Power Station Control and Optimisation

1
Power Station Control and Optimisation

• Anna Aslanyan
• Quantitative Finance Centre
• BP

2
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

3
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

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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

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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

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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

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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

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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

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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

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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

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Lagrangian relaxation (contd)

• Initialise and its range
• Update
• to move towards along a subgradient
• Anything more suitable for mixed-integer
  (non-smooth) problems?

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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

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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