A stochastic optimization model

for natural gas sale companies

M.T. Vespucci (*), F. Maggioni (*), E. Allevi (#), M. Bertocchi (*), M. Innorta (*)

(#) University of Brescia

(*) University of Bergamo

Structure of presentation

• description of some principles of gas market liberalization

• details of problem to be solved

• deterministic model

• stochastic model

1. unbundling of production and transportation activities both at national and local levels

national level: • shippers: production, import, re-gasification, wholesale commercialisation• national distributor: transport on national network and storage local level: • local distributors: transport on local networks• gas sellers: purchase gas from shippers and sell it to final consumers

2. protection of so called “small final consumers”realized by Regulatory Authority by setting a maximum price they may be required to pay

Basic principles of liberalized gas market

Small final consumersConsumers whose annual consumption does not exceed 8 · 106 MJoule

classes of consumption

annual consumption in MJoule

minimum maximum

1 1 20 000

2 20 001 35 000

3 35 001 100 000

4 100 001 160 000

5 160 001 800 000

6 800 001 8 000 000

7 8 000 001 80 000 000

8 80 000 001 800 000 000

9 800 000 001 8 000 000 000

10 8 000 000 001

domestic customers(cooking, cooking/heating)

commercial activities, crafts and small industries

medium and large industries

classes 1- 6 : high consumption proportion is for heating (depends on weather conditions)

industrial customers: consumption for production (independent on weather conditions)

Gas maximum price set (and periodically revised) by Regulatory Authority

on the basis of following splitting of cost

QE + QVI + QL + QT + QS + TD + QF + QVD

where

Maximum price for classes 1 - 6 set by Regulatory Authority

QE : raw material cost QVI : wholesale commercialization costQL : costs of rigassification of liquid gas

QT : trasportation costQS : storage cost

TD : distribution cost

QF : fixed retail commercialization costQVD : variable retail commercialization cost

shipper costs

national distributor costs

local distributor cost

gas seller costs

Gas seller purchases gas from shipper on the basis of a contract:

one contract for each citygate operated by gas seller for each thermal year (July 1st – June 30th)

In the contract are indicated

• gas volume required by gas seller for next thermal year (Va_citygate)

• gas volume required in particular in winter months (Vw_citygate)

• maximum daily consumption (capacity) requested by gas seller (Cg_citygate)

• purchase price fixed by shipper (P)

In the contract it is also specified how to compute penalties, which are due by gas seller if daily consumption exceeds daily capacity.

Shipper – gas seller interaction

Gas seller problem

Gas seller needs a model for determining optimal decisions about

1. number of final customers to supply in each consumption class

(so called citygate customer portfolio)

2. sell prices to apply to each consumption class

Different customer portfolios determine different citygate consumption patterns.

In a citygate where mainly industrial customers are served

• citygate consumption tends to be constant along the year,

since it is used more for production than for heating

• it is easier for gas seller to determine the citygate daily capacity to require

Shippers prefer a more constant citygate consumption along the year,

a lower purchase price for gas seller is set by shipper

In a citygate where mainly “small” customers are served

(consumption strongly dependent on weather conditions)

• citygate consumption may have big fluctuations in the year

• daily consumption may exceed daily capacity

a higher purchase price for gas seller is set by shipper

penalties are likely to be paid by gas seller

 

Revenue side:

gas seller can fix “freely” only prices for industrial consumers:

• these prices have to be set so as to maximize gas seller profits

• at the same time the price proposed to the industrial customer

does not have to result in the customer buying gas from another gas seller

Purchase price and citygate consumption profile

How does shipper fix price P paid by gas seller ?

• a mathematical relation is not known

• historical data show that better citygate consumption profiles correspond to lower prices P

Consumption profile indicators:

1. ratio among winter consumption and annual consumption

2. ratio of average daily gas consumption of gas seller and daily capacity Cg_citygate (citygate loading factor)

0.4eVa_citygat

eVw_citygatα_citygate constant consumption

in all months

eCg_citygat

eVa_citygateLF_citygat

365

0.70 _α : bad0.55 α_ :good citygatecitygate

• citygate consumption as constant as possible along the year

more than two thirds of gas consumptionconcentrated in winter months

0.40 : bad0.60 : good citygate_LFcitygate_LF

less than half of daily capacity used on average

• average daily use of “virtual pipeline” as high as possible

Shipper consumption preferences

Linear regression model of purchase price P onto LF_citygate:

P = QT + QS + intercept + slopeP · LF_citygate

Model based on both LF_citygate and _citygate not significant, _citygate and LF_citygate being highly correlated.

