Chapter 4idei.fr/sites/default/files/medias/doc/by/crampes/cramp... · 2014. 6. 12. · Box 4-2 for...
29
Chapter 4 COST RECOVERY AND SHORT-RUN EFFICIENCY Claude Crampes Gremaq and Idei, University of Toulouse, France When designing a tariff for the transport of electricity, the main difficulty is that the transport industry apparently incurs high fixed costs and no real variable cost. In effect, when the infrastructure is installed and when the operators are at their workplace, the only input which is necessary to deliver electricity at a given withdrawal node is electricity at some injection node since electricity is flowing by itself. Consequently, at first sight the problem is just to allocate fixed costs, mainly infrastructure maintenance costs, wages and financial charges, among the different types of users of the grid. As a matter of fact, the transport of electricity creates two significant variable costs. 1 One is an internal cost, that is, a cost in terms of electricity: a fraction of the energy which is injected into the grid will be lost during transport. It results that the consumption of 1 MWh of electricity requires the generation of (1+L) MWh, and this extra L MWh is a real cost for which producers must be compensated. Additionally, because the lines and nodes used for transport have a limited capacity, the optimal allocation of production and consumption is not as efficient as it would be, absent any grid constraint. And pricing must include this economic cost due to congestion. When transport prices are computed using the marginal values of these two costs (the so-called “nodal prices”), they provide a revenue larger than the losses of energy. This surplus can be used to pay for fixed costs of transport but, in most cases, it is not large enough to balance the budget of the operator. This explains why transport tariffs must be either second-best linear prices (Ramsey prices) or non linear prices. The chapter presents the normative principles of the economic analysis of pricing in electricity transport. It is exclusively dedicated to short-term
Transcript of Chapter 4idei.fr/sites/default/files/medias/doc/by/crampes/cramp... · 2014. 6. 12. · Box 4-2 for...
Cha
pter
4
CO
ST R
EC
OV
ER
Y A
ND
SH
OR
T-R
UN
E
FFIC
IEN
CY
C
laud
e C
ram
pes
Gre
maq
and
Idei
, Uni
vers
ity o
f Tou
lous
e, F
ranc
e
Whe
n de
sign
ing
a ta
riff f
or th
e tra
nspo
rt of
ele
ctric
ity, t
he m
ain
diff
icul
ty
is th
at th
e tra
nspo
rt in
dust
ry a
ppar
ently
incu
rs h
igh
fixed
cos
ts a
nd n
o re
al
varia
ble
cost
. In
eff
ect,
whe
n th
e in
fras
truct
ure
is i
nsta
lled
and
whe
n th
e op
erat
ors
are
at th
eir w
orkp
lace
, the
onl
y in
put w
hich
is n
eces
sary
to d
eliv
er
elec
trici
ty a
t a
give
n w
ithdr
awal
nod
e is
ele
ctric
ity a
t som
e in
ject
ion
node
si
nce
elec
trici
ty is
flow
ing
by it
self.
Con
sequ
ently
, at f
irst s
ight
the
prob
lem
is
just
to a
lloca
te fi
xed
cost
s, m
ainl
y in
fras
truct
ure
mai
nten
ance
cos
ts, w
ages
an
d fin
anci
al c
harg
es, a
mon
g th
e di
ffer
ent t
ypes
of u
sers
of t
he g
rid.
A
s a
mat
ter
of f
act,
the
trans
port
of e
lect
ricity
cre
ates
tw
o si
gnifi
cant
va
riabl
e co
sts.1 O
ne is
an
inte
rnal
cos
t, th
at is
, a c
ost i
n te
rms
of e
lect
ricity
: a
frac
tion
of t
he e
nerg
y w
hich
is
inje
cted
int
o th
e gr
id w
ill b
e lo
st d
urin
g tra
nspo
rt.
It re
sults
that
the
cons
umpt
ion
of 1
MW
h of
ele
ctric
ity r
equi
res
the
gene
ratio
n of
(1+L
) MW
h, a
nd th
is e
xtra
L M
Wh
is a
real
cos
t for
whi
ch
prod
ucer
s m
ust b
e co
mpe
nsat
ed.
Add
ition
ally
, bec
ause
the
lines
and
nod
es
used
fo
r tra
nspo
rt ha
ve
a lim
ited
capa
city
, th
e op
timal
al
loca
tion
of
prod
uctio
n an
d co
nsum
ptio
n is
not
as
effic
ient
as
it w
ould
be,
abs
ent
any
grid
con
stra
int.
And
pric
ing
mus
t in
clud
e th
is e
cono
mic
cos
t du
e to
co
nges
tion.
W
hen
trans
port
pric
es a
re c
ompu
ted
usin
g th
e m
argi
nal v
alue
s of
thes
e tw
o co
sts
(the
so-c
alle
d “n
odal
pric
es”)
, the
y pr
ovid
e a
reve
nue
larg
er th
an
the
loss
es o
f en
ergy
. T
his
surp
lus
can
be u
sed
to p
ay f
or f
ixed
cos
ts o
f tra
nspo
rt bu
t, in
mos
t cas
es, i
t is
not l
arge
eno
ugh
to b
alan
ce th
e bu
dget
of
the
oper
ator
. Th
is e
xpla
ins
why
tran
spor
t tar
iffs
mus
t be
eith
er s
econ
d-be
st
linea
r pric
es (R
amse
y pr
ices
) or n
on li
near
pric
es.
The
chap
ter p
rese
nts t
he n
orm
ativ
e pr
inci
ples
of t
he e
cono
mic
ana
lysi
s of
pric
ing
in e
lect
ricity
tra
nspo
rt.
It is
exc
lusi
vely
ded
icat
ed t
o sh
ort-t
erm
106
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
anal
ysis
, th
at i
s to
the
ope
ratio
n of
a g
iven
ele
ctric
ity n
etw
ork.
Th
e de
velo
pmen
t of t
he tr
ansp
ort i
nfra
stru
ctur
e is
ana
lyse
d in
Cha
pter
5.
In
Sec
tion
1, w
e de
fine
the
varia
ble
cost
of
trans
port
as th
e su
m o
f th
e co
st o
f con
gest
ion
and
the
cost
of o
hmic
loss
es.
In S
ectio
n 2,
we
show
how
no
dal
pric
es c
an b
e us
ed t
o de
sign
tar
iffs
for
trans
port.
Se
ctio
n 3
is
dedi
cate
d to
the
prob
lem
of f
und
risin
g in
ord
er to
bal
ance
the
budg
et o
f the
tra
nspo
rt op
erat
or.
Firs
tly,
we
focu
s on
Ram
sey
pric
es,
whi
ch a
re t
he
seco
nd-b
est
linea
r pr
ices
w
hen
one
tries
to
re
ach
effic
ienc
y w
ithou
t im
pairi
ng t
he b
udge
t eq
uilib
rium
of
the
trans
port
firm
. S
econ
dly,
we
cons
ider
a s
peci
al c
lass
of
non-
linea
r pr
ices
, nam
ely,
tw
o-pa
rt ta
riffs
. W
e co
nclu
de in
Sec
tion
4.
We
do n
ot d
iscu
ss h
ow t
o de
fine
trans
mis
sion
rig
hts
on t
he t
rans
port
infr
astru
ctur
e an
d th
e co
nflic
t bet
wee
n th
e su
ppor
ters
of p
hysi
cal r
ight
s an
d th
e su
ppor
ters
of
finan
cial
rig
hts.2
Nei
ther
do
we
cons
ider
the
ow
ners
hip
and
the
gove
rnan
ce o
f th
e tra
nspo
rt fir
m.
We
supp
ose
that
use
rs f
ace
no
barr
ier
to g
ain
acce
ss to
the
grid
. Th
e re
gula
tion
of th
e tra
nspo
rt op
erat
or
unde
r alte
rnat
ive
hypo
thes
es c
once
rnin
g ve
rtica
l int
egra
tion
and
com
petit
ion
betw
een
grid
use
rs is
scru
tinis
ed in
Cha
pter
6.
Whi
le i
n th
e fo
llow
ing
sect
ions
w
e ha
ve a
dopt
ed a
non
-tech
nica
l pr
esen
tatio
n, th
e re
ader
can
fin
d in
the
appe
ndix
a f
orm
al m
odel
ling
of th
e m
ain
resu
lts.
1.
FIR
ST-B
EST
DIS
PAT
CH
IN A
N E
LE
CT
RIC
ITY
N
ET
WO
RK
1.1
The
tran
spor
t of e
lect
rici
ty
In e
cono
mic
term
s, a
good
is d
efin
ed b
y: (i
) som
e in
trins
ic c
hara
cter
istic
s (w
eigh
t, si
ze, q
ualit
y, e
tc.);
(ii)
the
loca
tion;
(iii
) the
dat
e; a
nd (i
v) th
e st
ate
of n
atur
e w
here
it is
ava
ilabl
e. T
rans
port
is th
e ac
tivity
that
mai
nly
cons
ists
in
mod
ifyin
g at
tribu
te (
ii), e
ven
if, a
s si
de e
ffec
ts, t
he th
ree
othe
r at
tribu
tes
are
also
mod
ified
in
mos
t ca
ses.
It
resu
lts t
hat,
to a
naly
se t
he u
tility
of
trans
porti
ng a
spe
cific
goo
d, w
e ne
ed to
ana
lyse
the
utili
ty a
nd th
e co
st o
f th
at g
ood
at th
e de
partu
re a
nd a
rriv
al lo
catio
ns.
The
diff
eren
ce b
etw
een
the
net u
tility
of t
he g
ood
at th
e ar
rival
loca
tion
and
at th
e de
partu
re lo
catio
n is
th
e gr
oss
utili
ty fr
om tr
ansp
ortin
g it.
Thi
s di
ffer
ence
is to
be
com
pare
d w
ith
the
cost
of t
rans
port
in o
rder
to d
ecid
e if
the
good
is to
be
disp
lace
d or
if it
sh
ould
rem
ain
at th
e in
itial
loca
tion.
Th
is i
s th
e no
rmat
ive
prin
cipl
e th
at w
e ha
ve t
o ap
ply
whe
n an
alys
ing
elec
trici
ty t
rans
port.
Th
e st
artin
g po
int
is t
o de
term
ine
the
quan
titie
s to
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
10
7
gene
rate
and
to c
onsu
me
in o
rder
to m
axim
ise
the
wel
fare
of
all t
he a
gent
s th
at u
se t
he t
rans
port
infr
astru
ctur
e.
In t
he s
hort
run,
the
equ
ipm
ent
for
gene
ratio
n, t
rans
porta
tion
and
dist
ribut
ion
is f
ixed
. T
he p
refe
renc
es o
f co
nsum
ers
also
are
fixe
d. T
he o
ptim
al a
lloca
tion
is li
mite
d to
dec
idin
g ho
w
muc
h to
gen
erat
e at
eac
h no
de a
nd h
ow m
uch
to c
onsu
me
at e
ach
node
, gi
ven
the
rest
rain
ts im
pose
d by
the
topo
logi
cal c
hara
cter
istic
s of t
he n
etw
ork
and
the
tech
nica
l ca
pabi
lity
of e
ach
piec
e of
equ
ipm
ent
(see
Box
4-1
for
de
tails
on
the
obje
ctiv
e to
max
imis
e w
elfa
re u
nder
alte
rnat
ive
sets
of
cons
train
ts).
Box
4-1
: Fir
st-b
est,
seco
nd-b
est,
and
cons
trai
nts t
o m
axim
ise
wel
fare
In
the
shor
t run
, the
who
le g
ener
atio
n an
d tra
nspo
rtatio
n eq
uipm
ents
are
fixe
d, a
s w
ell a
s all
the
need
s.
Th
e “g
rid-f
ree”
firs
t be
st a
lloca
tion
is t
he s
et o
f qu
antit
ies
of e
lect
ricity
ge
nera
ted
and
cons
umed
at e
ach
node
that
max
imis
es th
e ne
t wel
fare
, tha
t is,
the
sum
of
the
diff
eren
ce b
etw
een
the
utili
ty o
f el
ectri
city
for
con
sum
ers
and
the
cost
to g
ener
ate
it, in
a fi
ctiti
ous
situ
atio
n w
here
ene
rgy
can
flow
from
one
nod
e to
oth
ers w
ithou
t any
con
stra
int a
nd w
ithou
t any
loss
.
Th
e “g
rid-c
onst
rain
ed”
first
bes
t allo
catio
n al
so m
axim
ises
the
net w
elfa
re, b
ut
taki
ng in
to a
ccou
nt th
e ph
ysic
al c
hara
cter
istic
s of t
he g
rid.
In th
is c
ase:
(i) s
ome
ener
gy i
s lo
st d
urin
g tra
nspo
rt; a
nd (
ii) b
ecau
se s
ome
lines
and
int
erm
edia
ry
node
s ha
ve l
imite
d ca
paci
ty,
the
“grid
-fre
e” o
ptim
al f
low
s ar
e no
lon
ger
feas
ible
, whi
ch c
reat
es a
“co
nges
tion
cost
”.
In
eco
nom
ic t
heor
y, “
seco
nd-b
est”
mai
nly
refe
rs t
o a
situ
atio
n w
here
the
be
nevo
lent
pla
nner
has
to
bala
nce
the
budg
et o
f th
e pr
oduc
ers
he s
uper
vise
s.
Act
ually
, th
is e
xpre
ssio
n ca
n be
use
d in
any
situ
atio
n w
here
a c
onst
rain
t is
ad
ded
to a
n in
itial
allo
catio
n pr
oble
m.
In t
hat
resp
ect,
the
“grid
-con
stra
ined
” fir
st-b
est i
s a se
cond
-bes
t with
resp
ect t
o th
e “g
rid-f
ree”
firs
t bes
t allo
catio
n.
W
hen
a co
nstra
int
is a
dded
to
a gi
ven
optim
isat
ion
prob
lem
, ei
ther
tha
t co
nstra
int
is n
ot b
indi
ng a
nd t
he i
nitia
l al
loca
tion
does
not
cha
nge,
or
it is
bi
ndin
g an
d it
resu
lts i
n a
decr
ease
of
the
initi
al p
erfo
rman
ce.
The
new
al
loca
tion
can
neve
r gi
ve a
hig
her
perf
orm
ance
sin
ce,
if fe
asib
le n
ow,
it w
as
feas
ible
bef
ore
the
new
con
stra
int i
s ad
ded
and
it w
ould
hav
e be
en c
hose
n. T
he
diff
eren
ce b
etw
een
the
perf
orm
ance
with
out
and
the
perf
orm
ance
with
the
co
nstra
int
is t
he e
cono
mic
cos
t of
the
con
stra
int.
For
exa
mpl
e, t
he e
cono
mic
co
st o
f th
e tra
nspo
rt gr
id is
the
diff
eren
ce b
etw
een
the
grid
-fre
e so
cial
wel
fare
an
d th
e gr
id-c
onst
rain
ed so
cial
wel
fare
.
In
the
sam
e w
ay, o
ne c
an m
easu
re th
e co
st o
f add
ition
al c
onst
rain
ts, s
uch
as:
– th
e ob
ligat
ion
to b
alan
ce b
udge
t;
108
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
– th
e pr
ohib
ition
to d
iscr
imin
ate
on p
rices
; –
rest
rictio
ns o
n ta
riff c
lass
es (l
inea
r, tw
o-pa
rt, e
tc.);
–
the
inab
ility
of t
he o
pera
tor t
o co
llect
info
rmat
ion
on p
refe
renc
es a
nd c
osts
; –
the
univ
ersa
l ser
vice
obl
igat
ion;
–
etc.
Sh
ort-r
un d
ecis
ions
are
con
stra
ined
by
the
inca
paci
ty t
o ad
apt
the
trans
port
infr
astru
ctur
e an
d th
e ge
nera
tion
plan
ts.
