1
Exploi ing he use o DC SCOPF app oxima ion o
imp o e i e a i e AC SCOPF algo i hms
A. Ma ano Ma colini, F. Capi anescu, J.L. Ma inez Ramos Senio Membe , IEEE, and L. Wehenkel
Abs ac —This pape ocuses on imp o ing he solu ion ech-
niques o he AC SCOPF p oblem o ac i e powe dispa ch
by using he DC SCOPF app oxima ion wi hin he SCOPF algo-
i hm. Ou app oach b ings wo bene i s compa ed o benchma k
SCOPF algo i hms: i speeds-up he solu ion o an i e a i e AC
SCOPF algo i hm hanks o a mo e e icien iden i ica ion o
binding con ingencies, and allows imp o ing he objec i e by an
app op ia e choice o a limi ed numbe o co ec i e ac ions o
each con ingency. The p oposed app oach is illus a ed on 5 es
sys ems o 60, 118, 300, 1203, and 2746 buses.
Index Te ms—secu i y-cons ained op imal powe low, DC
app oxima ion, con ingency il e ing, mixed-in ege linea p o-
g amming, nonlinea p og amming
I. INTRODUCTION
The Secu i y-Cons ained Op imal Powe Flow (SCOPF)
is a nonlinea , non-con ex, la ge-scale, s a ic op imiza ion
p oblem, wi h bo h con inuous and disc e e a iables [1], [2].
I compu es op imal p e en i e/co ec i e ac ions ha sa is y
cons ain s o bo h he p e-con ingency con igu a ion as well
as unde a se o pos ula ed con ingencies.
Powe sys em enginee s gene ally sol e sepa a ely and
sequen ially wo SCOPF p oblems, namely he ac i e powe
dispa ch and he eac i e powe dispa ch [3]. The main ea-
sons o his sepa a e compu a ion a e: hese p oblems ha e
di e en objec i e unc ions (e.g. minimum gene a ion cos s.
maximum eac i e powe ese es o minimum powe losses),
he wo p oblems a e be e posed sepa a ely as he con ol
a iables o one p oblem ha e gene ally li le impac on he
cons ain s and he objec i e unc ion o he o he p oblem,
and hey ha e a smalle size and a e hence mo e ac able.
This pape ocuses on he imp o emen o he solu ion ech-
niques o he AC SCOPF p oblem o ac i e powe dispa ch by
explo ing he use o a DC SCOPF. We use as a benchma k he
i e a i e AC SCOPF algo i hm p oposed in [4], [5]. Among
he challenges o he SCOPF compu a ions [3], [6], [8] we
deal wi h he iden i ica ion o he binding con ingencies and
wi h he selec ion o a limi ed numbe o co ec i e ac ions.
Re . [11] epo s a case whe e he e is a sa is ac o y ag ee-
men be ween he binding con ingencies a he AC and DC
SCOPF solu ion o a e y la ge eal-li e sys em in he con ex
o ene gy p icing by loca ional ma ginal p ices. Mo i a ed by
his encou aging esul and by he compu a ional speed o he
A Ma ano Ma colini and J.L. Ma inez Ramos a e wi h he Depa -
men o Elec ical Enginee ing, Uni e si y o Se ille, Spain (e-mail: ale-
jand [email p o ec ed]; [email p o ec ed]). F. Capi anescu and L. Wehenkel a e wi h
he Depa men o Elec ical Enginee ing and Compu e Science, Uni e si y
o Li`ege, B4000 Li`ege, Belgium (e-mail: capi ane@mon e io e.ulg.ac.be;
l.wehenke[email p o ec ed]).
linea p og amming sol e s, a i s con ibu ion o his pape is
o u he ca e ully assess he bene i o using he DC SCOPF
inside he i e a i e AC SCOPF algo i hms.
Inc easing le els o unce ain y in he con ex o day-
ahead ope a ional planning and in aday ope a ion oge he
wi h ongoing e o s o enhancing he ansmission sys em
lexibili y (e.g. WAMS and FACTS) lead o an inc eased use
o co ec i e con ol in e e y-day p ac ice, and he e o e yield
a g owing need o he e ec i e coo dina ion o co ec i e
and p e en i e con ols [8], [12], [13]. On he o he hand,
elying on co ec i e con ol inc eases signi ican ly he com-
plexi y o powe sys em ope a ion and also in oduces new
eliabili y issues, ela ed o he complexi y o implemen a ion
o co ec i e con ol and i s induced ailu e modes. Mos
mode n powe sys ems ha e deployed dedica ed communica-
ion channels be ween con ol cen es and con ol means (e.g.
powe plan s) ha may allow he au oma ic implemen a ion o
op imal solu ions in ol ing a la ge numbe o emedial ac ions.
Howe e , an op imal solu ion in ol ing many emedial ac ions
is di icul o unde s and (e.g. his is pa icula ly ue o
ac ions wi h small magni ude), in e p e , alida e by he TSO
expe ience, and hence us . Fu he mo e, du ing he au oma ic
implemen a ion, he TSO may lack ocusing on wha happens
on he g id (and is hence mo e p one o e o ) when dealing
wi h a la ge numbe o quickly mo ing powe injec ions.
Finally, i he numbe o ac ions o implemen is oo la ge
and some communica ion channels ail, he TSO may no be
able o use he ypical back-up solu ion based on phone calls.
Fo hese easons, o educe he complexi y o implemen a ion
and educe he p obabili y o ailu e o co ec i e con ol, one
possible app oach is o impose o each con ingency a bound
on he numbe o co ec i e ac ions used in he e en ha
i would happen. Indeed, he need o limi ing he numbe
o co ec i e con ol ac ions has al eady been pu o wa d by
many au ho s [6]–[10].
