Optimal battery charge/discharge strategies for consumers and suppliers Ben Mestel 4 April 2014
Major challenges • To understand the interplay between the electricity market (price) and electricity generation, and what role storage might have in management of the power system. • To understand the dynamics of smart grids consisting of a network of prosumers interacting in the physical, cyber and social layers • Prosumer = Producer/Consumer
Fundamental questions • How can a market be effectively regulated, controlled/stabilised and incentivised? • What is the effect of price on a power system? – Can price be used to regulate a complicated network containing a mix of generators, storers, prosumers? – Is price something we can impose on the system as an exogenous control variable or is it necessarily endogenous , a product of system dynamics?
Model (microeconomic) problem • Given a prosumer with a storage battery (perhaps in an electric/hybrid vehicle) and an exogenous price 𝑞(𝑢) what is the optimal charging/discharging to minimise cost? How should the price be chosen to induce a given behaviour in the prosumer? • In this talk we consider the charging problem and discuss the classical calculus of variations approach and its limitations
Battery types • Lead-acid • Li Ion e.g. LiCoO 2 • NaS • NiCd • ZnBr • LiO 2 Schematic of lithium-air battery charge • and discharge cycles LiS
Battery models • Batteries are complicated beasts ! • Modelling approaches include – Electrochemical and thermal modelling of the electrolyte, electrodes and their interaction – Dynamical modelling – macro modelling of battery behaviour, often involving equivalent circuit diagrams • Modelling of batteries is a well established and growing field. • Little analytical work possible, simulation (often through Matlab and Maple) is main approach
Charging regimes • Constant current • Constant voltage • High current decreased exponentially • Constant current/constant voltage • Constant current/constant voltage/constant current • Pulsed charging • Quick charging • Each regime has characteristic charging time and effect on battery temperature and lifetime • Each battery has its own attributes and manufacturer- recommended charging regimes
Battery models • State variable: 𝑇 = state of charge • 𝑇 = 0 battery fully discharged • 𝑇 = 1 battery fully charged • (Ignore battery temperature 𝜄 ) • Applied current and voltage 𝐽 𝑢 , 𝑊(𝑢) – 𝐽 𝑢 > 0 - charging – 𝐽 𝑢 < 0 - discharging – 𝐽(𝑢) , 𝑊(𝑢) are not independent • Time interval [𝑢 𝑡 , 𝑢 𝑓 ]
Single storage supplier fixed market • Supplier price taker • Prices: exogenous variables, set by power system operator • Two prices in, say, £ per KWh (i.e. money/energy): – 𝑞 𝑝 (𝑢) offer price i.e. price the storage supplier sells electricity – 𝑞 𝑐 (𝑢) bid price i.e. price the storage supplier buys electricity – In general 𝑞 𝑐 𝑢 ≥ 𝑞 𝑝 (𝑢) • Forward pricing on T = 𝑢 𝑡 , 𝑢 𝑓 • Here we consider 𝑞 𝑝 (𝑢) = 𝑞 𝑐 (𝑢) , which we write 𝑞(𝑢)
