Dual decomposition approach for dynamic
spatial equilibrium models
E. Allevi(1) , A. Gnudi(2) , I.V. Konnov(3) , G. Oggioni(1)
(1)
University of Brescia, Italy
University of Bergamo, Italy
Kazan Federal University, Russia
(2)
(3)
Computational Management Science Conference
(CMS2017)
May 30-31, June 1, 2017
Bergamo
Dual decomposition approach for dynamic spatial equilibrium models
Our aim...
We develop a dynamic equilibrium model for a system of auction markets joined
by transmission lines subject to joint balance and capacity flows constraints. The
model involves commodity storage within a given time period.
We propose the single-level model to represent this dynamic system of markets,
which are typical for energy systems.
We construct a single-level variational inequality problem whose solution yields an
equilibrium trajectory of this system.
We apply a dual decomposition method to find its solution.
The case study is based on the Italian day-ahead electricity market.
Dual decomposition approach for dynamic spatial equilibrium models
Main References
Harker, P.T., editor. Spatial Price Equilibrium: Advances in Theory, Computation
and Application. Springer-Verlag, Berlin, 1985.
Nagurney, A. Network Economics: A Variational Inequality Approach. Kluwer,
Dordrecht, 1999.
Konnov, I.V. Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam,
2007a.
Konnov, I.V. On variational inequalities for auction market problems. Optimization
Letters, 1: 155–162, 2007.
Konnov, I.V. Decomposition approaches for constrained spatial auction market
problems. Networks and Spatial Economics, 9: 505–524, 2009.
Dual decomposition approach for dynamic spatial equilibrium models
Static spatial equilibrium problem
Let us consider a system of K markets of a homogeneous commodity, which are
joined in a network by transmission lines.
Let Ik and Jk denote index sets of sellers and buyers at the k -th market,
respectively.
The i-th seller proposes his/her offer volume xi from the conditions:
0 ≤ α0i ≤ xi ≤ βi0 ≤ +∞,
i ∈ Ik ,
The j-th buyer announces his/her bid volume yj from the conditions:
00
0 ≤ α00
j ≤ yj ≤ βj ≤ +∞,
j ∈ Jk .
Together with offer/bid volumes, the sellers and buyers announce their prices
gi , i ∈ Ik and hj , j ∈ Jk , respectively.
Dual decomposition approach for dynamic spatial equilibrium models
Auction market problem
We define the offer/bid volumes vectors at the k -th market:
x(k ) = (xi )i∈Ik
and y(k ) = (yj )j∈Jk ,
The feasible sets of offer/bid volumes at the k -th market are:
Y
Y
00
X(k ) =
[α0i , βi0 ] and Y(k ) =
[α00
j , βj ];
i∈Ik
j∈Jk
Since prices depend on volumes, we have:
gi = gi (x(k ) , y(k ) ), i ∈ Ik , and hj = hj (x(k ) , y(k ) ), j ∈ Jk .
Assumption 1 (A1): The functions gi and −hj are monotone and continuous.
Dual decomposition approach for dynamic spatial equilibrium models
Auction market problem (2)
∗ ∈X
∗
The k -th market problem then consists in finding vectors x(k
(k ) , y(k ) ∈ Y(k ) , and a
)
number pk∗ such that:

 ≥ pk∗
= pk∗
≤ pk∗
∗
∗
gi (x(k
) , y(k ) ) 
and

∗
 ≤ pk
= pk∗
≥ pk∗
∗
∗
hj (x(k
) , y(k ) ) 
if xi∗ = α0i ,
if xi∗ ∈ (α0i , βi0 ),
if xi∗ = βi0 ,
i ∈ Ik ,
(1)
if yj∗ = α00
j ,
00
if yj∗ ∈ (α00
j , βj ),
if yj∗ = βj00 ,
j ∈ Jk ,
(2)
subject to the market balance condition
X
X
xi∗ −
yj∗ − uk = 0.
i∈Ik
j∈Jk
where
Let uk denote the excess supply (excess demand) volume for the k -th market.
Dual decomposition approach for dynamic spatial equilibrium models
(3)
Dynamic spatial equilibrium problem
Let us consider the k -th market model for the dynamic case.
For simplicity, we consider a discrete time; i.e., we divide the time interval into
subintervals t = 1, 2, . . . , T .
Let:
fkl,t be the flow from market k to market l at time interval t;
vk ,t be the volume of the commodity stored in the k -th market at the time
intervals from (t − 1) to t;
pk ,t be the price of the commodity at the k -th site at time interval t.
In addition, the following capacity bounds hold:
fkl,t ∈ [0, akl ], akl ≥ 0, k , l = 1, . . . , n,
vk ,t ∈ [0, bk ], bk ≥ 0, k , . . . , n, t = 1, 2, . . . , T .
Dual decomposition approach for dynamic spatial equilibrium models
Dynamic spatial equilibrium problem (2)
If the vectors
f = (f11,1 , . . . , fKK ,T )T , v = (v1,1 , . . . , vK ,T )T , p = (p1,1 , . . . , pK ,T )T
are known, one can determine
ckl,t : the transportation cost of a unit commodity from the k -th to the l-th market at
the t-th time interval,
rk ,t : the cost of storing a unit commodity at the k -th market at the time intervals
from (t − 1)-th to t-th,
uk ,t : the excess supply (excess demand) of the k -th auction market at time
interval t.
Assumption 2 (A2): The functions ckl,t and rk ,t are monotone and continuous.
Dual decomposition approach for dynamic spatial equilibrium models
Dynamic spatial equilibrium problem (3)
Let us consider the k -th market at the t-th time interval.
Each parameter above, including offer/bid volumes, their bounds, auction prices,
and export values, now depends on time interval t.
We can define the offer/bid vectors:
x(k ,t) = (xi,t )i∈Ik
and y(k ,t) = (yj,t )j∈Jk ,
and the feasible sets of offer/bid volumes:
Y
Y
0
00
X(k ,t) =
[α0i,t , βi,t
] and Y(k ,t) =
[α00
j,t , βj,t ];
i∈Ik
j∈Jk
Let uk ,t denote the export volume.
Together with offer/bid volumes, the sellers and buyers announce their prices
gi,t = gi,t (x(k ,t) , y(k ,t) ), i ∈ Ik , hj,t = hj,t (x(k ,t) , y(k ,t) ), j ∈ Jk .
Assumption 1 (A1) also applies to gi and −hj .
Dual decomposition approach for dynamic spatial equilibrium models
Dynamic auction market problem
At the t-th time interval, the k -th market problem consists in finding vectors
∗
∗
x(k
∈ X(k ,t) , y(k
∈ Y(k ) , and a number pk∗,t such that:
,t)
,t)
and

∗
 ≥ pk ,t
∗
∗
∗
=
p
gi,t (x(k
,
y
)
k ,t
,t) (k ,t) 
≤ pk∗,t
∗ = α0 ,
if xi,t
i,t
∗ ∈ (α0 , β 0 ),
if xi,t
i,t
i,t
∗ = β0 ,
if xi,t
i,t
i ∈ Ik ,
(4)