For
this
rea
son,
the
shor
t-run
cos
t, in
w
hich
the
re a
re s
igni
fican
t fix
ed c
osts
, is
hig
her
than
the
lon
g-ru
n co
st ex
clus
ivel
y m
ade
of o
ptim
ally
cho
sen
varia
ble
inpu
ts.
To u
nder
stan
d w
hy th
e op
timal
allo
catio
n ca
n be
def
ined
in th
at s
impl
e w
ay,
it is
int
eres
ting
to s
tress
som
e im
porta
nt d
iffer
ence
s be
twee
n th
e tra
nspo
rt of
ele
ctric
ity a
nd o
ther
tran
spor
t act
iviti
es, f
or e
xam
ple
frei
ght o
r pa
ssen
gers
tran
spor
t. F
irstly
, ele
ctric
ity is
hig
hly
stan
dard
ised
, whi
ch m
eans
cl
ose
subs
titut
abili
ty b
etw
een
gene
ratio
n no
des
for
a gi
ven
need
and
, sy
mm
etric
ally
, clo
se s
ubst
ituta
bilit
y be
twee
n co
nsum
ptio
n no
des
for a
giv
en
prod
uctio
n. T
his h
omog
enei
ty p
rope
rty a
llow
s to
pool
qua
ntiti
es.
Seco
ndly
, fo
r el
ectri
city
, the
tim
e at
tribu
te o
f th
e go
od i
s no
t m
odifi
ed b
y tra
nspo
rt.
Pow
er f
low
s in
stan
tane
ousl
y th
roug
h lin
es a
nd i
nter
med
iary
nod
es.
(Se
e B
ox 4
-2 fo
r som
e ill
ustra
tions
of n
etw
ork
topo
logi
es).
Thi
s exp
lain
s why
, in
the
optim
al d
ispa
tch,
inje
ctio
ns a
nd w
ithdr
awal
s ar
e co
ntem
pora
neou
s. F
or
the
sam
e re
ason
, the
dis
patc
her
can
know
for
sur
e th
e st
ate
of n
atur
e at
a
with
draw
al n
ode
whe
n in
ject
ing
pow
er a
t an
othe
r lo
catio
n, w
hich
gre
atly
re
duce
s th
e ra
ndom
ness
of n
et lo
catio
nal u
tiliti
es.
The
third
diff
eren
ce w
ith
mos
t tra
nspo
rt ne
twor
ks is
that
, for
ele
ctric
ity, t
he te
chno
logy
doe
s not
allo
w
to c
ontro
l the
phy
sica
l flo
w o
f en
ergy
thro
ugh
the
grid
(se
e B
ox 4
-2 f
or a
n ill
ustra
tion)
. C
onse
quen
tly, t
he a
ctua
l flo
w o
n ea
ch li
ne c
anno
t be
a co
ntro
l va
riabl
e.
Four
thly
, in
a g
iven
grid
, on
e ca
n pr
edic
t ve
ry p
reci
sely
the
am
ount
of t
rans
port
loss
es b
ecau
se th
ey fo
llow
wel
l kno
wn
phys
ical
law
s.
To s
um u
p, tr
ansp
ortin
g el
ectri
city
con
sist
s in
con
trolli
ng m
odifi
catio
ns
in it
s at
tribu
tes
(ii)
and
(iv)
with
out m
odify
ing
attri
bute
(iii
) an
d pr
ovok
ing
an u
ndes
irabl
e bu
t pr
edic
tabl
e ch
ange
in
attri
bute
(i).
Th
e ph
ysic
al p
ath
follo
wed
bec
ause
of t
he tr
ansf
orm
atio
n in
attr
ibut
e (ii
) can
not b
e co
ntro
lled;
it
resu
lts t
hat
trans
port
betw
een
two
node
s in
a m
eshe
d ne
twor
k cr
eate
s ex
tern
aliti
es o
n al
l lin
es a
nd n
odes
inte
ntio
nally
and
uni
nten
tiona
lly c
ross
ed
by th
e en
ergy
flow
. In
the
optim
al a
lloca
tion
defin
ed b
y a
bene
vole
nt s
ocia
l pla
nner
, at e
ach
node
, mar
gina
l ut
ility
is
equa
l to
mar
gina
l co
st.
If, a
t on
e no
de, m
argi
nal
cost
wer
e hi
gher
than
mar
gina
l util
ity, t
he la
st k
Wh
wou
ld b
e ge
nera
ted
at
loss
and
, sym
met
rical
ly, i
f m
argi
nal c
ost w
ere
less
than
mar
gina
l util
ity, i
t w
ould
mea
n th
at th
e en
tire
pote
ntia
l soc
ial s
urpl
us is
not
cre
ated
. A
t eac
h
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
10
9
node
, th
is e
qual
ity o
f m
argi
nal
cost
and
mar
gina
l ut
ility
will
mos
t lik
ely
requ
ire a
tra
nsfe
r of
ene
rgy.
So
me
node
s w
ill h
ave
to e
xpor
t en
ergy
and
ot
hers
will
be
net i
mpo
rters
, dep
endi
ng o
n th
e co
st s
truct
ure
of g
ener
atio
n an
d gi
ven
the
cons
umer
s' pr
efer
ence
s fo
r el
ectri
city
. W
hen
no p
iece
of
infr
astru
ctur
e ex
hibi
ts c
onge
stio
n, a
nd w
hen
ther
e is
no
ohm
ic l
oss,
the
optim
al a
lloca
tion
is s
uch
that
ene
rgy
has
one
sing
le v
alue
thr
ough
out
the
who
le n
etw
ork,
whi
ch c
an b
e vi
ewed
as
a gi
ant u
niqu
e no
de o
r as
a p
late
. To
see
this
, obs
erve
that
if th
ere
rem
aine
d a
diff
eren
ce b
etw
een
two
node
s, it
wou
ld b
e ea
sy to
incr
ease
the
glob
al s
urpl
us b
y tra
nsfe
rrin
g so
me
kWh
from
th
e no
de w
ith th
e lo
wer
mar
gina
l val
uatio
n to
war
ds th
e no
de w
ith th
e hi
gher
va
luat
ion.
Box
4-2
: Of n
odes
and
line
s
An
elec
tric
grid
can
be
view
ed a
s a
set
of n
odes
, ei
ther
fin
al (
inje
ctio
n an
d w
ithdr
awal
no
des)
or
in
term
edia
ry
(tran
sfor
mer
s, m
eter
s, co
ntro
llers
, et
c.),
inte
rcon
nect
ed b
y lin
es.
The
sim
ples
t ne
twor
k is
mad
e of
one
sin
gle
line
conn
ectin
g tw
o fin
al n
odes
. Th
e “n
orth
-sou
th”
netw
ork
repr
esen
ted
belo
w (s
ee F
igur
e 4-
1) is
a u
sefu
l the
oret
ical
co
nfig
urat
ion
to u
nder
stan
d co
nges
tion
and
loss
es,
but
it al
so g
ives
a r
easo
nabl
y go
od p
ictu
re o
f the
grid
in s
ome
coun
tries
.3 Sin
ce th
ere
exis
ts a
sin
gle
line,
ther
e is
on
e un
ique
pos
sibl
e pa
th f
or t
rans
porti
ng e
lect
ricity
fro
m n
orth
to
sout
h or
fro
m
sout
h to
nor
th.
Abs
ent
any
ener
gy l
oss,
the
phys
ical
equ
ilibr
ium
of
the
elec
tric
indu
stry
impo
ses
gg
ww
ns
ns
, whe
re
g nq (r
espe
ctiv
ely
w nq) s
tand
s for
the
quan
tity
gene
rate
d (r
espe
ctiv
ely
cons
umed
) at t
he n
orth
nod
e an
d g sq (r
espe
ctiv
ely
w sq) s
tand
s fo
r the
qua
ntity
gen
erat
ed (r
espe
ctiv
ely
cons
umed
) at t
he so
uth
node
. C
onse
quen
tly,
the
quan
tity
of e
lect
ricity
flow
ing
on th
e lin
e is
g
ww
gn
ns
sq
q.
But
in m
any
coun
tries
, par
ticul
arly
in c
ontin
enta
l Eur
ope,
ele
ctric
net
wor
ks a
re
mes
hed.
Th
e co
nseq
uenc
e is
that
ther
e is
not
one
uni
que
path
for
ele
ctric
ity to
go
from
one
nod
e to
ano
ther
. Th
is is
illu
stra
ted
in th
e th
ree-
node
net
wor
k he
reaf
ter (
see
Figu
re 4
-2).
Ene
rgy
flow
s fol
low
pat
hs o
f lea
st re
sist
ance
det
erm
ined
by
Kirc
hhof
f's
law
s. S
uppo
se a
gen
erat
ion
node
and
a c
onsu
mpt
ion
node
are
con
nect
ed b
y tw
o lin
es, o
ne w
ith a
res
ista
nce
twic
e th
e ot
her's
. W
hen
gene
rato
rs in
ject
qua
ntity
q a
t on
e no
de a
nd,
assu
min
g no
los
ses,
cons
umer
s w
ithdr
aw t
he s
ame
quan
tity
at t
he
othe
r no
de, t
he f
low
s on
the
low
res
ista
nce
line
and
on th
e hi
gh r
esis
tanc
e lin
e ar
e re
spec
tivel
y 2q
/3 a
nd q
/3.
In a
3-li
ne n
etw
ork
with
the
sam
e re
sist
ance
on
each
line
, lik
e in
Fig
ure
4-2,
if th
ere
are
two
gene
rato
rs in
stal
led
at n
odes
1 a
nd 2
resp
ectiv
ely,
an
d co
nsum
ers
are
loca
ted
at th
e th
ird n
ode,
the
sim
ulta
neou
s in
ject
ion
of
1g q a
nd
2g q w
ill g
ener
ate
a su
perp
ositi
on o
f flo
ws
on t
he l
ines
con
nect
ing
cons
umer
s to
ge
nera
tors
. Fo
r ex
ampl
e, th
e lin
e be
twee
n no
de 1
and
nod
e 3
trans
ports
two
third
s of
the
ener
gy in
ject
ed a
t nod
e 1
plus
one
third
of t
he e
nerg
y in
ject
ed a
t nod
e 2.
By
cont
rast
, on
ly o
ne t
hird
of
the
net
flow
circ
ulat
es o
n th
e lin
e be
twee
n th
e tw
o ge
nera
tors
.
110
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
nort
h
w sqg sq
w nq
sout
h
g nq
Figu
re 4
-1.
One
line
net
wor
k
12
3
gg
node
2
inje
ctio
n 2g q
node
1
inje
ctio
n 1g q
12
23g
gq
q1
22 3
gg
node
3
cons
umpt
ion
31
2w
gg
q
Figu
re 4
-2.
Thre
e-lin
e ne
twor
k B
ut, a
s ex
plai
ned
belo
w, b
ecau
se o
f lo
sses
and
bec
ause
of
som
e sc
arce
ca
paci
ty in
tran
spor
t, tra
nsfe
rs fr
om o
ne n
ode
to a
noth
er c
anno
t be
done
for
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
11
1
free
. It
mea
ns th
at th
e “v
ery-
first
-bes
t” a
lloca
tion
(or “
grid
-fre
e” a
lloca
tion)
is
not
feas
ible
. Th
e di
spat
cher
can
onl
y re
ach
the
“grid
-con
stra
ined
” op
timal
al
loca
tion
and,
con
sequ
ently
, ene
rgy
valu
atio
n re
sulti
ng f
rom
thi
s di
spat
ch
will
diff
er a
mon
g th
e no
des.
1.2
The
cos
t of c
onge
stio
n
Ass
ume
first
tha
t lo
sses
can
be
negl
ecte
d.
At
impo
rting
nod
es,
the
inco
min
g en
ergy
is li
mite
d by
the
capa
city
of
lines
and
tran
sfor
mer
s al
ong
the
phys
ical
pat
h fo
llow
ed b
y en
ergy
. A
s a
resu
lt, e
lect
ricity
is
rela
tivel
y sc
arce
and
it
is m
ore
valu
ed t
han
if th
ere
wer
e no
cap
acity
con
stra
int.
R
ecip
roca
lly,
at e
xpor
ting
node
s, en
ergy
is
rela
tivel
y in
exc
ess
and
low
va
lued
bec
ause
the
outg
oing
ene
rgy
is li
mite
d by
the
capa
city
of
lines
and
tra
nsfo
rmer
s. T
he m
ore
cong
este
d th
e ne
twor
k, th
e hi
gher
the
disc
repa
ncy
betw
een
noda
l val
uatio
ns.
The
limit
case
is th
e au
tark
y ca
se w
here
line
s ar
e cu
t sot
that
eac
h no
de is
isol
ated
from
the
othe
rs.
The
diff
eren
ce b
etw
een
two
noda
l val
uatio
ns o
f ene
rgy,
abs
ent a
ny lo
ss,
is a
n in
dex
of t
he t
ight
ness
of
the
trans
port
cons
train
t. I
t re
flect
s th
e in
capa
city
of
the
oper
ator
to
incr
ease
gen
erat
ion
at l
ow-c
ost
node
s an
d to
de
crea
se
it at
hi
gh-c
ost
node
s as
w
ell
as
its
inca
paci
ty
to
incr
ease
co
nsum
ptio
n at
hig
h-ut
ility
nod
es a
nd t
o de
crea
se i
t at
low
-util
ity n
odes
. Th
e “m
erit
orde
r” c
omm
ands
that
no
gene
rato
r sho
uld
be d
ispa
tche
d if
ther
e re
mai
ns s
ome
avai
labl
e ca
paci
ty w
ith a
low
er c
ost.
It
is n
o lo
nger
im
plem
enta
ble.
In
the
sim
ples
t ca
se o
f B
ox 4
-3,
with
one
sin
gle
line
conn
ectin
g ef
ficie
nt n
orth
ern
gene
rato
rs w
ith a
sou
th n
ode
whe
re th
ere
are
inef
ficie
nt g
ener
ator
s an
d th
e lo
ad, o
ne c
an e
asily
dra
w th
e gr
id-c
onst
rain
ed
optim
al q
uant
ities
and
mea
sure
how
they
dep
art f
rom
the
grid
-fre
e op
timal
qu
antit
ies.
The
diff
eren
ce in
nod
al v
alua
tions
exa
ctly
refle
cts
the
soci
al c
ost
of h
avin
g an
out
-of-
mer
it-or
der d
ispa
tch
beca
use
of th
e lim
ited
capa
city
for
trans
port
betw
een
north
and
sout
h. T
he d
iffer
ence
is th
e sh
adow
val
ue o
f the
tra
nspo
rt lin
e th
at s
igna
ls b
y ho
w m
uch
soci
al w
elfa
re w
ould
be
incr
ease
d if
the
cons
train
t cou
ld b
e re
laxe
d.
In m
eshe
d ne
twor
ks, e
nerg
y flo
ws
alon
g le
ast
resi
stan
ce p
aths
with
out
the
poss
ibili
ty to
con
trol t
hem
.4 As
a re
sult,
any
inje
ctio
n an
d w
ithdr
awal
of
a gi
ven
quan
tity
at t
wo
dist
inct
nod
es w
ill p
rovo
ke a
tra
nsit
of e
lect
ricity
th
roug
h al
l the
line
s of
the
netw
ork.
If
one
line
is c
onge
sted
, all
path
s w
ill
appe
ar c
onge
sted
. C
onse
quen
tly, i
n a
mes
hed
netw
ork,
con
gest
ion
on o
ne
line
is s
uffic
ient
for n
odal
val
ues
to d
iffer
thro
ugho
ut th
e ne
twor
k. B
ecau
se
of th
is “
cont
agio
n ef
fect
”, th
e du
al v
alue
of t
he c
onge
sted
line
is la
rger
than
th
e m
ere
diff
eren
ce b
etw
een
the
mar
gina
l va
lues
at
the
two
ends
of
the
cong
este
d lin
e.5
The
diff
eren
ce b
etw
een
thes
e tw
o va
lues
ref
lect
s th
e
112
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
nega
tive
exte
rnal
ities
in
el
ectri
city
tra
nspo
rt,
that
is
, th
e ad
ditio
nal
cong
estio
n co
st d
ue to
loop
-flo
ws.