In cu en p ac ice, he choice o an app op ia e subse o
co ec i e ac ions is ypically de ined in a heu is ic way based
on enginee ing judgemen and o -line s udies, yielding a ixed
lis o co ec i e ac ions ha a e hen plugged in o he sub-
sequen SCOPF calcula ions [6], [8]–[10]. Because he mos
e icien co ec i e ac ions o each con ingency may change
in unp edic able ways (e.g. due o he inc easing a iabili y
o load pa e ns, ne wo k opology, gene a o s dispa ch) and
because he se o possible co ec i e ac ions g ows (e.g. due
o he pene a ion o dispe sed gene a ion and demand side
managemen possibili ies) his app oach may lead o sub-
op imal esul s in e ms o ma ke e iciency and eliabili y.
The e o e, he limi ed lis o co ec i e ac ions used by he
2
SCOPF should ideally be compu ed in an au oma ic ashion
o each con ingency and o each ope a ing scena io.
Wi hin his con ex , a ew pape s ha e p oposed echniques
o limi ing he numbe o con ol ac ions in an OPF [8], i.e.
o one sys em s a e, whe eas only one app oach has been
epo ed o he SCOPF p oblem [15]. This app oach adop s
a DC g id model [16], [17] and looks o a limi ed numbe o
opological maneu e s as p e en i e ac ions only bu does no
ex end he analysis o he AC SCOPF. A second con ibu ion
o his pape is he ex ension o he concep s p oposed o
he OPF p oblem in [14] o he AC SCOPF p oblem wi h
limi ed numbe o co ec i e ac ions. We also explo e a DC
mixed-in ege linea p og amming (MILP) app oxima ion o
his p oblem in o de o iden i y app op ia e co ec i e ac ions
o each con ingency. We assess he in e es o ou app oach
o compu ing combina ions o p e en i e and co ec i e gen-
e a o e-dispa ches bu he app oach may be ex ended o
o he con ol ac ions (e.g. phase shi ing ans o me , ne wo k
swi ching, e c.). The ield o applica ion o ou app oach is
mainly day-ahead and in aday planning o ope a ion. I may
be used o ensu e he he mal secu i y o a uni commi men
solu ion o nex 12-24 hou s pe iod and also o eac o non-
an icipa ed changes in he in aday ope a ing scena io.
The es o he pape is o ganized as ollows. Sec ion II
p esen s he p oblem o mula ion and he solu ion echnique.
Sec ion III p o ides nume ical expe imen s wi h he p oposed
app oach. Sec ion IV concludes.
II. SECURITY-CONSTRAINED OPTIMAL POWER FLOW
WITH LIMITED NUMBER OF CORRECTIVE ACTIONS
A. S a emen o he p oblem
We conside he SCOPF p oblem o ac i e powe dispa ch
wi h limi ed numbe o co ec i e ac ions o ace line ou age
con ingencies. The p oblem is called he ea e SCOPF-LNCA,
and is o mula ed as ollows:
min
P0
gi ,P k
gi ,sk
i
X
i∈G
ciP0
gi (1)
subjec o:
Pk
gi −Pli −X
j∈Bk
i
Pk
ij(Vk
i, V k
j, θk
i, θk
j) = 0,
∀i∈ N,∀k∈ {0} ∪ C (2)
Qk
gi −Qli −X
j∈Bk
i
Qk
ij(Vk
i, V k
j, θk
i, θk
j) = 0,
∀i∈ N,∀k∈ {0} ∪ C (3)
Ik
ij (Vk
i, V k
j, θk
i, θk
j)≤Imax k
ij ,∀i, j ∈ N,∀k∈ {0} ∪ C (4)
Pmin
gi ≤Pk
gi ≤Pmax
gi ,∀i∈ G,∀k∈ {0} ∪ C (5)
Qmin
gi ≤Qk
gi ≤Qmax
gi ,∀i∈ G,∀k∈ {0} ∪ C (6)
−sk
i∆Pi≤Pk
gi −P0
gi ≤sk
i∆Pi,∀i∈ G,∀k∈ C (7)
X
i∈G
sk
i≤Nk,∀k∈ C (8)
sk
i∈ {0,1},∀i∈ G,∀k∈ C,(9)
whe e, supe sc ip 0 ( esp. k) e e s o he base case ( esp.
con ingency ks a e), Cis he se o pos ula ed con ingencies,
Gis he se o gene a o s, Nis he se o buses, Bk
iis he se o
b anches connec ed o bus iin s a e kand accoun s o he line
ou ages in he pos -con ingency s a es, ciis he ac i e powe
cos o gene a o i,Nkis he maximum numbe o co ec i e
ac ions ha he sys em ope a o wishes o implemen , sk
iis a
bina y a iable desc ibing he s a us o gene a o i o he pos -
con ingency s a e k( he gene a o can be used o co ec i e
con ol i sk
i= 1 and is o he wise ozen o i s p e-con ingency
alue), ∆Piis he maximum amoun o powe ha a gene a o
can edispa ch ollowing a con ingency, Vk
i( esp. θk
i) is he
ol age magni ude ( esp. angle) a bus iin s a e k, he o he
no a ions being sel -explana o y.
The objec i e unc ion (1) is minimum gene a ion cos in
he p e-con ingency s a e bu o he objec i es could be used
(e.g. minimum de ia ion om he ma ke solu ion).
Equali y cons ain s (2,3) a e powe low equa ions in p e-
con ingencyand pos -con ingencys a es. Inequali y cons ain s
(4) a e b anch cu en limi s in p e-con ingency and pos -
con ingency s a es. Inequali y cons ain s (5,6) a e physical
limi s o gene a o ’ ac i e and eac i e powe s.
The limi a ion o he numbe o co ec i e ac ions in pos -
con ingency s a es is modeled by cons ain s (7)-(9).
No e ha co ec i e ac ions compu ed wi h he SCOPF-
LNCA app oach o each con ingency me ely ensu e he
easibili y o all pos -con ingency con ol p oblems gi en he
esul ing p e en i e mode gene a ion eschedulings. Feasibil-
i y being g an ed by eezing hese p e en i e con ols, co -
ec i e con ols o each con ingency may hen be compu ed
sepa a ely by sol ing an app op ia e se o OPF p oblems ac-
co ding o any objec i e unc ion o in e es , while espec ing
he cons ain s on he numbe o co ec i e con ols.
