X MODELLING WEEKUNIVERSIDAD COMPLUTENSE DE MADRID

TRANSPORTATION OPTIMIZATION IN

EXTREME EUROPEAN DEMAND CONDITIONS

Giovanni Biagioli (Univ. of Florence, Italy)Michael Mark (Univ. of Leicester, United Kingdom)

Julia Martın Tortajada (UCM, Spain)David Peces Culebra (UCM, Spain)

Candela Retolaza Conde (UCM, Spain)

JUNE 2016

Indice general

1. Introduccion 2

2. Data processing 3

2.1. Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2. Define representative temperature . . . . . . . . . . . . . . . . . . . . . . . 5

2.3. Estimate the temperatures dynamic’s and simulate his evolve for this year 8

2.4. Compare the peak simulate demand with historical . . . . . . . . . . . . . 12

3. Optimize 14

3.1. Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4. Conclusions 17

1 Introduccion

The goal of this project is to optimize transportation in extrem European demand con-dition. We divided the problem into two steps:

The first step was to estimate the data: we recived historical temperature and de-mand data and there are two possible aproches:

• First one we took European temperature

• Secondly we grouped together the countries by temperature

We noticed negative correlation between temperature and demand. That is, the lo-wer is the temperature, the bigger is the gas demand.

We estimated the representative temperature in each cluster which minimize theerror between historical demand and the one we were estimating. Also, we analyzethe temperature and demand’s behaviour and identified the peak demand.

The second step was to optimize the cost of transport. To this, we built the graphmatrix simulating reality interconections and we also put he capacity of each edgeand the peak demand.

Then, we evaluated the strenght of European network in different cases:

• Historical peak demand.

• Peak demand from one temperature zone simulation.

• Peak demand from climate areas aggregate simulations.

2

2 Data processing

2.1. Clustering

Once we recived the historical data, we observed there exists negative a correlation bet-ween temperature and demand. In order to faciliate our work, we carried out clusteringanalysis and divided countries in similar groups.

The algorithm we are going to used is K-means (later, we are going to prove the bestvalue of K is K=4). K-means is a non parametric clustering methodan performed in twosteps: chose the initial centroids and recalculte it.

Firstly we chose K cluster centers (centroids) arbitrarily. We have to assign each ob-servation to the cluster whose mean yields the least within-cluster sum of squares. Thiseffectively means to assign each point to the closest centroid, since the sum of squares isthe euclidean distance.

Then, we update this step. We calculate new means of each cluster and set them asnew centroids. Now iterate until convergence. It is proven that K-means converges in fi-nite number of steps.

Heurestically, it can be deduced from following figure:

We can see on the figure there is a relative maximun in K=4. It means we have to usefour clusters.

3

K-means algorithm provide us this different climate areas in Europe:

Mid-East Northern Atlantic Mediterranean

Austria Belgium Croatia FyromCzech Republic Bulgaria Denmark Greece

Germany Estonia France IrelandHungary Finland Netherlands Italy

Luxembourg Latvia Slovenia PortugalPoland Lithuania United Kingdom Spain

Romania SwedenSlovakia

Switzerland

We check the clusters optained seem reasonable, with respect to the climate areas.

4

2.2. Define representative temperature

Now, we want to define the relationship between the demand and the temperature.

We know the temperature influences the demand. This two magnitudes act reversely.It seems, when temperature is higger, demand is lower (and the other way around).

Also, when temperature is very low, we have a lot of demand. Perhaps, there is a momentwhen demand doesn’t increase (even when temperature hits levels very low). This efectis known Saturation efect.

It seems we need a logitic function to explain the demand.

Dt = a1 +a2

1 + exp(a3 ∗ Tt)+ vt

Now, we have to estimate this parameters (a0, a1, a2). How will we do it? We are going toestime it minimizing the difference between the real demand and the estimated demand.

min(Dreal −Dt)

s.a. : ai ∈ R, ∀i ∈ {0, 1, 2}

5

To inicializate this process, we need initial values of each parameter a0, a1 and a2. Wetake this initial values:

a0 = Dreal, a1 = 1, a2 ∈ [0, 1]

The value of the real demand is known (like the sum of the countries’ demand). For theestimated demand, we will use the logistic function.

