Econometric modelling and forecasting of intraday electricity prices

E y y Xv:1812.09081v1 [q-.st] 21 D 2018 M Nw Uvy Duu-E F Z Uvy Duu-E D 24, 2018 A I w w y ID 3 -P G Iy Cuu Ey M u. A uv u uy qu-uy u y. W u qu u-- vy - uy. T w u. F u v w G Iy Cuu Ey M y ID 3 -P vu. T uy v. Kyw:, y, y,, v 1

1 Iu C v w- w y u Gy qu x, w w u, y u y, u w. Iu y y y uuy u 30 u vy w u 5 u vy w v z. T w w uy y w u. W y (EPF) y- wy,, u w, y w y y. T, A. (2017) y M. (2016) y y u u w. B S. Ry Uw. (2018) u y y, u. Ty u vy- ID 3 -P x uy u EPEX G Iy Cuu. T y u y y EPF. Z (2017), P. (2016) Gzáz-A Zu (2015) v u y. O, K Pv (2017) Aï. (2016) u vu y. T w y y. W w u v uy qu-uy Iy Cuu u. T, w u u ID 3 -P x. W uv, w w-w qu y, W (2014). W uz uv, u w u uy Iy Cuu. Ou v, v. Ay, x w u Dy-A Iy Au. L u, w 2

u y u,.. w. I x w y x y u u ID 3 - P x. W y yz ID 3 w y u, x ID y. I w u v z, w Uw. (2018), w qu,.. u () T (1996) uz Zu (2005). T w u. I u w uy, uz u, D M (1995) u Z (2016). I w u, w u - u. W w v. W w w u. 2 M D T G Iy Cuu M vy y 15:00 uy u 16:00 qu-uy u w y. T Iy Cuu y Dy-A Au Iy Au, w y 12:00 15:00, vy, EPEX (2018). F vu Fu 1. W y u, u y G y, w, u EXAA u. F V (2017). I Fw Dy-A P y w, u y w, w Iy P. C u u y. T vy y vu y vu,. T u u. Tu, v u w u y EPEX: P Ix, ID 3 -P ID 1 -P. T u u y, v u 24 vu uy u 96 vu qu-uy u. P Ix 3

vu-w v w, ID 3 - P vu-w v u 3 u ID 1 -P u ID 3 -P v u 3 u. T, v z 30 u vy, u u. 2.1 ID 3 -P I w w u u ID 3 -P, u x. I v uy G Iy C/F Fuu. W u v v G y, EEX (2018). L (, ) vy u y. By T, 3 = [(, ) 3, (, ) 0.5) w w 3 u 30 u vy. T EPEX ID 3 -P w. EPEX ID, 3 := 1 T, 3 T V,, V T, 3 T,, P,, (1) w T, u y, V, P, vu - w T, 3 T, vy. F u ID 3 -P - u, w - -,.. w uy, xu. I w T, 3, v ww x w E Dy-A Au Iy Au E w z Dvy Qu-uy Iy Cuu uy Iy Cuu 1, 1, 1,,,, 12:00 15:00 16:00 30. 5. Fu 1: Dy u G Ey M. y vy u vy. 4

u y. I, quuy u v Iy Au vu u uy u v Dy-A Au u u. I u u y w w u EPEX ID 3. Uuy,, v, w w -. I, w, w Iy Cuu y w vu y. S w vy -, w w u v. P u u y EPEX u v u w u. T w vu u u u, w x ID y u w. xid, y := 1 T V,, x,y T, V, T, x,y T, w T, x,y = [(, ) x y, (, ) x), x 0 y > 0. P,, (2) F u x ID y w u y EPEX u, u w vu T, x,y. T y, x v ww w, w x ID y u T, x,y. I, w u Iy Au Dy-A Au vu, y EPEX ID 3. O w (2) EPEX ID 3 = 0.5 ID 2.5 EPEX ID 1 = 0.5 ID 0.5. T wu y, w v. Tu u u y w ID 3, w w v 0.5 ID 2.5 u. L u x ID y - w vy y. T w T, x,y T, x,y = T, x,y = [(, ) x y, (, ) x ) (3) 5

B 1, 15:00 / 16:00 EPEX ID 3 = 0.5 ID 2.5 2.5ID 0.5 1.5 ID 1 0.5 ID 1... 2.5ID 0.5 2 ID 0.5 1.5 ID 0.5 1 ID 0.5 0.5 ID 0.5, 3, 2.5, 2, 1.5, 1, 0.5 Dvy, Fu 2: Iu x ID y u u x ID y. y vy u vy. {0, 1,..., J}, w x 0 + y 0 = x + y x J = x. T w V, x,y = T, V, x,y T, xid, y = x ID, y V, x,y V, x,y, (4). T u Ax. T y uu u z u u w x ID y x y. A x w Fu 2. Nuy, w T, x,y quy w u u x ID y u. 2.2 Dv I u u y w u Iy Cuu. T uy qu-uy u, y 01.01.2015 29.09.2018. I, w u Dy- A Au Iy Au x. Fu 3 w u y vy. L u, T u 6+06 4+06 2+06 0+00 Puy uy qu uy >10 9 8 7 6 5 4 3 2 1 0 T vy (u) Fu 3: Du y v vy. 6

0 0.020 Dy 250 0.000 ID3 P () 500 200 250 2015 2016 2017 100 0 2018 100 200 ID3 P () T Dy 250 0 0.000 0.015 ID3 P (q) 500 250 200 2015 2016 2017 0 2018 100 200 300 400 ID3 P (q) T 0 0.02 Dy 250 0.00 DA P () 500 250 100 2015 2016 2017 50 2018 0 50 100 150 DA P () T 0 0.020 Dy 250 0.000 IA P (q) 500 250 100 2015 2016 2017 0 100 200 300 400 2018 IA P (q) T Fu 4: ID3 -P uy u (), ID3 -P qu-uy u ( ), Dy-A P ( ) Iy Au P () v. R ww. T w y. y u vy u. T u v w qu-uy u. Ov 70% uy v 80% qu-uy u ID3. Tu, ID3 u y. Fu 4 ID3 -P, Dy-A P Iy Au P uy 7