∗
 ≤ pk ,t
= pk∗,t
≥ pk∗,t
∗ = α00 ,
if yj,t
j,t
∗ ∈ (α00 , β 00 ),
if yj,t
j,t
j,t
∗ = β 00 ,
if yj,t
j,t
j ∈ Jk ,
(5)
∗
∗
hj,t (x(k
,t) , y(k ,t) ) 
subject to the market balance condition
X
X
∗
∗
xi,t
−
yj,t
− uk ,t = 0;
i∈Ik
Dual decomposition approach for dynamic spatial equilibrium models
j∈Jk
(6)
Equilibrium conditions
We first write the volume balance condition at each node:
n
X
l=1
∗
−
fkl,t
n
X
flk∗,t − vk∗,t + vk∗,t+1 − uk ,t = 0,
l=1
(7)
k = 1, . . . , K ; t = 1, . . . , T ;
We define equilibrium conditions for costs and stored volumes as the following VI:
Find vk∗,t ∈ [0, bk ] such that
h
i
rk ,t (vk∗,t ) + pk∗,t−1 − pk∗,t (vk ,t − vk∗,t ) ≥ 0 ∀vk ,t ∈ [0, bk ],
(8)
k = 1, . . . , K ; t = 1, . . . , T ;
We define equilibrium conditions for network flows as the following VI: Find
∗ ∈ [0, a ] such that
fkl,t
kl
h
i
∗
∗
∗
ckl,t (fkl,t
) + pk∗,t − pl,t
(fkl,t − fkl,t
) ≥ 0 ∀fkl,t ∈ [0, akl ],
k , l = 1, . . . , K ; t = 1, . . . , T .
Dual decomposition approach for dynamic spatial equilibrium models
(9)
Equilibrium conditions (2)
∗ , y ∗ , f ∗ , v ∗ , p ∗ )}, k , l = 1, . . . , K , t = 1, . . . , T , is called
A trajectory {(x(k
k ,t
,t) (k ,t) kl,t k ,t
equilibrium if it satisfies conditions (4)–(9).
⇓
We reduce the problem of finding an equilibrium trajectory to a suitable VI.
⇓
Define the trajectory vectors
x = (x(k ,t) )k =1,...,K ;t=1,...,T , y = (y(k ,t) )k =1,...,K ;t=1,...,T ,
v = (vk ,t )k =1,...,K ;t=1,...,T , p = (pk ,t )k =1,...,K ;t=1,...,T ,
f = (fkl,t )k ,l=1,...,K ;t=1,...,T ;
and the corresponding feasible capacity sets
X =
V =
T Y
K
Y
t=1 k =1
T Q
K
Q
X(k ) ,
[0, bk ],
t=1 k =1
Dual decomposition approach for dynamic spatial equilibrium models
Y =
F =
T Y
K
Y
Y(k ) ,
t=1 k =1
T Q
K Q
K
Q
t=1 k =1 l=1
[0, akl ].
Equilibrium conditions (3)
We then define the general feasible set:
n

n
P
P


f
f
−
− vk ,t + vk ,t+1

lk
,t
kl,t


l=1
l=1

!
(x, y , v , f )
P
P
W =
∈
X
×
Y
×
V
×
F
−
xi,t −
yj = 0,




i∈Ik
j∈Jk ,t


k = 1, . . . , K ; t = 1, . . . , T







.
(10)






We define the problem of finding elements (x ∗ , y ∗ , v ∗ , f ∗ ) ∈ W such that


T X
K
X
X
X
∗
∗
∗
∗
∗
∗

gi,t (x(k ,t) , y(k ,t) )(xi,t − xi,t ) −
hj,t (x(k ,t) , y(k ,t) )(yj,t − yj,t )
t=1 k =1 i∈Ik
( K K
T
X
XX
+
t=1
j∈Jk
∗
ckl,t (fkl,t
)(fkl,t
k =1 l=1
−
∗
fkl,t
)
+
K
X
)
rk ,t (vk∗,t )(vk ,t − vk∗,t )
≥0
k =1
∀(x, y , v , f ) ∈ W .
(11)
Dual decomposition approach for dynamic spatial equilibrium models
Equilibrium conditions (4)
Theorem
(i) Let (x ∗ , y ∗ , v ∗ , f ∗ ) be a solution of VI (10)–(11). Then there exist a vector p∗ and
numbers uk∗,t , k = 1, . . . , n, t = 1, . . . , T , such that relations (4)–(6) for k = 1, . . . , n,
t = 1, . . . , T , and (7)–(9) are fulfilled.
(ii) If (x ∗ , y ∗ , v ∗ , f ∗ ) ∈ X × Y × V × F , a vector p∗ and numbers uk∗,t , k = 1, . . . , n,
t = 1, . . . , T , satisfy relations (4)–(6) for k = 1, . . . , n, t = 1, . . . , T , and (7)–(9), then
(x ∗ , y ∗ , v ∗ , f ∗ ) is a solution of VI (10)–(11).
This theorem implies that we can solve VI (10)–(11) instead of the systems (4)-(6),
(7)–(9).
Dual decomposition approach for dynamic spatial equilibrium models
Dual decomposition method
Under Assumptions A1 and A2, we can derive the equivalence to the (convex) optimization
problem:




T
n
X
X
X
X


min
µi,t (xi,t ) −
ηj,t (yj,t ) +
(x,y ,v ,f )∈X ×Y ×V ×F 
t=1
+
" n n
T
X
XX
t=1
subject to
k =1
i∈Ik
σkl,t (fkl,t ) +
k =1 l=1
n
X
(12)
j∈Jk
#
θk ,t (vk ,t )
k =1
(x, y , f , v ) ∈ W
where
0
µi,t : [α0i,t , βi,t
] → <,
i ∈ Ik ; t = 1, ..., T
00
ηi,t : [α00
i,t , βi,t ] → <,
i ∈ Ik ; t = 1, ..., T
σkl,t : [0, akl ] → <,
k , l = 1, ..., K ; t = 1, ..., T
θkl,t : [0, bk ] → <,
k = 1, ..., K ; t = 1, ..., T
are convex differentiable functions such that:
0
µi,t (xi,t ) = gi,t (xi,t ),
0
ηj,t (yj,t ) = hj,t (yj,t ),
Dual decomposition approach for dynamic spatial equilibrium models
0
σkl,t (fkl,t ) = ckl,t (fkl,t ),
0
θk ,t (vk ,t ) = rk ,t (vk ,t ).
Dual decomposition method (2)
(13)
maximize Ψ(p)
n
subject to p ∈ <
where
Ψ(p) =




T
K
X
X
X
X


µi,t (xi,t ) −
ηj,t (yj,t ) +
(x,y ,v ,f )∈X ×Y ×V ×F 
min
t=1
+
+
t=1 k =1

pk ,t 
n
X
l=1
i∈Ik
" K K
T
X
XX
t=1
T X
K
X
k =1
fkl,t −
K
X
k =1 l=1
!
flk ,t
l=1
σkl,t (fkl,t ) +
(14)
j∈Jk
K
X
#
θk ,t (vk ,t ) +
k =1



X
X
− vk ,t + vk ,t+1 − 
xi,t −
yj,t 

i∈Ik
j∈Jk
If the functions µi,t , ηj,t , σkl,t , and θk ,t are non strictly convex, it is possible to replace this problem
with a sequence of regularized problems, so that, at the s-iteration, we solve a dual regularized
problem.
Dual decomposition approach for dynamic spatial equilibrium models
Dual decomposition method (3)
(15)
maximize Ψs (p)
n
subject to p ∈ <
where



T
K
X
X
X
s−1 2


Ψs (p) =
min
µi,t (xi,t ) + 0.5λs (xi,t − xi,t )
+
(x,y ,v ,f )∈X ×Y ×V ×F 
t=1
−
k =1
(16)
i∈Ik

X
s−1 2 
+
ηj,t (yj,t ) − 0.5λs (yj,t − yj
)
j∈Jk
+
" K K
T
X
XX
t=1
+
s−1 2
σkl,t (fkl,t ) + 0.5λs (fkl,t − fkl,t )
k =1 l=1
T X
K
X
t=1 k =1

pk ,t 
+
K X
s−1 2
θk ,t (vk ,t ) + 0.5λs (vk ,t − vk ,t )
k =1
K
X
l=1
fkl,t −
K
X

!
flk ,t
− vk ,t + vk ,t+1 − 
l=1
Dual decomposition approach for dynamic spatial equilibrium models
X
i∈Ik
xi,t −
X
j∈Jk


yj,t  =

#
+


T
K
X
X
X


=
t=1
−
+
X
0
0
i∈Ik xi,t ∈[αi ,βi ]
max
00
00
j∈Jk yj,t ∈[αj ,βj ]
" K K
T
X
XX
t=1
+
k =1
K
X
k =1
vk ,t ∈[0,bk ]
min
s−1 2
µi,t (xi,t ) + 0.5λs (xi,t − xi,t ) − pk ,t xi,t
(17)
+