In the logistict function, it appears the temperature. As temperature is a linear com-bination of the countries’ temperatures, we have to estime this other parameters wi tofind out which country has more influence to the temperature.

In conclusion, if we considerer

Dt = a1 +a2

1 + exp (a3 ∗∑n

i=1wi ∗ Ti)

we have to resolve this problem:

min(Dreal −Dt)

s.a. :n∑

i=1

wi = 1 (2.1)

wi ≥ 0, ∀i ∈ {1, .., n}ai ∈ R, ∀i ∈ {0, 1, 2}

If we resolve it, we find the best temperature of each climate area.

The parameters we have obtained (for each climate area and Europe) are:

Mid-East Northern Atlantic Mediterranean Europe

a0 0,0371 0,0410 -0,0452 0,1537 -0,0160a1 1,2022 0,9177 1,6482 2,6971 1,5373a2 0,1145 0,1698 0,1218 0,2063 0,1317

6

Mid-East Northern Atlantic Mediterranean Europe

w1 0.0723 0.2482 0.0095 0.0000 0.0427w2 0.0000 0.1443 0.2736 0.2476 0.0000w3 0.1797 0.0275 0.2574 0.1362 0.0289w4 0.0010 0.0000 0.0570 0.2661 0.0000w5 0.1620 0.0600 0.1430 0.0677 -0.0000w6 0.1513 0.1904 0.2595 0.2824 0.0939w7 0.1623 0.3295 0.0000w8 0.1042 -0.0000w9 0.1672 0.0999w10 -0.0000w11 0.0259w12 0.0000w13 0.0000w14 0.0696w15 0.0317w16 0.0140w17 0.0181w18 0.0357w19 0.0710w20 0.0000w21 0.1054w22 0.0912w23 -0.0000w24 0.0788w25 0.0031w26 0.0842w27 0.0385w28 0.0674

7

2.3. Estimate the temperatures dynamic’s and simu-

late his evolve for this year

In this section, we want to analyze the behaviour of the temperature time series.

Obviously, temperature’s behaviour can be expressed as the sum of a seasonal compo-nent St (it’s understandable that the period will be one year) and a stationary stochasticprocess ut. That is, a continuous process at discrete parameter, since we have got tempe-rature datas at fixed time intervals (day by day).

Thus, we proposed the model

Tt = St + ut,

where

St = a+ b ∗ cos(w ∗ t).

Subsequently, we defined a function using nlinfit Matlab function for nonlinear regression,estimating the coefficients using iterative least squares estimation (i.e., minimizing the

function F (a, b, w) =∑size(T )

t=1 [Tt − (a+ b ∗ cos(w ∗ t))]2), with initial values

a = T , b = max(T )−mın(T ), w =2π

365

Using these Matlab routines for European temperature, we obtained the result shown inthe following figure. In particular, as you can see, the red line represents the st profileobtained with such Matlab codes.

8

We did the same for each cluster temperature, too, although the results (similar to theprevious) are not shown in this report.

Then, for both approaches (European and each cluster temperature), we calculated thestationary stochastic term ut, and used these time series to infer dynamic models genera-ting datas.

In particular, for European temperature, we identified an AR(2) model, whose expressionis

(1− L) ∗ (1− φ1L− φ2L2) ∗ ut = (1 + θ1L) ∗ at, at ∼ N(0, σ2).

We did the same for the climate areas using a VAR(2) model. The stationary stochasticterms u1t, ..., u4t can be written like

u1t.u4t

=

φ111 . φ1

14

. . .φ141 . φ1

44

u1t−1.u4t−1

+

φ211 . φ2

14

. . .φ241 . φ2

44

u1t−2.u4t−2

+Q, Q ∼ N(0,Σ).

Then, we estimated the parameters of the models and verified them for the Europeantemperature and the temperature of the four climate areas, validating the models throughautocorrelation and partial correlation function analysis.In the ACF and PACF plots (firstly, the European zone and, then, each climate area), wecan check we obtain white noise process in our models.