Computation of penalties

If daily consumption in month i exceeds daily capacity, penalties are applied by shipper to gas seller.

Percentages ik and unitary penalties ik (Euro/m3) for computing penalties are set in the contract.

Example: citygate Sotto il Monte (thermal year 2003 - ’04)

k 0 1 2

ik i = 5,…,9 10% 15% > 15%

ik i = 5, 6 0 3.54 3.94

ik i = 7, 8, 9 0 3.73 4.13

intervals with different unitary penalties are numbered from 0 to K

Gas seller commercial policies: sell prices

611 jsqvP jj'j

protected customers (classes 1 – 6):

maximum price, qvj , fixed by Regulatory Authority

with possible discount, sj, 0 sj < 1, fixed by gas seller

industrial customers (classes 7 – 10)

Pj” : fixed by gas seller, who relates selling price for class j to

• gas purchase price, P, paid to shipper

• consumption profile of customers of class j

Prices for industrial customers: customer consumption profiles

12

1

9

5

iij

iij

j

rVm_custome

rVm_customeα_customer

gas consumption from November to March

gas consumption of thermal year

On the basis of historical data of average monthly consumption

per customer of class j in month i (Vm_customerij)

2 indicators of consumption profiles of customer j are computed

1) average monthly consumption of customer j

2) average daily consumption of customer j

j

iij

j peak

rVm_customerLF_custome

365

12

1

n° of days in month i

γi

ij

ij days

rVm_customemaxpeak

||||||

|||||

> 1 coefficient set byRegulatory Authority

month 1 2 3 4 5 6 7

1 July 8.34 10.35 19.26 38.29 175.49 1 064.43 117 250.75

2 Aug. 8.37 10.57 19.75 39.07 75.18 435.14 40 801.00

3 Sept. 10.24 24.44 50.19 87.79 311.58 2 174.89 156 927.50

4 Oct. 14.19 53.65 114.34 216.33 593.79 2 037.68 154 460.25

5 Nov. 26.51 96.79 209.04 373.70 1 010.40 4 643.43 134 459.25

6 Dec. 34.56 127.01 275.41 528.49 1 302.36 5 674.78 135 771.75

7 Jan. 24.83 147.72 287.28 602.93 1 469.27 6 696.33 153 813.00

8 Feb. 23.50 112.58 265.67 518.42 1 259.50 5 523.37 172 910.60

9 Mar. 19.03 89.46 192.96 375.39 813.40 4 393.51 166 755.40

10 Apr. 24.50 42.54 81.70 214.90 718.95 3 129.60 138 427.75

11 May 10.76 28.29 58.66 108.15 248.84 1 351.53 140 747.00

12 June 9.01 15.32 30.17 37.13 223.48 988.28 138 472.75

Example: citygate Sotto il Monte - thermal year 2003-’04

Vm_customerij : average monthly consumption per customer of class j in month i

Example: ratio and loading factor per consumer class in citygate Sotto il Monte (thermal year 2003-04)

consumption class 1 2 3 4 5 6 7

_customerj 0.601 0.759 0.767 0.764 0.714 0.707 0.463

LF_customerj 0.404 0.340 0.356 0.340 0.365 0.372 0.628

First 6 consumption classes:• their consumption behaviours have a negative impact on citygate

consumption profile• therefore gas seller would tend to apply high price to counterbalance

the high purchase price due to bad citygate profile • but regulations protect these customers by setting a price cap

irVm_customej

ij in demand citygate 10

1

monthnceVm_citygat ji

Citygate gas demand depends on gas demand of each consumption classweigthed by the number of customers in each class (ncj)

j

j

j

j

recharge

_customercoeff

rLF_customecoeff

PP

α2

11

"

_citygate

eLF_citygat

α

Last 4 consumption classes:

• their consumption behaviours have a positive impact on citygate consumption profile

• gas seller should determine a sell price Pj” that attracts these customers,

in order to pay a lesser purchase price P to shipper

and therefore increase profits, in particular from first 6 classes

positive term subtracted if LF_customerj > LF_citygate (better than)

e.g.: coeff1 = 2.126, coeff2 = 2.554, rechargej = 1.1

positive term subtracted if _customerj < _citygate

(better than)

Data of deterministic model

• Vm_customerij : average monthly consumption per customer of class j

in month i of previous thermal year (historical data)

• : maximum number of customers available belonging to class j

• QT, QS, , qvj for 1 j 6 : set by Regulatory Authority

• intercept, slopeP : determined by regression on historical data

• sj for 1 j 6, rechargej for 7 j 10, coeff1 and coeff2 : set by gas seller

• ik and ik (widths of intervals from 0 to K–1 for computing penalties)

jnc

Assumption:average monthly consumption of each class may be estimated by previous year consumption

Therefore, following estimates are used for next thermal year:

• annual consumption of customer of class j

Consumption estimates

12

1iijj customer_Vmcustomer_Va

• peak consumption of customer of class j (for penalty computation)

γi

ijij days

customer_Vmpeak

The deterministic optimization model

Find values of

• number of customers per class

ncj , 1 j 10 (integer)

• contract parameters

Vm_citygatei , 1 i 12

Va_citygate, Vw_citygate and Cg_citygate

• variables related to penalties computation

surplusik , 5 i 9 and 0 k 2

that maximize gas seller profits

9

5 0

10

7

6

1

π

365

α2365

11

365

1

i

K

kik

jj

jj

j

jjjj

slopePerceptintQSQT

customer_Vaeargrech

_customercoeffrLF_custome

coeff

slopePerceptintQSQT

customer_Vasqvprofit

ik

j

j

surplus

eVa_citygateCg_citygat

eVa_citygat

nc

eVa_citygat

eVw_citygateCg_citygat

eVa_citygat

eCg_citygat

eVa_citygat

nc

Objective function: gas seller profits

revenues from classes 1 – 6

reve

nues

fro

m c

lass

es 7

– 1

0

penalties

costs

9

5

12

1

10

1

i

i

jijrVm_custome

i

i

ji

eVm_citygateVw_citygat

eVm_citygateVa_citygat

nceVm_citygat

(citygate annual demand)

(citygate winter demand)

(citygate demand in month i)

subject to

• constraints defining monthly, winter and annual citygate consumption

• constraints for computing penalties

• lower and upper bounds on number of customers per class, restricted to be integer

1 0and95ω

952

0

10

1

Kki

ipeak

ik

kjij

eCg_citygatsurplus

surpluseCg_citygatnc

ik

ikj

0

allfor integers and0

iki

j

surpluse,Cg_citygat e,Vw_citygat e,Va_citygateVm_citygat

nc

,

jnc j

The gas seller citygate model is a

nonlinear mixed integer model with linear constraints

nonlinearities, coming from definitions of LF_citygate and _citygate,

appear only in the objective function.

Simulation framework: based on

ACCESS 97, for database management

MATLAB, release 12, for data visualization

GAMS, release 21.5, for optimisation

Optimization solver: DICOPT (in GAMS framework) solves a sequence

of NLP subproblems, by CONOPT2,

and MIP subproblems, by CPLEX

citygate Sotto il Monte

recharge7 = 0 €cent/m3

recharge7 = 1.1 €cent/m3

nc7 not accepted accepted

profit 103 717 110 303 €

P 20.277 19.920 €cent/m3

24.727 24.727 €cent/m3

18.448 19.799 €cent/m3

Va_citygate 2.33 · 106 3.77 · 106 m3

Vw_citygate 1.76 · 106 2.42 · 106 m3

Cg_citygate 20 588 25 664 m3

_citygate 0.755 0.642

LF_citygate 0.310 0.402

1. analyse the impact on overall citygate management of industrial consumer

(_customer7 = 0.463 and LF_customer7 = 0.628)