s−1 2
ηj,t (yj,t ) − 0.5λs (yj,t − yj,t ) − pk ,t yj,t  +
f
∈[0,akl ]
k =1 l=1 kl,t
min
min
s−1 2
σkl,t (fkl,t ) + 0.5λs (fkl,t − fkl,t ) + (pk ,t − pl,t )fkl,t
s−1 2
θk ,t (vk ,t ) + 0.5λs (vk ,t − vk ,t ) + (pk ,t−1 − pk ,t )vk ,t
Dual decomposition approach for dynamic spatial equilibrium models
#
+
Case study: the Italian electricity market
Auction model on the Italian electricity market: this
market is subdivided into fifteen zones
(i, j = 1, ..., 15) that are classified into three main
groups:
1 Foreign countries represented by France
(i = 1), Switzerland (i = 2), Austria (i = 3),
Slovenia (i = 4), and Greece (i = 15) to which
Italy is connected;
2
3
Geographical zones represented by North
(i = 5), Central North (i = 6), Central South
(i = 7), Sardinia (i = 8), South (i = 9), and
Sicily (i = 10);
Virtual zones represented by Rossano
(i = 11), Foggia (i = 12), Brindisi (i = 13), and
Priolo (i = 14) defining areas with limited
power production.
Dual decomposition approach for dynamic spatial equilibrium models
FRANCE
SWITZERLAND
AUSTRIA
SLOVENIA
NORTH
CENTRAL NORTH
SARDINIA
CENTRAL SOUTH
SOUTH
ROSSANO
FOGGIA
SICILY
BRINDISI
PRIOLO
GREECE
Case study: the Italian electricity market (2)
Time interval: a day partitioned into 24 hours (t = 1, ..., 24).
The day is divided into daytime (t = 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) and
nighttime (t = 1, 2, 3, 4, 5, 6, 19, 20, 21, 22, 23, 24).
Flows from site i to site j at time intervals from (1 − t)-th to t-th is limited by
transfer capacity.
The transfer capacity data are provided by Terna1 and by ENTSO-E2 .
Nine different technologies: wind, photovoltaic, run-of-river, geothermal, nuclear,
coal, CCGT, other gas and oil based power plants.
Each of these technologies is characterized by variable costs based on fuel and
CO2 emission costs.
1
2
See http://www.terna.it
See https://www.entsoe.eu
Dual decomposition approach for dynamic spatial equilibrium models
Case study: the EU-ETS
The European Union Emissions Trading System (EU-ETS) is a cap and trade
system that limits and imposes a price on CO2 emissions generated by specific
production installations.
All network zones, except for Switzerland (i = 2), are directly involved in the
EU-ETS.
Zone i = 2, corresponding to the Switzerland, is not involved in the EU-ETS, but it
applies an own ETS, denoted as Swiss ETS.
In our numerical experiments, we assume a CO2 price of 5 e/ton.
Dual decomposition approach for dynamic spatial equilibrium models
Numerical experiments
The traders’ bid function gi,t is defined by the following affine function:
g
g
gi,t = ai,t + bi,t · xi,t
g
g
Where the parameters ai,t and bi,t are estimated using reference values for
generation and variable costs;
We consider 4 electricity companies participating to the different auction
markets.
The electricity buyers’ bid function hj,t is defined by the following affine function:
h
h
hj,t = aj,t − bj,t · yj,t
h and b h are
where the reference demand and price parameters aj,t
j,t
estimated using data taken from the Italian Market Operator website;
We consider 1 consumer group per auction market.
Dual decomposition approach for dynamic spatial equilibrium models
Results
0
Variable xi,t ranges between 0 (αi,t
) and the maximum available capacity
0
in each market (βi,t ).
We consider two possible intervals for variable yj,t :
00
Case 1 → α00
j,t = 0 and βj,t = peak load in market k ;
00
00 = peak load in
Case 2 → αj,t = average demand in market k and βj,t
market k increased by 20%;
Case2
#itera3ons
20
20
18
18
16
16
Seconds(execu3on3me)
Seconds(execu3on3me)
Case1
#itera3ons
14
12
10
8
6
4
14
12
10
8
6
4
2
2
0
0
1
2
3
1
2
3
lambda=0.02
lambda=2
lambda=3
lambda=0.02
lambda=2
lambda=3
lambda=4
lambda=5
lambda=100
lambda=4
lambda=5
lambda=100
Dual decomposition approach for dynamic spatial equilibrium models
Final remarks and conclusions
We conduct a sensitivity analysis on parameter λs ;
The analysis shows that the proposed methods is quite efficient (low number of
iterations and low computation time);
We are testing this approach on the gas market.
Dual decomposition approach for dynamic spatial equilibrium models
Thank you for your attention!
