Introduction Setting A non-convex control problem Overview of the solutions A model for optimal electricity storage Tiziano De Angelis School of Mathematics – University of Leeds Joint work with G. Ferrari (University of Bielefeld) J. Moriarty (Queen Mary University of London) Leeds 14th June 2016
Introduction Setting A non-convex control problem Overview of the solutions Outline 1 Introduction 2 Setting 3 A non-convex control problem 4 Overview of the solutions
Introduction Setting A non-convex control problem Overview of the solutions Optimal control for local electricity markets Motivation: Optimisation problems which may arise in power systems and energy markets Toy model: Imagine two agents and an electricity store one invests in power from spot market for storage - either slowly and e ffi ciently, or quickly and less e ffi ciently other consumes from the store Investor Consumer Yes No Grid connection Risk Financial Physical Example: Recharging an electric vehicle Investor: vehicle charger Consumer: driver
Introduction Setting A non-convex control problem Overview of the solutions A natural problem for the investor Statement of the problem : ( P ) What is the optimal way to buy electricity from the market for storage and committing to fully meet the demand (at a random future time)? The mathematical features Problem ( P ) is an optimal control problem with stochastic features (electricity price, time of physical demand, etc.). Electricity price can go negative Electricity does not have to be purchased at a rate (singular controls) and the investor enjoys limited storage (finite fuel) Random arrival of demand from the customer (random time-horizon)
Introduction Setting A non-convex control problem Overview of the solutions Basic model assumptions The spot price ( X t ) t ≥ 0 evolves according to a stochastic dynamic Demand for 1 unit of electricity arrives at a random time τ . We take τ indep. of X with τ ∼ Exp ( λ ) for some λ > 0. The storage level C cannot exceed 1 unit (limited storage) and cannot be decreased. It follows a purely controlled dynamics C ν C ν t = c + ν t , t > 0 , 0 = c ∈ [ 0 , 1 ] where ( ν t ) t ≥ 0 is the cumulative amount of electricity purchased from the grid. Notice that t �→ ν t is increasing and left-continuous with ν 0 = 0.
Introduction Setting A non-convex control problem Overview of the solutions The investor aims at minimising future, expected costs . Prior to time τ : cost of increasing the storage At time τ : terminal cost = X τ Φ ( C ν τ ) - proportional to the spot price through a function of the shortfall, An example: Φ ( c ) = α ( 1 − c ) β , α > 0, β ≥ 2 Market discount rate r = 0 (just for simplicity) Casting problem ( P ) The investor aims at finding an optimal strategy ν ∗ for the problem �� τ � U ( x , c ) = inf E X t d ν t + X τ Φ ( C τ ) (1) ν 0 where E [ · ] is the average over all possible scenarios for the spot price and τ .
Introduction Setting A non-convex control problem Overview of the solutions The investor aims at minimising future, expected costs . Prior to time τ : cost of increasing the storage At time τ : terminal cost = X τ Φ ( C ν τ ) - proportional to the spot price through a function of the shortfall, An example: Φ ( c ) = α ( 1 − c ) β , α > 0, β ≥ 2 Market discount rate r = 0 (just for simplicity) Casting problem ( P ) The investor aims at finding an optimal strategy ν ∗ for the problem �� τ � U ( x , c ) = inf E X t d ν t + X τ Φ ( C τ ) (1) ν 0 where E [ · ] is the average over all possible scenarios for the spot price and τ .
Introduction Setting A non-convex control problem Overview of the solutions The investor aims at minimising future, expected costs . Prior to time τ : cost of increasing the storage At time τ : terminal cost = X τ Φ ( C ν τ ) - proportional to the spot price through a function of the shortfall, An example: Φ ( c ) = α ( 1 − c ) β , α > 0, β ≥ 2 Market discount rate r = 0 (just for simplicity) Casting problem ( P ) The investor aims at finding an optimal strategy ν ∗ for the problem �� τ � U ( x , c ) = inf E X t d ν t + X τ Φ ( C τ ) (1) ν 0 where E [ · ] is the average over all possible scenarios for the spot price and τ .
