Power Station
Control and Optimisation
Anna Aslanyan
Quantitative Finance Centre
BP
Background
• Tolling (spark/dark spread) agreements widespread in power
industry
• Both physical and paper trades, usually over-the-counter
• Based on the profit margin of a power plant
• Reflect the cost of converting fuel into electricity
• Physical deals facility-specific
• Pricing often involves optimisation
Definitions
• Optimisation problem referred to as scheduling
(commitment allocation, economic dispatch)
• Profit is the difference between two prices (power
and fuel), less emissions and other variable costs
• The latter include operation and maintenance
costs, transmission losses, etc.
• Objective function similar to a spread option pay-off
P  max Power  Efficiency  Fuel  K ,0
Definitions (contd)
Examine power, fuel and CO2 price forecasts and choose
top N MWh to generate, subject to various constraints,
including
• volume (load factor) restrictions
• operational constraints
– minimum on and off times
– ramp-up rates
– outages
Apart from fuel and emissions costs, need to consider
• start-up costs
• operation and maintenance costs
Motivation
Trading of carbon-neutral spark spreads of interest to
anyone with exposure to all three markets
• Attractive as
– speculation
– basis risk mitigation
– asset optimisation
tools
• Modelling required to
– price contract/value power plant
– determine optimal operating regime and/or hedging strategy
Commodities to be modelled
• Electricity
– demand varies significantly
– sudden fluctuations not uncommon
– hardest to model
• Fuel (gas, coal, oil)
– sufficient historical data available
– stylised facts extensively studied
• Emissions
– new market, just entered phase two
– participants’ behaviour often unpredictable
– prices expected to rise
Methodology outline
• Given forward prices for K half-hours and a set of
operational constraints, allocate M generation halfhours, maximising profit or, equivalently, minimising
production costs C
• A. J. Wood, B. F. Wollenberg Power Generation,
Operation, and Control, 1996
•
S Takriti, J Birge, Lagrangian solution techniques and
bounds for loosely coupled mixed-integer stochastic
programs, Operations Research, 2000
– combination of two techniques, dynamic programming and
Lagrangian relaxation
Dynamic programming
• Forward recursive DP formalism implemented to
solve Bellman equation
• Given an initial state, consider an array of possible
states evolving from it
• States characterised by
– cost
– history
– status
– availability
Dynamic programming (contd)
• Ensure that only feasible transitions are permitted
– if the plant is on, it can
• stay on if allowed by availability
• switch off if reached minimum on time
– otherwise, it can
• stay off
• switch on if allowed by availability and reached minimum off
time
• Update the cost for each of these transitions
• Maximise the profit over all possible states at every
stage
Lagrangian relaxation
• Define L( , x)   (m  M )  C ( x) combining
– cost function C
– penalty  (Lagrangian multiplier)
– actual number of half-hours, m and maximum to be allocated, M
q( )  min L( , x) for a fixed 
x
*
• Update  to solve dual problem q  max q( )
• Solve primal problem
• Iterate until duality gap
min L  q
q*
vanishes
*
 0
Lagrangian relaxation (contd)
• Initialise  and its range [min , max ]
• Update 
  max  (min
M  m(max )
 max )
m(min )  m(max )
*
q
to move towards
along a subgradient
• Anything more suitable for mixed-integer (non-smooth)
problems?
Lagrangian relaxation (contd)
• Solution sub-optimal (optimal if using DP alone)
• Can be partly improved by redefining the ‘natural
undergeneration’ termination condition
  0, m  M
• Further optimisation may be required, for example
over outage periods
Summary
• Understanding of tolling deals provides market
players with
– alternatives to supply and/or purchase power
– risk-management instruments
– power plants valuation tools
– ability to optimise power plants
– competence necessary to participate in virtual power plant (VPP)
auctions
• Large dimensionality requires fast-converging
algorithms