Model validation

recharge7 = 1.1

class ncj

marginal

profit

1 278 8.16

2 256 22.97

3 618 74.29

4 139 72.91

5 49 646.96

6 2 1 679.60

7 1 11075.00

marginal profits give indications about

possible further reduction of price

for first 6 classes through parameter sj

61 j,P'j

"P7

2. investigate commercial policies towards customers of consumption class 6

(whose annual consumption is 73% of total annual consumption of first 6 classes)

Model validation

citygate Sotto il Monte

recharge7 = 1.1 s6 = 18.5 %

recharge7 = 1.1 s6 = 18.7 %

nc6 accepted not accepted

nc7 accepted not accepted

profit 106 984 100 366 €

P 19.920 20.282 €cent/m3

20.153 20.103 €cent/m3

19.799 19.550 €cent/m3

Va_citygate 3.77 · 106 2.26 · 106 m3

Vw_citygate 2.42 · 106 1.71 · 106 m3

Cg_citygate 25 664 20 037 m3

_citygate 0.642 0.758

LF_citygate 0.402 0.309

"P7

"P6

Dependence on temperature of consumption of classes 1 – 6

Gas consumption of first 6 classes strongly depends

on temperature variations along months

Investigate impact of weather conditions on optimal gas seller decisions

• build scenarios of future temperatures

• stochastic version of the model

• numerical experiments

Building scenarios of future temperatures

Data:

minimum and maximum daily temperature (degree Celsius) measured in Bergamo from 1.1.1994 to 30.11.2005

and maxt

mint TT

2 : day of raturemean tempe Compute

maxt

mint

t

TTTt

Observing historical data:temperature is a mean reverting process, reverting to some cyclical function

t

Tt

Histogram of daily temperature differences

temperature differences

between 2 subsequent days

approximate a normal distribution

temperature process to be modeled as a Brownian Motion

ttt

tttt dWdtdt

dTadT σ

at : speed of mean reversion

Tt : process to be modelled

mean value, which the process reverts to

:t t : processvolatility

dWt : Wienerprocess

Deterministic model of temperature

365

2πtsinCBtAt

it models cyclic

behaviourin the year

(phase angle): max and min temperaturesdo not necessarily occurat January 1st and July 1st

global warming trendassumed to be linear

365

2π

365

2π

4321

tcossinC

tsincosCtBA

aaaat

By using addition formulas for sin function,

a linear model in unknown parameters a1, a2, a3 and a4 is obtained

Values of a1, a2, a3 and a4 that

correspond to following parameter values in model of temperature

t

tt Ta,a,a,amin 24321

A = 13.33

B = 6.8891 · 10-5

C = 10.366

= – 1.7302

Estimation of volatility t

We only need a value of volatility for each month

t is taken as a piece-wise function, constant during each month

1

month of days allfor 1

σ1

1

21

2

iii

daysinit

initkkk

it

daysinittinit

itTTdays

ii

i

initi : number of the day in the year at which month i begins

mean of squared differencesbetween temperature values of two subsequent days

month Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

1.65 1.54 1.75 1.85 1.81 1.99 1.76 1.63 1.48 1.38 1.51 1.49tσ

Estimation of speed of reversion at

Efficient estimator of at is

1: month of days allfor

σ

σlogˆ

1

1121

11

1

21

11

iii

daysinit

initkkk

k

kk

daysinit

initkkk

k

kk

t

daysinittinit it

TT

TT

aii

i

ii

i

(Bibby and Sorensen, 1995)

ttttttttt ˆTaaT εσ1 11111

Process simulation

ttt

tttt dWdtdt

dTadT σ

From the differential equation

we obtain the approximation scheme

t, 1 t 365, independent standard normally distributedrandom variables by which nscen temperature scenarios

for 365 days ahead are built

st

T st

scenario of day in

Celsius) degree(in etemperatur

Scenarios are then expressed in Heating Degree Days

Tts max {18° – Tt

s , 0}

si

i

daysinit

initk

st

Tmdays

Tii

i

1

si

nscen

s

si

Tmnscen

Tm

1 expected value of random variable Tmi

s

over scenarios

mean temperature in month i,as available consumption data refer to months

si

si

si TmTmD deviation of mean temperature in month i

from mean value over scenarios

Representation of scenarios

Representation of scenarios: for each month i

• interval between is divided into nint sub-intervals

with middle point Dir being the representative point of sub-interval

(r : sub-interval index, 1 r nint)