We sol e he SCOPF-LNCA p oblem in wo s eps. We
i s sol e he MILP app oxima ion o he p oblem so as o
de e mine he op imal se {i∈ G :sk
i= 1}o co ec i e
ac ions o each con ingency k∈ C. Then we sol e a classical
SCOPF by allowing only hese la e con ol ac ions.
B. Fo mula ion o he DC MILP SCOPF app oxima ion o he
SCOPF-LNCA p oblem
We compu e he app op ia e co ec i e ac ions o be used
in he AC SCOPF by sol ing he ollowing DC MILP SCOPF
app oxima ion o he SCOPF-LNCA (1)-(9):
min
P0
gi ,P k
gi ,sk
i
X
i∈G
ciP0
gi (10)
3
Cpb
AC SCOPF
Ccyes
no
secu i y analysis
con ingency il e ing DC SCOPF
C;{˜sk
i}
secu i y analysis
AC OPF
C;{˜sk
i}
Cc;{˜sk
i}
Cpb;{˜sk
i}Cpb;{sk
i}o {˜sk
i}
con ingency il e ing
STOP
Cc=∅?
Cs
Cpb ← Cpb ∪ Cs
C Cpb
Fig. 1. Flowcha o he i e a i e SCOPF algo i hm a ian s
subjec o:
Pk
gi −Pli −X
j∈Bk
i
θk
i−θk
j
Xij
= 0,∀i∈ N,∀k∈ {0} ∪ C (11)
−Imax k
ij ≤θk
i−θk
j
Xij
≤Imax k
ij ,∀i, j ∈ N ,∀k∈ {0} ∪ C
(12)
Pmin
gi ≤Pk
gi ≤Pmax
gi ,∀i∈ G,∀k∈ {0} ∪ C (13)
−sk
i∆Pi≤Pk
gi −P0
gi ≤sk
i∆Pi,∀i∈ G,∀k∈ C (14)
X
i∈G
sk
i≤Nk,∀k∈ C (15)
sk
i∈ {0,1},∀i∈ G,∀k∈ C,(16)
whe e (11) a e he DC powe low equa ions [16], [17], Xij
is he b anch eac ance, he o he no a ions ha ing he same
meaning as in he SCOPF-LNCA p oblem. Because he DC
app oach assumes ha all he ol ages a e 1 p.u., hen he
alues in p.u. o b anch cu en s and ac i e powe lows
coincide.
C. I e a i e SCOPF algo i hms
Figu e 1 p o ides he lowcha o he i e a i e SCOPF
algo i hm a ian s compa ed in his pape . In his igu e he
p oposed app oach, called he ea e P-SCOPF, is depic ed wi h
con inuous lines, while he benchma k SCOPF algo i hm [4],
[5], called he ea e B-SCOPF, is depic ed by showing in
dashed lines i s i s s eps (uppe le pa o he lowcha ).
Figu e 1 also shows he a ious se s o con ingencies in
inpu and ou pu o each module. The no a ions o hese se s
o con ingencies a e u he explained in Table I.
TABLE I
DEFINITION OF VARIOUS CONTINGENCY SETS
Cbse o binding con ingencies
Cpb se o po en ially binding con ingencies
Ccse o c i ical con ingencies (i.e. ha iola e cons ain s)
Csse o con ingencies selec ed by he il e
C a se o alse ala ms in il e ing
Cni se o binding con ingencies no iden i ied by he il e
We dis inguish be ween wo uses o he DC SCOPF wi hin
he P-SCOPF app oach:
1) In p e en i e-only mode SCOPF 1o when he co ec-
i e ac ion se s a e chosen be o ehand (i.e. o ixed
alues o s a uses sk
i= ˜sk
iin (1)-(9)), he DC SCOPF
eplaces he h ee i s s eps o he B-SCOPF, namely
he OPF, he Secu i y Analysis (SA), and he Con-
ingency Fil e ing (CF). Thus a he i s i e a ion o
he algo i hm he AC SCOPF is ed wi h he binding
con ingencies om he DC SCOPF solu ion (se Cpb).
A he subsequen i e a ions bo h B-SCOPF and P-
SCOPF app oaches coincide. In pa icula hey use he
non-domina ed con ingency (NDC) echnique o con-
ingency selec ion [4], [5]. In his case he DC SCOPF
(10)-(16) is a linea p og amming p oblem.
2) In SCOPF applica ions in co ec i e-also mode wi h
an op imiza ion-based choice o a limi ed numbe o
co ec i e ac ions (i.e. he ull SCOPF-LNCA) he DC
SCOPF eeds he AC SCOPF also wi h he co ec i e
ac ion se s {sk
i}, chosen in a p ope way by sol ing he
MILP (10)-(16), in addi ion o he binding con ingencies
a he solu ion o his p oblem.
No e ha in o de o speed-up he compu a ions he DC
SCOPF in i s wo o ms, LP and MILP, i is implemen ed using
he same algo i hm as he B-SCOPF (i.e. elying on: OPF,
SA, CF, and DC SCOPF including only po en ially binding
con ingencies).
III. EXPERIMENTAL VALIDATION BY SIMULATION
A. Desc ip ion o he es sys ems
In his sec ion we p esen ep esen a i e nume ical esul s
ob ained wi h he p oposed app oaches on i e es sys ems:
a 60-bus sys em, which is a modi ied a ian o he No dic32
sys em [19], he IEEE118 and he IEEE 300 sys ems [20], a
modi ied old planning model o he RTE ( he F ench TSO) sys-
em o 1203 buses, and a win e peak load model o he Polish
powe sys em [21]. A summa y o hei cha ac e is ics is gi en
in Table II, whe e: |N|,|G|,|D|,|B|,|L|,|T |,|S|, and |C|
deno e he numbe o : buses, gene a o s, loads, b anches, lines,
all ans o me s, shun elemen s, and pos ula ed con ingencies,
espec i ely. We conside o each sys em a con ingency se
composed o all single line ou ages excluding hose ha would
lead o a spli ing o he sys em in o sepa a e islands.
