion
me hods o assu e a local con e gence, e c [28]–[31]. Fo cons ained op imiza ion, one o
he mos used is he GENOCOP algo i hm p oposed by Michakewicz [32]. The algo i hm
we used o he a ic applica ion is he simples one [24] and in he case a solu ion does
no sa is y a cons ain , we penalize wi h a high objec i e unc ion alue. Pa o he u he
esea ch is o implemen di e en adap a ions o GA, and o he algo i hms. In he simula ion
esul s we will compa e GA wi h exhaus i e enume a ion, so we can see how a he solu ions
o GA a e om he op imal solu ion.
V. ALTERNATING OPTIMIZATION
In he p e ious sec ion, he con inuous VSL gi en by a MPC con olle ha e been used
o he compu a ion o he disc e e ones. Howe e , hese disc e e VSL could be compu ed
di ec ly (i.e. wi hou using any con inuous solu ion) sa ing he compu a ion ime needed
o he con inuous op imiza ion (i.e. o sol e op imiza ion (17) wi hou using he ST-MPC
solu ion). To sol e he mixed-in ege op imiza ion p oblem wi h disc e e and con inuous
decision a iables, we decided o compu e he amp me e ing and he VSL i e a i ely in
o de o decompose he p oblem in one con inuous op imiza ion p oblem (22) and one disc e e
op imiza ion p oblem (23).
1) Ramp me e ing op imiza ion:
min
(k)
¯
J(k)(22)
s. : 0≤ i(k+ℓ)≤1 o i∈I , o ℓ= 0,1, ...Nu−1and wi h (k)∈RNu·N
whe e he VSL p o ile used V∗
c, (k) o he amp me e ing op imiza ion is he p o ile p oposed
in he p e ious i e a ion o , o he i s i e a ion, he p o ile p oposed in he p e ious sample
ime shi ed by one sample. In his pape , his con inuous amp me e ing op imiza ion p oblem
will be sol ed using an SQP algo i hm. The compu a ion ime needed is much lowe han he
compu a ion ime needed o he gene al con inuous op imiza ion p oblem because i does
no include he VSL signals as decision a iables.
2) VSL op imiza ion:
min
Vc, (k)
¯
J(k)(23)
s. : |Vc,i(k+ℓ)−Vc,i(k+ℓ−1)| ≤ γi|Vc,j+1(k+ℓ)−Vc,j(k+ℓ)| ≤ ζj
VSLmin ≤Vc,i(k+ℓ)≤VSLmax
o ℓ= 0,1, ...Nu−1, o i∈IVSL,and o jand j+ 1 ∈IVSL wi h Vc,i(k+ℓ)∈S
whe e he amp p o ile used is he one p oposed in he p e ious i e a ion ∗
(k). This op i-
miza ion (23) is equi alen o he op imiza ion done o he p e ious sec ion (20) bu , in his
case, we do no ha e any con inuous MPC solu ion o he VSL ha can be used o he
educ ion o he sea ch space o he p oblem. The ollowing subsec ions p opose some ways
o sol e he esul ing disc e e op imiza ion p oblem (23), as done o he p oblem (20) in
Sec ion IV.
A. Exhaus i e op imiza ion
In he same way ha i was explained in Sec ion IV, he easies way o sol e p oblem
(23) is o e alua e he cos unc ion o all he easible poin s. Again, he p oblem is he
compu a ion ime needed o he e alua ion o such a la ge numbe o possible combina ions
o disc e e VSL. The e o e, his solu ion is jus applicable o ela i ely small ne wo ks and
ho izons. Fo la ge ne wo ks and ho izons, a gene ic algo i hm is p oposed in he ollowing
subsec ion.
B. Gene ic op imiza ion
In he same way ha in Sec ion IV.B, a gene ic algo i hm is p oposed o sol e he disc e e
op imiza ion. The o mula ion is he same as in Sec ion IV bu wi hou using he cons ain
ela ed wi h a solu ion wi hin a dis ance θ om he con inuous one. Mo eo e , i will be
con enien o use a di e en GA pa ame e s uning due he g ea e numbe o easible
solu ions and he la ge compu a ion imes a ailable.
