=Paper=
{{Paper
|id=Vol-1623/paperme7
|storemode=property
|title=Dual Model of Power Market with Generation and Line Capacity Expansion
|pdfUrl=https://ceur-ws.org/Vol-1623/paperme7.pdf
|volume=Vol-1623
|authors=Svetlana Gakh
|dblpUrl=https://dblp.org/rec/conf/door/Gakh16
}}
==Dual Model of Power Market with Generation and Line Capacity Expansion==
https://ceur-ws.org/Vol-1623/paperme7.pdf
Dual Model of Power Market with Generation
and Line Capacity Expansion
Svetlana Gakh
Melentiev Energy Systems Institute SB RAS,
Lermontov str., 130, Irkutsk, Russia, 664033
Abstract. The investigated model is a mathematical model in which operating
power, installed power, power flows between nodes in electric power system
(EPS) are optimized for the last year of the calculation period. The model is
static, multinodal and it is represented as a large dimension linear programming
problem. The aim of this study is analysis of the relationships between dual
variables as the nodal and line prices.
Keywords: Electric Power System (EPS), primal and dual linear programming
problems, nodal prices, dual variables.
1 Introduction
In our paper we study a model which describes the development of Electric Power
System (EPS) in a long term period. From mathematical point of view the model is
represented by a linear programming problem. The model is static. The statistic for-
mulation have been used in Melentiev Energy Systems Institute SB RAS for 20 years.
It showed itself to good advantage.
There is Kirchhoff’s first law in the model, but there is no Kirchhoff’s second law. It
is due to the fact that the model is advanced. For this reason reactive energy is not
considered.
The problem has been successfully used for a quite long period of time [2]. In 2011 a
market version of the model was created and tested on real data from central Russia [5].
A necessity of investigation of power interstate connections caused a preliminary re-
search performed in [1], where only the primal model was used. The aim of this paper
consists in deriving the dual formulation and providing some interpretation of dual
variables.
2 Description of the Model
To describe the model we need first to describe sets, parameters, variables, objective
function and constraints.
Sets:
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Dual Model of Power Market with Generation and Line Capacity Expansion 607
J — set of nodes;
I — set of types of stations;
S — set of seasons (winter, spring, summer, autumn);
T — set of hours in days (0-23);
R⊂S — set of seasons, in which an annual maximum load is achieved;
Qs ⊂T — set of time intervals, in which an annual maximum load in season s∈S is
achieved.
Node parameters:
0
yji and y ji — initial and maximum installed powers of station of type i∈I in node
j∈J;
x0jist and xjist — minimum and maximum allowable operating powers of station of
type i∈I in node j∈J in hour t∈T of season s∈S;
vji — unit variable costs of station of type i∈I in node j∈J;
kji and bji — relative capital investments and unit fixed costs of station of type i∈I
in node j∈J;
Djst — consumer load in node j∈J in hour t∈T of season s∈S.
Line parameters:
κjj 0 and βjj 0 — relative capital investments and unit fixed costs for new and developing
lines between nodes j and j 0 ;
ajj 0 — maximum power line capacity between nodes j∈J and j 0 ∈J;
πjj 0 — unit line losses between nodes j∈J and j 0 ∈J.
Other parameters:
— power reserve ratio (i.e. this is a power reserve coefficient for stations repair, emer-
gency situations and etc.);
τsw (τsh ) — equivalent number of working days (holidays) in season s∈S (i.e. such a
number of days which when multiplied by season maximum load results in electrical
energy consumption equal to an accepted season value);
ρ(1+ρ)M
f — capital recovery factor (CRF), f = (1+ρ) M −1 , where ρ — discount rate, M —
number of years, in which the capital is returned.
Capital recovery factor is calculated on the condition of capital recovery in equal parts
G during M years with discount rate ρ.
Node variables:
yji — installed power of station of type i∈I in node j∈J;
xw h
jist (xjist ) — operating power of station of type i∈I in node j∈J in hour t∈T of season
s∈S on working days (holidays).
