Dynamics of supply-chain and market volatility of networks

Fernanda Strozzi

Cattaneo University-LIUC

Italy

WP5

WP5: Tasks overview

Supply chain Model

T5.1, T5.5Energy spot pricesVolatility

BlackoutsVolatility

Correlation(T5.2)Analysis(T5.3)

Electric power ModelT5.1

Electricity price ModelT5.1

EWDS of Blackouts T5.4

Coupling modelsTask5.5

Interaction RiskT5.6

Red=works to be presented

Contents• Data provision • Data treatment• Correlation analysis:

– Linear correlation coefficient– Cross Correlation function– Cross Recurrence Plots– Principal Component Analysis

• Conclusions

D5.3 (M24) Correlation analysis between electricity prices and

faults in electricity grid in the Nordic countries

• 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

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

Ele

ctri

city

pri

ces

Dis

turb

ance

sT

otal

Con

sum

ptio

n

Denmark(*), Finland(:), Norway(.-) and Sweden(-).

Data treatment

Tre

nds D

etre

nded

dat

a

Fir

st d

iffe

renc

es

Vol

atil

itie

s

1s m, 2 wm, 1

))(

)((ln)(

t

ttP

tPstdtVS

Data treatment

S Mean monthly spot prices *dt Detrend of *

D Monthly disturbances *fdFirst diff of *

T Monthly Total Consumption V* Volatilities of *

Data treatment

Window shift window shift

1 1 2 1

3 3 3 1

6 6 6 1

12 12 12 1

W=3, s=3

W=3, s=1

Linear Correlation Coefficient

ii

i

myiymxix

myiymxixr

22 )()(

)()([

The linear Correlation Coefficient r between x(i) and y(i) for i=1..Nwith mean mx and my

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

S 1 -0.2439 0.2014 0.7317 -0.1072 -0.0308 0.2651 -0.0307 0.0322 0.2568 -0.0484 0.0423

D 1 -0.6270 -0.0617 0.4828 -0.0885 -0.0491 0.5390 -0.0626 -0.1447 0.6106 -0.1054

T 1 -0.0227 -0.0311 0.2162 0.1139 -0.0941 0.3110 0.1349 -0.2259 0.3114

S_dt 1 -0.1259 -0.1649 0.2908 0.0018 -0.0833 0.2960 0.0267 -0.0747

D_dt 1 -0.1534 -0.0086 0.5122 -0.0238 -0.0396 0.5334 -0.0271

T_dt 1 0.2436 -0.0297 0.1539 0.2736 -0.0567 0.1535

S_fd 1 -0.0849 0.1476 0.8607 -0.0510 0.1495

D_fd 1 -0.1962 -0.1692 0.8761 -0.2584

T_fd 1 0.1485 -0.1922 0.9896

VS 1 -0.1373 0.1633

VD 1 -0.2568

VT 1

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)

Linear Correlation Coefficient

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)

Cross Correlation Function

-40 -20 0 20 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cro

ss C

orr

ela

tion

D-Sfd

X: -3Y: 0.2157

X: 4Y: -0.272

X: 8Y: 0.3529

-40 -20 0 20 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X: -2Y: 0.6896

Cro

ss C

orr

ela

tion

D-Tfd

X: 3Y: -0.6611

-40 -20 0 20 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X: -6Y: 0.671

Cro

ss C

orr

ela

tion

D-T

X: 0Y: -0.7354

X: 6Y: 0.6435

w=2,s=1

-40 -20 0 20 40-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X: 6Y: 0.2656

D-S

X: 0Y: -0.2692

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).

ji yx , where i=1, …, n, j=1, …m the CRP matrix is defined by

)()(, jiyxji yxji CR

t

t

time

for

y(t)

no embedding

x(t)

y(t)

x(t)y(t)

time for x(t)t

Cross Recurrence plot quantification

N

l

N

ll

llP

llP

DET

1

)(

)(min

N

jijiN

RR1,

,2)(

1)( R

N

v

N

vv

vvP

vvP

LAM

1

)(

)(min

% Determinism % Recurrence % Laminarity

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

Lines diagonally oriented

Cross Recurrence plot quantification:Line Of Synchronization (LOS)

50 100 150 200 250 300 350 400

-1

0

1

50 100 150 200 250 300 350 400-1

0

1

Underlying Time Series

Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

50 100 150 200 250 300 350 400

-1

0

1

50 100 150 200 250 300 350 400-1

0

1

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

50 100 150 200 250 300 350 400

-1

0

1

50 100 150 200 250 300 350 400-1

0

1

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

50 100 150 200 250 300 350 400

-1

0

1

50 100 150 200 250 300 350 400-1

0

1

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

50 100 150 200 250 300 350 400

-1

0

1

50 100 150 200 250 300 350 400-1

0

1

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.01 (fixed distance euclidean norm)