1The p e en i e-only mode is a pa icula case o he SCOPF-LNCA (1)-(9)
ob ained o sk
i= 0,∀i∈ G,∀k∈ C, excep o he gene a o s pa icipa ing
in equency egula ion.
4
TABLE II
TEST SYSTEMS SUMMARY
sys em |N | |G| |D| |B| |L| |T | |S| |C|
No dic32 60 23 22 81 57 31 12 33
IEEE118 118 54 91 186 175 11 14 166
IEEE300 300 69 198 411 282 129 14 174
1203-bus 1203 177 767 1797 1394 403 11 1029
2746-bus 2746 370 2024 3279 3107 172 0 2468
B. Sol e s used
The AC SCOPF o mula ion is handled by using he
in e io -poin based NLP sol e desc ibed in [22] and ollows
he i e a i e app oach desc ibed in [4], [5].
The DC SCOPF in ei he linea p og amming o m o
mixed-in ege linea p og amming o m is sol ed on he
GAMS pla o m [23] using he CPLEX sol e . CPLEX im-
plemen s a dual simplex algo i hm o sol ing he linea
p og amming p oblem and a b anch and cu algo i hm o he
mixed-in ege p oblem [24].
All es s ha e been pe o med on a PC Pen ium IV, 1.9-
GHz, 2-GB RAM.
C. Compa isons o he P-SCOPF and B-SCOPF app oaches
We conside he SCOPF-LNCA p oblem (1)-(9) o mula ed
in p e en i e mode o all sys ems bu he 2746-bus sys em,
whe e we use he co ec i e-also mode.
We conside a ew cases (deno ed o “case 1” o “case 4”)
o each es sys em which di e by he o al sys em load (e.g.
no mal load, peak load, e c.) and/o b anch he mal limi s.
Table III p o ides a compa ison be ween he P-SCOPF and
B-SCOPF app oaches in e ms o hei abili y o iden i y, a
he i s i e a ion o he i e a i e SCOPF algo i hm, he binding
con ingencies a he AC SCOPF solu ion. In he P-SCOPF
( esp. B-SCOPF) he po en ially binding con ingencies a e
p o ided by he DC SCOPF ( esp. non-domina ed con ingency
(NDC) il e ing echnique [4], [5]). The a ious se s o con-
ingencies ha e been explained in Table I.
To compensa e possible non-iden i ied con ingencies, due
o he DC model app oxima ion, he DC SCOPF app oach
also includes in he po en ially binding con ingencies se some
nea -binding con ingencies (e.g. ha would lead o a line
loading o mo e han 98%).
We can obse e ha he DC SCOPF app oach p o ides
excellen esul s, he binding con ingencies being co ec ly
iden i ied in 10 ou o 11 cases (i.e. when |Cni|= 0) while
in oducing a easonably small numbe o alse ala ms. On
he o he hand, in he case 4 o he IEEE118 sys em, he DC
SCOPF app oach does no iden i y 2 binding con ingencies.
Acco ding o he DC SCOPF app oach he loading o he
binding line he mal limi , a he AC SCOPF solu ion, o hese
con ingencies is o 97.48 %and 94.49 %, espec i ely. This
misma ch be ween he wo app oaches is due o h ee ac o s:
hese con ingencies lead o a signi ican amoun o losses, he
losses a e compensa ed by he slack gene a o only, and hese
wo lines a e loca ed e y closely o he slack gene a o .
Anyway, despi e hese excellen esul s, cases whe e no
all binding con ingencies a e iden i ied a e o be expec ed as
TABLE III
COMPARISON OF THE ACCURACY OF DC SCOPF AND NDC APPROACHES
TO IDENTIFY THE BINDING CONTINGENCIES AT THE SCOPF SOLUTION
es AC SCOPF DC SCOPF NDC
case |Cb| |Cpb| |Cni| |C a| |Cpb| |Cni| |C a| |Cc|
No dic32 sys em
case 1 5 12 0 7 7 1 3 16
case 2 3 9 0 6 7 0 4 9
case 3 5 11 0 6 7 2 4 18
case 4 4 7 0 3 9 0 5 18
IEEE118 sys em
case 1 9 13 0 4 19 3 13 107
case 2 10 13 0 3 12 3 5 87
case 3 8 13 0 5 17 1 10 104
case 4 8 9 2 3 11 1 4 90
IEEE300 sys em
case 1 4 7 0 3 9 0 5 11
case 2 3 6 0 3 5 0 2 6
1203-bus sys em
case 1 4 6 0 2 4 1 1 8
case 2 5 10 0 5 6 1 2 29
2746-bus sys em
case 1 4 5 0 1 6 0 2 8
ound ou in [11], e.g. due o he eac i e powe lows which
also con ibu e o b anches cu en a e neglec ed, lossless g id
assump ion o he DC model, e c.
On he o he hand he NDC app oach ails in 6 ou o
11 cases o iden i y all he binding con ingencies a he i s
i e a ion o he i e a i e SCOPF algo i hm. This is mos o
he imes due o a binding con ingency does no iola e
any cons ain a ha s age o he algo i hm and o a less
ex en due o a binding con ingency is il e ed ou by mis ake.
The la e si ua ion a ises especially when he numbe o
c i ical con ingencies |Cc|is la ge (e.g. as is he case in he
IEEE118 sys em). No e ha i e a i e NDC SCOPF app oach
assumes ha ew loops may be needed o iden i y all binding
con ingencies especially when hese a e ha mless a he i s
i e a ion.
We compa e bo h app oaches in e ms o o e all SCOPF
CPU ime solu ion. By looking a Table IV we no ice ha
he be e accu acy o iden i y binding con ingencies o he
DC SCOPF app oach leads o a smalle numbe o loops on
he i e a i e algo i hm, and hence a smalle numbe o calls
o he secu i y analysis module, ansla es consequen ly in o
a signi ican gain o compu a ional ime in almos all cases.