Box
4-3
: Out
-of-
mer
it-or
der
optim
al d
ispa
tch
Con
side
r th
e no
rth-s
outh
net
wor
k in
Box
4-2
(see
Fig
ure
4-1)
. A
ssum
e th
at a
ll co
nsum
ers
are
in ‘
sout
h’ a
nd t
heir
mar
gina
l ut
ility
fro
m w
ithdr
awin
g th
e qu
antit
y w sq
of
elec
trici
ty i
s gi
ven
by t
he d
ecre
asin
g fu
nctio
n '(
)10
ww
ss
Uq
q (
see
Figu
re
4-3)
. In
‘nor
th’,
elec
trici
ty c
an b
e ge
nera
ted
at a
con
stan
t mar
gina
l cos
t equ
al to
1¢
and,
in ‘s
outh
’, th
ere
exis
t pla
nts w
ith a
con
stan
t mar
gina
l cos
t equ
al to
4¢.
The
re is
no
con
stra
int o
f cap
acity
for g
ener
atio
n an
d th
ere
is n
o en
ergy
loss
on
the
line.
Let
K
den
ote
the
capa
city
of t
he li
ne.
12
3
gg
node
2
inje
ctio
n 2g q
node
1
inje
ctio
n 1g q
12
23g
gq
q1
22 3
gg
node
3
cons
umpt
ion
31
2w
gg
q
Figu
re 4
-3.
Out
-of-
mer
it-or
der o
ptim
al d
ispa
tch
If K
is v
ery
larg
e, th
e op
timal
dis
patc
h co
nsis
ts in
pro
duci
ng n
othi
ng in
‘so
uth’
w
here
gen
erat
ion
is v
ery
cost
ly.
The
who
le e
nerg
y co
mes
fro
m ‘
north
’. T
he
optim
al c
onsu
mpt
ion
is s
uch
that
'(
)1
Uq
¢, th
at is
the
grid
-fre
e op
timal
qua
ntity
w sq
=9.
If
K <
9, t
he c
apac
ity o
f the
line
doe
s no
t allo
w to
impo
rt th
is q
uant
ity o
f ene
rgy
from
‘no
rth’.
The
dis
patc
h fir
st c
onsi
sts
in s
atur
atin
g th
e lin
e to
tran
sfer
as
muc
h en
ergy
as
poss
ible
fro
m ‘
north
’, w
hich
is K
. A
fter t
hat,
ther
e ar
e tw
o po
ssib
ilitie
s:
(i) if
the
mar
gina
l util
ity fr
om c
onsu
min
g K
is s
till h
ighe
r tha
n th
e m
argi
nal c
ost o
f ge
nera
tion
in ‘s
outh
’, us
e th
e so
uth
plan
t up
to th
e po
int w
here
the
mar
gina
l util
ity
of e
lect
ricity
is
equa
l to
the
mar
gina
l co
st o
f ge
nera
tion,
tha
t is
'(
)4
g sU
Kq
¢.
Oth
erw
ise,
do
not d
ispa
tch
the
cost
ly p
lant
. In
the
first
cas
e, th
e gr
id-c
onst
rain
ed
optim
al
allo
catio
n is
g nq
K,
6g sq
K
(as
long
as
6
K)
and
the
tota
l co
nsum
ptio
n is
6
w sq.
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
11
3
In t
he s
econ
d ca
se,
that
is
for
K b
etw
een
6 an
d 9,
the
tot
al o
utpu
t is
the
co
nstra
ined
flo
w c
omin
g fr
om ‘
north
’, g nq
K,
0g sq
. T
hese
qua
ntiti
es a
re
grap
hed
in th
e m
iddl
e pa
nel o
f Fig
ure
4-3
as fu
nctio
ns o
f the
cap
acity
of t
he li
ne K
.
Bec
ause
of
the
inef
ficie
nt d
ispa
tch
crea
ted
by t
he l
imite
d ca
paci
ty o
f th
e lin
e,
wel
fare
can
not b
e as
hig
h as
it w
ould
be
if lo
w-c
ost g
ener
ator
s w
ere
loca
ted
at th
e so
uth
node
with
con
sum
ers.
The
low
er p
anel
of F
igur
e 4-
3 re
pres
ents
the
mar
gina
l va
lue
of th
e ca
paci
ty c
onst
rain
t, w
hich
rep
rese
nts
the
mar
gina
l co
st o
f co
nges
tion
. W
hen
K i
s sm
alle
r th
an 6
, on
e ad
ditio
nal
unit
of c
apac
ity w
ould
allo
w t
o su
bstit
ute
one
north
kW
h to
one
sou
th k
Wh,
that
is to
sav
e =
4¢
- 1¢
= 3¢
for a
n un
chan
ged
tota
l out
put.
Whe
n K
is b
etw
een
6 an
d 9,
one
add
ition
al u
nit o
f cap
acity
al
low
s to
incr
ease
the
cons
umpt
ion
by m
eans
of m
ore
impo
rts fr
om ‘n
orth
’, so
that
its
val
ue is
the
net
mar
gina
l util
ity o
f el
ectri
city
; =
U'(K
) - 1
¢ =
9 - K
. Fi
nally
, w
hen
K is
larg
er th
an 9
, any
dev
elop
men
t wou
ld b
e us
eles
s, w
hich
is s
igna
lled
by
= 0
. A
s sh
own
in C
hapt
er 5
, the
sha
dow
pric
e of
the
lines
is to
be
com
pare
d w
ith th
e re
al p
rice
of o
ne u
nit o
f eq
uipm
ent t
o kn
ow w
heth
er th
e tra
nspo
rt ca
paci
ty i
s to
be
incr
ease
d or
dec
reas
ed.
Whe
n th
e du
al v
alue
of
the
cons
train
t is
hig
her
than
the
cos
t of
one
add
ition
al u
nit
of e
quip
men
t, th
e tra
nspo
rt lin
e sh
ould
be
deve
lope
d.
And
it
shou
ld b
e do
wns
ized
in
the
oppo
site
cas
e. I
n ac
tual
net
wor
ks, t
he in
fras
truct
ure
is a
lmos
t alw
ays
larg
er
than
its
optim
al s
ize.
The
con
sequ
ence
is th
at th
e sh
adow
pric
e of
line
s an
d tra
nsfo
rmer
s is
less
than
thei
r un
it co
st.
And
it is
eve
n eq
ual t
o ze
ro w
hen
ther
e is
no
cong
estio
n at
all.
1.3
The
cos
t of l
osse
s
Supp
ose
now
tha
t th
ere
is n
o co
nges
tion.
Th
e m
ain
cost
of
deliv
erin
g 1
MW
h at
one
nod
e st
artin
g fr
om a
spe
cific
inje
ctio
n no
de r
esul
ts f
rom
the
fact
that
a q
uant
ity L
of t
he e
nerg
y in
ject
ed w
ill b
e lo
st in
tran
spor
t. It
mea
ns
that
1+ L
MW
h ar
e to
be
gene
rate
d. A
s a
resu
lt, in
an
optim
ised
net
wor
k th
e m
argi
nal v
alua
tion
of 1
MW
h w
ill d
iffer
fro
m o
ne n
ode
to th
e ot
her
by th
e va
lue
of th
e lo
st e
nerg
y. N
ote
that
it is
not
a c
ost i
ncur
red
by th
e tra
nspo
rt gr
id it
self.
It i
s a
cost
due
to th
e di
stan
ce b
etw
een
inje
ctio
n an
d w
ithdr
awal
no
des.
It d
irect
ly c
once
rns g
ener
ator
s and
con
sum
ers.
In e
lect
ric n
etw
orks
, los
ses
incr
ease
pro
porti
onal
ly t
o th
e sq
uare
of
the
ener
gy i
njec
ted.
Th
e co
nseq
uenc
e of
thi
s pr
ecis
e fu
nctio
nal
form
is
that
m
argi
nal l
osse
s are
twic
e hi
gher
than
ave
rage
loss
es.6
114
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
1.4
The
shor
t-ru
n m
argi
nal c
ost o
f tra
nspo
rt
To s
um u
p, b
ecau
se m
aint
aini
ng a
nd d
evel
opin
g th
e in
fras
truct
ure
is
cost
ly, i
t is
optim
al to
kee
p so
me
cong
estio
n in
mos
t pie
ces
of th
e tra
nspo
rt gr
id.
As
a re
sult,
the
dis
patc
h th
at w
ould
max
imis
e ne
t so
cial
wel
fare
w
ithou
t any
refe
renc
e to
the
grid
(lik
e if
all g
ener
ator
s an
d co
nsum
ers
wer
e lo
cate
d at
the
sam
e pl
ace)
is n
ot fe
asib
le.
The
actu
al d
ispa
tch
is s
ub-o
ptim
al
as c
ompa
red
with
the
fic
titio
us o
ne-n
ode
indu
stry
. T
he c
ost
due
to
cong
estio
n is
equ
al to
the
diff
eren
ce b
etw
een
the
max
imum
wel
fare
obt
aine
d w
ithou
t tra
nspo
rt co
nstra
ints
and
the
wel
fare
tha
t re
sults
fro
m t
he a
ctua
l di
spat
ch.
In
addi
tion
to c
onge
stio
n co
sts,
only
a f
ract
ion
of t
he q
uant
ity
inje
cted
can
be
cons
umed
. Th
e cu
mul
ativ
e ef
fect
s of
thes
e el
emen
ts is
that
in
a
netw
ork
built
fo
r el
ectri
city
tra
nspo
rt,
the
optim
al
allo
catio
n of
qu
antit
ies
to g
ener
ate
and
to w
ithdr
aw a
t ea
ch n
ode
is s
uch
that
mar
gina
l va
luat
ions
will
be
diff
eren
t fro
m o
ne n
ode
to o
ther
s. T
he n
odal
val
uatio
n of
en
ergy
is a
nat
ural
by-
prod
uct o
f the
opt
imis
atio
n al
gorit
hms
used
by
syst
em
oper
ator
s. W
hen
an I
ndep
ende
nt S
yste
m O
pera
tor
com
pute
s th
e fe
asib
ility
of
a g
iven
dis
patc
h on
the
grid
he
cont
rols
, the
val
ue o
f ene
rgy
at e
ach
node
ca
n be
pub
lishe
d in
stan
tane
ousl
y. T
he d
iffer
ence
bet
wee
n no
dal v
alua
tions
th
at i
nclu
des
a re
al c
ost
(the
valu
e of
los
t en
ergy
) an
d a
shad
ow c
ost
(the
valu
e of
lost
eff
icie
ncy)
, is
to b
e vi
ewed
as
the
shor
t-run
mar
gina
l cos
t of
trans
port.
Y
et,
note
tha
t no
ne o
f th
e tw
o co
mpo
nent
s of
the
sho
rt-ru
n m
argi
nal
cost
ca
n be
di
rect
ly
rela
ted
to
an
econ
omic
or
ac
coun
ting
expe
nditu
re in
curr
ed b
y th
e op
erat
or o
f the
tran
spor
t inf
rast
ruct
ure.
1.5
Tim
e va
riat
ion
The
need
s fo
r el
ectri
city
are
stro
ngly
var
iabl
e in
tim
e.
They
are
bot
h cy
clic
al (
acco
rdin
g to
wel
l kno
wn
daily
, wee
kly
and
year
ly v
aria
tions
) an
d ra
ndom
(for
exa
mpl
e be
caus
e of
cha
nges
in te
mpe
ratu
re).
The
re o
ccur
oth
er
type
s of
tim
e va
riabi
lity
on th
e ge
nera
tion
side
, due
to th
e sc
arci
ty o
f hyd
ro
reso
urce
s, th
e flu
ctua
tions
of
fuel
pric
es a
nd t
he a
vaila
bilit
y of
pla
nts
(mai
nten
ance
and
rep
airin
g).
In
cont
rast
, th
e tra
nspo
rt in
fras
truct
ure
is
alm
ost
fixed
for
the
med
ium
run
. Th
e op
timis
atio
n of
a s
trong
ly v
aria
ble
wel
fare
func
tion
unde
r inv
aria
ble
trans
port
cons
train
ts o
bvio
usly
resu
lts in
a
cont
inuo
usly
var
ying
opt
imal
allo
catio
n an
d, c
onse
quen
tly, a
con
tinuo
usly
va
ryin
g m
argi
nal v
alue
of e
nerg
y at
eac
h no
de.
Dur
ing
low
act
ivity
per
iods
, th
e ca
paci
ty o
f th
e lin
es a
nd t
rans
form
ers
is n
ot b
indi
ng.
The
nod
al
valu
atio
ns d
iffer
onl
y by
the
mar
gina
l val
ue o
f lo
sses
, whi
ch a
re r
athe
r lo
w
sinc
e th
e le
vel o
f con
sum
ptio
n is
low
. In
con
trast
, dur
ing
peak
per
iods
, the
di
ffer
ence
bet
wee
n no
dal v
alua
tions
is v
ery
high
. N
ote
that
the
last
ass
ertio
n is
not
alw
ays
true.
In
som
e ci
rcum
stan
ces,
the
diff
eren
ce b
etw
een
peak
load
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
11
5
and
off-
peak
loa
d ca
n be
larg
er a
t an
exp
ortin
g no
de t
han
at a
n im
porti
ng
node
. In
that
cas
e th
e el
ectri
city
flow
ing
on th
e lin
e at
pea
k pe
riods
can
be
smal
ler
than
at
off-
peak
per
iods
and
the
nod
al d
iffer
ence
in
elec
trici
ty
valu
atio
n ca
n be
hig
her a
t off
-pea
k pe
riods
.
2.
NO
DA
L PR
ICE
S
2.1
Ene
rgy
pric
es
The
first
-bes
t allo
catio
n ca
n be
dec
entra
lised
by
mea
ns o
f pr
ices
whi
ch
refle
ct t
he m
argi
nal
valu
e of
ele
ctric
ity.
It
mea
ns t
hat
the
first
-bes
t ge
nera
tion
leve
ls a
nd c
onsu
mpt
ion
leve
ls w
ould
be
free
ly c
hose
n by
in
divi
dual
gen
erat
ors
and
indi
vidu
al c
onsu
mer
s if
they
wer
e fa
cing
pric
es
equa
l to
the
mar
gina
l va
lue
of e
lect
ricity
in
the
optim
al a
lloca
tion.
A
nd
sinc
e, a
s sh
own
form
erly
, mar
gina
l val
uatio
ns c
hang
e fr
om o
ne n
ode
to th
e ot
her
(and
fro
m ti
me
to ti
me)
, the
dec
entra
lisat
ion
of f
irst-b
est n
eces
sita
tes
noda
l pr
ices
. I
f pe
rfec
t co
mpe
titio
n m
echa
nism
s pr
evai
l at
eac
h no
de
(rou
ghly
sai
d, if
ther
e ex
ists
a la
rge
num
ber o
f sm
all b
uyer
s an
d su
pplie
rs o
f el
ectri
city
at
each
nod
e w
ho b
ehav
e as
pric
e-ta
kers
), th
e eq
uilib
rium
will
na
tura
lly d
eter
min
e en
ergy
pric
es e
qual
to
the
mar
gina
l va
lues
obt
aine
d fo
rmer
ly.
Con
sequ
ently
, or
gani
zing
com
petit
ion
is a
goo
d w
ay t
o re
ach
effic
ienc
y.