0 20 40 00:00 06:00 12:00 18:00 00:00 u ID3 P () Wy My Tuy Wy Tuy Fy Suy Suy () 0 20 40 00:00 06:00 12:00 18:00 00:00 u ID3 P (q) () 0 20 40 00:00 06:00 12:00 18:00 00:00 u DA P () Wy My Tuy Wy Tuy Fy Suy Suy () 0 20 40 00:00 06:00 12:00 18:00 00:00 u IA P (q) () Fu 5: Wy : () ID 3 -P uy u, () ID 3 -P qu-uy u, () Dy-A P () Iy Au P. qu-uy u v. L u, v u u u qu. S, qu-uy u x v u quy uy u. T 8

vu v v Fu 5. W wy : ID 3 -P uy, Dy-A P Iy Au P. L u, Fu 5 Fu 5. T x w, w y x y K Pv (2017). B vu ID 3 uy u v. I w, wy, y y y-. T w- u,.. Z. (2015). O Fu 5 w v v v y Tuy Fy, uy. T My vy w Tuy Fy, u. Auy y-, w v w, w, w Suy Suy y v w Suy. 3 M 3.1 D T ID 3 x vy x, w Fu 4. T y u. Uw. (2018) v w u v z u quy. Tu,, w y z v z. T w y u y. M z P, v y u, = 1 MAD(P )/z 0.75 (P, M(P )), (5) w M(P ) P, D = 365-y, MAD(P ) u v u z 0.75 75% qu u. w u MAD v y z 0.75 u y y v. W u z u u, w uu w w vy. 9

T y () v y u ( ) Y, = (, ) =, + 2, + 1, (6) w z. I w w w y v y, wu u. S G y w v,. T vu vu, u v v vu. Tu, v u vy. I Uw. (2018) w,, quy, y w v z. T y w - w vu uu. T, Uw. (2018), (E(Y, )) E((Y, )) = E(, ). I u (.. Uw., 2018; Z W, 2018) u, vu Y, 0, w xy. I w w : w y wy,.. w u Ê(P, ) = Ê(,) + â = (Ê(Y,)) + â, (7) w Ê(Y,) Y,, â u MAD qu (5), vy. I w wy Ê(P, ) = Ê(,) + â = (x) F Y, + â, (8) w F Y, uuv u u Y,. W uuv u u y F Y, () = 1 1 D = D+ 1 E(Y,, w ε )+ ε,, - u. T, w (8) Ê(P, ) = (x) F Y, + â = 1 D 3.2 E qu 1 = D+ ) (Ê(Y, ) + ε, + â. (9) I w w y, u w uz 3 : y qu (OLS), u () 10

, w. W u OLS y w vy, w w x, vy u. T OLS, w-w, w u u. T, w w u y T (1996), uz qu. I u u v,.. y Z (2016), Uw W (2018) Uw. (2018). T y w y w y. L u u w OLS w Y, = X,β + ε,, (10) w X, v u β v. L u, w z u X, w u MAD u xu v, qu. T u uu v, u v uy v. S qu qu y z,.. w 0 v qu 1, w u w Y, = X, β + ε,. (11) W u v. v β, w y u β (10) y. L z u L 1 y u. L D u v y. T β v y { D } β = (Y, X β,β) 2 + λ β, (12) =1 =1 w λ u u. T, w w u y Zu (2005),, v. 11

T y L 1 L 2. T β { D β = β =1 (Y, X,β) 2 + αλ =1 β + 1 α λ 2 =1 β 2 v y }, (13) w α x. L u w α = 1, β. Suquy, w x α 0.5, u vy. A u λ. T vu, v u, u x. I u y, u w uz y Z (2016). T y, vy, w uz x Λ = {λ = 2 G}, w G qu -15 4 100 u-- w u λ. I w u λ Λ z By (BIC). A BIC vv, u - w. F w u R v y F. (2010). 3.3 Fu A w, u u u v, w u 3 u 15 u vy. T, w w vu ID 3 u v. F u w u u ID, 3 = 24 { 1,0,1} =1 x I (,) + + 96 β (1),,x { 1,0,1} =1 x I Q (,) 14 24 = 1 =1 β (5), DA, + x ID, 0.25 + β (3),,x 14 14 24 =2 Q x ID, 0.25 + 96 = 1 =1 =1 =2 β (2), 14 96 β (6), IA, + =1 7 =1 ID, 3 β (4), Q ID, 3 β (7) DW + ε,, (14) 12

w I (, ) = {0, 0.25, 0.5,..., (, ) ( ) 0.25, (, ) ( )} {, Q} () uy y. M (14) 7, xu. T x ID 0.25 vu w. T y, w, y Fu 2, u w u, 15 u. W uy u vu y, u y x y, w y u y u,.. 15:00 u y. Iu w Fu 6. T u w ID 3 uy u w vy 20:00 v vu v x ID 0.25. W w u, w y u. W -, w y. I x w wu 16:45, 3 u 15 u vy. T w - v. Evy,, w u. L u x w y u. Tu, w u vu-w w y. L u qu, y w vu u w vy 20:00, w 3.25 ID 0.25. T (14) vu ID 3 uy u y w w w w y u y. T vu uy u w wy -wy. T u qu-uy quv, vy. T Dy-A Au u. I y- uy u w w, u y w y, y y u. T x Iy Au quv v y w u. E u u w vy u, y x ID 0.25. Du, (14) 15250 xy v ( u w vy 00:00) 24834 13

