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Power Station Control and OptimisationAnna 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 0,max KFuelEfficiencyPowerP
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 half-hours, 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 combining
– cost function C
– penalty (Lagrangian multiplier)
– actual number of half-hours, m and maximum to be allocated, M
• Solve primal problem for a fixed
• Update to solve dual problem
• Iterate until duality gap
vanishes
)()(),( xCMmxL
),( min)( xLqx
)(max0
*
*min
q
qL
Lagrangian relaxation (contd)
• Initialise and its range
• Update
to move towards along a subgradient
• Anything more suitable for mixed-integer (non-smooth) problems?
],[ maxmin
*q
)()(
)()(
maxmin
maxmaxminmax
mm
mM
Lagrangian relaxation (contd)
• Solution sub-optimal (optimal if using DP alone)
• Can be partly improved by redefining the ‘natural undergeneration’ termination condition
• Further optimisation may be required, for example over outage periods
Mm ,0
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