How
ever
, eve
n w
hen
the
num
ber
of g
ener
ator
s an
d th
e nu
mbe
r co
nsum
ers
are
reas
onab
ly l
arge
, th
e de
cent
ralis
atio
n of
dec
isio
ns r
equi
res
som
e pu
blic
inte
rven
tion,
for
exa
mpl
e to
org
aniz
e th
e m
atch
ing
of d
eman
d an
d su
pply
in w
hole
sale
mar
kets
. In
con
trast
, if
ther
e is
onl
y a
smal
l nu
mbe
r of
lar
ge s
uppl
iers
and
/or
buye
rs,
the
equi
libriu
m p
rice
wou
ld r
efle
ct t
he m
arke
t po
wer
of
thes
e ag
ents
.7 To
impl
emen
t the
firs
t-bes
t, no
dal p
rices
are
to b
e re
gula
ted
by th
e go
vern
men
t or
, m
ore
prec
isel
y, b
y a
regu
latio
n en
tity
or a
n an
titru
st
auth
ority
.
In a
ny c
ase,
whe
n co
nsum
ers
at n
ode
i can
buy
ele
ctric
ity a
nyw
here
and
pa
y el
ectri
city
com
ing
from
any
nod
e j a
t a p
rice
ip e
qual
to
the
optim
al
mar
gina
l val
uatio
n of
nod
e i,
they
con
sum
e th
e op
timal
qua
ntity
. A
nd w
hen
gene
rato
rs a
t nod
e i a
re a
utho
rised
to s
ell e
lect
ricity
any
whe
re a
nd re
ceiv
e p i
fo
r any
kW
h so
ld to
any
nod
e j,
they
pro
duce
the
optim
al q
uant
ity.8
2.2
Tra
nspo
rt p
rice
s
For
a gi
ven
trans
actio
n, w
hen
the
buye
r an
d th
e se
ller
are
loca
ted
at th
e sa
me
node
i, th
ey tr
ansa
ct a
t the
sam
e pr
ice
p i th
at c
over
s on
ly g
ener
atio
n
116
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
cost
s. If
the
selle
r is
at n
ode
i and
the
buye
r at n
ode
j, th
e fo
rmer
rece
ives
pi
and
the
latte
r pa
ys p
j. T
he p
robl
em i
s to
kno
w w
hat
to d
o w
ith t
he
diff
eren
ce.
To g
ive
an a
nsw
er, n
ote
that
ano
ther
way
to d
ecen
tralis
e th
e fir
st-b
est i
s to
dis
tingu
ish
the
pric
e of
ene
rgy
and
the
pric
e of
tran
spor
t. W
hen
ther
e ar
e tw
o se
para
te b
ills,
the
pric
e fo
r tra
nspo
rting
1 M
Wh
from
nod
e i t
o no
de j
is
to b
e t ij=
pj -
pi.
As
a m
atte
r of
fac
t, th
e co
nsum
er a
t no
de j
mus
t be
in
diff
eren
t bet
wee
n bu
ying
its
ener
gy a
t nod
e j a
t pric
e p j
on
the
one
hand
an
d, o
n th
e ot
her h
and,
buy
ing
it at
nod
e i a
t pric
e p i
and
then
pay
ing
t ij fo
r its
tra
nspo
rt to
nod
e j.
Sim
ilarly
, the
gen
erat
or a
t no
de i
mus
t be
ind
iffer
ent
betw
een
selli
ng it
s en
ergy
at n
ode
i at p
rice
p i o
n on
e ha
nd a
nd, o
n th
e ot
her
hand
, sel
ling
it at
nod
e j a
t pric
e p j
whi
le p
ayin
g t ij
for i
ts tr
ansp
ort t
o no
de j.
C
onse
quen
tly, w
e ca
n co
nclu
de th
at, f
rom
the
poin
t of
view
of
cons
umer
s an
d pr
oduc
ers,
the
diff
eren
ce t i
j= p
j - p
i re
ally
app
ears
as
a tra
nspo
rt fe
e. F
or
this
reas
on, i
t is n
atur
al to
pay
it to
the
oper
ator
of t
he tr
ansp
ort i
nfra
stru
ctur
e ev
en if
the
cost
s cov
ered
are
on
the
user
s' si
de.
N
ote
that
if th
e ne
t flo
w o
f ene
rgy
is fr
om i
(the
expo
rting
nod
e) to
j (th
e im
porti
ng n
ode)
, any
indi
vidu
al tr
ansa
ctio
n fr
om i
to j
incr
ease
s co
nges
tion
and
mus
t be
char
ged
t ij >
0. I
n co
ntra
st, a
ssum
e p j
< p
i so
that
nod
e j i
s a
net
expo
rter
to n
ode
i an
d th
ere
occu
rs a
n in
divi
dual
tra
nsac
tion
betw
een
a ge
nera
tor
loca
ted
at i
and
a co
nsum
er lo
cate
d at
j.
How
can
it b
e po
ssib
le
sinc
e th
e ge
nera
tor
rece
ives
mor
e th
an w
hat
is p
aid
by t
he c
onsu
mer
? B
ecau
se t
he i
ndiv
idua
l tra
nsac
tion
crea
tes
a co
unte
r-flo
w,
it al
levi
ates
the
co
nges
tion
on t
rans
port
lines
. C
onse
quen
tly,
the
trans
port
pric
ing
syst
em
shou
ld p
rom
ote
this
type
of
trans
actio
n by
rew
ardi
ng th
e pa
rties
inst
ead
of
char
ging
them
for t
rans
port.
And
this
act
ually
occ
urs w
ith n
odal
pric
es si
nce
the
trans
port
of 1
MW
h fr
om i
to j
wou
ld th
en b
e ch
arge
d t ij=
pj -
pi <
0. I
n ot
her w
ords
, the
cou
nter
-flo
w tr
ansa
ctio
ns w
ould
be
enco
urag
ed.
The
tota
l val
ue o
f th
e flo
ws
of e
nerg
y us
ing
the
trans
port
pric
es d
eriv
ed
from
nod
al p
rices
is th
e “m
erch
andi
zing
sur
plus
”. E
ven
if co
unte
r-flo
ws
are
to b
e re
war
ded,
the
mer
chan
dizi
ng s
urpl
us i
s ob
viou
sly
posi
tive9 f
or t
he
follo
win
g re
ason
. R
ecal
l th
at n
odal
pric
es r
efle
ct o
ptim
al v
alua
tions
of
ener
gy.
Con
sequ
ently
, the
mer
chan
dizi
ng s
urpl
us is
mad
e of
two
elem
ents
. Th
e fir
st p
iece
is
a ne
t su
rplu
s du
e to
the
spe
cific
fun
ctio
nal
form
of
the
loss
es o
f ene
rgy
and
thei
r cov
erag
e by
a m
argi
nal p
ricin
g ru
le.
Act
ually
, the
ge
nera
tor
mus
t be
paid
for
the
elec
trici
ty h
e pr
oduc
es, e
ven
if it
cann
ot b
e co
nsum
ed.
But
bec
ause
los
ses
incr
ease
with
the
squ
are
of t
he l
oad,
the
ir m
argi
nal
valu
e is
tw
ice
thei
r av
erag
e va
lue.
Con
sequ
ently
, af
ter
com
pens
atin
g th
e ge
nera
tor f
or it
s los
ses,
it re
mai
ns a
n eq
ual s
um th
at c
an b
e us
ed to
pay
for t
rans
port.
The
sec
ond
part
is th
e co
st o
f con
gest
ion,
and
its
over
all v
alue
can
not b
e ne
gativ
e. E
ither
som
e lin
es o
r no
des
are
cong
este
d an
d th
en t
he s
urpl
us i
s po
sitiv
e, o
r th
ere
is n
o co
nges
tion
and
then
the
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
11
7
surp
lus
is z
ero.
B
y its
ver
y de
finiti
on, t
he m
erch
andi
sing
sur
plus
doe
s no
t co
rres
pond
to th
e pa
ymen
t of c
osts
incu
rred
by
the
oper
ator
. Th
e tra
nspo
rt pr
ices
cal
cula
ted
usin
g no
dal p
rices
var
y w
ith th
e da
te, t
he
initi
al n
ode
and
the
term
inal
nod
e of
the
tra
nsac
tion.
A
ll th
e us
ers
who
tra
nsac
t at t
he s
ame
date
, the
sam
e in
itial
nod
e an
d th
e sa
me
term
inal
nod
e sh
ould
pay
(or
sho
uld
be p
aid
for
coun
ter-
flow
s) t
he s
ame
unit
pric
e.
Ther
efor
e, w
e ca
n as
sert
that
the
nod
al p
rice
syst
em i
s ob
ject
ivel
y no
n-di
scrim
inat
ory
(see
Cha
pter
1).
3.
CO
NST
RA
INE
D A
ND
UN
CO
NST
RA
INE
D
PRIC
ING
3.1
Eff
icie
ncy
conc
ern
vs.
fund
-rai
sing
con
cern
Firs
t-bes
t pric
ing
allo
ws
payi
ng fo
r all
the
cost
s of
an
indu
stry
whe
n th
e eq
uipm
ent i
s op
timal
ly d
esig
ned
and
ther
e ar
e no
incr
easi
ng re
turn
s to
sca
le
in th
e lo
ng ru
n. I
n th
is h
ypot
hetic
al s
ituat
ion,
on
the
one
hand
the
shor
t-run
m
argi
nal
cost
and
the
lon
g-ru
n m
argi
nal
cost
are
equ
al a
nd,
on t
he o
ther
ha
nd,
the
long
-run
mar
gina
l co
st i
s at
lea
st e
qual
to
the
long
-run
ave
rage
co
st.
Con
sequ
ently
, a p
rice
equa
l to
mar
gina
l cos
t can
not b
e be
low
the
long
-ru
n av
erag
e co
st.
B
ut th
ese
optim
al c
ondi
tions
are
pra
ctic
ally
impo
ssib
le to
mee
t in
actu
al
netw
orks
, bec
ause
of t
he fo
llow
ing
reas
ons:10
(i)
dis
crep
ancy
bet
wee
n lo
ng-r
un a
nd sh
ort-r
un n
eeds
Th
e op
timal
dyn
amic
exp
ansi
on o
f th
e tra
nspo
rt in
fras
truct
ure
does
not
am
ount
to
the
mer
e co
nnec
tion
of m
any
stat
ic p
lans
. M
ost
netw
ork
faci
litie
s ar
e bu
ilt fo
r 20
or 3
0 ye
ars
and
it is
unl
ikel
y fo
r any
one
of t
he
faci
litie
s to
hav
e ex
actly
the
opt
imal
sta
tic c
apac
ity f
or a
giv
en y
ear.
A
dditi
onal
ly, d
evia
tions
fro
m t
he o
ptim
al p
lan
are
due
to e
rror
s in
the
fo
reca
st o
f dem
and
and
gene
ratio
n co
sts.
(ii) t
echn
ical
non
con
vexi
ties
For
trans
port
faci
litie
s, de
cisi
ons
are
disc
rete
op
tions
ra
ther
th
an
cont
inuo
us v
aria
bles
. Fo
r a g
iven
rein
forc
emen
t, th
e fe
asib
le s
et is
ver
y sm
all,
e.g.
, lin
es o
f 220
kV
or 4
00 k
V.
Con
sequ
ently
, as
com
pare
d w
ith
the
“mar
gina
l opt
imum
”, th
e re
sult
of p
rogr
amm
ing
with
inte
ger n
umbe
r w
ill b
e an
app
aren
tly e
ither
ove
rsiz
ed o
r un
ders
ized
net
wor
k, w
ith a
na
tura
l te
nden
cy
tow
ards
ov
ersi
zing
w
hen
dem
and
is
expe
cted
to
in
crea
se.
118
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
The
tend
ency
tow
ards
ove
r in
vest
men
t is
int
ensi
fied
beca
use
of s
cale
ec
onom
ies:
the
lar
ger
the
capa
city
of
the
inve
stm
ent
the
smal
ler
the
capi
tal c
ost p
er u
nit o
f cap
acity
. (ii
i) ad
ditio
nal c
onst
rain
ts
Net
wor
k ex
pans
ion
plan
ning
is
subj
ect
to r
elia
bilit
y co
nstra
ints
whi
ch
crea
te th
e ne
ed f
or e
xtra
cap
acity
with
res
pect
to th
e op
timal
pla
n un
der
stric
t ec
onom
ic t
erm
s. F
inan
cial
, en
viro
nmen
tal,
tech
nica
l or
eve
n po
litic
al re
stric
tions
are
als
o im
pose
d on
the
expa
nsio
n of
the
netw
ork
in
gene
ral o
r of a
giv
en c
orrid
or.
W
e co
nclu
de t
hat,
beca
use
the
infr
astru
ctur
e is
ove
rsiz
ed a
s co
mpa
red
with
sho
rt-ru
n ne
eds
and
beca
use
ther
e ex
ist
long
-run
eco
nom
ies
of s
cale
, co
nges
tion
rent
s ca
lcul
ated
on
the
basi
s of
sho
rt-ru
n m
argi
nal c
osts
will
be
rath
er lo
w, a
nd th
e re
nts
from
loss
es w
ill n
ot b
e hi
gh e
noug
h to
cov
er a
ll th
e fix
ed c
osts
incu
rred
by
the
grid
ope
rato
r.
3.2
Bud
get c
onst
rain
t
Seve
ral s
olut
ions
are
ava
ilabl
e to
bal
ance
the
budg
et o
f the
tran
spor
t firm
. If
pub
lic s
ubsi
dies
are
allo
wed
, th
ey c
an b
e us
ed t
o pa
y th
e di
ffer
ence
be
twee
n th
e m
erch
andi
sing
sur
plus
and
the
fixed
cos
ts o
f the
infr
astru
ctur
e.
But
the
taxe
s lev
ied
to fu
nd th
e su
bsid
ies c
reat
e ec
onom
ic d
isto
rtion
s and
the
resu
lting
allo
catio
n of
con
sum
ptio
n an
d pr
oduc
tion
cann
ot b
e th
e fir
st-b
est.
B
ecau
se o
f th
e co
st o
f pu
blic
fun
ds,
we
can
only
rea
ch a
sec
ond
best
. A
dditi
onal
ly,
with
in t
he E
urop
ean
liber
alis
ed f
ram
ewor
k, t
he S
tate
-aid
so
lutio
n is
not
aut
horis
ed.
Ther
efor
e, p
rices
are
to b
e ad
just
ed to
cov
er a
ll th
e co
sts.
Whe
n co
nstra
ined
to
bala
nce
the
budg
et o
f th
e tra
nspo
rt op
erat
or,
the
optim
al ta
riffs
will
dep
art f
rom
firs
t-bes
t if s
peci
fic re
stric
tions
are
impo
sed
in th
eir c
ompu
tatio
n. W
hen
any
func
tiona
l for
m c
an b
e im
plem
ente
d, th
at is
w
hen
the
bill
is n
ot n
eces
saril
y pr
opor
tiona
l to
the
quan
tity
of e
nerg
y th
at is
tra
nspo
rted
(non
lin
ear
pric
es),
the
first
-bes
t al
loca
tion
rem
ains
fea
sibl
e.11
A
mon
g no
n-lin
ear
tarif
fs w
e w
ill o
nly
disc
uss
two-
part
pric
es.
On
the
cont
rary
, if p
rices
are
rest
ricte
d to
be
linea
r, th
e be
st o
nes a
re R
amse
y pr
ices
, an
d th
e re
sulti
ng a
lloca
tion
of g
ener
atio
n an
d co
nsum
ptio
n w
ill o
nly
be a
se
cond
bes
t. W
e fir
st c
onsi
der t
he c
ase
whe
re o
nly
linea
r pric
es a
re fe
asib
le.
