WP5
Dynamics of supply-chain and
market volatility of networks
Fernanda Strozzi
Cattaneo University-LIUC
Italy
WP5: Tasks overview
Electricity price
Model
T5.1
Coupling
models
Task5.5
EWDS of
Blackouts T5.4
Electric power
Model
T5.1
Energy spot prices
Volatility
Correlation(T5.2)
Analysis(T5.3)
Blackouts
Volatility
Red=works to be presented
Interaction Risk
T5.6
Supply chain
Model
T5.1, T5.5
D5.3 (M24)
Correlation analysis between electricity prices and
faults in electricity grid in the Nordic countries
•
•
•
Contents
Data provision
Data treatment
Correlation analysis:
–
–
–
–
•
Linear correlation coefficient
Cross Correlation function
Cross Recurrence Plots
Principal Component Analysis
Conclusions
Data provision
• Monthly Disturbances
• Monthly Total Consumption
http://www.nordel.org
• Monthly Electricity prices
http://www.nordpool.com
in Denmark, Finland, Norway and Sweden
from January 2000 until December 2006
Data provision
• Nordel is the collaboration organisation of
the Transmission System Operators (TSOs)
( Denmark, Finland, Iceland, Norway and Sweden).
• Nord Pool is the Nordic Power Exchange
Market
Norway(1993), Sweden(1996), Finland (1997),
W Denmark (1999), E Denmark (2000), Kontek (2005)
Data provision
• Nordel annual report:
Disturbance is an outage,
forced or unintended
disconnection or failed
reconnection as a results of
faults in the power grid
• A disturbance may consist of
a single fault but it can also
contain many faults, typically
consisting of an initial fault
followed by some secondary
faults.
• The grid considered is the
100-400kV network
Total Consumption
Electricity prices
Disturbances
Data treatment
Denmark(*), Finland(:),
Norway(.-) and Sweden(-).
Detrended data
Volatilities
First differences
Trends
Data treatment
 P(t ) 
)
VS (t )  std (ln 
 P(t  t ) 
t  1 m, w  2 m, s  1
Data treatment
S
Mean monthly spot prices
*dt
D
Monthly disturbances
*fd
Detrend of *
First diff of *
T
Monthly Total Consumption
V*
Window shift
window shift
1
3
6
12
2
3
6
12
1
3
6
12
1
1
1
1
Volatilities of *
W=3, s=3
W=3, s=1
Linear Correlation Coefficient
[x(i)  mx y(i)  my
r
i
2
2




x
(
i
)

mx
y
(
i
)

my


i
S
D
T
S_dt
D_dt
T_dt
S_fd
D_fd
T_fd
VS
VD
VT
The linear Correlation Coefficient r
between x(i) and y(i) for i=1..N
with mean mx and my
i
Correlation matrix.
w=1, s=1,
yellow if |r|>0.7071 (r2>0.5)
confidence level of 95%
S
D
T
S_dt
D_dt
T_dt
S_fd
D_fd
T_fd
VS
VD
VT
1
-0.2439
0.2014
0.7317
-0.1072
-0.0308
0.2651
-0.0307
0.0322
0.2568
-0.0484
0.0423
1
-0.6270
-0.0617
0.4828
-0.0885
-0.0491
0.5390
-0.0626
-0.1447
0.6106
-0.1054
1
-0.0227
-0.0311
0.2162
0.1139
-0.0941
0.3110
0.1349
-0.2259
0.3114
1
-0.1259
-0.1649
0.2908
0.0018
-0.0833
0.2960
0.0267
-0.0747
1
-0.1534
-0.0086
0.5122
-0.0238
-0.0396
0.5334
-0.0271
1
0.2436
-0.0297
0.1539
0.2736
-0.0567
0.1535
1
-0.0849
0.1476
0.8607
-0.0510
0.1495
1
-0.1962
-0.1692
0.8761
-0.2584
1
0.1485
-0.1922
0.9896
1
-0.1373
0.1633
1
-0.2568
1
Linear Correlation Coefficient
r values between Std (for VS, VD, VT) and the mean (for the others
time series), |r|>0.7071 (r2>0.5), confidence level of 95%
w=2; s=1
w=3; s=1
w=6; s=1
w=12; s=1
T,D (-0.7354)
S, Sdt(0.7195)
T,D (-0.8057)