Dual decomposition approach for dynamic spatial equilibrium models
Reference marginal cost
Reference generation
20.14
3.53
25.55
18.56
61.93
40.57
79.99
58.87
48.39
81.60
55.52
55.52
27.01
55.52
41.41
61642
7744
8199
1548
15409
2679
4833
1359
5106
1866
630
145
247
480
5755
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Table 8. Reference marginal production costs and reference generation level in zone i
Dual decomposition approach for dynamic spatial equilibrium models
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
g
a
i
1.83
0.32
2.32
1.69
5.63
3.69
7.27
5.35
4.40
7.42
5.05
5.05
2.46
5.05
3.76
g
g
b
i
0.000297
0.000414
0.002833
0.010901
0.003654
0.013767
0.015047
0.039371
0.008615
0.039750
0.080123
0.348133
0.099493
0.105097
0.006541
g
Table 9. Values of parameters ai and bi of the offer price function gi
Dual decomposition approach for dynamic spatial equilibrium models
MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
38256
37176
35052
33760
34206
36426
38960
42713
46285
48645
49786
50953
51556
50657
49816
48524
47365
46779
47831
46924
44600
45297
48412
46397
3891
3896
3823
3784
3731
3972
3980
3937
4297
4445
4670
4836
4630
4661
4564
4467
4361
4176
4139
3976
3919
4179
4171
3802
5292
5034
4926
4853
5134
5816
6926
7885
8420
8653
8866
9049
8637
8521
8416
8280
8128
8160
7946
7507
7416
7075
6726
6089
1027
1006
1139
1137
1157
1208
1200
1355
1390
1397
1399
1431
1462
1433
1430
1397
1412
1398
1424
1445
1468
1452
1359
1251
17996
17000
16328
16105
15955
16672
18258
20958
24201
26001
26764
27229
26587
26671
27157
27305
27209
26596
25376
24757
24561
24541
22940
21315
3459
3267
3138
3095
3066
3204
3509
4028
4651
4997
5144
5233
5110
5126
5219
5248
5229
5112
4877
4758
4720
4717
4409
4097
4945
4671
4486
4425
4384
4581
5017
5759
6650
7145
7354
7482
7306
7328
7462
7503
7476
7308
6973
6803
6749
6743
6303
5857
1113
1051
1010
996
987
1031
1129
1296
1496
1608
1655
1684
1644
1649
1679
1688
1682
1644
1569
1531
1519
1517
1418
1318
2958
2794
2684
2647
2622
2740
3001
3445
3978
4273
4399
4475
4370
4383
4463
4488
4472
4371
4171
4069
4037
4033
3770
3503
2019
1907
1831
1806
1790
1870
2048
2351
2715
2916
3002
3054
2982
2992
3046
3063
3052
2983
2846
2777
2755
2753
2573
2391
46
44
42
41
41
43
47
54
62
67
69
70
68
69
70
70
70
68
65
64
63
63
59
55
205
194
186
184
182
190
208
239
276
296
305
310
303
304
310
311
310
303
289
282
280
280
261
243
217
205
197
194
192
201
220
253
292
313
322
328
320
321
327
329
328
320
306
298
296
296
276
257
142
134
129
127
126
131
144
165
191
205
211
214
209
210
214
215
214
209
200
195
193
193
181
168
7097
6741
6475
6314
6263
6223
6614
7328
7972
8293
8549
8833
9123
9187
8975
8825
8792
8675
8546
8519
8825
8502
8183
7792
Table 10. Reference demand per time segment t and zone i
Dual decomposition approach for dynamic spatial equilibrium models
e/MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Reference price
62.72
48.10
47.88
41.80
38.00
42.00
71.53
83.98
90.71
90.92
87.94
85.84
78.34
77.38
81.10
85.12
91.76
119.87
119.55
93.39
87.19
83.25
79.91
71.89
Table 11. Reference price per time segment t
Dual decomposition approach for dynamic spatial equilibrium models
MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2415
2358
2445
2540
2595
2540
2513
2745
2755
2754
2755
2755
2730
2680
2755
2755
2755
2755
2729
2754
2743
2745
2745
2540
2962
3026
3172
3266
3159
3060
3195
4194
4077
4085
4059
4077
3638
3703
3862
4160
4185
4252
4177
4227
4185
4182
3850
3316
296
296
294
296
296
296
291
311
306
306
311
311
315
316
316
316
316
316
316
316
316
316
316
296
729
689
750
770
698
709
583
796
751
758
665
592
588
587
580
641
627
681
673
615
757
570
587
671
9237
8843
8724
8339
8417
9371
10881
14020
17261
18056
17832
17610
15421
15218
15893
16843
17755
19667
19822
18993
17121
14856
12256
10259
1838
1831
1731
1731
1742
1746
1888
2154
2429
2724
2909
2955
2882
2822
2799
2557
2448
2468
2458
2417
2240
2173
2095
1879
3331
3331
3298
3270
3229
3210
3231
3666
4764
5122
5037
5058
4802
4954
4743
4743
4806
4476
4792
4453
4255
3709
3640
3612
1361
1372
1365
1348
1330
1328
1380
1371
1398
1456
1506
1572
1637
1632
1583
1503
1450
1580
1530
1231
1228
1233
1242
1238
1490
1512
1556
1544
1495
1463
1467
1490
1812
2423
2838
3101