Fu 6: C w ID3 uy u w vy 20:00 xid0.25 uy u w vy Y-x. T X-x y vu x. E. 14

( qu-uy u w vy 23:45). Tu, u w uz. W x u,.. 3.25 ID, 0.25 v, w w vu. T w wy. T wy w z z β () 0,,3.25, w {1, 3}, uy. T wy w x 1,.. β () 0,,3.25 = 1. T wy w. T uz, w (14) 3 wy, u 2 2 w. T v u 12 v. W v w FI.X.Y.Z, w FI u, X {, }, Y {,, x} wy Z {IC, C} w. F, FI...C u u w y y -. 3.4 B T, w uz y-,.. ÎD, 3 = DA, ÎD, 3 = IA,, (15) uy. W Nv.DA. T u y y- y. I y u y y Uw. (2018). T w, u vy uv. T, w y 15-u u,.. ÎD, 3 = 3.25 ID, 0.25. (16) B Fu 6 w x w Nv.MR1. T. W u 2.5 u 15 u. W 15

y Nv.MR2 u v y ÎD, 3 = 3.25 ID, 2.5. (17) T u ARX y Uw. (2018), w w x DW, Z W (2018) v y u ID, 3 =β 1 ID 1 ( 4),( 4) S+1 3 + β 2 ID 1, 3 + β 3 ID 2, 3 + β 4 ID 7, 3 + β 5 3.25ID, 0.25 7 + β 6 DA, + β 6+ DW + ε,, =1 (18) w S u u v y. T (18) uy u, qu-uy u w w DA, w IA,. L u ID 1 ( 4),( 4) S+1 3 v ID 3 vu. ID 1, 3, ID 2, 3 ID 7, 3 u uv vu y,.. u yy, w y w. T w u ARX u y Uw. (2018) vu. W v ARX. I u u, w y - - w w. W y ARX., ARX.IC ARX.C, vy. T, u y Uw. (2018),.. LASSO(λ 7 ), w λ 7 = 0.01. T v qu-uy u u. T Uw. (2018) w y uy u, qu-uy. S qu y u, u w uy, w,. T u w ID, 3 =, 4,= 7, +, 13,= 7, β (1), ID, 3 + β (4) Q, ID, 3 +,+8,= 7,,+32,= 7, β (2), DA, + 7 =1 β (5), IA, + ε,. β (3) DW (19) 16

T wu, u v. W L.AR. 4 F Suy Evu S w w, w uz ww. T y y u. F u vu w D = 365-y ww z. T w Fu 4. I u w y y uv, w u u w 120 (24 uy 96 qu-uy u). T vy,.. w w ID 3 -P 3 u 15 u vy u. T u w 01.01.2015 31.12.2015 w ID 3 -P x y. T w v u ww w y y x u u--. Ou u-- 01.01.2016 29.09.2018, w v u N = 1003 y u. W uz u (MAE) qu (RMSE) u. RMSE qu u, y v u. Tu, w y MAE, w u, u u, w w. MAE RMSE v y MAE = 1 S N RMSE = 1 S N N =1 S ID D+, 3 ÎDD+, 3 =1 N =1 S ID D+, 3 ÎDD+, 3 =1, (20) 2. (21) L u, w vu y uy qu-uy u. Mv, u u v y, w u MAE RMSE u. W y MAE 17

RMSE. Ty w MAE = 1 N RMSE = 1 N N ID D+, 3 ÎDD+, 3 =1 N ID D+, 3 ÎDD+, 3 =1, (22) 2. (23) RMSE MAE wy u EPF u v,.. y Z (2016) y Uw. (2018). wv, w y u u. T, w u D M (1995), w A B. I w w u uv v DM Z W (2018). T y EPF u, w DM y u, W (2014). Muv DM u y, u S- v y, w S {24, 96} uy qu-uy u, vy. T, ε A, = [ ε A,,1, ε A,,2,..., ε A,,S ] ε B, = [ ε B,,1, ε B,,2,..., ε B,,S ] v u-- y A B, vy. T uv A,B, = ε A, ε B, (24),.. ε A, = ( S =1 ε A,, ) 1/, w {1, 2}. F w u -vu w - DM. T w u y 0 : E( A,B, ) 0,.. u B y A. T w v u y 0 R : E( A,B, ) 0,.. u A y B. L u y, w u w : 1 2. Nuy, w u v y. F u u (14), w y DM. W y uy v uz. 18

A u z β, w uz u uz Z (2016) ι, = β, =1 β,. (25) Nuy, 0 ι, 1 =1 ι, = 1. T ι,, v ID 3 -P u. W vu ι, y y u- β u ww. 5 Ru Du 5.1 F vu I T 1, w MAE RMSE vu. W u y uy qu-uy u. Fu 7 8 -vu DM uy qu-uy u, vy. T u uy u qu u. Nv (16) v w MAE RMSE. I y y, DM u w Fu 7. L u, Nv.MR1 ID 3 w w-v 15 u. T y, -u, y. T vu w u y Z. (2015) G-Au y-. T, y u v EXAA u EPEX Dy-A Au. T v EXAA y u EXAA, w u u 1 u 40 u u EPEX Dy-A Au. T u u w, y v, w z u vu v x 1. O, FI...C 19