9

In the successive time, we forecasted temperature for a 458-days period, from Oct, 1st,2015 (note that we have datas till Sep, 30th, 2015) to Dec, 31st, 2016, via Montecarlosimulation methods, as we can see in the following figure, showing feasible behaviours. Infact, all the plots show a periodic profile, as we expected.

10

11

2.4. Compare the peak simulate demand with histo-

rical

With this simulation data and the relationship between the demand and the temperaturewe estimated before, we obtend the simulated demand’s behaviour in two escenarios: fourclimate areas (like the agregated demand) and the European demand.

If we simulate the demand as the same process we used with the temperature, we ob-tend this graphs:

It’s clear to see the demand’s graphs are inverses than the temperature’s graphs. It reaf-firms the decision we took at the begining of the report to use the logistics function toexplain the relationship between the demand and the temperature.

12

In this other graphs, we can apreciate the hisotrical demands (as the blue serie) and thesimulated demands (represented by the multicolor series). Also, we can see a horizontalred line which represents the historical peack demand.

In both simulations graphs, it appears the number of simultations which exceed the his-torical peack deamand. In fact, there’re less simulations in the firts graph. Also, we canfind much extreme temperatures in the second scenario.

In conclusion, everything we have told seems the first situation (four climate areas, usingthe agregated demand) is more credible and realistic than the second. It facts it’s a greatidea choose the four climate areas to explain the demand.

13

3 Optimize

Once we have obtained the peak demand from both points of view we have considered, wemust assess whether it is possible to transport by pipeline network and European sourcesof supply Europe has different types of supplies.

Europe has different types of supplies, which are:

Producing countries (Norway, Algeria, Russia, Turkey and Libya)

LNG regasification plants (LNG terminal)

Underground storage (UGS)

In addition the European pipeline flow allows the interconnection of different countriesfollowing the following graph:

The objective is to analyze whether these points can be transported demands obtainedby minimizing the cost of transport.

14

We consider the costs of supplies stated above as constants:

15 euros producing countries

19 euros LNG

14 euros UGS

However the price of interconnection is calculated as the weighted average of the pricesof two countries, acting as weights demand without interconnections, that is:

pAB =xAPA + xBpBxA + xB

pA =

∑s∈S(A) p(S)x(S)∑

S∈S(A) x(S), S(A): supplies of country A

The price of each country is a weighting of the prices of each of its supplies.

3.1. Modelling

Therefore, as we have indicated above, the target will be to evaluate the following twoproblems where C denotes the set of countries, S(C) the set of procurement associatedwith a country and I the set of interconnections within the network that would the edgesof the graph between European countries.

Min Cost =n∑

C=1

∑S∈S(C)

pSxSC +

n(I)∑i∈I

pixi

(P1):[As AI ] [xS xl] = peakDE

L ≤ [xSxl] ≤ U

15

Min Cost =n∑

C=1

∑S∈S(C)

pSxSC +

n(I)∑i∈I

pixi

(P2):

[As AI ] [xS xl] = peakDClimate Area

L ≤ [xSxl] ≤ U

ASC =

{1 , S(C)0 , otherwise

Aij =

{−1 , Cap(i, j) > 00 , Cap(i, j) = 0

Aji =

{−Aij , Cap(j, i) 6= 0 ∧ Cap(i, j) 6= 0

0 , otherwise

Additionally, this problem is also evaluated for historical peak.

The historical peak obtained from the cluster analysis is more moderate than the onewe obtained considering a temperature which are involved in all European countries, withtheir growth compared to the historical peak 1,8 % and 6,8 % respectively.

The result of the optimization is that the system could carry the historical demand andoptimal cost is 459K euros, could also convey the peak obtained from climatic areas withthe cost of 465K euros. Hhowever could not transport the demand for the peak calculatedwith only one European temperature.

16

4 Conclusions

We have identified four climate areas in Europe: mediterranean, atlantic, northern andmid-east.

We get lower peaks if we consider climate areas temperature simulations. The proba-bility of pass the historical peak are 0,4 % and 1,2 % from climates area and all Europe,respectively.

The optimal costs in historical peak demand is 459K euros.

Peak demand estimation from climate areas could be transported by the actual network(465K euros). However, the result from only one climate area needs the capacity to beincreased.

17