50 100 150 200 250 300 350 400

50

100

150

200

250

300

350

400

)sin( t )sin( t

)sin( t

)sin( t

))5.1sin(sin( tt )3sin( t

)cos( t )cos( t

)sin( t )3sin( t

t=[-2:0.01:2]

LOS calculation for electricity prices, disturbances and Total Consumption w=2,s=1,

=0.5

Tfd

Sfd

Q=80.40 Q=64.32

10 20 30 40 50 60 70 80

-5

0

5

10 20 30 40 50 60 70 80-5

0

5

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.5 (fixed distance euclidean norm)

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

D D

10 20 30 40 50 60 70 80

-4

-2

0

2

4

10 20 30 40 50 60 70 80-4

-2

0

2

4

Underlying Time Series

Cross Recurrence Plot Dimension: 1, Delay: 1, Threshold: 0.5 (fixed distance euclidean norm)

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

LOS Quality

100*NgNt

NtQ

Nt is the number of targeted pointsNg the number of gap points. The larger is Q the better is LOS

LOS Algorithm

1. Find the recurrence point next to the origin2. 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 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 intervalsconsidered

r_t r_i using CRP

Note

D-S 69.32 D(10:20); S(1:11);

-0.2692 0.6304 Interval +shift

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

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

Principal Component Analysis

•The data have very different mean and variancesCorrelation matrix

•Eigenvectors loadings

•Corresponding eigenvaluesPercent of variance explained on that direction = 100*eigenvalue/sum(eigenvalues);

•Percent of variance Cumulative sum of variance explained

0.1801 -0.4014 -0.2349 -0.4308 0.0655 -0.2196 -0.2880 -0.6285 0.1153 -0.1032 0.1062 0.0023

-0.3934 -0.1531 0.2379 0.0802 -0.3920 -0.1733 -0.1521 0.1371 0.6040 -0.2497 0.3226 0.0436 0.2872 0.0083 0.1344 -0.2054 0.6711 0.2206 0.0899 0.2632 0.4220 -0.0910 0.3051 0.0378 0.1018 -0.4453 -0.3041 -0.3593 -0.1610 -0.1850 -0.0466 0.6879 -0.0987 0.1073 -0.1129 -0.0156 -0.2924 -0.1836 0.3101 -0.1076 0.1262 0.5012 -0.6616 0.0590 -0.1768 0.1009 -0.1439 -0.0518 0.1527 -0.0640 0.2032 0.4514 0.3460 -0.6499 -0.3977 0.1075 -0.0781 0.0744 -0.0624 -0.0133 0.2213 -0.4827 0.0824 0.3576 -0.0895 0.2244 0.1905 -0.1379 0.3781 0.5055 -0.2666 0.0098 -0.3963 -0.2527 0.1831 -0.1117 0.3116 -0.1412 0.3549 -0.0538 0.0358 -0.3832 -0.5822 0.0093 0.2760 0.0306 0.5399 -0.2630 -0.1996 -0.1090 0.1178 -0.0161 -0.0529 0.0168 -0.0135 -0.7023 0.2610 -0.4511 0.0570 0.3734 -0.0783 0.2393 0.0953 0.0227 -0.3546 -0.5614 0.2681 -0.0044 -0.4144 -0.2761 0.1856 -0.0793 0.1824 -0.1403 0.3181 -0.0775 -0.3328 0.4165 0.5145 0.0476 0.2985 0.0421 0.5203 -0.2574 -0.2139 -0.0787 0.0375 -0.0069 -0.1150 0.0149 -0.0860 0.7056

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12

S D T Sdt Ddt Tdt Sfd Dfd Tfd VS VD VT

Loadings for w=1, s=1Principal Component Analysis

eigenvalues Cumulative sum of % variances explained

3.4124 2.2086 1.9863 1.3426 1.1692 0.8048 0.4502 0.2249 0.1709 0.1215 0.1021 0.0065

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

#varfor at least 50% var

% var

#varfor at least 90% var

% var

83 3 63.39 6 91.03

Principal Component Analysis

w s # points #var at least 50% var

% var #varat least 90% var

% 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 -0.26920.6304; D-Sfd 0.0702 0.5466

Principal Component Analysis:2-3 variables at least 50% variance explainedMore than 3 variables at least 90% variance explained

• Qeen Mary (Physica A)• JRC (Physica A, Physica D)• COLB (under discussion)• MASA (defined)

2 female PhD students started to work on:• Models of Supply Chain• Ranking Risk in Supply Chain

1 female student for the final project

LIUC Colaborations

LIUC Gender Action

Conferences DISSEMINATION1_ 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, D5.2Papers1_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