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Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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Page 1: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

Power Station Control and OptimisationAnna Aslanyan

Quantitative Finance Centre

BP

Page 2: 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

Page 3: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 4: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 5: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 6: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 7: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 8: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 9: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 10: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

* qq

*

*min

q

qL

Page 11: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 12: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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

Page 13: Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

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