Line variables:
ajj 0 — power line capacity between nodes j∈J and j 0 ∈J;
0
uw w
jj 0 st (uj 0 jst ) — operating power flow in the set from node j∈J to node j ∈J (from
0
node j ∈J to node j∈J) in hour t∈T of season s∈S on working days;
uhjj 0 st (uhj 0 jst ) — operating power flow in the set from node j∈J to node j 0 ∈J (from
node j 0 ∈J to node j∈J) in hour t∈T of season s∈S on holidays;
u
ejj 0 st (euj 0 jst ) — ”emergency” power flow in the set from node j∈J to node j 0 ∈J (from
node j 0 ∈J to node j∈J) in hour t∈Qs of season s∈R.
608 S. Gakh
The objective function:
The objective function is a function of the total costs of the whole EPS and it has the
form of:
XXXX XXXX
τsw vji xw
jist + τsh vji xhjist + (1)
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
XX XX
0
+f kji (yji − yji )+ bji yji + (2)
j∈J i∈I j∈J i∈I
XX XX
+f κjj 0 (ajj 0 − a0jj 0 ) + βjj 0 ajj 0 → min . (3)
j∈J j 0 ∈J j∈J j 0 ∈J
j 0 >j j 0 >j
The components of the objective function are total (annual) costs for operating pow-
er (1), costs for introduction of new capacities and fixed costs for its maintenance (2),
costs for line capacity development and corresponding fixed costs (3).
Constraints:
On electric power station development:
0
yji ≤ yji ≤ y ji , j∈J, i∈I . (4)
On power line development:
a0jj 0 ≤ ajj 0 ≤ ajj 0 , j∈J, j 0 ∈J j0 > J . (5)
On operating power on working days and holidays respectively:
x0jist ≤ xw
jist ≤ xjist , j∈J, i∈I, s∈S, t∈T ; (6)
x0jist ≤ xhjist ≤ xjist , j∈J, i∈I, s∈S, t∈T . (7)
On flows in power lines on working days and holidays respectively:
0 ≤ uw
jj 0 st ≤ ajj 0 , j∈J, j 0 ∈J, j 0 6=J, s∈S, t∈T ; (8)
0 0
0 ≤ uhjj 0 st ≤ ajj 0 , j∈J, j ∈J, j 6=J, s∈S, t∈T . (9)
On ”emergency” flows in power lines:
0≤u
ejj 0 st ≤ ajj 0 , j∈J, j 0 ∈J, j 0 6=J, s∈R, t∈Qs . (10)
On operating power balance of power stations on working days and holidays (it is based
on Kirchhoff’s first law):
X X X
xw
jist − uwjj 0 st + uw
j 0 jst (1−πj 0 j )=Djst , j∈J, s∈S, t∈T ; (11)
i∈I j 0 ∈J j 0 ∈J
j 0 6=j j 0 6=j
X X X
xhjist − uhjj 0 st + uhj 0 jst (1−πj 0 j )=Djst , j∈J, s∈S, t∈T . (12)
i∈I 0 0
j ∈J j ∈J
j 0 6=j j 0 6=j
Dual Model of Power Market with Generation and Line Capacity Expansion 609
On installed power in peak load hours:
X X X
yji − u
ejj 0 st + ej 0 jst (1 − πj 0 j ) ≥ Djst + · Djst ,
u (13)
i∈I j 0 ∈J j 0 ∈J
j 0 6=j j 0 6=j
j∈J, s∈R, t∈Qs .
The mathematical model is a linear programming problem. One needs to find a mini-
mum of the objective function (1)–(3) with the constraints (4)–(13).
3 The Dual Problem
We derive the dual problem using technique from [7] based on the Lagrange function
for problem (1)–(13). First, a part of Lagrange function, corresponding to considered
group of constraints, is written. Then, the resulting constraint will be converted by
rearrangement of the summands.