MAE RMSE MAE Q RMSE Q Nv.DA 5.004 7.965 7.643 11.701 Nv.MR1 3.334 5.278 7.706 11.714 Nv.MR2 3.501 5.588 7.51 11.434 ARX. 3.503 5.442 6.977 10.529 ARX.IC 3.442 5.493 6.858 10.627 ARX.C 3.436 5.422 6.896 10.462 L.AR 4.695 7.289 7.76 11.715 FI...IC 3.372 5.415 6.992 10.859 FI...C 3.353 5.319 6.871 10.513 FI..x.IC 3.387 5.333 7.599 11.566 FI..x.C 3.393 5.334 7.579 11.533 FI...IC 3.728 6.249 7.025 11.224 FI...C 3.575 5.983 6.718 10.685 FI...IC 3.376 5.42 7.346 11.364 FI...C 3.356 5.324 7.234 11.019 FI..x.IC 3.447 5.391 7.731 11.722 FI..x.C 3.453 5.392 7.723 11.704 FI...IC 3.663 6.052 7.044 11.143 FI...C 3.547 5.832 6.81 10.7 T 1: MAE RMSE vu. B vu w u. u DM. T x y w, w Nv.DA L.AR w. L u, y, vu, u u. I w, wy w u w y, wy. Fu 9 w uy u y v y. W y, Nv.MR1, w vy u. T, u vu, u z u, v u, u y 20

Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% vu Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% vu () () Fu 7: Ru D-M uy u. () -vu 2, () vu 1. T u u -vu. T y z ( ), w X-x () Y-x (w). w vu vu Nv.MR1. A y u Fu 9. Mv, u u y u, y. T u y qu-uy u. B T 1 Nv.MR1. A DM u, Fu 8,. T u, qu-uy y uv uu u. I MAE w v w u u w y y -,.. FI...C. I RMSE u u - y x. B Fu 8,, w y u, FI...C. I, y w FI...C, FI...C ARX, 21

Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% vu Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C 10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% vu () () Fu 8: Ru D-M qu-uy u. () -vu 2, () vu 1. T u u -vu. T y z ( ), w X-x () Y-x (w). w w v, L.AR u, v vu x 1. Fu 10 w qu-uy u v y. L u, u x, Fu 9. T vu x w, qu-uy Fu 5. I w. C u w qu-uy uy u. T y vy qu-uy, y u u. A y w quy 3 u 15 u vy. Fu 3 w y vy u. T y x Nv.MR1. 22

RMSE 10 8 6 4 0 5 10 15 20 B vy Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C MAE 5 4 3 0 5 10 15 20 B vy Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C () () Fu 9: P u uy u RMSE 18 15 12 9 0 25 50 75 B vy Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C MAE 10 9 8 7 6 5 0 25 50 75 B vy Nv.DA Nv.MR1 Nv.MR2 ARX. ARX.IC ARX.C L.AR FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C FI...IC FI...C FI..x.IC FI..x.C FI...IC FI...C () () Fu 10: P u qu-uy u 5.2 V W v y u uy u,.. FI...C qu-uy u,.. FI...C. T 2 w u v FI...C uy u. L u, vy u vu 23

00:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 I: 1 I: 2 I: 3 I: 4 3.25ID,00:00 0.25 (96.8) 3.25ID,01:00 0.25 (98.5) 1.25ID 1,22:00 0.25 (0.7) Q 0.5 ID 1,22:15 0.25 (0.4) 0.25ID 1,21:00 0.25 (0.4) 3.25ID,02:00 0.25 (99.9) IA 3,23:15 (0) DA 13,06:00 (0) 3.25ID,03:00 0.25 (99.9) 3.25ID,04:00 0.25 (99.9) 3.25ID,05:00 0.25 (100) 3.25ID,06:00 0.25 (99.9) 3.25ID,07:00 0.25 (100) 3.25ID,08:00 0.25 (99.6) 3.25ID,09:00 0.25 (99.7) 3.25ID,10:00 0.25 (99.9) 3.25ID,11:00 0.25 (99.8) 3.25ID,12:00 0.25 (100) 3.25ID,13:00 0.25 (99.9) 3.25ID,14:00 0.25 (100) Q 1.75 ID 1,19:30 Q 3.5 ID 1,18:45 0.25 (0) 0.25 (0.1) Q ID 7,21:00 3 (0) Su (0) Q 12.75 ID 1,23:15 0.25 (0) Q ID 9,19:00 3 (0) Q 0 ID,02:15 0.25 (0.3) Q ID 6,21:30 3 (0.2) Q ID 10,11:15 2.25ID 1,23:00 0.25 (0.3) Q 0.75 ID 1,21:15 0.25 (0.2) IA 2,08:00 (0.2) Q 0 ID 1,07:00 0.25 (0) Q 26.5 ID 1,18:45 0.25 (0) Q ID 10,19:30 3 (0) Q 8.5 ID,02:30 0.25 (0) Q 23.25 ID,20:30 0.25 (0) Q ID 3,22:15 3 (0.1) 3 (0) DA 2,12:00 (0) Q 3 ID,00:00 0.25 (0.1) Q ID 9,00:15 3 (0) Q ID 9,09:30 3 (0) Q 8 ID 1,21:15 0.25 (0) Q 0.75 ID 1,13:15 Q 17.25 ID 1,21:30 0.25 (0) Q 12.75 ID 1,06:45 0.25 (0) Q 10.25 ID,04:15 0.25 (0) Q ID 4,07:30 3 (0) Q 14.25 ID 1,23:15 0.25 (0) Q ID 9,05:15 3 (0) Q ID 5,08:00 3 (0) Q 30.5 ID,22:45 0.25 (0) Q ID 10,21:30 3 (0) 0.25 (0) IA 8,17:00 (0) Q ID 9,07:45 3 (0) Q ID 9,07:15 3 (0) 3.25ID,15:00 0.25 (99.7) IA 9,06:00 (0.1) IA 9,05:15 (0.1) 3.25ID,16:00 0.25 (99.9) IA 4,18:00 (0.1) IA 9,00:45 (0) 3.25ID,17:00 0.25 (99.8) 3.25ID,18:00 0.25 (99.9) 3.25ID,19:00 0.25 (99.9) 3.25ID,20:00 0.25 (100) Q 5.25 ID,06:15 0.25 (0.2) Q 2 ID 1,16:45 0.25 (0) Q 6.75 ID 1,14:00 0.25 (0.1) Q 0 ID 1,15:15 Q 0 ID,00:15 0.25 (0) Q 0 ID,00:15 0.25 (0) Q 25.75 ID 1,23:45 0.25 (0) Q ID 9,09:45 3 (0) Q ID 2,06:45 3 (0) Q ID 10,20:45 3 (0) Q ID 4,13:45 3 (0) 0.25 (0) IA 8,04:45 (0) Q 0.25 ID 1,15:30 0.25 (0) Q 12.75 ID,18:00 0.25 (0) Q ID 3,16:45 3 (0) IA 5,18:00 (0) IA 8,23:00 (0) 3.25ID,21:00 0.25 (99.6) IA 10,18:00 (0.2) IA 1,07:00 (0.2) IA 3,18:00 (0) 3.25ID,22:00 0.25 (99.7) 3.25ID,23:00 0.25 (100) Q 25.5 ID1,20:00 0.25 (0.2) Q 2.5 ID,19:45 0.25 (0) IA 1,12:00 (0) T 2: M v FI...C uy u 99%. T x wy y y w, Nv.MR1. T y,, w, vu. T v, y vu vy. Oy w u, w, w v u vu y y. O, u w vy 23:00 y v,, 24