T,D(-0.9044)
Tfd,Dfd(-0.8010)
T,D(-0.7807)
D,Tdt(-0.7586)
D,Ddt(0.8060)
T,Tdt(0.9904)
w=1, s=1
w=3; s=3
w=6; s=6
w=12; s=12
S,Sdt (0.7317)
Sfd,Vs(0.8607)
Dfd,VD(0.8761)
Tfd,VT(0.9896)
D,T (-0.8154)
Dfd,D(-0.8503)
Tfd,T(-0.8686)
VD,Dfd(0.7698)
D,T(-0.8594)
D, Tfd(0.776)
T,Dfd(0.7752)
Tdt,T(0.9842)
VD,T(-0.9057)
VD,Sdt(0.8138)
VD,Tdt(-0.9014)
D-Sfd
1
Cross Correlation Function
0.8
Cross Correlation
0.6
D-T
1
0.8
X: -6
Y: 0.671
w=2,s=1
X: 6
Y: 0.6435
X: 8
Y: 0.3529
0.4
X: -3
Y: 0.2157
0.2
0
-0.2
X: 4
Y: -0.272
-0.4
0.6
D-Tfd
0.2
1
0
0.8
-0.2
0.6
-0.4
-0.6
-0.8
-1
-40
X: 0
Y: -0.7354
-20
0
Cross Correlation
Cross Correlation
-0.6
0.4
-0.8
D-S
1
-1
-40
0.8
X: -2
Y: 0.6896
-20
0
20
40
0.6
0.4
X: 6
Y: 0.2656
0.4
0.2
0.2
0
0
-0.2
20
-0.4
-0.2
40
-0.6
X: 0
Y: -0.2692
-0.4
X: 3
Y: -0.6611
-0.6
-0.8
-0.8
-1
-40
-20
0
20
40
-1
-40
-20
0
20
40
Cross Recurrence plot, (Marwan, 2007)
CRP is a bivariate extension of RP and is a tool to analyse the dependencies between two
different time systems by comparing their states (Marvan and Kurths, 2002).
It can be considered as a generalization of the linear cross-correlation function (Marwan et al. 2007).
where i=1, …, n, j=1, …m the CRP matrix is defined by
xi , y j
CR i ,i j j ( )  (  xi  y j )
x y
x(t)
y(t)
t
no embedding
x(t)
y(t)
time for y(t)
t
t
time for x(t)
Cross Recurrence plot quantification
% Determinism
N
DET 
 lP (l )
l lmin
N
 lP (l )
% Laminarity
% Recurrence
1
RR( )  2
N
N
N
R
i , j 1
i, j
( )
LAM 
 vP(v)
v  vmin
N
 vP(v)
v 1
l 1
Lines diagonally oriented
They represent segments of both trajectories running parallel for some
time. Frequency and length of these lines are related to the similarity
between the two dynamical systems not always detected by cross
correlation function.
One trajectory main black diagonal (LOI)
If the values of the second trajectory are modified LOI becomes
LOS
Cross Recurrence plot quantification:
Line Of Synchronization (LOS)
1
50
100
Underlying Time Series
150
200
250
300
350
400
1
1
0
50
100
Underlying Time Series
150
200
250
300
t=[-2:0.01:2]
0.01
350
400
1
1
0
0
50
100
Underlying Time Series
150
200
250
300
350
400
1
0
0
-1
Recurrence
50
100
150
200 Plot
250
300
350
400
Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)
400
0
-1
-1
-1
Plot
50
100 Cross
150 Recurrence
200
250
300
350
400
Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)
400
-1
-1
Plot
50
100 Cross
150 Recurrence
200
250
300
350
400
Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)
400
350
350
350
sin( t )
300
sin(  3t )
300
sin( t  sin( 1.5t ))
250
300
250
250
200
200
200
150
150
150
100
100
100
50
50
50
50
100
150
200
250
300
350
50
400
50
sin( t )
1
50
100
Underlying Time Series
150
200
250
300
350
400
1
0
100
150
200
250
300
350
350
1
200
250
300
50
100
Underlying Time Series
150
200
250
300
350
400
1
0
0
-1
-1
Plot
50
100 Cross
150 Recurrence
200
250
300
350
400
Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)
400
350
300
cos(t )
150
250
cos(t )
200
300
250
200
150
150
100
100
50
50
50
100
150
200
250
sin( t )
300
350
350
sin( t )
sin( t )
0
-1
-1
Plot
50
100 Cross
150 Recurrence
200
250
300
350
400
Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)
400