E en i he SA is implemen ed using pa allel compu a ions,
as is he case on con ol cen es, he compu a ional ad an age
o he p oposed app oach will s ill pe sis , as demons a ed in
Tables V and VI. This gain is less impo an only in cases
whe e ei he bo h app oaches iden i y all binding con ingen-
cies a he i s i e a ion (e.g. in cases 2 and 4 o he No dic32
sys em) o whe e he DC SCOPF ails iden i ying all binding
con ingencies (e.g. his happens only in case 4 o he IEEE118
bus sys em). Fu he mo e bo h app oaches may bene i om a
u he educ ion o he CPU imes o he AC SCOPF module
i sel hanks o a mo e e icien implemen a ion.
In o de o enable he compa ison be ween bo h app oaches
conce ning he compu a ional e o o each ask o he SCOPF
algo i hm p esen ed in Fig. 1, Table V ( esp. Table VI)
p o ides he samples o CPU imes o each module o he
5
TABLE IV
OVERALL CPU TIMES (S)OF BOTH APPROACHES AND TIME REDUCTION
THANKS TO THE USE OF THE PROPOSED APPROACH
es case B-SCOPF P-SCOPF ime educ ion (%)
No dic32 sys em
case 1 9.17 5.53 39.7
case 2 4.03 4.48 -11.2
case 3 9.36 5.27 43.7
case 4 4.57 4.03 11.8
IEEE118 sys em
case 1 51.4 18.9 63.2
case 2 58.6 19.3 67.1
case 3 45.3 19.2 57.6
case 4 35.09 30.19 14.0
IEEE300 sys em
case 1 57.2 37.6 34.3
case 2 48.9 33.16 32.2
1203-bus sys em
case 1 1102.4 554.3 49.7
case 2 1667.4 824.8 50.5
2746-bus sys em
case 1 2456.1 1488.1 39.4
TABLE V
SAMPLE OF CPU TIMES (S)FOR THE P-SCOPF APPROACH
es case i e DC SCOPF AC SCOPF SA CF ime (s)
No dic32 sys em
case 1 1 0.12 5.2 0.21 - 5.53
IEEE118 sys em
case 4 1 0.35 10.2 3.06 0.0 30.19
2 - 13.6 2.98 -
IEEE300 sys em
case 2 1 0.46 32.6 10.10 - 33.16
1203-bus sys em
case 1 1 8.4 238.7 307.2 - 554.3
2746-bus sys em
case 1 1 12.4 611.9 863.8 - 1488.1
i e a i e P-SCOPF ( esp. B-SCOPF) algo i hm o one case o
each es sys em.
D. Choosing a limi ed numbe o co ec i e ac ions by he
MILP DC SCOPF
1) Mo i a ion o he need o upda e au oma ically he se
o co ec i e ac ions: Figu e 2 p o ides he alue o he
AC SCOPF objec i e o a ious se s o co ec i e ac ions
p oposed by he MILP DC SCOPF. We conside wo ope a ing
poin s unde no mal load and peak load, espec i ely. The
AC SCOPF objec i e e e s o no mal load condi ions. The
SCOPF objec i e equal o 1 is ob ained using he whole
se o 22 co ec i e ac ions allowed o each among he 33
con ingencies. This igu e shows ha i he se o co ec i e
ac ions de i ed o he peak load condi ions is also used o
eed he SCOPF p oblem o he no mal load condi ions he
solu ions ob ained a e sys ema ically sub-op imal. In pa icula
he solu ion sub-op imali y is unaccep able o N= 2. These
expe imen s suppo he need o upda e au oma ically he se
o co ec i e ac ions o each an icipa ed ope a ing poin .
2) Illus a ion o he app oach: Figu e 3 plo s he alue
o he AC SCOPF objec i e o inc easing alues o he
numbe o co ec i e ac ions and o wo anges o co ec i e
ac ions. We do no ex end he analysis beyond N= 7 as
TABLE VI
SAMPLE OF CPU TIMES (S)FOR THE B-SCOPF APPROACH
es case i e AC SCOPF SA CF ime (s)
No dic32 sys em
case 1 1 0.17 0.33 0.0 9.172 3.9 0.24 0.0
3 4.3 0.23 -
IEEE118 sys em
case 4 1 0.37 3.32 0.0 35.092 11.2 3.10 0.0
3 14.1 3.00 -
IEEE300 sys em
case 2 1 1.17 10.64 0.0 48.9
2 26.9 10.28 -
1203-bus sys em
case 1 1 2.3 310.5 0.0 1102.42 51.3 306.1 0.0
3 126.9 305.3 -
2746-bus sys em
case 1 1 4.1 863.8 0.0 2456.1
2 724.4 863.8 0.0
1
1.01
1.02
1.03
1.04
1.05
2 3 4 5 6 7
SCOPF objec i e o no mal load (pu)
numbe o co ec i e ac ions allowed N
co ec i e ac ions de i ed o no mal load
co ec i e ac ions de i ed o peak load
Fig. 2. No dic32 sys em: AC SCOPF objec i e o a ious se s o co ec i e
ac ions de i ed unde wo ope a ing condi ions
he objec i e o he p oposed app oach becomes p ac ically
equal o he objec i e o SCOPF ha employs all possible
co ec i e ac ions. The SCOPF objec i e wi h he whole se
o 22 co ec i e ac ions allowed o each con ingency is 1.000
( esp. 1.0128) o he la ge ( esp. smalle ) ange o co ec i e
ac ions.
We can obse e ha in bo h cases he objec i e unc ion
dec eases as he numbe o co ec i e ac ions inc eases which
demons a es he e ec i eness o he app oach. Fu he mo e,
he choice o co ec i e ac ions by he MILP DC SCOPF is
consis en as he la ge he amoun o co ec i e ac ions he
be e he objec i e.