Dual function consists of several parts. The dual part corresponding to constraints on
electric power station development, left part of constraints (4), dual variables ξ ji :
XX XXh i XX
0 0
ξ ji (yji − yji ) = −ξ ji yji + ξ ji yji . (14)
j∈J i∈I j∈J i∈I j∈J i∈I
The dual part corresponding to constraints on electric power station development, right
part of constraints (4), dual variables ξ ji :
XX XX XX
ξ ji (yji − y ji ) = ξ ji yji − ξ ji y ji . (15)
j∈J i∈I j∈J i∈I j∈J i∈I
The dual part corresponding to constraints on power line development, left part of
constraints (5), dual variables ν jj 0 :
XX XX XX
ν jj 0 (a0jj 0 − ajj 0 ) = ν jj 0 a0jj 0 .
−ν jj 0 ajj 0 + (16)
j∈J j 0 ∈J j∈J j 0 ∈J j∈J j 0 ∈J
j 0 >j j 0 >j j 0 >j
The dual part corresponding to constraints on power line development, right part of
constraints (5), dual variables ν jj 0 :
XX XX XX
ν jj 0 (ajj 0 − ajj 0 ) = ν jj 0 ajj 0 − ν jj 0 ajj 0 . (17)
j∈J j 0 ∈J j∈J j 0 ∈J j∈J j 0 ∈J
j 0 >j j 0 >j j 0 >j
The dual part corresponding to minimum power loading on working days, left part of
constraints (6), dual variable µw
jist
:
XXXX
µw (x0 − xw
jist jist jist ) = (18)
j∈J i∈I s∈S t∈T
610 S. Gakh
XXXXh i XXXX
= −µw
jist
xw
jist + µw x0 .
jist jist
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
The dual part corresponding to maximum power loading on working days, right part
of constraints (6), dual variables µw
jist :
XXXX
µw w
jist (xjist − xjist ) = (19)
j∈J i∈I s∈S t∈T
XXXX XXXX
= µw w
jist xjist − µw
jist xjist .
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
The dual part corresponding to minimum power loading on holidays, left part of con-
straints (7), dual variable µhjist :
XXXX
µhjist (x0jist − xhjist ) = (20)
j∈J i∈I s∈S t∈T
XXXXh i XXXX
= −µhjist xhjist + µhjist x0jist .
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
The dual part corresponding to maximum power loading on holidays, right part of
constraints (7), dual variables µhjist :
XXXX
µhjist (xhjist − xjist ) = (21)
j∈J i∈I s∈S t∈T
XXXX XXXX
= µhjist xhjist − µhjist xhjist .
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
The dual part corresponding to constraints on flows in power lines on working days,
left part of constraints (8), dual variables γ w
jj 0 st
:
X X XXh i
−γ w 0
jj st
uw
jj 0 st . (22)
j∈J j 0 ∈J s∈S t∈T
j 0 6=j
The dual part corresponding to constraints on flows in power lines on working days,
right part of constraints (8), dual variables γ w jj 0 st :
X X XX
γw w
jj 0 st (ujj 0 st − ajj 0 ) = (23)
j∈J j 0 ∈J s∈S t∈T
j 0 6=j
X X XX X X XX
= γw w
jj 0 st ujj 0 st − γw
jj 0 st ajj 0 .
0
j∈J j ∈J s∈S t∈T 0
j∈J j ∈J s∈S t∈T
j 0 6=j j 0 6=j
The dual part corresponding to constraints on flows in power lines on holidays, left
part of constraints (9), dual variables γ hjj 0 st :
X X XXh i
−γ hjj 0 st uhjj 0 st . (24)
j∈J j 0 ∈J s∈S t∈T
j 0 6=j
Dual Model of Power Market with Generation and Line Capacity Expansion 611
The dual part corresponding to constraints on flows in power lines on holidays, right
part of constraints (9), dual variables γ hjj 0 st :
X X XX
γ hjj 0 st (uhjj 0 st − ajj 0 ) = (25)
j∈J j 0 ∈J s∈S t∈T
j 0 6=j
X X XX X X XX
= γ hjj 0 st uhjj 0 st − γ hjj 0 st ajj 0 .