3.3
Seco
nd-b
est l
inea
r pr
ices
Ram
sey
pric
es a
re th
e be
st li
near
pric
es w
hen
the
bala
ncin
g of
bud
get i
s m
anda
tory
. Th
ey c
over
mar
gina
l cos
t (co
nges
tion
and
loss
es) p
lus
a m
argi
n co
mpu
ted
to p
ay f
or t
he f
ixed
cos
ts.
In
met
hods
for
pric
ing
appl
ied
by
prac
titio
ners
, fix
ed c
osts
are
allo
cate
d ac
cord
ing
to so
me
prop
ortio
nalit
y ru
le
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
11
9
(for
exa
mpl
e th
e co
st s
hare
pai
d by
age
nt i
whe
n co
nsum
ing
the
quan
tity
iq
is:
/i
jj
, or
()/
()
ii
jj
jC
qC
q,
whe
re
()
ii
Cq
is th
e di
rect
cos
t to
prod
uce
iq, o
r an
y ot
her
ratio
12).
In
Ram
sey
pric
es, o
ne ta
kes i
nto
acco
unt t
he re
actio
n of
the
infr
astru
ctur
e us
ers
whe
n th
ey a
re b
illed
or
paid
. P
ricin
g ab
ove
mar
gina
l co
st i
s in
effic
ient
be
caus
e it
prov
okes
a d
ecre
ase
in c
onsu
mpt
ion.
Sim
ilarly
, whe
n a
prod
ucer
is
pai
d le
ss th
an m
argi
nal u
tility
, he
decr
ease
s his
pro
duct
ion.
Con
sequ
ently
, th
e ai
m o
f sec
ond-
best
pric
ing
is to
lim
it th
is b
ias
in q
uant
ities
pro
duce
d an
d co
nsum
ed.
It is
eas
y to
und
erst
and
that
, if
som
e co
nsum
ers
reac
t to
pric
e in
crea
ses
less
than
oth
ers,
it is
opt
imal
to m
ake
them
pay
mor
e th
an m
ore
reac
tive
cons
umer
s. A
nd s
ymm
etric
ally
, if
som
e ge
nera
tors
dec
reas
e th
eir
outp
ut m
ore
than
oth
er p
rodu
cers
whe
n th
e se
lling
pric
e is
dec
reas
ed, i
t is
optim
al to
mod
ify th
eir s
ellin
g pr
ice
less
. Th
is e
xpla
ins
that
Ram
sey
pric
es
are
inve
rsel
y re
late
d to
the
ela
stic
ity o
f de
man
d an
d to
the
ela
stic
ity o
f su
pply
. A
s a
cons
eque
nce,
the
y ar
e di
scrim
inat
ory:
eac
h se
gmen
t of
use
rs t
hat
can
be is
olat
ed f
rom
the
othe
rs w
ill p
ay o
r w
ill b
e pa
id a
diff
eren
t Ram
sey
noda
l pric
e. I
n th
e si
mpl
e ca
se w
here
, at e
ach
node
, one
can
not d
istin
guis
h be
twee
n di
ffer
ent
type
s of
con
sum
ers
and
betw
een
diff
eren
t ty
pes
of
prod
ucer
s, it
rem
ains
true
that
one
can
dis
tingu
ish
a gr
oup
of c
onsu
mer
s on
on
e si
de a
nd a
gro
up o
f pr
oduc
ers
one
the
othe
r si
de.
Con
sequ
ently
the
R
amse
y de
man
d pr
ice
w ip w
ill b
e di
ffer
ent
from
the
Ram
sey
supp
ly p
rice
g ip:
beca
use
of t
he n
eed
to l
evy
fund
s to
cov
er f
ixed
cos
ts,
we
lose
the
pr
oper
ty o
f pric
e un
ique
ness
at o
ne n
ode.
W
hen
ener
gy a
nd tr
ansp
ort o
f ene
rgy
from
i to
j ar
e bi
lled
sepa
rate
ly, t
he
equi
libriu
m c
ondi
tion
is t
hat
payi
ng
w jp t
o co
nsum
e at
nod
e j
is t
o be
eq
uiva
lent
to b
uyin
g at
pric
e g ip a
t nod
e i a
nd tr
ansp
ortin
g to
war
ds j,
that
is
gw j
iij
pp
t.
It re
sults
that
the
Ram
sey
pric
e fo
r tra
nspo
rt g
wij
ji
tp
p is
th
e w
eigh
ted
sum
of t
wo
elem
ents
: (i)
the
cost
in te
rms
of e
ffic
ienc
y du
e to
en
ergy
loss
es a
nd c
onge
stio
n w
e al
read
y ha
d in
Sec
tion
1; a
nd (i
i) th
e co
st in
te
rms
of e
ffic
ienc
y du
e to
the
bud
get
cons
train
t. T
he f
irst
part
can
be
posi
tive
or n
egat
ive
depe
ndin
g on
the
dire
ctio
n of
indi
vidu
al o
pera
tions
as
com
pare
d w
ith th
e ne
t flo
w b
etw
een
i and
j. T
he s
econ
d pa
rt is
nec
essa
rily
posi
tive:
it
depe
nds
on t
he e
last
icity
of
the
ener
gy s
uppl
y fu
nctio
n at
the
in
ject
ion
node
and
on
the
elas
ticity
of
the
ener
gy d
eman
d fu
nctio
n at
the
w
ithdr
awal
nod
e. T
he s
mal
ler t
hese
ela
stic
ities
, the
hig
her t
he R
amse
y pr
ice
for
trans
port
betw
een
i an
d j,
wha
teve
r th
e co
sts
of o
hmic
los
ses
and
cong
estio
n.
120
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
Not
e th
at R
amse
y pr
icin
g co
mm
ands
a g
loba
l al
loca
tion
of f
ixed
cos
ts
base
d on
dem
and
elas
ticity
, but
tran
spar
ency
and
pol
itica
l arg
umen
ts im
pose
so
me
frag
men
tatio
n of
fix
ed c
osts
. A
ctua
lly, t
he o
blig
atio
n to
bal
ance
the
budg
et c
an b
e so
lved
in m
any
diff
eren
t way
s. O
ne p
ossi
bilit
y is
to im
pose
a
glob
al b
udge
t co
nstra
int
to t
he t
rans
port
oper
ator
. I
n th
is c
ase,
Ram
sey
pric
es a
lloca
te a
ll th
e fix
ed c
osts
of
infr
astru
ctur
e to
all
user
s. L
ines
and
tra
nsfo
rmer
s in
low
vol
tage
are
pai
d by
all
user
s, in
clud
ing
larg
e co
nsum
ers
and
gene
rato
rs w
ho u
se o
nly
high
or
med
ium
vol
tage
equ
ipm
ent.
Ano
ther
so
lutio
n is
to e
xem
pt la
rge
user
s fr
om p
ayin
g fo
r th
e fix
ed c
ost o
f th
e lo
w
volta
ge s
yste
m.
In
this
cas
e, t
he o
ptim
al a
lloca
tion
of c
onsu
mpt
ion
and
gene
ratio
n am
ong
node
s an
d th
e re
sulti
ng f
low
s on
line
s is
to b
e ca
lcul
ated
su
bjec
t to
seve
ral b
udge
t con
stra
ints
. A
s exp
lain
ed in
Box
4-1
, the
larg
er th
e nu
mbe
r of
con
stra
ints
, the
hig
her
the
effic
ienc
y lo
ss.
It m
eans
tha
t, w
hen
pric
e di
scrim
inat
ion
is a
llow
ed, d
isco
mpo
sing
the
cos
t re
cove
ry c
onst
rain
t in
to
seve
ral
user
-targ
eted
co
nstra
ints
ca
nnot
be
ju
stifi
ed
in
term
s of
ef
ficie
ncy.
3.4
Tw
o-pa
rt ta
riff
s
The
draw
back
s of
Ram
sey
pric
es a
re o
bvio
us:
(i) a
lar
ge q
uant
ity o
f in
form
atio
n is
nec
essa
ry t
o co
mpu
te t
he e
last
iciti
es;
(ii) d
iscr
imin
atio
n is
fo
rbid
den
by l
aw, (
iii) t
he r
esul
ting
quan
titie
s de
viat
e fr
om f
irst
best
. Fo
r th
ese
reas
ons,
mul
ti-pa
rt pr
ices
can
be
pref
erre
d.
In th
e si
mpl
e ca
se o
f tw
o-pa
rt pr
ices
, the
con
sum
er p
ays
a fix
ed a
mou
nt
of m
oney
inde
pend
ent o
f the
qua
ntity
of t
rans
port
he w
ill re
quire
and
, the
n,
he p
ays
a “m
argi
nal
pric
e” f
or e
ach
unit
he w
ants
to
trans
port.
Si
nce
the
mar
gina
l pric
e ca
n be
set e
qual
to th
e m
argi
nal c
ost o
f con
gest
ion
and
loss
es,
this
tar
iff a
llow
s im
plem
entin
g th
e fir
st-b
est
allo
catio
n, p
rovi
ded
that
the
fix
ed p
art o
f the
tarif
f doe
s no
t exc
lude
any
use
r with
a m
argi
nal w
illin
gnes
s to
pay
hig
her t
han
the
mar
gina
l cos
t. In
eff
ect,
the
risk
with
one
uni
que
two-
part
pric
e is
tha
t th
e fix
ed f
ee m
ust
be h
igh
enou
gh t
o co
ver
all
the
fixed
co
sts,
so th
at u
sers
with
low
inco
me
will
not
be
able
to p
ay f
or it
. If
thes
e lo
w-in
com
e us
ers
can
be e
asily
ide
ntifi
ed, a
sol
utio
n is
to
prop
ose
them
a
spec
ific
two-
part
pric
e, fo
r ins
tanc
e w
ith a
zer
o fix
ed p
art.
The
flaw
is th
at
we
are
back
to d
iscr
imin
atio
n. E
ven
a si
ngle
two-
part
pric
e is
dis
crim
inat
ory
sinc
e th
e un
it pr
ice
paid
is d
ecre
asin
g w
ith th
e qu
antit
y co
nsum
ed.
Inso
far
as tw
o-pa
rt pr
ices
are
acc
epta
ble
on le
gal g
roun
ds, i
t is
effic
ient
to p
ropo
se
to u
sers
not
one
but
a w
hole
set
of t
wo-
part
pric
es, l
ettin
g ea
ch u
ser c
hoos
e w
ithin
the
men
u th
e ta
riff
he p
refe
rs.
It is
a v
ery
effic
ient
way
to
colle
ct
fund
s fo
r fix
ed c
osts
cov
erin
g.
It is
als
o a
good
way
to o
rgan
ize
“sec
ond
degr
ee p
rice
disc
rimin
atio
n” w
hen
the
pric
e m
aker
lack
s in
form
atio
n ab
out
the
will
ingn
ess t
o pa
y of
use
rs.13
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
12
1
App
lyin
g th
ese
prin
cipl
es to
the
trans
port
of e
nerg
y, th
e ta
riff t
o tra
nspo
rt a
give
n qu
antit
y fo
rm n
ode
i to
node
j sh
ould
hav
e a
first
par
t pro
porti
onal
to
the
quan
tity
(whe
re t
he p
ropo
rtion
ality
coe
ffic
ient
is
the
mar
gina
l co
st o
f co
nges
tion
and
loss
es)
and
a se
cond
par
t, fix
ed,
com
pute
d af
ter
the
fixed
pa
rts o
f th
e en
ergy
tarif
fs f
or c
onsu
mer
s at
nod
e j a
nd g
ener
ator
s at
nod
e i.
Th
e fix
ed p
art o
f the
tarif
fs fo
r ene
rgy
mus
t be
adju
sted
not
onl
y to
pay
for
the
fixed
cos
ts o
f ge
nera
tors
but
als
o to
pay
for
the
fixed
cos
ts o
f th
e gr
id
oper
ator
. B
y de
finiti
on, t
he fi
xed
part
of tw
o-pa
rt ta
riffs
doe
s not
dis
tort
effic
ienc
y.
Ther
efor
e, th
e ta
riff
mak
er h
as s
ome
degr
ees
of f
reed
om to
cal
cula
te th
em.
He
mai
nly
has
to a
void
the
exc
lusi
on o
f so
me
agen
ts,
indi
vidu
ally
or
colle
ctiv
ely.
In
elec
trici
ty
trans
port,
th
e m
ain
conc
ern
is
that
la
rge
cons
umer
s ha
ve a
ltern
ativ
e op
portu
nitie
s to
the
use
of th
e gr
id, f
or e
xam
ple
by i
nsta
lling
gen
erat
ion
plan
ts a
t th
eir
loca
tion.
To
pre
vent
thi
s by
-pas
s, se
vera
l tw
o-pa
rt ta
riffs
mus
t be
prop
osed
. Ea
ch it
em in
the
men
u is
tailo
red
so th
at n
o on
e is
exc
lude
d (p
artic
ipat
ion
cons
train
t) an
d ea
ch a
gent
cho
oses
th
e ta
riff
calc
ulat
ed f
or h
im (
self-
sele
ctio
n co
nstra
int).
W
hen
also
tak
ing
into
acc
ount
the
poss
ibili
ty o
f co
aliti
on b
etw
een
seve
ral p
rodu
cers
or
larg
e co
nsum
ers,
or b
etw
een
a pr
oduc
er a
nd o
ne o
f its
clie
nts
in o
rder
to b
ypas
s th
e tra
nspo
rt in
fras
truct
ure,
it is
nec
essa
ry to
hav
e re
cour
se to
coo
pera
tive-
gam
es th
eory
to o
ptim
ally
des
ign
the
fixed
par
t of t
he ta
riff.14
Fi
nally
, no
te t
hat
devi
ces
able
to
cont
inuo
usly
met
er t
he f
low
of
elec
trici
ty o
n a
line
are
very
exp
ensi
ve a
nd,
in m
any
coun
tries
, th
ey a
re
inst
alle
d on
ly o
n th
e hi
gh v
olta
ge g
rid.
Add
ition
ally
, w
hen
avai
labl
e, i
t w
ould
be
very
cos
tly b
oth
at t
he o
pera
tor
leve
l an
d at
the
use
r le
vel
to
proc
ess
all
the
info
rmat
ion
they
col
lect
. A
s a
resu
lt, i
nste
ad o
f se
ndin
g in
voic
es b
ased
on
the
time
prof
ile o
f th
e de
man
d fo
r tra
nspo
rt, i
n m
ost
coun
tries
one
use
s to
cal
cula
te t
he a
ggre
gate
qua
ntity
tra
nspo
rted
durin
g a
give
n pe
riod
(for
exa
mpl
e, o
ne y
ear)
and
to c
harg
e a
unifo
rm m
argi
nal p
rice
of tr
ansp
ort a
ll th
e ye
ar lo
ng.
The
draw
back
is th
at th
e sa
me
tota
l qua
ntity
of
ener
gy c
an b
e tra
nsfe
rred
with
stro
ng t
ime
regu
larit
y or
with
a h
igh
irreg
ular
ity.
And
the
irreg
ular
load
pro
file
is m
uch
mor
e de
man
ding
in te
rms
of tr
ansp
ort c
apac
ity th
an th
e re
gula
r one
. O
ne s
olut
ion
is to
dis
tingu
ish
an
ener
gy c
ompo
nent
and
a c
apac
ity c
ompo
nent
in
the
dem
and
for
trans
port,
ea
ch w
ith a
spe
cific
con
stan
t mar
gina
l pric
e.
But
it r
emai
ns tr
ue th
at th
ese
two
elem
ents
are
to b
e de
sign
ed to
sen
d ac
cura
te s
igna
ls o
f sca
rcity
to u
sers
. A
fixe
d pa
rt re
mai
ns n
eces
sary
to c
over
the
fixed
cos
ts o
f the
infr
astru
ctur
e w
ithou
t dis
torti
ng th
e de
cisi
on o
f the
use
rs.
122
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
4.