vu. T FI...C qu-uy u u Ax. T 3 w y- u Iy Au vu vy v v. T, u T 4 5. T vu u vu uy u v y u. T u, vu v v, v u, uy u. 6 Suy Cu W u y uy G Iy Cuu M. W uz w, u uy u w-w qu. W w v u. W y y uy quuy u. T u uy u u, w,, w w w. T, v vu. T vy w G-Au EXAA EPEX y-, x y Z. (2015). O, u qu-uy u v w, v. F vu v y u. T y w u y, w uy u. I qu-uy u u u w y y. V y w, v : Iy Au, vu u vu uy u. Fuu. Ly v u. T x u w u 25

u v,.. w u w. A, u vy y. I, u w y y u u, u. Aw T w y u y G R Fu (DFG, Gy) N S C (NCN, P) u BEETOVEN. 2016/23/G/S4/01005. Ax T x ID y w-v vy y P. L u T, x,y T, x,y = T, x,y = [(, ) x y, (, ) x ), w {0, 1,..., J}, x 0 + y 0 = x + y x J = x. T w V, x,y = T, x,y T, V,. xid, y = x ID, y V, x,y V, x,y, (26) P. T v y, w y v T, x,y 26

(2) x ID y. xid, y := = 1 T V,, x,y T, 1 ( V, T, x,y ) T, V, T, x,y T, P, V, ( T, x,y ) T, 1 = (T, x,y T, ) V, (T, 1 = T, x,y T V,, T, 1 x =,y = V, x,y 1 V, x,y V, V, x,y V, x,y x ID, y = x,y T, ) V, x,y T, V, T, x,y T, x ID, P, V, P, P, P, y V, x,y. V, x,y (27) Qu-uy u v I w v FI...C qu-uy u. 27

