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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 l.jpg

Power Station Control and Optimisation

Anna Aslanyan

Quantitative Finance Centre

BP


Background l.jpg
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


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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