0
j∈J j ∈J s∈S t∈T 0
j∈J j ∈J s∈S t∈T
j 0 6=j j 0 6=j
The dual part corresponding to constraints on ”emergency” flows in power lines, left
part of constraints (10), dual variables η jj 0 st :
XXX Xh i
−η jj 0 st uejj 0 st . (26)
j∈J j 0 ∈J s∈R t∈Qs
j 0 6=j
The dual part corresponding to constraints on ”emergency” flows in power lines, right
part of constraints (10), dual variables η jj 0 st :
XXX X
ujj 0 st − ajj 0 ) =
η jj 0 st (e (27)
j∈J j 0 ∈J s∈R t∈Qs
j 0 6=j
XXX X XXX X
= ejj 0 st −
η jj 0 st u η jj 0 st ajj 0 .
j∈J j 0 ∈J s∈R t∈Qs j∈J j 0 ∈J s∈R t∈Qs
j 0 6=j j 0 6=j
The dual part corresponding to constraints on operating power balance of power sta-
tions on working days (11), dual variables λw
jst :
XXX X w X X
λw uw uw
jst xjist − jj 0 st + j 0 jst (1 − πj 0 j ) − Djst =
(28)
j∈J s∈S t∈T i∈I j 0 ∈J j 0 ∈J
j 0 6=j j 0 6=j
XXX X X X XX
= λw
jst xw
jist − (λw w
jst − (1 − πj 0 j )λj 0 st )×
j∈J s∈S t∈T i∈I j∈J j 0 ∈J s∈S t∈T
j 0 6=j
XXX
×uw
jj 0 st − Djst λw
jst .
j∈J s∈S t∈T
The dual part corresponding to constraints on operating power balance of power sta-
tions on holidays (12), dual variables λhjst :
XXX X h X X
λhjst uhjj 0 st + uhj 0 jst (1 − πj 0 j ) − Djst
xjist − = (29)
j∈J s∈S t∈T i∈I j 0 ∈J j 0 ∈J
j 0 6=j j 0 6=j
612 S. Gakh
XXX X X X XX
= λhjst xhjist − (λhjst − (1 − πj 0 j )λhj 0 st )×
j∈J s∈S t∈T i∈I j∈J j 0 ∈J s∈S t∈T
j 0 6=j
XXX
×uhjj 0 st − Djst λhjst .
j∈J s∈S t∈T
The dual part corresponding to constraints on installed power in peak load hours (13),
dual variables σjst :
XX X X X X
σjst Djst +·Djst − yji + ejj 0 st −
u ej 0 jst ×
u (30)
j∈J s∈R t∈Qs i∈I j 0 ∈J j 0 ∈J
j 0 6=j j 0 6=j
!
XX X X XXX X
×(1−πj 0 j ) = [−σjst ] yji + (σjst −
j∈J s∈R t∈Qs i∈I j∈J j 0 ∈J s∈R t∈Qs
j 0 6=j
XX X
−(1 − πj 0 j )σj 0 st )e
ujj 0 st + σjst (Djst + ·Djst ) .
j∈J s∈R t∈Qs
Thus, the dual Lagrange function consists of the following components: the objective
function (1)–(3), the dual parts corresponding to all constraints of the problem, (14)–
(30).