I: 1 I: 2 I: 3 I: 4 00:00 IA,00:00 (40.1) 00:15 00:30 1 ID 1,22:00 0.25 (11.8) 2.75ID,00:00 0.25 (14.3) 00:45 IA,00:45 (39.6) 01:00 IA,01:00 (25.1) 01:15 01:30 01:45 02:00 02:15 02:30 2.25ID,00:00 0.25 (42.7) 2.75ID,01:00 0.25 (25.7) Q 3.25 ID,00:00 0.25 (13.9) Q 0.5 ID 1,21:15 0.25 (8.9) 3 ID,00:00 0.25 (11.7) IA,00:15 (11.3) 4 ID,01:00 0.25 (12.9) 2.5ID,00:00 0.25 (11.6) 2.25ID,00:00 0.25 (11.3) 3 ID,01:00 0.25 (13.2) 2.25ID,00:00 0.25 (10.1) 2.5ID,01:00 0.25 (32.8) IA,01:45 (31.9) 1.25ID,00:00 0.25 (17.9) IA,02:00 (17.2) 1 ID,00:00 0.25 (19.5) 1.75ID,01:00 0.25 (26.9) 02:45 IA,02:45 (13.6) 03:00 IA,03:00 (21.9) 03:15 03:30 1 ID,01:00 0.25 (32.4) 2.75ID,03:00 0.25 (26.7) 03:45 IA,03:45 (15.2) 04:00 IA,04:00 (40.1) 04:15 04:30 2 ID,03:00 0.25 (24.8) 2.75ID,04:00 0.25 (22.3) 04:45 IA,04:45 (17.1) 05:00 IA,05:00 (24.9) 05:15 05:30 3 ID,05:00 0.25 (19.4) 3.75ID,06:00 0.25 (17.9) 05:45 IA,05:45 (14.3) 06:00 06:15 06:30 2 ID,01:00 0.25 (18.2) 6 ID,05:00 0.25 (7.2) Q 1.25 ID,00:45 0.25 (12) 1.25ID,01:00 0.25 (12.9) 1.25ID,01:00 0.25 (12.5) 5 ID,05:00 0.25 (7.5) Q 2.25 ID,02:45 0.25 (9.3) Q 3.25 ID,04:00 0.25 (10.9) 3 ID,04:00 0.25 (10) 4.25ID,05:00 0.25 (9.7) Q 3.25 ID,04:45 0.25 (12.2) 1.25ID,03:00 0.25 (17.3) 2 ID,04:00 0.25 (10.8) 3.25ID,05:00 0.25 (10.1) Q 3.25 ID,05:45 0.25 (13.7) Q 3.25 ID,06:00 0.25 (24.7) IA,06:00 (20.5) 3.25ID,06:00 0.25 (30.9) 2.75ID,06:00 0.25 (15.6) 06:45 IA,06:45 (27) 07:00 IA,07:00 (38.8) 07:15 07:30 07:45 3 ID,07:00 0.25 (42.2) 3 ID,07:00 0.25 (25.6) 3 ID,06:00 0.25 (28.1) Q 3.25 ID,06:30 Q 3.25 ID,06:45 0.25 (21) 0.25 (12.8) 2.25ID,06:00 0.25 (24.8) 2 ID,06:00 0.25 (14.9) 2.75ID,07:00 0.25 (23.7) 2.5ID,07:00 0.25 (27.2) IA,07:45 (20.8) 1.75ID 1,23:00 0.25 (9.8) 3.5ID,01:00 0.25 (10.3) Q 3.25 ID,01:00 0.25 (7.3) 3.25ID,01:00 0.25 (7.2) 1.75ID,00:00 0.25 (6.7) Q 4.25 ID,02:45 0.25 (3.4) 2.25ID,01:00 0.25 (9.1) 1.25ID 1,22:00 0.25 (5.4) 2 ID 1,23:00 0.25 (10.8) 3.75ID,01:00 0.25 (7.5) Q 3.25 ID,00:45 0.25 (6.2) Q 0.75 ID 1,22:30 0.25 (4.4) 2 ID,00:00 0.25 (6.6) 3 ID,01:00 0.25 (4.3) 1.5ID,00:00 0.25 (3.2) Q 9.75 ID,02:00 0.25 (4.7) 3 ID,02:00 0.25 (13.1) IA,02:15 (5) 2.75ID,02:00 0.25 (6.6) IA,02:30 (5.8) 2.5ID,02:00 0.25 (9.4) Q 0.5 ID,00:15 0.25 (7.6) 2 ID,01:00 0.25 (7.6) 4.75ID,05:00 0.25 (5.6) Q 3.25 ID,03:45 0.25 (9.2) 2.75ID,03:00 0.25 (8.3) 1 ID,02:00 0.25 (9.6) 3 ID,04:00 0.25 (8.9) 3.5ID,05:00 0.25 (8.6) Q 3.25 ID,05:00 0.25 (10.4) Q 3.25 ID,05:15 0.25 (10.2) 3 ID,05:00 0.25 (8.6) 3.5ID,06:00 0.25 (10.1) 3.25ID,06:00 0.25 (7.1) Q 3.25 ID,06:15 0.25 (11) 3.75ID,07:00 0.25 (9.8) 2.5ID,06:00 0.25 (12.3) Q 3.25 ID,07:00 0.25 (17.4) 3.25ID,07:00 0.25 (9) 1.75ID,06:00 0.25 (7.5) Q 3.25 ID,07:45 0.25 (13.9) Q 3.25 ID,02:45 0.25 (8.5) Q 3.25 ID,03:00 0.25 (6.4) 3 ID,03:00 0.25 (7.1) 5 ID,04:00 0.25 (5.5) 4.5ID,05:00 0.25 (8.3) 1.25ID,02:00 0.25 (4.2) 2.75ID,03:00 0.25 (7) Q 3.25 ID,04:30 0.25 (7.8) Q 2.25 ID,03:45 0.25 (4.1) Q 12.75 ID,05:00 0.25 (3.1) 1.75ID,01:00 0.25 (5.8) Q 3.25 ID,05:30 0.25 (6.9) Q 4.25 ID,05:45 0.25 (8.9) Q 4.75 ID,06:00 0.25 (4.7) Q 4.75 ID,06:15 0.25 (5.6) 3 ID,06:00 0.25 (7.4) Q 3.75 ID,06:45 0.25 (8.8) 3.25ID,07:00 0.25 (5.8) Q 3.25 ID,07:15 0.25 (4.6) 4.75ID,09:00 0.25 (5) 4.5ID,09:00 0.25 (4.2) T 3: M v FI...C qu-uy u 00:00 07:45 28