Figu e 4 p o ides a compa ison be ween he p oposed
app oach and an al e na i e app oach in e ms o quali y o
he objec i e. The di e en ea u e o he la e app oach is
ha he se o co ec i e ac ions o an i e a ion includes he
se o co ec i e ac ions a he p e ious i e a ion. This means
ha he se o co ec i e ac ions o N= 4 is de e mined
by sol ing successi ely he MILP p oblems o N= 2,
N= 3, and N= 4 while looking only o he nex con ol
ac ion o be added o he exis ing se . The igu e shows ha

6
1
1.01
1.02
1.03
1.04
1.05
2 3 4 5 6 7
SCOPF objec i e (pu)
numbe o co ec i e ac ions allowed N
co ec i e ac ions dP=0.2*(Pmax-Pmin)
co ec i e ac ions dP=0.1*(Pmax-Pmin)
Fig. 3. No dic32 sys em: AC SCOPF objec i e e sus he numbe o
co ec i e ac ions allowed
1.005
1.01
1.015
1.02
1.025
1.03
2 3 4 5 6 7
SCOPF objec i e (pu)
numbe o co ec i e ac ions allowed N
p oposed app oach
al e na i e app oach
Fig. 4. No dic32 sys em: AC SCOPF objec i e e sus he numbe o
co ec i e ac ions allowed by wo app oaches
he p oposed app oach sligh ly ou pe o ms he al e na i e
app oach. On he o he hand, Figu e 5 shows ha he p oposed
app oach becomes slowe han he al e na i e echnique when
Ninc eases while, as expec ed, he ime equi ed o he
al e na i e app oach is a he insensi i e o N. Figu es 4 and 5
aken oge he hus highligh he ade-o be ween he solu ion
quali y and he compu a ional ime o hese wo app oaches.
Figu e 4 allows he sys em ope a o o assess he bes ade-
o be ween he objec i e and he numbe o co ec i e ac ions
allowed, o in o he wo ds, he sub-op imali y implied by
using smalle numbe s o con ol ac ions and whe he he e
is enough oom o maneu e in he case whe e some con ol
ac ions would ail.
3) Discussion abou he solu ion sub-op imali y: Since we
use a linea app oxima ion o he o iginal MINLP AC SCOPF
p oblem we could expec ha he p o ided se s o co ec i e
ac ions lead o sub-op imal solu ions.
Ve y ecen esea ch [18] epo s ha nowadays some
signi ican MINLP sol e s a e unable o sol e o op imali y
p oblems simila o ou o a la ge eal-li e sys em gi en ha
he ime cons ain o p o iding he solu ion o an AC SCOPF
in day-ahead planning is a mos a ew hou s. As a ma e o
1
1.5
2
2.5
3
3.5
4
4.5
5
2 3 4 5 6 7
CPU ime (s)
numbe o co ec i e ac ions allowed N
p oposed app oach
al e na i e app oach
Fig. 5. No dic32 sys em: CPU ime (s) e sus he numbe o co ec i e
ac ions allowed by wo app oaches
ac , e en o he No dic32 sys em he combina o ial p oblem
is e y la ge since we would ha e o ix |G|×|C| = 23×33 =
759 bina y a iables which leads o explo e a signi ican sub-
space o he whole space o 2759 possible combina ions o
co ec i e ac ions s a uses.
In o de o assess he deg ee o sub-op imali y o he p ob-
lem we ha e ied a ious heu is ics in he MILP DC SCOPF
(e.g. sligh ly lowe ing he MVA limi s, sligh ly inc easing he
load o compensa e o he lossless assump ion o he DC
model, e c.) and sol ed he AC SCOPF wi h di e en se s o
co ec i e ac ions. We ha e no iced ha o a gi en numbe
o co ec i e ac ions he p oposed app oach may lead o
sligh ly di e en solu ions bu no heu is ic leads o consis en ly
be e solu ions. Ne e heless hese solu ions do no di e
signi ican ly in e ms o he objec i e unc ion. Fu he mo e,
as shown in Fig. 3, using me ely 5 co ec i e ac ions allows
one o ob ain a alue o he objec i e unc ion which is al eady
e y close o ha o he SCOPF ob ained when using he whole
se o 22 co ec i e ac ions o each con ingency.
Since adequa e MINLP sol e s canno comply wi h ou
compu a ional ime cons ain s, especially on la ge eal-li e
sys ems, we belie e ha ou app oach o selec au oma ically
he se o co ec i e ac ions o each con ingency p o ides
easonable esul s.
4) Assessmen o compu a ional ime o he DC MILP
SCOPF app oach: In o de o assess in ealis ic condi ions he
compu a ional ime equi ed by he DC MILP SCOPF solu ion
we conside he 1203-bus sys em and he 2746-bus sys em. In
hese simula ions we assume ha he se o he po en ially
mos e icien possible co ec i e ac ions can be educed, e.g.
hanks o he TSO expe ise, o 15 candida e gene a o shi s
among he 143 ( espec i ely 71) dispa chable gene a o s in he
1203-bus sys em ( espec i ely he 2746-bus sys em).
Table VII epo s he CPU imes o he DC MILP SCOPF
o inc easing numbe s No allowed co ec i e ac ions among
hose 15 candida es. The SCOPF includes only he six binding
con ingencies a he DC SCOPF solu ion (see Table III) o
he 1203-bus sys em, and he eigh c i ical con ingencies o
he 2746-bus sys em.
7
TABLE VII
SAMPLE OF CPU TIMES (S)FOR THE DC MILP SCOPF FOR VARIOUS
NUMBERS NOF ALLOWED CORRECTIVE ACTIONS
sys em N= 2 N= 3 N= 4 N= 5
1203-bus 17.4 19.0 17.2 14.6
2746-bus 23.3 27.4 25.8 28.5
TABLE VIII
INFLUENCE OF THE NUMBER OF CANDIDATE CORRECTIVE CONTROLS (K)
AND OF ALLOWED ONES (N)ON CPU TIME (S)OF THE DC MILP SCOPF
N=K= 15 K= 30 K= 45 K= 60 K= 75
0 11.3 11.4 11.4 11.4 11.4
5 14.6 25.6 30.3 44.6 349.9
10 12.5 17.8 25.0 26.3 27.5
To gain u he insigh , Table VIII epo s he CPU imes
on he 1203-bus sys em o g owing numbe s o candida e
co ec i e con ol ac ions (K) and o allowed ones (N). In
o de o assess he ex a compu a ional ime in ol ed by he
MILP combina o ial sea ch we also p o ide he CPU imes
o he SCOPF p e en i e mode ( he la e co esponds o he
case N= 0, whe e he MILP educes o a LP p oblem).