Introduction Setting A non-convex control problem Overview of the solutions The investor aims at minimising future, expected costs . Prior to time τ : cost of increasing the storage At time τ : terminal cost = X τ Φ ( C ν τ ) - proportional to the spot price through a function of the shortfall, An example: Φ ( c ) = α ( 1 − c ) β , α > 0, β ≥ 2 Market discount rate r = 0 (just for simplicity) Casting problem ( P ) The investor aims at finding an optimal strategy ν ∗ for the problem �� τ � U ( x , c ) = inf E X t d ν t + X τ Φ ( C τ ) (1) ν 0 where E [ · ] is the average over all possible scenarios for the spot price and τ .
Introduction Setting A non-convex control problem Overview of the solutions A SSC problem with non-convex costs Mathematical context and challenges . Problem (1) is a singular stochastic control (SSC) problem with 2-dimensional state space. Cost of using control is linear in ν while ν �→ X Φ ( c + ν ) is not convex . We want to minimise a functional ν �→ J ( ν ) which is neither convex nor concave !
Introduction Setting A non-convex control problem Overview of the solutions Existing mathematical literature and novelty mathematics of SSC problems dates back to 1966 (Bather and Cherno ff ) and it has been widely developed since then by many authors a key ingredient is often the convexity of the performance criterion w.r.t. the control variable (concavity for maximisation problems) under convexity assumption (and some other standard conditions): (i) there exists a unique optimal control (ii) the state spaces is split in action and inaction region, and (iii) this is of reflecting type What kind of solution should we expect when convexity breaks down?
Introduction Setting A non-convex control problem Overview of the solutions Existing mathematical literature and novelty mathematics of SSC problems dates back to 1966 (Bather and Cherno ff ) and it has been widely developed since then by many authors a key ingredient is often the convexity of the performance criterion w.r.t. the control variable (concavity for maximisation problems) under convexity assumption (and some other standard conditions): (i) there exists a unique optimal control (ii) the state spaces is split in action and inaction region, and (iii) this is of reflecting type What kind of solution should we expect when convexity breaks down?
Introduction Setting A non-convex control problem Overview of the solutions It turns out that the structure of our problem depends on the choice of the price dynamics and on the sign of k ( c ) := λ + λ Φ ′ ( c ) for c ∈ [ 0 , 1 ] (related to marginal cost of investment) In many practical examples there exists a unique ˆ c ∈ ( 0 , 1 ) s.t. k ( c ) > 0 , c > ˆ c k ( c ) < 0 , c < ˆ c
Introduction Setting A non-convex control problem Overview of the solutions It turns out that the structure of our problem depends on the choice of the price dynamics and on the sign of k ( c ) := λ + λ Φ ′ ( c ) for c ∈ [ 0 , 1 ] (related to marginal cost of investment) In many practical examples there exists a unique ˆ c ∈ ( 0 , 1 ) s.t. k ( c ) > 0 , c > ˆ c k ( c ) < 0 , c < ˆ c
Introduction Setting A non-convex control problem Overview of the solutions Two examples Example 1 . Spot prices with mean reversion, e.g. Ornstein-Uhlenbeck model dX t = θ ( µ − X t ) dt + σ dB t , X 0 = x θ force of mean reversion, µ long-term average, σ di ff usion coe ffi cient. Intuition: harmonic oscillator + Gaussian noise √ Example 2 . Spot prices with X t ∼ N ( x , t ) for some given x ∈ R , i.e. Brownian motion model X t = x + B t see http://stuartreid.co.za/interactive-stochastic-processes/ One may choose other models to fit real data - here we were interested in mathematical tractability and explicit solutions.
Introduction Setting A non-convex control problem Overview of the solutions Two examples Example 1 . Spot prices with mean reversion, e.g. Ornstein-Uhlenbeck model dX t = θ ( µ − X t ) dt + σ dB t , X 0 = x θ force of mean reversion, µ long-term average, σ di ff usion coe ffi cient. Intuition: harmonic oscillator + Gaussian noise √ Example 2 . Spot prices with X t ∼ N ( x , t ) for some given x ∈ R , i.e. Brownian motion model X t = x + B t see http://stuartreid.co.za/interactive-stochastic-processes/ One may choose other models to fit real data - here we were interested in mathematical tractability and explicit solutions.
Introduction Setting A non-convex control problem Overview of the solutions What do we mean by “solution”? An explicit characterisation of the optimal storage strategy - in terms of simple algebraic equations An analytical expression of the value function U - this can be interpreted as the minimum price for a recharge Remark : These expressions can be used to implement real strategies and calculate real prices after the model has been calibrated on real data.
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