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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

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power station control and optimisation

Power Station Control and Optimisation

Anna Aslanyan

Quantitative Finance Centre


  • 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
  • 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
definitions contd
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

Trading of carbon-neutral spark spreads of interest to anyone with exposure to all three markets

  • Attractive as
    • speculation
    • basis risk mitigation
    • asset optimisation


  • Modelling required to
    • price contract/value power plant
    • determine optimal operating regime and/or hedging strategy
commodities to be modelled
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
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
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
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
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


lagrangian relaxation contd
Lagrangian relaxation (contd)
  • Initialise and its range
  • Update

to move towards along a subgradient

  • Anything more suitable for mixed-integer (non-smooth) problems?
lagrangian relaxation contd12
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
  • 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