CO
NC
LU
DIN
G C
OM
ME
NT
S
To c
alcu
late
the
tar
iffs
to t
rans
port
elec
trici
ty,
the
trade
-off
bet
wee
n ef
ficie
ncy,
bud
get r
equi
rem
ents
, leg
al c
onst
rain
ts a
nd p
ract
icab
ility
, res
ults
in
a n
on li
near
tarif
f ba
sed
on th
e no
dal p
rices
of
ener
gy a
nd o
n th
e fix
ed
cost
s of t
he in
fras
truct
ure.
The
sim
ples
t is t
wo-
part.
The
tota
l bill
to b
e pa
id
to th
e tra
nspo
rt fir
m is
mad
e of
an
“eff
icie
ncy-
orie
nted
” pa
rt, v
aria
ble
with
th
e qu
antit
y tra
nspo
rted,
and
a “
cost
-rec
over
y” p
art,
inde
pend
ent
of t
he
quan
tity
trans
porte
d.
The
first
par
t is
aim
ed a
t si
gnal
ling
the
cost
of
phys
ical
los
ses
and
the
cost
of
cong
estio
n.
It sh
ould
var
y w
ith t
ime
and
loca
tion
and
shou
ld b
e in
crea
sing
with
the
quan
tity
of e
nerg
y tra
nspo
rted.
15 P
ract
ical
ly, i
t sho
uld
be
a fu
nctio
n of
the
inje
ctio
n an
d w
ithdr
awal
nod
es (n
ot o
n th
e ph
ysic
al p
ath
of
ener
gy s
ince
it
actu
ally
can
not
be c
ontro
lled)
and
on
the
dire
ctio
n of
the
flo
w.
For
prac
tical
rea
sons
, thi
s fir
st p
art o
f th
e ta
riff
can
be d
isco
mpo
sed
into
an
ener
gy te
rm a
nd a
cap
acity
term
, bot
h lin
ear f
unct
ions
, pro
vide
d th
at
the
scar
city
sig
nals
the
y tra
nsm
itted
are
sim
ilar
to t
he o
nes
that
allo
w a
n ef
ficie
nt u
se o
f the
grid
. Th
e fix
ed p
art o
f the
tarif
f is
aim
ed a
t pay
ing
for t
he in
fras
truct
ure
cost
s.
Bec
ause
its
pur
pose
is
pure
ly b
udge
tary
, it
shou
ld n
ot i
nter
fere
with
the
sc
arci
ty s
igna
ls s
ent b
y th
e va
riabl
e pa
rt(s)
. In
par
ticul
ar, w
hen
the
trans
port
tarif
f dis
tingu
ishe
s bet
wee
n an
ene
rgy
com
pone
nt a
nd a
cap
acity
com
pone
nt,
the
latte
r sho
uld
not b
e vi
ewed
as t
he fi
xed
part
of th
e ta
riff.
In m
ost c
ount
ries,
actu
al tr
ansp
ort t
ariff
s ar
e tw
o-pa
rt bu
t the
way
they
ba
lanc
e th
e va
riabl
e pa
rt(s)
and
the
fixed
par
t is v
ery
hete
roge
neou
s.16
APP
EN
DIX
MO
DEL
SET
TIN
G
Two
node
s, n
(for
nor
th) a
nd s
(for
sout
h), a
re c
onne
cted
by
a lin
e of
cap
acity
K.
The
unit
cost
of
capa
city
is r
. Th
e qu
antit
y of
ele
ctric
ity tr
ansp
orte
d th
roug
h th
e lin
e is
mea
sure
d in
th
e sa
me
unit
as K
. W
e no
te
ii
Uq
the
utili
ty d
eriv
ed fr
om th
e co
nsum
ptio
n of
a q
uant
ity
iq o
f ele
ctric
ity a
t nod
e i a
nd
ii
Cq
the
cost
of g
ener
atin
g iq a
t nod
e i
(,
)i
ns
.
Let
,,
,,
wg
wg
wg
wg
nn
ss
nn
nn
ss
ss
Wq
qK
Uq
Cq
Uq
Cq
rK s
tand
for t
he
wel
fare
func
tion
of th
is in
dust
ry, w
here
sup
ersc
ript w
sta
nds
for “
with
draw
al”
and
supe
rscr
ipt
g st
ands
for
“ge
nera
tion”
. W
e co
nsid
er o
nly
shor
t-run
dec
isio
ns, t
hat i
s, K
is n
ot a
con
trol
varia
ble.
The
opt
imal
allo
catio
n is
the
solu
tion
to:
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
12
3
,
,,
max
,,
,,
wg
wg
nn
ss
gg
ww
nn
ss
Wq
qK
(1
)
s.t.
g
ww
gn
ns
sq
q
(2)
de
f
gw
nsn
nz
K
(3)
Prov
isio
nally
, we
assu
me
ther
e is
no
loss
, whi
ch is
refle
cted
by
(2).
To
sim
plify
not
atio
ns,
we
assu
me
that
pre
fere
nces
and
cos
ts a
re s
uch
that
the
north
nod
e w
ill a
lway
s ex
port
a ne
t flo
w o
f ele
ctric
ity
nszto
war
ds so
uth.
For
this
reas
on,
0nsz
.
THE
SOC
IAL
CO
ST O
F C
ON
GES
TIO
N
Let
den
ote
the
dual
var
iabl
e as
soci
ated
to c
onst
rain
t (3)
. Th
e fir
st-o
rder
con
ditio
ns th
at
char
acte
rise
the
solu
tion
(,
,,
)go
gow
ow
on
sn
sq
qto
the
abov
e pr
oble
m a
re:
'w
ogo
ss
ss
Uq
Cq
(4
) '
wo
gon
nn
nU
qC
q
(5)
ogo
gos
sn
nC
qC
q
(6)
gogo
wo
wo
ns
ns
(7
)
0,,
.0
ogo
wo
ogo
wo
nn
nn
Kq
qK
. (8
)
By
(4)
and
(5),
at e
ach
node
mar
gina
l ut
ility
and
mar
gina
l co
st a
re t
o be
equ
al.
If t
he
capa
city
of t
he li
ne is
ver
y la
rge,
we
obta
in th
e “g
rid-f
ree”
allo
catio
n. I
n th
is c
ase,
o nsz
K
so t
hat
0o
by
the
com
plem
enta
ry s
lack
ness
con
ditio
n (8
). T
hen,
by
(6),
the
mar
gina
l va
luat
ion
of e
lect
ricity
is th
e sa
me
at b
oth
node
s.
The
serie
s of
eq
ualit
ies
''
wo
gow
ogo
ss
ss
nn
nn
Uq
Cq
Uq
Cq
, to
geth
er
with
co
nditi
on (7
), fu
lly d
escr
ibe
the
optim
al a
lloca
tion.
(se
e Fi
gure
4-4
).
impo
rts=
o nsz
expo
rts=
o nsz
' nU
nC
wo
nq
go nq
nq
go sqw
osq
sq
' sU
sC
Nor
th
Sout
h
Figu
re 4
-4.
No
capa
city
con
stra
int,
sam
e va
luat
ion
at b
oth
node
s
124
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
Con
vers
ely,
if K
is s
mal
l, th
e co
nstra
int (
3) is
bin
ding
and
we
obta
in th
e gr
id-c
onst
rain
ed
allo
catio
n.
The
optim
al p
rodu
ctio
n in
‘no
rth’
is g
iven
by
'go
gon
nn
nU
qK
Cq
and
, in
‘sou
th’,
by
'go
gos
ss
sU
qK
Cq
. N
ow,
the
shad
ow
pric
e '
'o
gogo
gogo
ss
nn
ss
nn
Uq
KU
qK
Cq
Cq
is
st
rictly
pos
itive
. It
sign
als
that
a l
arge
r K
wou
ld a
llow
to
gene
rate
mor
e en
ergy
in ‘
north
’ w
here
the
cost
of o
ne a
dditi
onal
uni
t is s
mal
ler t
han
in ‘s
outh
’ and
to c
onsu
me
mor
e in
‘sou
th’
whe
re th
e ut
ility
of t
his a
dditi
onal
uni
t is l
arge
r tha
n in
‘nor
th’ (
see
Figu
re 4
-5).
o
K K
' nU
nC
wo
nq
go nq
nq
go sqw
osq
sq
' sU
sC
Nor
th
Sout
h
Figu
re 4
-5.
Bin
ding
tran
spor
t cap
acity
, nod
al m
argi
nal v
alua
tions
of e
nerg
y di
verg
e
THE
SOC
IAL
CO
ST O
F LO
SSES
A
ssum
e no
w t
hat
a fr
actio
n of
the
ele
ctric
ity i
njec
ted
at t
he n
orth
nod
e is
los
t be
fore
ar
rivin
g at
the
sout
h no
de.
In th
e pr
ogra
m to
obt
ain
the
first
-bes
t allo
catio
n, w
e m
ust r
epla
ce
(2) w
ith:
wg
gw
wg
ss
nn
ss
Lq
q
(9)
sinc
e tra
nsm
issi
on lo
sses
.
L in
crea
se w
ith th
e po
wer
con
sum
ed b
y th
e lo
ad.
The
first
-or
der c
ondi
tions
rem
ain
(4),
(5) a
nd (8
). B
ut (7
) is
repl
aced
by
(9) w
ith th
e co
nseq
uenc
e th
at
(6) i
s rep
lace
d by
:
1o
gogo
ss
nn
Cq
Cq
L
(10)
whe
re L
stan
ds fo
r mar
gina
l los
ses.
In t
he n
o-co
nges
tion
case
her
eafte
r ill
ustra
ted
by F
igur
e 4-
6 (
0o
), w
e se
e th
at t
he
optim
al a
lloca
tion
sets
:
(1)
gogo
gos
sn
nn
Cq
Cq
LC
q.
This
refle
cts t
hat i
t tak
es (1
+L')
MW
h ge
nera
ted
in ‘n
orth
’ to
mat
ch th
e ne
ed fo
r 1 M
Wh
in
‘sou
th’.
As
com
pare
d w
ith t
he c
ase
whe
re
0,L
the
vol
ume
of t
rade
is
redu
ced
and
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
12
5
cons
umpt
ion
in ‘
sout
h’ i
s sm
alle
r. B
ut g
ener
atio
n in
‘no
rth’
can
be l
arge
r in
ord
er t
o co
mpe
nsat
e fo
r the
loss
es.
'(1
)n
LU
(1)
gon
nC
qL
go
ss
Cq
nq
sq
' nU
sC
' sU
(1)
nL
C
nC
gon
nC
q
wo nq
go nq
go sqwo sq
Figu
re 4
-6.
Opt
imal
allo
catio
n w
ith e
nerg
y lo
sses
TIM
E V
AR
IATI
ON
Fo
r a g
iven
tran
spor
t inf
rast
ruct
ure,
the
optim
al d
ispa
tch
is ti
me
depe
nden
t, fo
llow
ing
the
chan
ges
in c
onsu
mer
s pr
efer
ence
s of
and
in
gene
ratio
n pl
ants
ava
ilabi
lity.
Th
e ob
viou
s co
nseq
uenc
e is
that
nod
al v
alua
tions
of e
nerg
y ar
e al
so ti
me-
depe
nden
t. A
ssum
e th
ere
are
no
loss
es a
nd th
e tra
nspo
rt lin
e of
cap
acity
K is
use
d du
ring
two
perio
ds o
f equ
al d
urat
ion:
a p
eak
perio
d (la
belle
d D
for d
ay) a
nd a
n of
f-pe
ak p
erio
d (la
belle
d N
for n
ight
). T
he p
robl
em is
to
max
imis
e:
,
,[
]w
g
ii
ii
DN
in
sU
qC
qrK
,
subj
ect t
o tw
o co
nstra
ints
of
type
(2)
and
tw
o co
nstra
ints
of
type
(3)
, one
pai
r fo
r ea
ch
perio
d.17
Fro
m th
e op
timal
allo
catio
n, w
e de
rive
the
shad
ow p
rice
of th
e lin
e du
ring
the
day
ogo
goD
sDsD
nDnD
Cq
Cq
an
d th
e sh
adow
pr
ice
of
the
line
durin
g th
e ni
ght
ogo
goN
sNsN
nNnN
Cq
Cq
. A
s the
nee
d fo
r tra
nspo
rtatio
n is
hig
her d
urin
g th
e da
y, w
e ha
ve
oo
DN
, w
here
the
equ
ality
hol
ds o
nly
in t
he c
ase
0o
oD
N,
that
is
whe
n th
e lin
e is
pe
rman
ently
non
-bin
ding
(w
hich
sho
uld
neve
r oc
cur
sinc
e th
e un
it co
st o
f ca
paci
ty i
s po
sitiv
e).
Dep
endi
ng o
n th
e di
scre
panc
y be
twee
n th
e ni
ght
and
day
char
acte
ristic
s of
co
nsum
ptio
n an
d ge
nera
tion,
w
e ca
n ha
ve
eith
er
a pe
rman
ently
bi
ndin
g ca
paci
ty
(0
oo
DN
) or c
onge
stio
n on
ly d
urin
g th
e pe
ak p
erio
d (
0o
oD
N).
In
any
cas
e, b
ecau
se th
e sa
me
line
is u
sed
to sa
tisfy
diff
eren
t nee
ds, i
t has
the
feat
ures
of a
“p
ublic
goo
d”:18
thi
s ex
plai
ns w
hy t
he t
est t
o kn
ow w
heth
er t
he c
apac
ity o
f th
e lin
e is
too
la
rge
or to
o sm
all c
onsi
sts n
ow in
com
parin
g th
e un
it co
st o
f cap
ital r
with
the
sum
(o
oD
N).
IMPL
EMEN
TATI
ON
Th
e fir
st-b
est s
olut
ion
can
be d
ecen
tralis
ed b
y m
eans
of
noda
l pric
es.
At t
he s
outh
nod
e an
y tra
nsac
tion
will
cos
t '
wo
gos
ss
ss
pU
qC
q p
er u
nit t
o a
cons
umer
and
will
retu
rn
sp
126
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
to a
pro
duce
r, w
hate
ver t
he lo
catio
n of
the
othe
r sid
e in
the
trans
actio
n. S
imila
rly, a
t the
nor
th
node
'
wo
gon
nn
nn
pU
qC
q w
ill b
e ch
arge
d fo
r any
uni
t of c
onsu
mpt
ion
and
will
be
paid
fo
r any
uni
t of p
rodu
ctio
n.
It is
eas
y to
che
ck th
at c
onsu
mer
s who
solv
e:
max
ii
ii
iq
Uq
pq
will
cho
ose
wo
iq, a
nd p
rodu
cers
who
solv
e:
max
ii
ii
iq
pq
Cq
will
cho
ose
go iq a
t ,
in
s.
Ther
efor
e no
dal
pric
es,
eith
er r
esul
ting
from
com
petit
ive
noda
l mar
kets
or i
mpl
emen
ted
by a
regu
latio
n en
tity,
allo
w to
reac
h th
e fir
st-b
est a
lloca
tion.
W
hen
ener
gy a
nd tr
ansp
ort a
re b
illed
sep
arat
ely,
the
equi
libriu
m p
rice
for
trans
port
from
no
rth to
sou
th m
ust b
e su
ch th
at
sn
nsp
pt
; the
n ns
sn
tp
p.
We
conc
lude
that
, whe
n lo
sses
can
be
negl
ecte
d, fr
om (6
) the
firs
t-bes
t tra
nspo
rt pr
ice
is:
oo
nst
(11)
In w
ords
, ab
sent
any
ohm
ic l
osse
s, tra
nspo
rt sh
ould
be
free
if
and
only
if
ther
e is
no
cong
estio
n. W
here
ther
e is
con
gest
ion,
a p
ositi
ve p
rice
is to
be
char
ged.