The dual Lagrange function is the following:
Θ(λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h , γ h , η, η) = (31)
n
= min L(x, y, u, u e, a, λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h ,
x,y,u,eu,a
o
γ h , η, η) = Θ1 (λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h , γ h , η, η)+
+Θ2 (λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h , γ h , η, η),
where
Θ1 (λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h , γ h , η, η) = (32)
(
XXXX
= min (τsw vji − λw w w
jst − µjist + µjist )xjist +
w
x,y,u,e
u,a
j∈J i∈I s∈S t∈T
XXXX
+ (τsh vji − λhjst − µhjist + µhjist )xhjist +
j∈J i∈I s∈S t∈T
XX X X
+ (f kji + bji − σjst − ξ ji + ξ ji )yji +
j∈J i∈I s∈R t∈Qs
XX XX XX
+ (f κjj 0 + βjj 0 − ν jj 0 + ν jj 0 − γw
jj 0 st − γ hjj 0 st −
j∈J j 0 ∈J s∈S t∈T s∈S t∈T
j 0 >j
X X X X XX
− η jj 0 st )ajj 0 + (λw w w
jst − λj 0 st (1 − πj 0 j ) − γ jj 0 st +
s∈R t∈Qs j∈J j 0 ∈J s∈S t∈T
j 0 6=j
Dual Model of Power Market with Generation and Line Capacity Expansion 613
X X XX
+γ w w
jj 0 st )ujj 0 st + (λhjst − λhj 0 st (1 − πj 0 j ) − γ hjj 0 st + γ hjj 0 st )uhjj 0 st +
j∈J j 0 ∈J s∈S t∈T
j 0 6=j
)
XXX X
+ (σjst − σj 0 st (1 − πj 0 j ) − η jj 0 st + η jj 0 st )e
ujj 0 st ,
0
j∈J j ∈J s∈R t∈Qs
j 0 6=j
Θ2 (λw , λh , σ, ξ, ξ, ν, ν, µw , µw , µh , µh , γ w , γ w , γ h , γ h , η, η) = (33)
XXX XXX
= λw
jst Djst + λhjst Djst +
j∈J s∈S t∈T j∈J s∈S t∈T
XX X XX XX
0
+ σjst Djst (1 + ) + ξ ji yji − ξ ji y ji −
j∈J s∈R t∈Qs j∈J i∈I j∈J i∈I
XX XX XX
0
−f kji yji + ν jj 0 a0jj 0 − ν jj 0 ajj 0 −
j∈J i∈I j∈J j 0 ∈J j∈J j 0 ∈J
j 0 >j j 0 >j
XX XXXX XXXX
−f κjj 0 a0jj 0 + µw x0 −
jist jist
µw
jist xjist +
0
j∈J j ∈J j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
j 0 >j
XXXX XXXX
+ µhjist x0jist − µhjist xjist .
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
To get the dual problem expressions-cofactors before every primal variable could be ze-
ro. Thus one get constraints of the dual problem. The remaining part without variables
is an objective function of the dual problem. Also conditions for the nonnegativity are
written. As a result we get the following representation of the dual problem:
XXX XXX
λw
jst Djst + λhjst Djst + (34)
j∈J s∈S t∈T j∈J s∈S t∈T
XX X XX XX
0
+ σjst Djst (1 + ) + ξ ji yji − ξ ji y ji −
j∈J s∈R t∈Qs j∈J i∈I j∈J i∈I
XX XX XX
0
−f kji yji + ν jj 0 a0jj 0 − ν jj 0 ajj 0 −
j∈J i∈I j∈J j 0 ∈J j∈J j 0 ∈J
j 0 >j j 0 >j
XX XXXX XXXX
−f κjj 0 a0jj 0 + µw x0 −
jist jist
µw
jist xjist +
j∈J j 0 ∈J j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
j 0 >j
XXXX XXXX
+ µhjist x0jist − µhjist xjist → min,
j∈J i∈I s∈S t∈T j∈J i∈I s∈S t∈T
τsw vji − λw w w
jst − µjist + µjist = 0, j∈J, i∈I, s∈S, t∈T ; (35)
τsh vji − λhjst − µhjist + µhjist = 0, j∈J, i∈I, s∈S, t∈T ; (36)
X X
f kji + bji − σjst − ξ ji + ξ ji = 0, j∈J, i∈I; (37)
s∈R t∈Qs
614 S. Gakh
XX XX
f κjj 0 + βjj 0 − ν jj 0 + ν jj 0 − γw
jj 0 st − γ hjj 0 st − (38)