We obse e ha o a gi en alue o N, he compu a ional
ime inc eases gene ally slowly as he numbe Ko candida e
co ec i e con ols g ows. On he o he hand, o a ixed
numbe Ko candida e con ols, he CPU ime dec eases when
we inc ease he numbe No allowed con ols om N= 5
o N= 10, which sugges s ha he “p ac ical” complexi y o
he MILP p oblem is no di ec ly linked o he numbe o can-
dida e combina ions ha would be explo ed by an exhaus i e
sea ch p ocedu e (in ou case, all possible subse s o size N
chosen among Kcandida es). A possible explana ion is ha
s a e-o - he-a MILP b anch-and-cu algo i hm has a much
be e han wo s -case complexi y in ou p ac ical con ex .
These expe imen s indica e ha he addi ional compu a-
ional ime o sol ing he DC MILP SCOPF on op o he
AC SCOPF is gene ally accep ably small. Ne e heless cases
whe e he MILP solu ion ime becomes la ge due o he
combina o ial explosion migh appea (e.g. as sugges ed by
he las column o N= 5) bu o una ely hey can be kep
ac able hanks o he p e-selec ion, on a long e m ho izon,
o a se o candida e emedial ac ions o app op ia e size based
on he TSO expe ience.
IV. CONCLUSION
This pape has p oposed wo me hods o imp o e he
solu ion echniques o he AC SCOPF p oblem o ac i e
powe dispa ch. These me hods ely on he solu ion o he
DC SCOPF app oxima ion o he o iginal p oblem.
Using hese me hods in an i e a i e AC SCOPF algo i hm
exhibi s wo ad an ages:
•a signi ican speed-up o he solu ion hanks o a mo e
e icien iden i ica ion o he binding con ingencies a he
op imum;
• he au oma ic and p ope choice o a limi ed numbe o
co ec i e ac ions o each con ingency a a gene ally low
compu a ional cos is a be e solu ion han he cu en
sys em ope a o p ac ice which do no adap hese subse s
o con ol a iables o he si ua ion a hand.
The e ec i eness o he p oposed app oaches has been
ex ensi ely alida ed on a ious es sys ems up o 2746 buses.
Al hough we illus a ed he app oach only o line ou ages
and o gene a o edispa ch as emedial ac ions, his app oach
cons i u es a gene ic amewo k ha may include o he ypes
o con ingencies as well as any o he ype o use ul eme-
dial ac ions (e.g. opological swi ching, phase shi e angle
changes, e c.). This app oach is lexible in he sense ha i
o e s he use he possibili y o de ine he maximum numbe
o sough emedial ac ions ha she/he conside s easible o
i s p oblem o in e es .
Fu u e wo k could add ess he inco po a ion o o he in ege
a iables in o he op imiza ion p oblem in a simila ashion
(e.g. ne wo k swi ching among he se o allowed co ec i e
ac ions and gene a o s a -up decisions among he se o
allowed p e en i e mode con ol ac ions), and o he ypes o
secu i y conce ns (e.g. ol age and ansien s abili y).
ACKNOWLEDGMENTS
We hank RTE-F ance o allowing us o use and publish
esul s wi h hei da a. The p ojec is a esul o a collabo a ion
spawn by Eu opean FP7 p ojec PEGASE, whose unding is
kindly acknowledged. A. Ma ano Ma colini and J.L. Ma -
inez Ramos like o hank he inancial suppo p o ided by
he Go e nmen o Andalusia, Spain, unde g an TEP-5170.
This pape p esen s esea ch esul s o he Belgian Ne wo k
DYSCO, unded by he In e uni e si y A ac ion Poles P o-
g amme, ini ia ed by he Belgian S a e, Science Policy O ice.
The scien i ic esponsibili y es s wi h he au ho s.
REFERENCES
[1] O. Alsac and B. S o , “Op imal load low wi h s eady-s a e secu i y”,
IEEE T ans. PAS, ol. PAS-93, no. 3, 1974, pp. 745-751.
[2] A.J. Mon icelli, M.V.P. Pe ei a, and S. G an ille, “Secu i y-cons ained
op imal powe low wi h pos -con ingency co ec i e escheduling”, IEEE
T ans. Powe Sys ., ol. PWRS-2, no. 1, Feb ua y 1987, pp. 175-182.
[3] B. S o , O. Alsac, and A.J. Mon icelli, “Secu i y analysis and op imiza-
ion” (In i ed Pape ), IEEE P oc., ol. 75, no. 12, 1987, pp. 1623-1644.
[4] F. Capi anescu, M. Gla ic, D. E ns , and L. Wehenkel, “Con ingency
il e ing echniques o p e en i e secu i y-cons ained op imal powe
low”, IEEE T ans. Powe Sys ., ol. 22, no. 4, 2007, pp. 1690-1697.
[5] F. Capi anescu and L. Wehenkel, “A new i e a i e app oach o he
Co ec i e Secu i y-Cons ained Op imal Powe Flow P oblem”, IEEE
T ans. Powe Sys ., ol. 23, no. 4, 2008, pp. 1533-1541.
[6] J.A. Momoh e al., “Challenges o op imal powe low”, IEEE T ans.
Powe Sys ., ol. 12, no. 1, 1997, pp. 444-455.
[7] B. Delou me, A. Lasnie , H. Le e b e, and G. Simean , “Minimizing he
cos o gene a ion edispa ching aking in o accoun emedial ac ions”,
pape C2-103, CIGRE con e ence, F ance, 2006.
[8] F. Capi anescu, J.L. Ma inez Ramos, P. Pancia ici, D. Ki schen, A.