VI. SIMULATION
A. Se -up and scena io
In o de o simula e he analyzed con olle s, he benchma k ne wo k in Fig. 7 has been
used. The ne wo k has been aken om [1]. The eeway has N= 6 segmen s wi h a leng h
o Li= 1000 m and wi h λ= 2 lanes. The e a e 3 con ol signals: wo VSL ( o segmen s
3 and 4) and a amp me e ing a e. All he METANET pa ame e s (which can be seen in
Table I) a e conside ed o be he same o all he segmen s.
VSL 1 VSL 2
60 50
Ramp
me e ing
Fig. 7. S e ch used as example.
TABLE I
MODEL PARAMETERS
τ18 s
K40 eh/(km·lane)
ρc i 33.5 eh/(km·lane)
α0.1
a1.867
ee 102 km/h
µ60 km2/h
δ0.0122
ρmax 180 eh/(km·lane)
Co4000 eh/h
C amp 2000 eh/h
Thi een a iables (densi y and speed o each segmen and queue o he amp) a e sup-
posed o be measu ed a each con olle sample s ep and used o he compu a ion o
he con ol signals. The simula ion ime chosen is wo and hal hou co esponding o
75 con olle sample s eps (T= 120s) and 900 simula ion s eps (Tm= 10s). See [1]
[8] o u he de ails abou he chosen benchma k. The se o allowed VSL is supposed
o be S={20,30,40,50,60,70,80,90,100,110,120}and he implemen a ion cons ain s
pa ame e s a e ζi= 10, and γi= 10.
B. Con inuous MPC con olle s esul s
The esul s o he closed-loop simula ions o he VSL compu ed by each con inuous MPC
con olle and he ounding disc e iza ion o ST-MPC can be seen in Fig. 8. I can be seen
ha VSL1o uncons ained MPC suddenly changes om 76 km/h o 32 km/h a minu e
12. Howe e , his change is done a a a e o 10 km/h o T-MPC. Mo eo e , he absolu e
di e ence be ween VSL1and VSL2(i.e. |VSL1−VSL2|) keeps be ween 40 km/h and 60
km/h o uncons ained MPC and T-MPC du ing a long pe iod o ime. When his di e ence
is cons ained in ST-MPC, he alue o VSL2is s ongly dec eased in o de o ob ain a
di e ence o 10 km/h wi h VSL1(which is also sligh ly inc eased).
0 50 100 150
20
30
40
50
60
70
80
90
Va iable Speed Limi 1
Minu es
km/h
0 50 100 150
20
30
40
50
60
70
80
90
Va iable Speed Limi 2
Minu es
km/h
Uncons ained MPC
Tempo ally Cons ained MPC
Spa ially and Tempo ally Cons ained MPC
Fig. 8. VSL o segmen 3 and 4 o con inuous MPC and ounding disc e iza ion
In he nume ical esul s on Table II can be seen how ST-MPC ( he mos ealis ic one)
esul s on a TTS educ ion ha is he 36.7% lowe han he TTS educ ion co esponding
o uncons ained MPC (12.47% e sus 8.14%) concluding han he spa ial and empo al
cons ain s subs an ially dec ease he po en ial con olle pe o mance. Howe e , in eal im-
plemen a ions, he lack o sa e y cons ain s may inc ease he ac ual TTS educ ion due o
sudden b aking and acciden s.
TABLE II
MPC PERFORMANCES
TTS Reduc ion (%)
Uncon olled Sys em 0 %
Uncons ained MPC 12.87 %
Tempo ally cons ained MPC (T-MPC) 10.48 %
Spa ially and empo ally cons ained MPC (ST-MPC) 8.14 %
C. Rounding, Floo ing and Ceiling
0 50 100 150
20
40
60
80
Minu es
km/h
Va iable Speed Limi 1
Rounding
Ceiling
Floo ing
Spa ially and Tempo ally Cons ained MPC
0 50 100 150
20
40
60
80
Va iable Speed Limi 2
Minu es
km/h
Fig. 9. Rounding, Ceiling, Floo ing and con inuous VSL (ST-MPC) o segmen 3 and 4
The closed loop VSL disc e iza ion by ounding, loo ing, and ceiling, and he con inuous
ST-MPC solu ion a e shown in Fig. 9. I can be seen han di e en VSL choices du ing a ew
sample s ep du ing he beginning o he conges ion can cause a di e en simula ion du ing a
long pe iod o ime ( he ceiling VSL a e 20 km/h and 30 km/h o e he ounding and ceiling
VSL du ing mo e han 90 minu es).