Sym
met
rical
ly, t
he
equi
libriu
m p
rice
for
trans
port
from
sou
th t
o no
rth m
ust
be s
uch
that
n
ssn
pp
t.
We
dedu
ce t
hat
reve
rse
flow
s sh
ould
be
char
ged
0o
osn
ns
tp
p.
Ind
eed,
as
long
as
0g
ww
gns
nn
ss
zq
q, i
ndiv
idua
l tra
nsac
tions
from
sou
th to
nor
th s
houl
d be
rew
arde
d si
nce
they
redu
ce th
e co
nges
tion
of th
e lin
e.
Sim
ilarly
, ta
king
los
ses
into
acc
ount
, fr
om (1
0) w
e ob
tain
'
'0
oo
nsn
tL
C a
nd
'(
')
0o
osn
nt
LC
. Th
e la
tter i
s on
ce m
ore
just
ified
bec
ause
any
net
inje
ctio
n in
‘sou
th’
decr
ease
s th
e qu
antit
y of
ene
rgy
that
mus
t be
inje
cted
at t
he n
orth
nod
e an
d, c
onse
quen
tly,
decr
ease
s bot
h th
e lo
sses
of e
nerg
y an
d th
e co
nges
tion
on th
e lin
e.
BA
LAN
CIN
G T
HE
BU
DG
ET O
F TH
E TR
AN
SPO
RT
OPE
RA
TOR
Is
thi
s fir
st-b
est
sust
aina
ble
on f
inan
cial
gro
unds
? O
n th
e ge
nera
tor’
s si
de,
if th
ere
are
incr
easi
ng re
turn
s to
scal
e, m
argi
nal-c
ost p
ricin
g is
not
suff
icie
nt to
bre
ak e
ven.
But
sinc
e th
is
repo
rt is
ded
icat
ed to
tran
spor
t, w
e as
sum
e th
at g
ener
ator
s do
not l
ose
mon
ey w
hen
gene
ratio
n is
val
ued
at fi
rst b
est19
.
Wha
t abo
ut th
e re
sour
ces
of th
e gr
id o
pera
tor?
The
mar
ket e
quili
briu
m u
nder
nod
al p
rices
cr
eate
s a “
mer
chan
disi
ng su
rplu
s”:
wo
wo
gogo
ss
nn
ss
nn
MS
pq
pq
pq
pq
.o
n
LC
LK
K
if th
e co
nstra
int i
s bin
ding
'
.o
nns
o nsLC
Lz
z
othe
rwis
e.
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
12
7
This
surp
lus i
s pos
itive
for t
wo
reas
ons:
be
caus
e of
the
cong
estio
n re
nt
oK
whe
n th
e lin
e is
con
gest
ed,
beca
use
loss
es a
re in
crea
sing
with
the
squa
re o
f the
ene
rgy
inje
cted
. It
resu
lts th
at:
2o nsL
Lz
,
so th
at th
e su
rplu
s due
to lo
sses
is
.n
CL
In
sum
mar
y, u
nder
nod
al p
ricin
g th
e di
ffer
ence
MS
betw
een
the
expe
nditu
res o
f cus
tom
ers
and
the
gain
s of
pro
duce
rs i
s po
sitiv
e an
d it
can
be u
sed
to c
over
the
cos
t of
the
tra
nspo
rt op
erat
or.
But
is it
true
that
MS
rK?
If t
he s
ize
of t
he l
ine
coul
d be
opt
imal
ly a
djus
ted
to t
rans
port
need
s, w
e w
ould
hav
e o
r a
nd th
e M
S w
ould
leav
e th
e op
erat
or w
ith a
net
pro
fit e
qual
to th
e va
lue
of lo
sses
(p
rovi
ded
ther
e ar
e no
-incr
easi
ng re
turn
s to
scal
e).
But
for t
echn
ical
, eco
nom
ical
and
pol
itica
l re
ason
s, in
mos
t tra
nspo
rt ne
twor
ks, t
here
is e
xces
s cap
acity
so th
at
or
, and
eve
n 0
o
for a
bsol
ute
exce
ss c
apac
ity.
In m
ost t
rans
port
infr
astru
ctur
e, it
resu
lts th
at th
e m
erch
andi
sing
su
rplu
s doe
s not
cov
er th
e fix
ed c
osts
.
We
succ
essi
vely
con
side
r sec
ond-
best
line
ar p
rices
and
non
line
ar p
rices
.
RA
MSE
Y N
OD
AL
PRIC
ES
To s
impl
ify o
ur a
naly
sis,
we
com
e ba
ck to
the
hypo
thes
is o
f no
loss
es.
Then
, the
pro
blem
is
to
max
imiz
e (1
) w
ith r
espe
ct t
o pr
ices
(,
,,
)w
wg
gn
sn
sp
pp
p s
ubje
ct t
o (2
) an
d (3
), pl
us t
he
addi
tiona
l obl
igat
ion
to b
alan
ce th
e bu
dget
of t
he tr
ansp
ort f
irm, t
hat i
s:
0w
ww
wg
gg
gn
ns
sn
ns
sp
qp
qp
qp
qrK
, (1
2)
know
ing
that
pric
e-ta
king
con
sum
ers
beha
ve o
ptim
ally
'
ww
ii
iU
qp
and
pric
e-ta
king
ge
nera
tors
beh
ave
optim
ally
g
gi
ii
Cq
p a
t eac
h no
de i=
n,s.
D
enot
ing
by
the
mul
tiplie
r as
soci
ated
with
the
bud
get
cons
train
t (1
2),
by
the
m
ultip
lier a
ssoc
iate
d w
ith th
e flo
w c
onst
rain
t (2)
and
by
the
mul
tiplie
r ass
ocia
ted
with
the
capa
city
con
stra
int
(3),
from
the
firs
t-ord
er c
ondi
tions
, it
is s
traig
htfo
rwar
d to
writ
e th
e se
cond
-bes
t pric
es a
s:20
11
1
wg
nn
wg
nn
wg
nn
pp
pp
in
‘nor
th’
11
1
wg
ss
wg
ss
wg
ss
pp
pp
in
‘sou
th’.
Whe
re:
128
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
/ /
def
ww
ii
w iw
wi
i
pp
is th
e pr
ice
elas
ticity
of d
eman
d an
d
/ /
def
gg
ii
g ig
gi
i
pp
is th
e pr
ice
elas
ticity
of s
uppl
y at
nod
e i.
Bot
h co
nsum
ers
and
gene
rato
rs a
t eac
h no
de h
ave
to c
ontri
bute
to c
over
the
fixed
cos
t of
trans
port.
At n
ode
i we
obse
rve
that
:
()
1
wg
ii
wg
ii
wg
ii
pp
pp
.
It is
no
long
er tr
ue th
at th
ere
exis
ts o
ne u
niqu
e pr
ice
at e
ach
node
. Th
e di
ffer
ence
bet
wee
n de
man
d pr
ice
and
supp
ly p
rice
incr
ease
s w
ith th
e va
lue
of th
e fix
ed c
ost o
f tra
nspo
rt to
be
paid
(w
hich
app
ears
in
wg
ii
pp
thr
ough
the
dua
l va
riabl
e )
and
it de
crea
ses
whe
n th
e de
man
d an
d su
pply
ela
stic
ities
incr
ease
.
RA
MSE
Y T
RA
NSP
OR
T PR
ICES
It
obvi
ousl
y re
mai
ns t
rue
that
the
equ
ilibr
ium
tra
nspo
rtatio
n ch
arge
s ap
plie
d to
the
do
min
ant f
low
(fr
om n
orth
to s
outh
) ar
e su
ch th
at
wg
sn
nsp
pt
. U
sing
the
Ram
sey
noda
l pr
ices
, we
obta
in:
wg
nss
nt
pp
11
wg
sn
wg
sn
pp
(1
3)
As c
ompa
red
with
(11)
, we
obse
rve
two
dist
ortio
ns:
—
first
, to
the
mar
gina
l co
st o
f co
nges
tion21
we
have
to
add
the
cont
ribut
ion
to t
he
reco
very
of t
he tr
ansp
ort f
ixed
cos
ts;
—
seco
nd, t
he m
argi
nal c
ost o
f con
gest
ion
is it
self
dist
orte
d.
An
incr
ease
in
K b
oth
incr
ease
s f
or f
inan
cial
rea
sons
and
dec
reas
es
for
tec
hnic
al
reas
ons.
It
conf
irms
that
the
pric
ing
of t
rans
port
mai
nly
appe
ars
as t
he c
over
age
of f
ixed
co
sts,
exce
pt w
hen
the
capa
city
is v
ery
smal
l.
Fina
lly, n
ote
that
for r
ever
se tr
ansa
ctio
ns (f
rom
sout
h to
nor
th) t
he R
amse
y pr
ice
shal
l be:
wg
snn
st
pp
11
wg
ns
wg
ns
pp
(1
4)
It m
eans
that
the
rew
ard
for a
llevi
atin
g lo
sses
and
con
gest
ion
will
pro
babl
y be
mor
e th
an
com
pens
ated
by
the
part
of t
he t
ariff
tha
t pa
rtici
pate
s to
the
cov
erag
e of
fix
ed c
osts
. I
f de
man
d is
muc
h m
ore
elas
tic i
n ‘s
outh
’ th
an i
n ‘n
orth
’ an
d su
pply
muc
h m
ore
elas
tic i
n ‘n
orth
’ th
an in
‘so
uth’
, it c
an e
ven
be th
e ca
se th
at
snns
tt
, des
pite
the
coun
ter-
flow
eff
ect.
Th
is is
bec
ause
fina
ncia
l con
cern
s app
ear a
s dom
inan
t as c
ompa
red
with
eff
icie
ncy
conc
erns
. In
any
cas
e, c
ompa
ring
(13)
and
(14
), w
e ob
serv
e th
at
nssn
tt
onl
y by
coi
ncid
ence
. A
ctua
lly, s
econ
d-be
st p
rices
are
dire
ctio
nal b
oth
for t
echn
ical
and
fina
ncia
l rea
sons
. A
lso,
it
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
12
9
mus
t be
reca
lled
that
in (
13) a
nd (
14)
the
dual
var
iabl
es, t
he n
odal
pric
es a
nd th
e el
astic
ities
ar
e al
l tim
e de
pend
ent.
We
conc
lude
that
Ram
sey
trans
port
pric
es m
ust b
e tim
e de
pend
ent.
TWO
-PA
RT
TAR
IFFS
W
e co
nsid
er n
ow t
he c
ase
whe
re p
rices
are
not
con
stra
ined
to
be l
inea
r. W
e lim
it th
e an
alys
is to
the
two-
part
tarif
f w
here
(
)w
wi
iT
q =
w
ww
ii
ia
qb
is th
e ex
pend
iture
for
con
sum
ing
w iq a
t nod
e i a
nd
()
gg
ii
Tq
=
gg
gi
ii
aq
b is
the
reve
nue
whe
n ge
nera
ting
g iq a
t nod
e i.
The
ob
ligat
ion
to b
alan
ce th
e bu
dget
of t
he tr
ansp
orta
tion
firm
read
s:
()
0w
ww
wg
gg
gw
wg
gn
ns
sn
ns
sn
sn
sa
qa
qa
qa
qb
bb
brK
(1
5)
By
fixin
g '
'(
)(
)w
wo
ggo
ii
ii
ii
aU
qa
Cq
at e
ach
node
, we
obvi
ousl
y im
plem
ent t
he f
irst-
best
allo
catio
n of
con
sum
ptio
n an
d ge
nera
tion
at e
ach
node
. G
iven
thes
e pr
ices
and
giv
en th
e flo
w c
onst
rain
t (2
) (a
ssum
ing
L=0)
, (15
) re
ads
()
()
0w
wg
go
ns
ns
bb
bb
rK
. O
ne
obta
ins a
n in
finite
set o
f sol
utio
ns fo
r the
fixe
d pa
rt of
the
loca
tiona
l ene
rgy
tarif
fs, e
ven
whe
n ta
king
int
o ac
coun
t th
e pa
rtici
patio
n co
nstra
ints
of
cons
umer
s (
)(
)w
ww
wi
ii
ii
Uq
Tq
u a
nd
gene
rato
rs
()
()
gg
gg
ii
ii
iT
qC
qg
, w
here
iu (
resp
ectiv
ely
ig)
stan
ds f
or t
he r
eser
vatio
n va
lue
of c
onsu
mer
s (r
espe
ctiv
ely,
gen
erat
ors)
at n
ode
i. A
ny ru
le to
sha
re (
)o
rK
am
ong
the
agen
ts b
y m
eans
of
the
fixed
par
t of
the
tar
iffs
that
sat
isfie
s th
e ab
ove
parti
cipa
tion
cons
train
ts,
plus
add
ition
al c
onst
rain
ts d
ue t
o th
e in
form
atio
nal
disa
dvan
tage
of
the
pric
e m
aker
22, i
s per
mitt
ed.
Whe
n th
e bu
yer a
nd th
e se
ller o
f a v
olum
e q
are
loca
ted
at th
e sa
me
node
i, th
ey c
reat
e a
surp
lus:
()
()
wg
ww
gg
ii
ii
ii
Tq
Tq
aq
ba
qb
=
wg
ii
bb
(1
6)
To p
reve
nt a
ny a
rbitr
age
betw
een
the
two
node
s, th
e tra
de o
f a q
uant
ity q
of e
nerg
y fr
om
north
to so
uth
is to
be
bille
d (
)(
)(
)w
gw
wg
gns
sn
ss
nn
Tq
Tq
Tq
aq
ba
qb
. Th
eref
ore:
()
ow
gns
sn
Tq
qb
b
(17)
Sym
met
rical
ly, a
n in
divi
dual
reve
rse
trans
actio
n sh
ould
be
bille
d:
()
()
()
wg
ow
gsn
ns
ns
Tq
Tq
Tq
qb
b
(18)
For t
he sa
ke o
f tra
nspa
renc
y, o
ne c
an p
refe
r a u
nifo
rm
w b fo
r all
cons
umer
s and
a u
nifo
rm
g b f
or a
ll ge
nera
tors
wha
teve
r th
eir
loca
tion,
but
the
ris
k of
exc
lusi
on (
parti
cipa
tion
cons
train
ts th
at w
ould
not
be
satis
fied)
shou
ld n
ot b
e ne
glec
ted.
Fo
r th
e sa
me
reas
ons
as th
e on
es m
entio
ned
form
erly
, bot
h th
e m
argi
nal p
rice
a an
d th
e fix
ed fe
e b
shou
ld b
e tim
e de
pend
ent.
As l
ong
as p
artic
ipat
ion
cons
train
ts a
re n
ot b
indi
ng, t
he
varia
tion
of b
with
the
dat
e is
sup
erflu
ous.
In
cont
rast
, th
e va
riabl
e pa
rt of
the
tar
iff f
or
ener
gy a
and
, con
sequ
ently
, the
var
iabl
e pa
rt of
the
tar
iff f
or t
rans
port
o s
houl
d be
tim
e de
pend
ent i
n or
der t
o se
nd g
ood
scar
city
sign
als t
o us
ers.
MU
LTIP
RO
DU
CT
TWO
-PA
RT
TAR
IFFS
A
s al
read
y m
entio
ned,
ene
rgy
pric
es a
nd c
onse
quen
tly tr
ansp
ort p
rices
sho
uld
be v
aryi
ng
with
tim
e. F
or e
xam
ple,
the
hour
ly tw
o-pa
rt ta
riff d
efin
ed in
(17)
shou
ld re
ad:
130
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
(,
())
()
()
()
nsns
nsns
nsT
qA
qB
(1
8)
and
the
tota
l exp
endi
ture
s pa
id to
the
trans
port
oper
ator
by
an a
gent
tran
sfer
ring
ener
gy
from
nor
th to
sout
h, le
t us s
ay d
urin
g on
e ye
ar, w
ould
be:
8760 1
()
()
()
nsns
nsns
TA
qB
.
