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.
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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
Basic principles of liberalized gas market
both at national and local levels
realized by Regulatory Authority by setting a maximum price
they may be required to pay
Consumers whose annual consumption does not exceed 8 · 106 MJoule
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)
Maximum price for classes 1  6 set by Regulatory Authority
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
QE : raw material cost
QVI : wholesale commercialization cost
QL : costs of rigassification of liquid gas
QT : trasportation cost
QS : storage cost
TD : distribution cost
QF : fixed retail commercialization cost
QVD : variable retail commercialization cost
shipper costs
national distributor costs
local distributor cost
gas seller costs
Shipper – gas seller interaction
Purchase price and citygate consumption profile served
constant consumption
in all months
Shipper consumption preferences served
more than two thirds of gas consumption
concentrated in winter months
less than half of daily capacity
used on average
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 served
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)
intervals with different unitary penalties are numbered from 0 to K
Gas seller commercial policies: sell prices served
maximum price, qvj , fixed by Regulatory Authority
with possible discount, sj, 0 sj < 1, fixed by gas seller
Pj” : fixed by gas seller, who relates selling price for class j to
Prices for industrial customers: customer consumption profiles






gas consumption from November to March
gas consumption of thermal year
2) average daily consumption of customer j
Regulatory Authority





n° of days in month i
Example: citygate Sotto il Monte  thermal year 2003’04
Vm_customerij : average monthly consumption per customer of class j in month i
Example: 2003’04 ratio and loading factor per consumer class
in citygate Sotto il Monte (thermal year 200304)
Citygate gas demand depends on gas demand of each consumption class
weigthed by the number of customers in each class (ncj)
positive term subtracted if
LF_customerj > LF_citygate
(better than)
positive term subtracted if
_customerj < _citygate
(better than)
e.g.: coeff1 =2.126, coeff2 =2.554, rechargej = 1.1
Data of deterministic model 2003’04
Consumption estimates 2003’04
The deterministic optimization model 2003’04
Objective function: gas seller profits 2003’04
revenues from classes 1 – 6
revenues from classes 7 – 10
costs
penalties
(citygate demand in month i)
(citygate annual demand)
(citygate winter demand)
The gas seller citygate model is a 2003’04
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
Model validation 2003’04
marginal profits give indications about
possible further reduction of price
for first 6 classes through parameter sj
Model validation 2003’04
Dependence on temperature of consumption of classes 1 2003’04– 6
Building scenarios of future temperatures 2003’04
Data:
minimum and maximum daily temperature (degree Celsius)
measured in Bergamo from 1.1.1994 to 30.11.2005
Tt
Observing historical data:
temperature is a
mean reverting process,
reverting to some cyclical function
t
Histogram of daily temperature differences 2003’04
dWt : Wiener
process
at : speed of mean
reversion
mean value, which the
process reverts to
t : process
volatility
Tt: process to be
modelled
Deterministic model of temperature 2003’04
max and min temperatures
do not necessarily occur
at January 1st and July 1st
it models cyclic
behaviour
in the year
global warming trend
assumed to be linear
By using addition formulas for sin function,
a linear model in unknown parameters a1, a2, a3 and a4 is obtained
Values of 2003’04a1, a2, a3 and a4 that
correspond to following parameter values in model of temperature
A = 13.33
B = 6.8891 · 105
C = 10.366
= – 1.7302
Estimation of volatility 2003’04t
We only need a value of volatility for each month
t is taken as a piecewise function,constant during each month
initi : number of the day in the year at which month i begins
mean of squared differences
between temperature values of two subsequent days
Process simulation 2003’04
From the differential equation
we obtain the approximation scheme
t, 1 t 365, independent
standard normally distributed
random variables
by which nscen temperature scenarios
for 365 days ahead are built
Representation of scenarios 2003’04
Scenarios are then expressed in Heating Degree Days
Tts max {18° – Tts , 0}
mean temperature in monthi,
as available consumption data refer to months
expected value of random variable Tmis
over scenarios
deviation of mean temperature in month i
from mean value over scenarios
Monthly consumption, dependent on temperature, of first 6 classes
random
variable
slopeSij : consumption variation
for unitary temperature variation
in month i for class j
Objective function of stochastic model classes
Expected value of profits, given by
revenues from classes 1 – 6
revenues from classes 7 – 10
costs
penalties
Expected value of profits classes
C
P
where
Model validation classes
Solution of stochastic model: 1000 scenarios classes
5
Solution of stochastic model: 10000 scenarios classes
5
Computed solutions classes
Case study: citygate Sotto il Monte (thermal year 200405)
optimal values of deterministic model
optimal values of stochastic model
Future work classes