The nume ical esul s o he ounding, ceiling, and loo ing cases can be seen in Table
III. In his simula ion, he h ee op ions gi e a simila pe o mance wi h a sligh ad an age
o he ounding case. Howe e , he TTS educ ion o he ounding case is 46% lowe han
he TTS educ ion co esponding o ST-MPC (8.14% e sus 4.40%). This shows ha he
supposi ion done in many p e ious pape s abou he con inuous implemen a ion o he VSL
can en ail a la ge loss o pe o mance o he con olled sys em o some ne wo ks and some
a ic condi ions.
TABLE III
MPC PERFORMANCES
TTS Reduc ion (%)
Spa ially and empo ally cons ained MPC 8.14 %
Rounding disc e iza ion 4.40 %
Ceiling disc e iza ion 4.37 %
Floo ing disc e iza ion 4.22 %
D. θ-Gene ic and θ-Exhaus i e Op imiza ions
The VSL disc e iza ion o ST-MPC by θ-Gene ic and θ-Exhaus i e op imiza ion a e shown
in Fig.10. I can be seen han he solu ions o bo h he gene ic and he exhaus i e op imiza ions
keep a ound he con inuous solu ion du ing he whole simula ion. Du ing he ime ha he
con inuous solu ion s ays ela i ely cons an , he disc e e solu ions app oxima e he ST-MPC
swi ching be ween wo disc e e alues in he same way as in a pulse-wid h modula ion.
The nume ical esul s can be seen in Table IV. I can be seen ha alues o θhighe han
10 do no b ing much inc ease in he pe o mance o he con olled sys em as can be seen
in TTS educ ion o he case wi h θ= 18 and θ= 25. Mo e conc e ely, he θ-exhaus i e
op imiza ion wi h θ= 10 is only 1.6% wo se han o θ= 25. Howe e , he compu a ion
0 50 100 150
20
30
40
50
60
70
80
90
Minu es
km/h
Va iable Speed Limi 1
Exhaus i e disc e iza ion
Gene ic disc e iza ion
ST−MPC
0 50 100 150
20
30
40
50
60
70
80
90
Va iable Speed Limi 2
Minu es
km/h
Fig. 10. Exhaus i e and gene ic op imiza ions
TABLE IV
θ-GENETIC AND θ-EXHAUSTIVE PERFORMANCES
θTTS Reduc ion Disc e e op imiza ion ime
θ-Exhaus i e op imiza ion 10 7.22% 1.85 s
θ-Exhaus i e op imiza ion 14 7.24% 14.12 s
θ-Exhaus i e op imiza ion 18 7.25% 16.26 s
θ-Exhaus i e op imiza ion 25 7.34% 26.70 s
θ-Gene ic op imiza ion 10 6.98% 0.4 s
ime is inc eased om 1.85s o 26.70s. The e o e, we decided o use a alue o θ= 10 o
he θ-Gene ic op imiza ion algo i hm.
A e a uning p ocess (equi alen o he one shown in he ollowing subsec ion o he
al e na ing gene ic op imiza ion), 20 indi iduals wi h 100 genes we e chosen o he GA. A
c osso e pa ame e o 0.8 and a mu a ion pa ame e o 0.01 a e used. I can be seen han he
θ-Gene ic op imiza ion gi es a solu ion ha is close o he θ-Exhaus i e one (6.98% e sus
7.22%) bu wi h a lowe compu a ion imes (0.4 s e sus 1.85 s).