For t
he re
ason
exp
lain
ed fo
rmer
ly, t
he fi
xed
part
of th
e ta
riff c
an b
e ch
osen
qui
te e
asily
23,
for
exam
ple
it ca
n be
a c
onst
ant.
The
diff
icul
ty is
that
(18
) re
quire
s to
met
er a
nd to
bill
the
flow
s fr
om n
orth
to s
outh
alm
ost c
ontin
uous
ly.
Bec
ause
the
trans
actio
n co
sts
wou
ld b
e ve
ry
high
, the
ope
rato
r w
ill in
stal
l sim
ple
met
erin
g an
d bi
lling
dev
ices
that
do
not d
istin
guis
h th
e da
te o
f ea
ch f
low
. T
his
mea
ns t
hat
the
mar
gina
l pr
ice
()
nsA w
ill n
ot b
e a
cont
inuo
us
func
tion
of ti
me.
In
mos
t cas
es, i
t will
be
a pi
ecew
ise
func
tion
with
two
seas
onal
(su
mm
er
and
win
ter)
and
two
daily
(da
y an
d ni
ght)
valu
es.
At w
orst
it w
ill b
e un
iform
all
the
year
lo
ng.
Ass
ume
the
latte
r.
In th
is c
ase
()
nsns
AA
so th
at:
8760 1
(,
())
()
()
ens
nsns
nsns
nsns
nsT
qA
qB
Aq
B,
whe
re
8760 1
()
e nsns
is th
e to
tal e
nerg
y co
nsum
ed a
ll th
e ye
ar lo
ng.
The
obvi
ous
draw
back
of
this
tw
o-pa
rt pr
icin
g is
tha
t it
does
not
allo
w t
o di
scrim
inat
e be
twee
n re
gula
r use
rs w
ho c
an b
e sa
tisfie
d w
ith a
“m
ediu
m s
ize”
grid
on
one
side
and
, on
the
othe
r si
de, i
rreg
ular
use
rs w
ho n
eed
larg
e ca
paci
ty o
f tra
nspo
rt fo
r sh
ort p
erio
ds o
f tim
e.
A
solu
tion
is t
o co
mbi
ne t
he t
rans
port
dem
and
for
ener
gy
e nsq a
nd a
pro
xy f
or t
he t
rans
port
dem
and
for c
apac
ity, f
or e
xam
ple
max
()
K nsns
, and
to b
ill th
em se
para
tely
in a
way
that
ap
prox
imat
es th
e op
timal
tim
e-de
pend
ent t
wo-
part
tarif
f. W
e ob
tain
a m
ulti-
prod
uct t
wo-
part
tarif
f:
ee
KK
nsns
nsns
nsns
TA
qA
qB
(1
9)
LOO
P FL
OW
S C
onsi
der
the
thre
e-no
de n
etw
ork
of B
ox 4
-2 a
nd s
uppo
se t
hat
the
line
betw
een
the
two
inje
ctio
n no
des
is t
he o
nly
one
with
a l
imiti
ng c
apac
ity K
. N
egle
ctin
g lo
sses
, the
opt
imal
di
spat
ch is
the
solu
tion
to:
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
13
1
3
11
22
12
3,
,m
axw
gg
gg
wq
Uq
Cq
Cq
rK
s.t.
3
12
wg
gq
1
2
3
gg
K.
Whe
n th
e ca
paci
ty c
onst
rain
t on
this
line
is b
indi
ng, e
ach
gene
rato
r cre
ates
a c
ount
er-f
low
th
at a
llevi
ates
the
load
pro
voke
d by
the
inje
ctio
n of
pow
er fr
om th
e ot
her p
lant
. B
ecau
se o
f th
is p
ositi
ve e
xter
nalit
y, t
he o
ptim
al d
ispa
tch
can
com
man
d th
at a
hig
h-co
st p
lant
sho
uld
gene
rate
pow
er d
espi
te th
e ex
iste
nce
of a
vaila
ble
gene
ratio
n ca
paci
ty a
t low
cos
t som
ewhe
re
else
in th
e ne
twor
k.
The
leve
l of
the
ther
mal
con
stra
int o
n th
e lin
e be
twee
n no
des
1 an
d 2
affe
cts
all
noda
l pr
ices
. If
pla
nt 1
is
mor
e ef
ficie
nt t
han
plan
t 2,
i.e
, if
''
12
Cq
Cq
, pr
ices
:
'2
22
3'
3
def
og
wp
Cq
Uq
>
33
'de
f
wp
Uq
>
'1
11
3'
3
def
og
wp
Cq
Uq
allo
w to
enc
oura
ge (d
isco
urag
e) g
ener
atio
n at
nod
e 2
(nod
e 1)
.
If e
nerg
y an
d tra
nspo
rt ha
ve t
o be
inv
oice
d se
para
tely
, fro
m t
he e
quili
briu
m c
ondi
tions
j
iij
pp
t, w
e ob
tain
the
set o
f lin
ear t
ariff
s for
tran
spor
t:
13/3
ot
, 23
/3o
t,
122
/3o
t.
They
exp
licitl
y en
cour
age
cons
umer
s to
tran
sact
with
exp
ensi
ve g
ener
ator
s (n
ode
2) a
nd
they
dis
suad
e th
em t
o tra
nsac
t w
ith l
ow-c
ost
gene
rato
rs (
node
1).
The
se p
rices
sen
d th
e ac
cura
te s
igna
ls to
pre
vent
jeop
ardi
sing
the
safe
ty o
f the
grid
, nam
ely
the
ohm
ic c
onst
rain
t on
the
line
betw
een
node
s 1 a
nd 2
. N
ote
that
, in
cont
rast
to th
e on
e-lin
e m
odel
, in
a m
eshe
d ne
twor
k th
e sh
adow
cos
t of
the
ther
mal
con
stra
int
on t
he l
ine
betw
een
two
node
s is
no
long
er e
qual
to
the
diff
eren
ce i
n m
argi
nal v
alua
tions
at t
he n
odes
. In
our
illu
stra
tion,
'
'2
21
13
2g
go
Cq
Cq
. Th
is is
be
caus
e al
l con
nect
ed li
nes a
re a
ffec
ted
by a
bila
tera
l tra
nsac
tion.
Con
sequ
ently
, the
mar
gina
l va
lue
of c
onge
stio
n re
flect
s al
l th
e ne
gativ
e ex
tern
aliti
es c
reat
ed t
hrou
gh t
he g
rid b
y th
e lim
ited
capa
city
of e
ach
piec
e of
equ
ipm
ent.
RE
FER
EN
CE
S
Cha
o, H
.P.
and
S.Pe
ck (
1996
), “A
Mar
ket
Mec
hani
sm f
or E
lect
ric P
ower
Tra
nsm
issi
on”,
Jo
urna
l of R
egul
ator
y Ec
onom
ics,
vol 1
0, p
p 25
-60.
C
hao,
H.P
., S.
Peck
, S.
Ore
n an
d R
.Wils
on (
2000
), “F
low
-Bas
ed T
rans
mis
sion
Rig
hts
and
Con
gest
ion
Man
agem
ent”
, Ele
ctric
ity J
ourn
al, O
ctob
er, p
p 38
-58.
C
ram
pes
C.
and
J.J.L
affo
nt (
2001
) “T
rans
port
Pric
ing
in t
he E
lect
ricity
Ind
ustry
” O
xfor
d Re
view
of E
cono
mic
Pol
icy,
Aut
umn,
vol
17,
no
3, p
p 31
3-32
8.
Cre
mer
, H
. an
d J.J
.Laf
font
(20
02),
“Com
petit
ion
in G
as M
arke
ts”,
Eur
opea
n Ec
onom
ic
Revi
ew, v
ol 4
6, n
o 4-
5, p
p 92
8-93
5.
132
Tran
spor
t pri
cing
of e
lect
rici
ty n
etw
orks
Hog
an, W
.W. (
1992
) “C
ontra
ct N
etw
orks
for
Ele
ctric
ity P
ower
Tra
nspo
rt”,
Jour
nal
of
Regu
lato
ry E
cono
mic
s vol
4, p
p 21
1-24
2.
Hog
an, W
.W. (
1998
) “N
odes
an
d Zo
nes
in
Elec
trici
ty
Mar
kets
: Se
ekin
g Si
mpl
ified
C
onge
stio
n Pr
icin
g”, i
n H
ung-
Po C
hao
and
H.G
. Hun
tingt
on (E
ds.)
Des
igni
ng C
ompe
titiv
e El
ectr
icity
Mar
kets
, Klu
wer
Aca
dem
ic P
ublis
hers
, Lon
don,
pp
33-6
2.
Hsu
, M. (
1997
) “A
n In
trodu
ctio
n to
the
Pric
ing
of E
lect
ric P
ower
Tra
nsm
issi
on”,
Util
ities
Po
licy,
vol
6, n
o 3,
pp
257-
270.
Jo
skow
, P.L
. and
J.T
irole
(200
0) “
Tran
smis
sion
Rig
hts
and
Mar
ket P
ower
on
Elec
tric
Pow
er
Net
wor
ks”,
RAN
D J
ourn
al o
f Eco
nom
ics,
vol 3
1, n
o 3,
Aut
umn,
pp
450-
487.
Pé
rez-
Arr
iaga
I.J.
, F.J.
Rub
io, J
.F.P
uerta
, J.A
rcel
uz a
nd J
.Mar
ín (
1995
) “M
argi
nal p
ricin
g of
tra
nsm
issi
on s
ervi
ces:
an
anal
ysis
of
cost
rec
over
y”,
IEEE
Tra
nsac
tions
on
Pow
er
Syst
ems,
vol 1
0, n
o 1,
Feb
ruar
y, p
p 54
6-55
3.
Schw
eppe
F.,
M.C
aram
anis
, R.T
abor
s an
d R
.Boh
n (1
988)
Spo
t pric
ing
for
elec
tric
ty, K
luw
er
Aca
dem
ic P
ublis
hers
. St
oft S
. (2
002)
Pow
er S
yste
m E
cono
mic
s, IE
EE/W
iley.
W
u, F
., P.
Var
aiya
, P.
Spill
er a
nd S
.Ore
n (1
996)
, “F
olk
Theo
rem
s on
Tra
nsm
issi
on O
pen
Acc
ess:
Pro
ofs a
nd C
ount
er e
xam
ples
”, J
ourn
al o
f Reg
ulat
ory
Econ
omic
s, pp
5-2
3.
NO
TE
S
1 Th
is is
a g
ood
illus
tratio
n of
the
diff
icul
ty to
giv
e a
prec
ise,
obj
ectiv
e de
finiti
on o
f cos
ts,
as e
xpla
ined
in C
hapt
er 2
. 2
As
show
n in
Cha
o an
d Pe
ck (
1996
), ph
ysic
al r
ight
s an
d fin
anci
al r
ight
s ar
e eq
uiva
lent
w
hen
the
ener
gy m
arke
ts a
nd th
e rig
hts
mar
kets
are
per
fect
ly c
ompe
titiv
e.
Josk
ow a
nd
Tiro
le (2
000)
pro
vide
an
anal
ysis
of v
ario
us n
on-c
ompe
titiv
e co
nfig
urat
ions
. 3
See
Josk
ow a
nd T
irole
(200
0), p
. 45
2.
4 Fo
r an
illus
tratio
n se
e B
ox 4
-2.
For m
ore
deta
ils, s
ee fo
r exa
mpl
e H
su (1
997)
. 5
Schw
eppe
et a
l. (1
988)
. 6
See
for e
xam
ple
Stof
t (20
02),
p. 4
17.
7 St
udie
s of
mar
ket
pow
er a
re b
ased
on
calib
rate
d si
mul
atio
n m
odel
s. S
ee r
efer
ence
s in
C
ram
pes a
nd L
affo
nt (2
001)
. 8
We
do n
ot c
onsi
der t
he is
sue
of n
odal
-pric
e ra
ndom
ness
. To
hed
ge a
gain
st p
rice
vola
tility
, us
ers c
an si
gn fi
nanc
ial c
ontra
cts.
See
Hog
an (1
992)
and
Cha
pter
5 in
fra.
9
See
Ore
n et
al.
(199
6).
10
See
Pére
z A
rria
ga e
t al.
(199
5).
11
Prov
ided
the
auth
ority
in c
harg
e of
pric
ing
is n
ot c
onst
rain
ed b
y a
lack
of i
nfor
mat
ion
on
the
will
ingn
ess t
o pa
y of
the
user
s. F
or a
n ill
ustra
tion,
see
Cha
pter
6.
12
Se
e C
hapt
er 3
. 13
Fo
r det
ails
on
econ
omic
and
lega
l pric
e di
scrim
inat
ions
, the
read
er is
refe
rred
to C
hapt
er 1
. 14
O
n th
e pr
inci
ples
of c
oope
rativ
e-ga
mes
theo
ry, s
ee C
hapt
er 3
. 15
In
the
tw
o-pa
rt ca
se, t
he v
aria
ble
part
is i
ncre
asin
g lin
early
with
the
qua
ntity
of
ener
gy
trans
porte
d.
In m
ulti-
part
tarif
fs,
the
varia
ble
part
is a
pie
cew
ise
linea
r in
crea
sing
fu
nctio
n.
16
Det
ails
on
actu
al ta
riffs
are
pre
sent
ed in
Cha
pter
8.
17
We
do n
ot d
iscu
ss t
he a
dditi
onal
int
erte
mpo
ral
cons
train
ts d
ue t
o th
e m
anag
emen
t of
hy
drop
lant
s or t
o th
e ob
ligat
ion
to sa
tisfy
hea
ting
cond
ition
s in
ther
mal
pla
nts.
Cos
t rec
over
y an
d sh
ort-r
un e
ffici
ency
13
3
18
A
ctua
lly, D
and
N a
re s
ucce
ssiv
e, b
ut th
e ar
gum
ent i
s th
e sa
me
as if
ther
e w
ere
seve
ral
non-
rival
sim
ulta
neou
s ne
eds
to s
atis
fy w
ith th
e sa
me
equi
pmen
t: it
is th
e w
illin
gnes
s to
pa
y of
all
user
s tha
t mus
t be
take
n in
to c
onsi
dera
tion.
19
Th
is i
s tru
e w
hen
the
long
run
mar
gina
l co
st o
f ge
nera
tion
is n
on d
ecre
asin
g an
d ge
nera
tors
hav
e op
timiz
ed th
e si
ze o
f the
ir pl
ants
. 20
Se
e C
rem
er a
nd L
affo
nt 2
001.
21
To
whi
ch th
e m
argi
nal c
ost o
f los
ses s
houl
d be
add
ed.
22
Self-
sele
ctio
n co
nstra
ints
are
to
be a
dded
to
desi
gn t
he m
enu
of t
ariff
s w
hen
the
pric
e m
aker
can
not o
bser
ve s
ome
indi
vidu
al c
hara
cter
istic
s of
the
pric
e ta
kers
, for
exa
mpl
e th
e co
nsum
ers'
will
ingn
ess
to p
ay o
r th
e ge
nera
tion
cost
of
prod
ucer
s. T
he p
robl
em i
s to
pr
even
t an
opp
ortu
nist
ic s
witc
h of
som
e us
ers
tow
ards
tar
iffs
that
are
des
igne
d fo
r so
meb
ody
else
. C
hapt
er 6
pro
vide
s an
illu
stra
tion
of th
e de
sign
of
cont
rol m
echa
nism
s un
der i
nfor
mat
ion
asym
met
ry.
23
Exce
pt w
hen
ther
e is
a se
rious
info
rmat
iona
l gap
that
impo
ses f
ine
tuni
ng to
resp
ect a
ll th
e pa
rtici
patio
n an
d se
lf se
lect
ion
cons
train
ts.
