Constrained Functional Time Series: an Application to the Italian - - PowerPoint PPT Presentation

1 Università di Torino, Italy 2 Collegio Carlo Alberto, Italy 3 MOX – Department of Mathematics, 


Politecnico di Milano, Italy Mercati energetici e metodi quantitativi un ponte tra Università e Aziende Padova, 8 Ottobre, 2015

Constrained Functional Time Series: an Application to the Italian Natural Gas Balancing Platform Antonio Canale 1,2 e Simone Vantini 3

The Infrastructure and the Shippers

SNAM Rete Gas Gas Shippers One Large Few Medium-size Many Small

Gas does not travel at “speed of light” Possible mismatch between the daily demand and the daily availability Risk of unbalance of the entire pipeline system Mitigation / adaptation and physical / normative strategies are used

Main Issues

For example:

• Injection / ejection of gas in geological traps (i.e., former oil or gas fields)
• Liquefying / gasifying stations
• Economic penalties for shippers based on their daily and monthly unbalance
• Gas balancing platform or Mercato del Bilanciamento (i.e., a system in which 


gas shippers sell and buy natural gas already in the national pipeline)

Gas Balancing Platform: a toy example

Six Buying Bids Four Selling Offers Demand Curve Supply Curve p* = 4.6 q* = 4.0

Why predict tomorrow exchange price p* ? Because it provides a threshold for having buying bids and selling offers accepted. Shippers want indeed to buy at low price or to sell at high price. [Interested actors: Shippers] Why predict tomorrow demand and supply curves pD(q) and pS(q) ? Because they give the possibility of introducing “non-standard” competitive strategies. By means of non-standard bids or offers, “big” shippers could manipulate p* and q*
 for saving/earning money and/or preventing competitors’ bids/offers to be accepted. [Interested actors: Large and Medium-size Shippers]

Why one-day ahead prediction?

Non-standard strategies: a toy example

Standard 
 buying bids and selling

• ffers

p* = 4.6 q* = 4.0 p* = 10.0 q* = 6.0 I can buy 3.0 GJ at 10 €/GJ p* = 7.0 q* = 6.0 I can buy 3.0 GJ at about 7 €/GJ

Standard 
 buying bids and 
 selling offers + my non standard buying bid (p = 10, q = 3.5) Standard 
 buying bids and 
 selling offers + my non standard buying bid (p = 7.0, q = 3.5)

V X V X

Dataset

Supply Curves Demand Curves

• 395 demand and supply curves available (1st Dec 2011 - 31st Dec 2012)
• monotonic non-decreasing / non-increasing
• bounded at the right-edge of the domain
• constrained at the left-edge of the domain
• Possible covariates:
• predicted unbalance
• predicted temperature (…)
• calendar date
• Possible trend (first months removed)
• Possible seasonality
• Large derivatives
• Strong temporal dependence

Dataset Description and Challenge

One-day ahead prediction of functions subjected (i) to monotonicity constraint,
 (ii) to an equality constraint at the left-edge of the domain, and 
 (iii) to an inequality constraint at the right-edge of the domain

The Method at a Glance

Constrained Space (i.e., M2(a,b) ) Unconstrained Sub-space of L2(a,b) (i.e., logH(M2(a,b)) )

transf.

• inv. tran

FAR(p) M2-FAR(p)

Functional Autoregressive Models (i.e., FAR). 
 This are autoregressive models in which the sequence of (1d) random variables is replaced by a sequence of functional random variables:

Dealing with 
 prediction of time-dependent functions

I. Exponential Smoother or similar: Ψj is a real number. II. Concurrent FAR(p): Ψj is a function.

• III. Non-concurrent FAR(p): Ψj is a Hilbert-Schmidt operator.

Results

M2 – Root Mean Square Error Scalar Root Mean Square Error

Reference Paper

Canale, A., Vantini, S. (2014): "Constrained Functional Time Series: an Application to Demand and Supply Curves in the Italian Natural Gas Balancing Platform" Mox Report 42/2014, Department of Mathematics, Politecnico di Milano