s∈S t∈T s∈S t∈T
X X
− η jj 0 st = 0, j∈J, j 0 ∈J, j 0 >J, s∈S, t∈T ;
s∈R t∈Qs
λw w w w
jst − λj 0 st (1 − πj 0 j ) − γ jj 0 st + γ jj 0 st = 0, (39)
0 0
j∈J, j ∈J, j 6=J, s∈S, t∈T ;
λhjst − λhj 0 st (1 − πj 0 j ) − γ hjj 0 st + γ hjj 0 st = 0, (40)
0 0
j∈J, j ∈J, j 6=J, s∈S, t∈T ;
σjst − σj 0 st (1 − πj 0 j ) − η jj 0 st + η jj 0 st = 0, (41)
0 0
j∈J, j ∈J, j 6=J, s∈R, t∈Qs ;
σjst ≥ 0, j∈J, s∈R, t∈Qs , (42)
ξ ji ≥0, ξ ji ≥0, j∈J, i∈I, (43)
ν jj 0 ≥0, ν jj 0 ≥0, j∈J, j 0 ∈J, j0 > j (44)
w w h h
µ ≥0, µ ≥0, µ ≥0, µ ≥0, j∈J, i∈I, s∈S, t∈T, (45)
γw
jj 0 st
≥0, γw
jj 0 st ≥0, γ hjj 0 st ≥0, γ hjj 0 st ≥0, (46)
j∈J, j 0 ∈J, j 0 6=j, s∈S, t∈T,
η jj 0 st ≥0, η jj 0 st ≥0, j∈J, j 0 ∈J, j 0 6=j, s∈S, t∈T . (47)
4 Example
To give an interpretation of the above approach we consider an illustrative example.
The network consists of three nodes: J = {”V olga”, ”Center”, ”South”}. There are
two periods of time: T = {0, 1}. Period 0 is a peak period. Assume that f =0.02,
π=0.025, =0.1. There are connections between the lines (”Volga” – ”Center”, ”Cen-
ter” – ”South”, ”South” – ”Volga”). The node and lines parameters of the model are
presented in Tab. 1–2. The values of the primal and dual variables are presented in
Tab. 3–6.
Table 1. The node parameters y j , y j , kj , bj , vjt , Djt
Node, j y j yj kj bj vj0 vj1 Dj0 Dj1
Volga 10 100 500 30 640 960 120 50
Center 12 120 200 17 1536 1600 100 90
South 20 200 350 21 1280 1408 100 45
It is shown in [3, 4] that the dual variables corresponding to every constraint in the
primal problem have a certain conceptual meaning. Let us consider some of the most
Dual Model of Power Market with Generation and Line Capacity Expansion 615
Table 2. Lines parameters ajj 0 , κjj 0 , βjj 0
Connection ajj 0 κjj 0 βjj 0
Volga – Center 35 200 50
Center – South 50 190 76
South – Volga 78 180 91
Table 3. The values of primal variables xjt , yj and dual variables µjt , λjt , σjt , ξ j
Node, j xj0 xj1 yj µj0 µj1 λj0 λj1 σj0 ξj
Volga 100 100 100 857.6 412.80 1497.60 1372.80 20.48 1250.88
Center 23.22 7.13 54.43 0 0 1536 1600 21 0
South 200 80.38 200 85.56 0 1365.56 1408 19.96 77.52
Table 4. The values of primal variables ujj 0 t and dual variables γ jj 0 t , γ jj 0 t
Flow ujj 0 0 ujj 0 1 γ jj 0 0 γ jj 0 1 γ jj 0 0 γ jj 0 1
Volga → Center 28.75 35 0 0 0 187.2
Center → Volga 0 0 75.84 261.52 0 0
Center → South 0 0 204.58 227.2 0 0
South → Center 50 50 0 0 132.04 152
South → Volga 50 0 0 69.52 94.6 0
Volga → South 0 15 166.18 0 0 0
Table 5. The values of primal variables u
ejj 0 t and dual variables η jj 0 t
Flow u
ejj 0 0 η jj 0 0 η jj 0 0
Volga → Center 7 0 0
Center → Volga 0 1.04 0
Center → South 0 1.54 0
South → Center 50 0 0.51
South → Volga 40 0 0
Volga → South 0 0 0
Table 6. The values of primal variables ajj 0 and dual variables ν jj 0
Connection ajj 0 ν jj 0
Volga – Center 35 133.20
Center – South 50 204.75
South – Volga 50 0
important of them.