100
400
400
50
100
150
200
250
sin(  3t )
300
350
400
400
LOS calculation for electricity prices,
disturbances and Total Consumption
w=2,s=1,
=0.5
4
10
20
Underlying Time Series
30
40
50
5
60
70
80
10
20
Underlying Time Series
30
40
50
60
70
80
2
2
0
0
-2
-2
-4
-4
0
0
-5
-5
Cross
Plot 60
10
20
30 Recurrence
40
50
70
80
Dimension: 1, Delay: 1, Threshold: 0.5 (fixed distance euclidean norm)
Cross
Plot 60
10
20
30 Recurrence
40
50
70
80
Dimension: 1, Delay: 1, Threshold: 0.5 (fixed distance euclidean norm)
80
80
Sfd
70
70
60
Tfd
60
50
50
40
40
30
30
20
20
10
10
10
10
20
30
40
50
60
70
20
30
40
50
60
80
D
Q=80.40
5
4
D
Q=64.32
70
80
LOS Algorithm
1. Find the recurrence point next to the origin
2. Find the next point by looking for recurrence points in a squared window
of size w=2, If the edge of the window find a recurrence point we go to step 3,
else we iteratively increase the size of the window.
3. If there are subsequent recurrence points in y-direction (x-direction), the size w
of the window is iteratively increased in y-direction (x-direction) until a predefined
size or until no new recurrent points are met. When a new recurrence point is found
we return to step 2
LOS Quality
Nt
Q
*100
Nt  Ng
Nt is the number of targeted points
Ng the number of gap points.
The larger is Q the better is LOS
LOS calculation in CRP between Disturbances and the other time
series. Only the CRP with at least a part of the LOS parallel to the
main diagonal is considered.
Figure
Q
Temporal intervals
considered
r_t
r_i
using CRP
D-S
69.32
D(10:20);
S(1:11);
-0.2692
D-T
69.89
D(1:20);
T(1:20)
-0.7354
-0.8087
interval
D-Sfd
64.32
D(1:30);
Sfd(5:34)
0.0702
0.5466
interval+ shift
D-Dfd
81.69
D(1:19);
Dfd(2:20)
-0.4119
-0.7021
Interval+ shift
D-Tfd
80.40
D(1:60);
Tfd(3:62)
0.2429
0.7455
Interval +shift
0.6304
Note
Interval +shift
Principal Component Analysis
Principal Component Analysis
how many independent
variables
Principal Factor Models
Which are the independent
variables and how we have to
use them to build the model
Principal Component Analysis
•The data have very different mean and
variancesCorrelation matrix
•Eigenvectors loadings
•Corresponding eigenvalues
Percent of variance explained on that direction =
100*eigenvalue/sum(eigenvalues);
•Percent of variance 
Cumulative sum of variance explained
Loadings for w=1, s=1
Principal Component Analysis
PC1
S
D
T
Sdt
Ddt
Tdt
Sfd
Dfd
Tfd
VS
VD
VT
0.1801
-0.3934
0.2872
0.1018
-0.2924
0.1527
0.2213
-0.3963
0.2760
0.2610
-0.4144
0.2985
eigenvalues
3.4124
2.2086
1.9863
1.3426
1.1692
0.8048
0.4502
0.2249
0.1709
0.1215
0.1021
0.0065
PC2
PC3
PC4
PC5
-0.4014
-0.1531
0.0083
-0.4453
-0.1836
-0.0640
-0.4827
-0.2527
0.0306
-0.4511
-0.2761
0.0421
-0.2349
0.2379
0.1344
-0.3041
0.3101
0.2032
0.0824
0.1831
0.5399
0.0570
0.1856
0.5203
-0.4308
0.0802
-0.2054
-0.3593
-0.1076
0.4514
0.3576
-0.1117
-0.2630
0.3734
-0.0793
-0.2574
0.0655
-0.3920
0.6711
-0.1610
0.1262
0.3460
-0.0895
0.3116
-0.1996
-0.0783
0.1824
-0.2139
PC6
-0.2196
-0.1733
0.2206
-0.1850
0.5012
-0.6499
0.2244
-0.1412
-0.1090
0.2393
-0.1403
-0.0787
PC7
-0.2880
-0.1521
0.0899
-0.0466
-0.6616
-0.3977
0.1905
0.3549
0.1178
0.0953
0.3181
0.0375
PC8
-0.6285
0.1371