I: 1 I: 2 I: 3 I: 4 08:00 IA,08:00 (32.6) 08:15 08:30 08:45 09:00 09:15 09:30 09:45 10:00 10:15 10:30 10:45 11:00 11:15 11:30 11:45 12:00 12:15 12:30 12:45 13:00 13:15 13:30 13:45 14:00 14:15 14:30 3 ID,08:00 0.25 (33.7) 3.75ID,09:00 0.25 (21.6) 3.5ID,09:00 0.25 (13.2) 2.25ID,08:00 0.25 (33.3) 3 ID,09:00 0.25 (41.3) 2.75ID,09:00 0.25 (31.7) 3.5ID,10:00 0.25 (28.4) 2.25ID,09:00 0.25 (29.4) 3 ID,10:00 0.25 (35.5) 3.75ID,11:00 0.25 (16.1) 2.25ID,07:00 0.25 (31.3) 2 ID,07:00 0.25 (32.9) 2.75ID,08:00 0.25 (20) Q 3.25 ID,08:45 0.25 (11.8) Q 3.25 ID,08:00 0.25 (6.9) 3.25ID,08:00 0.25 (8.8) 1.75ID,07:00 0.25 (10.1) 4.5ID,10:00 0.25 (8.9) 1.25ID,07:00 0.25 (18.5) IA,09:00 (15.6) 2 ID,08:00 0.25 (16.7) 3.75ID,10:00 0.25 (18.2) 4.5ID,11:00 0.25 (15.8) 1.25ID,08:00 0.25 (9.7) 2 ID,09:00 0.25 (24.2) 2.75ID,10:00 0.25 (14) 3.5ID,11:00 0.25 (18.8) IA,10:45 (16.5) 2.25ID,10:00 0.25 (30.4) 3 ID,11:00 0.25 (48.5) 2.75ID,11:00 0.25 (24.1) Q 3.25 ID,11:45 0.25 (12.6) 2.25ID,11:00 0.25 (19.6) 3 ID,12:00 0.25 (24.7) 3.75ID,13:00 0.25 (41.5) 3.5ID,13:00 0.25 (28.1) 2.25ID,12:00 0.25 (20.8) 3 ID,13:00 0.25 (48.6) 2.75ID,13:00 0.25 (26.8) Q 3.25 ID,13:45 0.25 (19.1) 1.25ID,12:00 0.25 (14) 3 ID,14:00 0.25 (37.9) 3.75ID,15:00 0.25 (10.2) 14:45 IA,14:45 (18.5) 15:00 IA,15:00 (14.4) 15:15 15:30 15:45 3 ID,15:00 0.25 (45.1) 3.75ID,16:00 0.25 (31.9) Q 3.25 ID,15:45 0.25 (17.8) 3.25ID,11:00 0.25 (12.4) 2 ID,10:00 0.25 (10) 3.75ID,12:00 0.25 (18.2) 3.5ID,12:00 0.25 (12) Q 3.25 ID,12:00 0.25 (11.8) Q 1.75 ID,10:45 0.25 (13.4) 2.75ID,12:00 0.25 (7.8) 4.5ID,14:00 0.25 (13.5) Q 3.25 ID,13:00 0.25 (12.1) 2 ID,12:00 0.25 (9.7) 4.75ID,15:00 0.25 (12.1) Q 2.25 ID,12:45 0.25 (13) Q 3.25 ID,14:00 0.25 (13.7) Q 1.75 ID,12:45 0.25 (10.9) 4.75ID,15:00 0.25 (8) Q 3.25 ID,14:45 0.25 (15.5) Q 3.25 ID,15:00 0.25 (10.6) 2 ID,14:00 0.25 (10.3) Q 4.75 ID,17:00 0.25 (6.8) Q 1.75 ID,07:45 0.25 (3.9) 5 ID,11:00 0.25 (9.4) Q 2.25 ID,08:45 Q 3.25 ID,10:00 Q 3.25 ID,10:15 0.25 (4.4) 4 ID,11:00 0.25 (13.4) 1.25ID,06:00 0.25 (6.5) 2.25ID,07:00 0.25 (4.9) Q 3.5 ID,08:45 0.25 (6.8) Q 2.25 ID,07:45 0.25 (6.1) Q 3.25 ID,09:00 0.25 (5.4) 3.25ID,09:00 0.25 (3.8) 4.75ID,11:00 0.25 (8.5) Q 3.25 ID,09:45 0.25 (5.7) 0.25 (9.5) IA,10:00 (6.3) 0.25 (10.3) Q 3.25 ID,10:45 0.25 (11.6) 1.25ID,09:00 0.25 (9.2) Q 1.75 ID,09:45 0.25 (5.3) Q 2.5 ID,10:45 0.25 (11.3) 4.5ID,13:00 0.25 (10.5) 1.25ID,10:00 0.25 (10.5) 3.75ID,12:00 0.25 (8.7) Q 1.5 ID,10:45 0.25 (6.6) Q 2.25 ID,11:45 Q 0.75 ID,10:30 0.25 (6.6) Q 1.5 ID,11:30 0.25 (4.5) Q 2.5 ID,12:45 0.25 (9.5) 3.5ID,14:00 0.25 (11.8) 0.25 (10.9) Q 1 ID,11:45 0.25 (12.8) 2 ID,13:00 0.25 (6.7) 4.75ID,16:00 0.25 (6.6) 4.5ID,16:00 0.25 (12.3) Q 1 ID,12:45 0.25 (9.7) Q 4 ID,16:00 0.25 (3.8) 4.75ID,17:00 0.25 (5.4) Q 2.25 ID,14:45 0.25 (13.2) IA,15:45 (9.6) Q 1.25 ID,07:45 0.25 (4.4) Q 1.5 ID,08:45 0.25 (7.4) 4.5ID,12:00 0.25 (7.5) Q 3.25 ID,11:00 0.25 (4.6) 4 ID,12:00 0.25 (3.5) Q 3.25 ID,11:30 0.25 (9) 4.75ID,13:00 0.25 (7.5) Q 1.75 ID,10:15 0.25 (5.9) 2 ID,11:00 0.25 (6.1) Q 2.5 ID,11:45 0.25 (6.5) Q 3.5 ID,12:45 0.25 (4.9) Q 2.25 ID,12:00 0.25 (4.8) 1 ID,11:00 0.25 (4.2) 3.75ID,14:00 0.25 (7.8) 4.5ID,15:00 0.25 (8.2) Q 4.25 ID,14:00 0.25 (5.7) Q 0.75 ID,11:45 0.25 (4.4) Q 2.5 ID,13:45 0.25 (6.6) Q 2.25 ID,13:45 0.25 (7.4) Q 3.5 ID,15:00 0.25 (9.3) Q 1.75 ID,13:45 0.25 (3.7) Q 2.25 ID,14:30 0.25 (5.3) Q 23.25 ID,15:45 0.25 (6.2) T 4: M v FI...C qu-uy u 08:00 15:45 29