3241
2953
2527
2067
1663
1497
1543
1439
1373
1372
1320
1363
1473
1400
1333
1392
1417
1393
1548
1865
2180
2310
2420
2393
2376
2298
2343
2342
2350
2327
2282
2301
2231
2210
1963
1659
930
945
962
981
992
1014
1031
1123
1195
1195
1195
1189
1183
1181
1178
967
1136
1205
1203
1191
1113
1037
778
782
317
316
317
316
307
310
309
374
374
309
310
312
312
313
315
314
310
681
748
876
511
309
310
311
2648
2162
1930
1936
1965
2485
2618
2877
2635
2216
1913
1909
1905
1905
1906
2242
2389
2312
2283
2311
2263
2574
2222
2326
370
235
220
86
89
219
370
371
372
370
369
370
370
370
369
367
366
543
732
734
718
549
370
370
450
306
306
306
306
306
500
500
500
500
334
300
300
360
421
400
360
345
370
341
289
354
468
498
Table 12. Values of parameters α0i,t corresponding to the reference offer values submitted
by market i in time t to the Italian day-ahead market
Dual decomposition approach for dynamic spatial equilibrium models
MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
73130
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
6941
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
10435
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
1860
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
28908
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
5026
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
9066
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
2549
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
9579
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
3501
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
1182
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
272
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
463
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
901
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
7381
0
Table 13. Values of parameters βi,t
corresponding
to the total available capacity in market i and in time t
Dual decomposition approach for dynamic spatial equilibrium models
MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1047
763
713
703
603
748
197
0
200
200
200
200
460
831
200
200
200
200
200
200
0
0
99
573
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
209
169
230
250
178
189
269
326
140
138
211
213
190
169
148
107
131
165
167
143
235
244
240
219
15822
15841
16082
16028
16049
16682
18198
21834
24530
25228
25042
25031
22390
22470
23417
23727
23524
24138
24177
22894
21197
19713
17667
16112
2745
2537
2459
2436
2446
2604
2962
3654
4275
4525
4520
4495
4098
3889
4151
4287
4439
4662
4681
4495
4130
3798
3329
2967
4271
3927
3723
3642
3655
3874
4418
5351
6156
6686
6647
6570
6467
6203
6279
6399
6765
7344
7558
7501
7041
6451
5651
4901
1235
1172
1135
1130
1135
1167
1273
1406
1505
1550
1520
1487
1476
1428
1412
1428
1472
1567
1645
1650
1621
1533
1421
1305
2206
2044
1963
1930
1946
2051
2324
2803
3181
3289
3230
3187
3189
3023
2992
3075
3300
3786
3871
3836
3691
3345
2978
2564
2143
1935
1853
1777
1806
1912
2018
2336
2652
2781
2890
2862
2846
2768
2812
2809
2816
2970
3114
3135
3048
2859
2633
2329
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
20
85
100
85
85
85
0
0
35
55
60
324
451
389
55
55
135
140
110
210
235
100
0
0
Table 14. Values of parameters α00
j,t corresponding to reference bid
demand values submitted by market i and time t to the Italian day-ahead market
Dual decomposition approach for dynamic spatial equilibrium models
MWh
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
55719
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7372
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
7885
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
1449
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
28240
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
5487
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
8258
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
2256
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
4446
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
3445
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
5927
00
Table 15. Values of parameters βj,t
corresponding to the average
power demand in market i and time t
Dual decomposition approach for dynamic spatial equilibrium models