Ma ano Ma colini, L. Pla b ood, and L. Wehenkel, “Secu i y-cons ained
op imal powe low: s a e-o - he-a , challenges, and u u e ends”, Elec-
ic Powe Sys . Resea ch, ol. 81, no. 8, 2011, pp 1731-1741.
[9] R. Bache (Edi o s: K. F auendo e , H. Gla i sch, and R. Bache ),
“Powe sys em models, objec i es and cons ain s in op imal powe low
calcula ions” (chap e o he book “Op imiza ion in Planning and Ope a-
ion o Elec ic Powe Sys ems”), Physica Ve lag (Sp inge ), Heidelbe g,
Ge many, 1993, pp. 217-264.
[10] A. Papalexopoulos, “Challenges o On-Line OPF Implemen a ion”,
IEEE/PES Win e Mee ing, New Yo k (USA), 1995.
8
[11] T.J. O e bye, Xu Cheng, and Yan Sun, “A compa ison o he AC and
DC powe low models o LMP calcula ions”, P oc. o he 37 h Annual
HICSS con e ence, Hawaii, 2004.
[12] FERC Con e ence on Enhanced Op imal Powe Flow Models,
Washing on, USA, June 2010, p esen a ions a ailable on-line a
h p://www. e c.go .
[13] P. Pancia ici, Y. Hassaine, S. Fliscounakis, L. Pla b ood, M. O ega-
Vazquez, J.L. Ma inez-Ramos, L. Wehenkel, “Secu i y managemen
unde unce ain y: om day-ahead planning o in aday ope a ion”, IREP
Symposium, Buzios (B azil), 2010.
[14] F. Capi anescu and L. Wehenkel, “Re-dispa ching ac i e and eac i e
powe s using a limi ed numbe o con ol ac ions”, IEEE T ans. Powe
Sys ., ol. 26, no. 3, 2011, pp. 1221-1230.
[15] K.W. Hedman, R.P. O’Neill, E.B. Fishe , and S.S. O en, “Op imal
ansmission swi ching wi h con ingency analysis”, IEEE T ans. Powe
Sys ., ol. 24, no. 3, 2009, pp. 1577-1586.
[16] B. S o and O. Alsac, “Fas decoupled load low”, IEEE T ans. Powe
Appa a us and Sys ., May-June 1974, pp. 859-869.
[17] J.L. Ma inez Ramos and V.H. Quin ana (Edi o s: A. Gomez-Exposi o,
A. Conejo and C. Caniza es), “Op imal and Secu e Ope a ion o T ans-
mission Sys ems” (chap e o he book “Elec ic Ene gy Sys ems: Anal-
ysis and Ope a ion”), CRC P ess, 2009, pp. 211-264.
[18] L. Pla b ood, S. Fliscounakis, F. Capi anescu, P. Pancia ici, C. Me ckx,
and M. O ega-Vazquez, “Deli e able D3.2: De elopmen o p o o ype
so wa e o sys em s eady-s a e op imisa ion o he Eu opean ans-
mission sys em”, PEGASE p ojec , a ailable on-line a h p://www. p7-
pegase.eu/, 2011.
[19] CIGRE Task Fo ce 38.02.08, “Long-Te m Dynamics, Phase II”, 1995.
[20] Da a o IEEE118 sys em and IEEE300 sys em, a ailable online a
h p://www.ee.washing on.edu, 1996.
[21] Da a o he Poland powe sys em, a ailable online a he MAT-
POWER (“A MATLAB Powe Sys em Simula ion Package” by R.D.
Zimme man, C.E. Mu illo-Sanchez, and Deqiang Gan) web-page
h p://www.pse c.co nell.edu/ma powe /.
[22] F. Capi anescu, M. Gla ic, D. E ns , and L. Wehenkel, “In e io -poin
based algo i hms o he solu ion o op imal powe low p oblems”, Elec.
Powe Sys . Resea ch, ol. 77, no. 5-6, Ap il 2007, pp. 508-517.
[23] B.A. McCa l, “GAMS Use Guide”, Ve sion 23.6, 2011. A ailable on-
line: www.gams.com.
[24] CPLEX 10 Use Manual, A ailable on-line: www.gams.com.
Alejand o Ma ano Ma colini was bo n in A gen ina in 1977. He ecei ed he
elec ical enginee ing deg ee om he Uni e si y o Malaga, and he Ph.D.
deg ee om he Uni e si y o Se ille, Spain, in 2010. He is cu en ly an
assis an p o esso a he Uni e si y o Se ille. His p ima y a eas o in e es
a e ol age s abili y, powe sys em con ol and ope a ion, and op imiza ion
applied o powe sys em enginee ing.
Flo in Capi anescu was bo n in Romania in 1973. He g adua ed in Elec ical
Powe Enginee ing om he Uni e si y “Poli ehnica” o Bucha es in 1997.
He ob ained he Ph.D. deg ee om he Uni e si y o Li`ege in 2003. His main
esea ch in e es s a e in powe sys ems ope a ion, planning, and con ol wi h
pa icula emphasis on op imiza ion me hods and ol age s abili y.
Jose Luis Ma inez Ramos (SM’04) was bo n in Dos He manas, Spain, in
1964. He ecei ed his Ph.D. deg ee in elec ical enginee ing in 1994. Since
1990 he has been wi h he Depa men o Elec ical Enginee ing, Uni e si y
o Se ille, whe e he is cu en ly ull P o esso . His p ima y a eas o in e es
a e ac i e and eac i e powe op imiza ion, powe sys em analysis and con ol,
and elec ici y ma ke s.
Louis Wehenkel g adua ed in Elec ical Enginee ing in 1986 and ecei ed
he Ph.D. deg ee in 1990, bo h om he Uni e si y o Li`ege, whe e he is
ull P o esso o Elec ical Enginee ing and Compu e Science. His esea ch
in e es s lie in he ields o s ochas ic me hods o sys ems and modeling,
op imiza ion, machine lea ning and da a mining, wi h applica ions in complex
sys ems, in pa icula la ge scale powe sys ems planning, ope a ion and
con ol, indus ial p ocess con ol, bioin o ma ics and compu e ision.