16:00 16:15 16:30 16:45 17:00 17:15 17:30 17:45 I: 1 I: 2 I: 3 I: 4 Q 3.25 ID,16:00 0.25 (16.4) 3 ID,16:00 0.25 (47.2) 3.75ID,17:00 Q 3.25 ID,16:45 0.25 (18.8) 0.25 (29.4) 2.25ID,15:00 0.25 (15) IA,16:00 (9) 2 ID,15:00 0.25 (20.7) 2.75ID,16:00 0.25 (17) Q 3.25 ID,17:00 0.25 (24) IA,17:00 (12.8) 3 ID,17:00 0.25 (32.1) 2.75ID,17:00 0.25 (22.9) Q 3.25 ID,17:45 0.25 (31.4) 18:00 IA,18:00 (19.4) 18:15 18:30 18:45 19:00 19:15 19:30 19:45 20:00 20:15 20:30 3 ID,18:00 0.25 (45.2) 3.75ID,19:00 0.25 (24.8) Q 3.25 ID,18:45 0.25 (26.6) Q 3.25 ID,16:15 0.25 (2.8) Q 2.5 ID,15:45 0.25 (10.7) 3.5ID,17:00 0.25 (10.7) IA,16:45 (8.8) 2 ID,16:00 0.25 (17.6) Q 3.25 ID,17:30 0.25 (16) 3.5ID,18:00 0.25 (14.8) Q 3.25 ID,18:00 0.25 (18.3) 3.25ID,18:00 0.25 (7.8) 2.75ID,18:00 0.25 (24.6) 3.5ID,19:00 0.25 (21.1) 2.25ID,18:00 0.25 (28) IA,19:00 (20.8) 3 ID,19:00 0.25 (66.7) 2.75ID,19:00 0.25 (23.4) Q 3.25 ID,19:45 0.25 (31.1) 2.25ID,19:00 0.25 (28.3) 3 ID,20:00 0.25 (37.3) 3.75ID,21:00 0.25 (25.5) 20:45 IA,20:45 (21.3) 21:00 21:15 21:30 21:45 3.25ID,21:00 0.25 (31.7) 3 ID,21:00 0.25 (48.3) 2.75ID,21:00 0.25 (15.5) Q 3.25 ID,21:45 0.25 (11.6) 22:00 IA,22:00 (16.2) 22:15 22:30 22:45 23:00 23:15 23:30 23:45 3 ID,22:00 0.25 (41.6) 2.75ID,22:00 Q 3.25 ID,22:45 0.25 (33.7) 0.25 (16.3) Q 3.25 ID,19:15 0.25 (4.1) 3.75ID,20:00 0.25 (15.5) 3.5ID,20:00 Q 3.25 ID,20:00 0.25 (21.1) 2 ID,19:00 2.25ID,16:00 0.25 (11.9) 3.5ID,17:00 0.25 (7.2) 3.75ID,18:00 Q 2.25 ID,16:45 Q 3.5 ID,18:00 Q 0.5 ID,15:30 Q 3.5 ID,18:45 0.25 (14.8) Q 2.25 ID,15:00 0.25 (8.3) Q 0.75 ID,13:45 0.25 (2.7) Q 3.25 ID,16:30 0.25 (8.3) Q 2.25 ID,15:45 0.25 (5.5) Q 0.75 ID,14:30 0.25 (6.5) Q 3.25 ID,17:15 0.25 (6.3) Q 1.5 ID,15:45 0.25 (8.1) 0.25 (7.5) IA,17:45 (7.3) 0.25 (10.7) 0.25 (5.4) 0.25 (8.7) 2.25ID,17:00 0.25 (9.7) Q 0.75 ID,15:45 0.25 (4.5) Q 2.5 ID,17:45 0.25 (6.2) Q 3.5 ID,18:45 0.25 (10.6) IA,18:45 (7.3) 3.25ID,19:00 0.25 (10.1) 3.75ID,19:00 0.25 (2.9) Q 2.5 ID,18:45 0.25 (8.2) 0.25 (15.2) IA,19:45 (11.1) 0.25 (24.8) Q 2.5 ID,19:45 0.25 (17.2) Q 2.25 ID,19:45 0.25 (15.6) 2.25ID,20:00 Q 3.25 ID,21:15 0.25 (5.5) 3 ID,21:00 3.25ID,20:00 Q 3.25 ID,20:15 0.25 (8.1) Q 3.5 ID,19:00 0.25 (9.1) Q 3.5 ID,19:00 0.25 (2.9) 4.5ID,20:00 0.25 (6.1) 4.5ID,21:00 0.25 (5.8) 0.25 (10.9) IA,20:00 (10) 2.75ID,20:00 0.25 (11.3) 3.5ID,21:00 0.25 (15) 0.25 (6.8) IA,21:00 (6.5) 0.25 (14.7) Q 2.25 ID,20:45 0.25 (10.1) 3.5ID,22:00 0.25 (13.5) Q 3.25 ID,22:15 0.25 (4.6) Q 3.25 ID,22:30 0.25 (8.7) 2.5ID,22:00 0.25 (16.3) Q 3.25 ID,23:00 0.25 (28.5) IA,23:00 (9.7) 3 ID,23:00 0.25 (18) 3 ID,23:00 0.25 (26) Q 3.25 ID,23:45 0.25 (20.1) 2 ID,22:00 0.25 (15.1) 2.75ID,23:00 0.25 (16.8) 2.5ID,23:00 0.25 (16.5) 2 ID,20:00 0.25 (4.3) 3.75ID,22:00 0.25 (12.9) Q 3.5 ID,21:45 0.25 (9.8) 3.25ID,22:00 0.25 (12.2) Q 3.5 ID,22:15 0.25 (3.7) 3.75ID,23:00 0.25 (6.6) 3.5ID,23:00 0.25 (13.9) 2.25ID,22:00 0.25 (9.6) Q 3.25 ID,23:15 0.25 (9.9) Q 3.25 ID,23:30 0.25 (9.6) 3.25ID,20:00 0.25 (5.6) Q 2.75 ID,19:45 0.25 (6.7) Q 3.25 ID,20:45 0.25 (7.2) Q 0.75 ID,18:30 0.25 (6.5) Q 3.75 ID,21:15 0.25 (4.1) 4 ID,22:00 0.25 (6) 4.5ID,23:00 0.25 (7.4) Q 3.25 ID,22:00 Q 1 ID,19:30 0.25 (3.2) 4 ID,23:00 0.25 (11.9) 0.25 (4.1) Q 3.5 ID,22:45 0.25 (9.6) Q 4 ID,23:00 0.25 (6.3) 3.5ID,23:00 0.25 (8.3) 3.5ID,23:00 0.25 (8.7) 2.75ID,23:00 0.25 (14.4) IA,23:45 (5.7) T 5: M v FI...C qu-uy u 16:00 23:45 30

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