0 50 100 150
20
40
60
80
Minu es
km/h
Va iable Speed Limi 1
0 50 100 150
20
40
60
80
Va iable Speed Limi 2
Minu es
km/h
Al e na ing Explici Op imiza ion
Al e na ing Gene ic Op imiza ions
Fig. 11. Exhaus i e op imiza ion e sus gene ic op imiza ions
E. Al e na ing Op imiza ion esul s
The VSL ob ained by Al e na ing Exhaus i e Op imiza ion and six examples o he VSL
ob ained by using Al e na ing Gene ic Op imiza ion a e shown on Fig.11. I can be seen han
o he majo i y o he p o iles ob ained by he GA, he VSL s ay e y close o he exhaus i e
solu ions.
TABLE V
TUNING OF ALTERNATING EXHAUSTIVE OPTIMIZATION
Ramp I e a ions VSL I e a ions TTS Reduc ion (%)
Al e na ing Exhaus i e Op imiza ion 1 1 7.4229
Al e na ing Exhaus i e Op imiza ion 2 1 7.4252
Al e na ing Exhaus i e Op imiza ion 3 3 7.4252
The nume ical esul s o he exhaus i e op imiza ion o di e en numbe s o i e a ions
a e shown in Table V. I can be seen ha he i e a ions almos do no inc ease he TTS
educ ion (7.4229% wi hou i e a ions e sus 7.4252% wi h h ee i e a ions o he amp
me e ing and he VSL). In ac , in his simula ion he algo i hm con e ges in one i e a ion
o he amp me e ing o 72 ou he 75 sample imes. The e o e, we decided o use only one
i e a ion o he exhaus i e and gene ic op imiza ions (i.e. amp me e ing op imiza ion and
VSL op imiza ion will be un jus one ime each sample ime). As he algo i hm con e ges
wi h h ee i e a ions o all he sample imes and assuming ha he amp me e ing op imiza ion
is using enough ini ial poin s o a oid local minima, i can be said ha 7.4252% is likely o
be he maximum TTS educ ion ha can be achie ed wi h he used ho izons and cos unc ion
pa ame e s. I will be di icul o implemen he exhaus i e algo i hm in eal ime o la ge
ne wo ks due he compu a ion imes equi ed. Howe e , in his case i is possible o compu e
he con ol inpu s in eal ime. Fo he exhaus i e op imiza ion wi h jus one i e a ion, he
mean compu a ion ime o one sample s ep is 31.04 s wi h a maximum o 40.64 s which is
less han he 2 minu es con olle sample ime. Using 3 i e a ions o he amp me e ing and
o he VSL, he mean compu a ion ime is 95.43 s wi h a maximum o 125.78 s.
TABLE VI
TUNING OF ALTERNATING GENETIC OPTIMIZATION
Ind Gen Mu a C os MTR(%) SD-TR Mean CT SD-CT
20 100 0.01 0.8 6.99 % 0.29 6.14 s 1.71
20 100 0.001 0.8 5.81 % 1.30 7.22 s 1.81
40 200 0.01 0.8 7.37 % 0.14 12.55 s 2.65
40 200 0.001 0.8 7.13 % 0.38 9.85 s 1.93
60 100 0.01 0.6 7.27 % 0.178 8.45 s 1.01
30 100 0.01 0.6 7.11 % 0.36 6.26 s 1.91
Some o he nume ical esul s ob ained du ing he uning o he Al e na ing Gene ic
Op imiza ion a e shown on Table VI. In he Table, Ind, Gen a e he numbe o indi iduals and
genes in he GA, Mu a and C os a e he Mu a ion and C osso e pa ame e o he GA. MCT
is he Mean Compu a ion Time, and SDCT is he S anda d De ia ion o he Compu a ional
Time. In o de o e alua e he pe o mance o he gene ic algo i hm, he simula ions ha e
been un 10 imes and he mean TTS educ ion (MTR) and he s anda d de ia ion o his
educ ion (SDTR) ha e been used as pe o mance measu es.
The con olle s show a good beha io o he majo i y o he pa ame e s se s e alua ed,
usually wi h mo e han a 7% TTS educ ion (close o he maximum o 7.43 %). Mo eo e ,
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