• to temperature difference Dir we associate

– monthly gas consumption

– probability PRir based on frequency

(n° of scenarios of month i belonging to sub-interval r)

si

s

si

sDmaxDmin and

slopeSij : consumption variation for unitary temperature variationin month i for class j

Monthly consumption, dependent on temperature, of first 6 classes

siijij

sij DslopeScustomer_Vmcustomer_Vm

irijijijr DslopeScustomer_Vmcustomer_VmR

randomvariable

9

5 0

10

7

6

1

π

365

α2365

11

365

1

i

K

kik

jj

jj

j

j

sjj

slopePerceptintQSQT

customer_Vaeargrech

_customercoeffrLF_custome

coeff

slopePerceptintQSQT

rVa_customesqvprofitj

s

ss

j

s

s

s

s

j

iksurplus

eVa_citygateCg_citygat

eVa_citygat

nc

eVa_citygat

eVw_citygateCg_citygat

eVa_citygat

eCg_citygat

eVa_citygat

nc revenues from classes 1 – 6

reve

nues

fro

m c

lass

es 7

– 1

0

penalties

costs

Objective function of stochastic modelExpected value of profits, given by

By computing the expected value of profits we obtain that

• terms representing expected values of revenues from classes 1 – 6 and 7 – 10: coincide with corresponding terms of deterministic model

• terms representing expected values of costs and of penalties: computed under the hypothesis that random variables Di

s and Djs , i j , are independent

Expected value of profits

9

5 0 1

12

1 1

12

1 1

2

π

2

365

12

21

2211

i

2

k

intn

ririk

iii,i

intn

rriri

i

intn

rir

PR

PRPRPR

slopePerceptintQSQT

ikr

ririir

surplusR

teVmR_citygateVmR_citygateVmR_cityga

eCg_citygateVa_citygat

jjir ncncteVmR_cityga

10

7

6

1

j

ijj

irijij customer_VmDslopeScustomer_Vmwhere

C

P

Model validation

• sensitivity of solution to generation of different numbers of scenarios

(a) 1000 scenarios

(b) 10000 scenarios

• given a number of scenarios, how solution changes as representations

of a given number of scenarios become more and more refined

i.e. number of sub-intervals increases: nint = 5, 10, 15, ……

Note:

both in (a) and in (b),

as scenario representation becomes more and more refined (nint increases),

optimal profit converges to a value between 151 700 and 151 710,

Such value is lower than the value (154 265) obtained in the deterministic case,

as expected, because cold scenarios have been taken into account

Solution of stochastic model: 1000 scenarios

5

Solution of stochastic model: 10000 scenarios

5

optimal values of deterministic model

profit 154 265 €

P 19.67 €cent/m3

Va_citygate 4 484 406 m3

Cg_citygate 26 399 m3

LF_citygate 0.4654

Computed solutions

Case study: citygate Sotto il Monte (thermal year 2004-05)

optimal values of stochastic model

profit 151 713 €

P 19.67 €cent/m3

Va_citygate 4 484 725 m3

Cg_citygate 26 319 m3

LF_citygate 0.4669

• in the deterministic solution no penalties are paid

• solution of stochastic model: by requiring in the contract 4 484 725 m3 gas

(expected value of annual volume), gas seller has an expected value of profit

lower than in the deterministic case, since the expected value of penalties is

positive. This solution, though, allows gas seller to have the same purchase price

of the deterministic case and therefore the same selling price for the industrial

customer.

Future work

1. As number of scenarios and number of sub-intervals increase,

complexity of the problem also increases (numerical difficulties)

devise a new algorithm that decouples computation of Cg_citygate

from all other decision variables, so that reduced problem is linear

2. Scenarios not represented on a monthly basis, but treated as vectors of 12

realizations scenario reduction techniques

3. There exists a relation between purchase price P and international price indeces,

since gas seller must choose the index of reference among a certain number of

admitted choises (e. g. oil price index, etc.)

Study the influence on P of future variations of these indices to help gas seller

in taking his decision.