First, it is necessary to say about the dual variables λjt . They are nodal prices. These
616 S. Gakh
prices vary in time. The concept of nodal prices is well studied. In the very beginning
the nodal prices were proposed and developed by Schweppe and his collaborators in [6].
For this example the equations (35)–(36) are wrote in the following form:
vj − λjt − µjt + µjt = 0 . (48)
Substitute the value of primal and the dual variables in the equation (48):
Node ”Volga” at 0 and 1 hours respectively:
640 − 1497.60 + 857.60 = 0;
960 − 1372.80 + 412.8 = 0 .
Node ”Center” at 0 and 1 hours respectively:
1536 − 1536 = 0;
1600 − 1600 = 0 .
Node ”South” at 0 and 1 hours respectively:
1280 − 1365.56 + 85.56 = 0;
1408 − 1408 = 0 .
The prices in the node ”Volga” at 0 and 1 hours are higher than unit variable costs.
Producers get producer surplus: 857.60 at 0 hour and 412.80 at 1 hour per unit. Also
producers get producer surplus in the node ”South” at 0 hour and it equals 85.56 per
unit. In the node ”Center” at 0 and 1 hours, in the node ”South” at 1 hour there is no
producer surplus, the electricity hour price equals unit variable costs.
Now we consider equations (39)–(40). They are wrote in the following form:
λjt − λj 0 t (1 − π) − γ jj 0 t + γ jj 0 t = 0 . (49)
Substitute the value of primal and the dual variables in the equation (49):
Flow from the node ”Volga” to the node ”Center” at 0 and 1 hours:
1497.60 − 1536(1 − 0.025) = 0;
1372.80 − 1600(1 − 0.025) = 0 .
Flow from the node ”South” to the node ”Center” at 0 and 1 hours:
1365.56 − 1536(1 − 0.025) + 132.04 = 0;
1408 − 1600(1 − 0.025) + 152 = 0 .
Flow from the node ”South” to the node ”Volga” at 0 hours:
1365.56 − 1497.60(1 − 0.025) + 94.60 = 0 .
Flow from the node ”Volga” to the node ”South” at 1 hours:
1372.80 − 1408(1 − 0.025) = 0 .
Dual Model of Power Market with Generation and Line Capacity Expansion 617
We can see that the price in the node where power come is higher than the price in
another node for the value of line losses. Moreover, if the line is used on maximum ca-
pacity the consumer pays for the line. When transferring power from the node ”South”
to the node ”Center” the consumers pay 132.04 at 0 hour and 152 at 1 hour per unit
for the use of line.
5 Conclusion
In conclusion it is possible to state the following:
1. The dual variables corresponding to constraints on the operating power balance of
power stations (11)–(12) are electricity hour prices in the model. These electricity
hour prices take into account line losses, and use of maximum line capacity.
2. More effective producers will get consumer surplus, which is defined by the dual
variables corresponding to the operating power µjist . Also it is necessary to point
that consumers don’t compensate fixed costs of power plants.
3. The example in this paper has small dimension and it is illustrative. It is necessary
to point that there was solved model of large dimension for the central part of
Russia, which united five regions: the North-West, Central, Volga, South and Ural.
Also there are 24 hour in a day, 4 seasons, working days and holidays, peak hours
in the model. The model not inclusive in this paper because of large dimension.
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