0.2632
0.6879
0.0590
0.1075
-0.1379
-0.0538
-0.0161
0.0227
-0.0775
-0.0069
PC9
PC10 PC11
0.1153 -0.1032
0.6040 -0.2497
0.4220 -0.0910
-0.0987 0.1073
-0.1768 0.1009
-0.0781 0.0744
0.3781 0.5055
0.0358 -0.3832
-0.0529 0.0168
-0.3546 -0.5614
-0.3328 0.4165
-0.1150 0.0149
0.1062
0.3226
0.3051
-0.1129
-0.1439
-0.0624
-0.2666
-0.5822
-0.0135
0.2681
0.5145
-0.0860
PC12
0.0023
0.0436
0.0378
-0.0156
-0.0518
-0.0133
0.0098
0.0093
-0.7023
-0.0044
0.0476
0.7056
Cumulative sum of
% variances explained
28.4363
46.8410
63.3939
74.5818
84.3254
91.0324
94.7836
96.6580
98.0824
99.0951
99.9458
100.0000
#
points
#var %
for at var
least
50%
var
#var %
for at var
least
90%
var
83
3
6
63.39
91.03
Principal Component Analysis
w
s
# points
#var
at least
50% var
% var
#var
% var
at least
90% var
1
1
83
3
63.39
6
91.03
2
1
82
3
53.08
8
91.08
3
1
81
3
56.35
8
92.67
6
1
78
3
62.62
7
92.85
12
1
72
2
54.80
6
93.26
3
3
27
3
56.55
7
91.69
6
6
13
2
61.44
5
94.38
12
12
6
2
65.63
4
96.75
Conclusions
Linear Correlation Coefficient:
•For near all the windows w and time shifts s we found a
high linear correlation between D and T or their modified versions.
Exception w=1, s=1.
•For w=12 s=12 a new correlation appears between VD, Sdt(0.8138)
Cross Recurrence Plot (LOS):
We can detect windows and shifts to increase linear correlation:
D-S -0.26920.6304; D-Sfd 0.0702 0.5466
Principal Component Analysis:
2-3 variables  at least 50% variance explained
More than 3 variables  at least 90% variance explained
LIUC Colaborations
•
•
•
•
Qeen Mary (Physica A)
JRC (Physica A, Physica D)
COLB (under discussion)
MASA (defined)
LIUC Gender Action
2 female PhD students started to work on:
• Models of Supply Chain
• Ranking Risk in Supply Chain
1 female student for the final project
Conferences
DISSEMINATION
1_ Analysis of complex systems by means of mathematical and simulation methods (Noè,
Rossi) . International Conference on applied simulation and modeling, Corfù (June 2008)
2_ Quantifying and ranking risks. IPMA world congress Rome 9-11 Nov 2008. Colicchia,
Sivonen, Noè, Strozzi.
3_Application of RQA to Financial Time Series, F. Strozzi, J.M. Zaldivar, J. Zbilut,
Second International workshop on Recurrence Plot, Siena, 10-12 September 2007.
Reports
-Application of non-linear time series analysis techniques to the Nordic spot electricity market
F. Strozzi, E.Gutiérrez, C. Noè, T. Rossi, M.Serati and J.M.Zaldívar. LIUC Paper
200, october 2007
-Deliverables D5.1, D5.2
Papers
1_Time series analysis and long range correlations of Nordic spot electricity market data,
H.Erzgraber, F. Strozzi, J.M. Zaldivar, H.Touchette, E. Gutierrez,
D.K.Arrowsmith, submitted to Physica A
2_ Measuring volatility in the Nordic spot electricity market using Recurrence Quantification
Analysis. F. Strozzi, E.Gutiérrez, C. Noè, T. Rossi, M.Serati and J.M. Zaldívar .
Accepted in EPJ Special Topics.
3_ A supply chain as a serie of filter or amplificators of the bullwhip effect . Caloiero, G.,
Strozzi, F., Zaldívar, J.M., 2007. International Journal of Production Economics
(Accepted).
4_Control and on-line optimization of one level supply chain, F. Strozzi, C.Noè, J.M. Zaldivar,
submitted to IJPE, 2008