Charging optimisation • 𝑇(𝑢) state of charge, 0 ≤ 𝑇 𝑢 ≤ 1 , 𝑅 𝑢 = 𝑅 𝑁𝐵𝑌 𝑇 𝑢 𝑢 𝑓 • 𝐷 = 𝑞 𝑢 𝑋 𝑢 𝑒𝑢, 𝑋 𝑢 = 𝐻(𝑇 𝑢 , 𝑇 𝑢 ) 𝑢 𝑡 • For a simple-battery – 𝑋 𝑢 = 𝐽 𝑢 𝑊 𝑢 – 𝑊 𝑢 = 𝑊 𝑃𝐷 + 𝑆 𝑐 𝐽 𝑢 – 𝐽 𝑢 = 𝑅 𝑁𝐵𝑌 𝑇 𝑢 – 𝐻 = 𝐻(𝑇 𝑢 ) = ( 𝑊 𝑃𝐷 + 𝑆 𝑐 𝑅 𝑁𝐵𝑌 𝑇 𝑢 )𝑅 𝑁𝐵𝑌 𝑇 𝑢 – 𝐻 convex for 𝑆 𝑐 > 0
Classical Calculus of Variations Approach • Theorem. Let 𝑀(𝑢, 𝑇, 𝑊) be a twice continuously differentiable function with respect to (𝑢, 𝑇, 𝑊) which is convex with respect to 𝑇, 𝑊 . Then the functional 𝑢 𝑓 𝑀 𝑢, 𝑇(𝑢), 𝑇 𝑢 𝑒𝑢, 𝑇 𝑢 𝑡 = 𝑇 𝑡 , 𝑇 𝑢 𝑓 = 𝑇 𝑓 has a 𝑢 𝑡 minimum path S 𝑢 that satisfies the Euler-Lagrange 𝑒 𝑒𝑢 𝑀 𝑊 𝑢, 𝑇 𝑢 , 𝑇 𝑢 = 𝑀 𝑇 (𝑢, 𝑇 𝑢 , 𝑇 𝑢 ) with equation boundary conditions 𝑇 𝑢 𝑡 = 𝑇 𝑡 , 𝑇 𝑢 𝑓 = 𝑇 𝑓 • Note: The Euler-Lagrange equation may be written as 𝑍 (𝑢) = 𝑀 𝑇 (𝑢, 𝑇 𝑢 , 𝑇 𝑢 ) , 𝑍 𝑢 = 𝑀 𝑊 𝑢, 𝑇 𝑢 , 𝑇 𝑢
Q1: optimal charging • For a given price 𝑞 𝑢 , what is the optimal 𝑇 𝑢 ( within a given class of approved charging regimes)? Is the charging regime unique, and, if not, can we characterize the degree of degeneracy? • Euler-Lagrange equation 𝑒 𝑒𝑢 𝑞 𝑢 𝐻 𝑇 − 𝑞 𝑢 𝐻 𝑇 = 0, 𝑇 𝑢 𝑡 = 𝑇 𝑡 , 𝑇 𝑢 𝑓 = 𝑇 𝑓 Note: 𝐻 𝑇 = 𝜖𝐻/𝜖𝑇 etc • For simple battery 𝑢 𝑒𝑡 𝑊 𝑃𝐷 𝑇 𝑢 = 𝑇 𝑡 + 𝐿 − 2 𝑅 𝑁𝐵𝑌 𝑆 𝑐 t − t s , S t e = S e 𝑢 𝑡 𝑞(𝑡)
Q2: Optimal price • For a given 𝑇 𝑢 what is the price 𝑞(𝑢) for which S(t) is optimal? And is 𝑞(𝑢) unique and, if not, can we characterize the degree of degeneracy? 𝐻 𝑇 t s 𝑢 𝐻 𝑇 (𝑡) • 𝑞(𝑢) = 𝑞 𝑢 𝑡 𝑢 exp [ 𝑡 𝑒𝑡] 𝑢 𝑡 𝐻 𝑇 𝐻 𝑇 • For simple battery 𝑊 𝑃𝐷 +2 𝑅 𝑁𝐵𝑌 𝑆 𝑐 𝑇 𝑢 𝑡 𝑞 𝑢 = 𝑞(𝑢 𝑡 ) 𝑊 𝑃𝐷 +2 𝑅 𝑁𝐵𝑌 𝑆 𝑐 𝑇 𝑢
Q3: Specifying power 𝑋(𝑢) • For a given power 𝑋(𝑢) , what is 𝑇(𝑢) (and is it unique and in a given class of approved charging functions)? • 𝐻(𝑇, 𝑇 ) = 𝑋(𝑢) is an implicit differential equation for 𝑇(𝑢) , which can be solved with one boundary condition 𝑇 𝑢 𝑡 = 𝑇 𝑡 . It’s then possible in principle to determine the price function inducing this charging function providing 𝑋(𝑢) is compatible with 𝑇 𝑢 𝑓 = 𝑇 𝑓 . • Simple battery: 1 2 + 4 𝑆 𝑐 𝑋 𝑡 𝑢 1 • 𝑇 𝑢 = 𝑇 𝑡 + 𝑊 2 − 𝑊 𝑃𝐷 𝑒𝑡 2 𝑅 𝑁𝐵𝑌 𝑆 𝑐 𝑃𝐷 𝑢 𝑡
Q3 cont’d • For a given power 𝑋(𝑢) , what is the price 𝑞(t) that induces 𝑋 𝑢 ? • Apply the previous theory. • For a simple battery: 1 2 +4 𝑆 𝑐 𝑋 𝑢 𝑡 𝑊 2 𝑞(𝑢) = 𝑞 𝑢 𝑡 𝑃𝐷 2 +4 𝑆 𝑐 𝑋 𝑢 𝑊 𝑃𝐷 1 2 𝑋 𝑡 𝑒𝑡 2 +4 𝑆 𝑐 𝑋 𝑢 𝑡 𝑢 𝑓 𝑊 𝐷 = 𝑞 𝑢 𝑡 𝑃𝐷 2 +4 𝑆 𝑐 𝑋 𝑡 𝑢 𝑡 𝑊 𝑃𝐷
Example 𝑇 𝑋 𝑞 𝑇 𝑊
Advantages of the modern theory • Relaxation in the smoothness requirements for 𝑀 𝑢, 𝑇, 𝑊 and 𝑇 𝑢 to account for non-smoothness in price, battery models • Natural incorporation of constraints and penalties • But possibly including unphysical solutions, mathematically oversophisticated, and computations may require classical methods
Modern theory of CoV (after Rockafellar) 𝑢 𝑓 • Functional J = 𝑀 𝑢, 𝑇 𝑢 , 𝑇 𝑢 dt + ℓ 𝑇 𝑢 𝑡 , 𝑇 𝑢 𝑓 where 𝑢 𝑡 𝑀 and ℓ may take value ∞ to incorporate constraints • Technical assumptions: • ℓ is lower semi continuous (lsc), proper • 𝑀 is lsc, proper, a normal integrand and • 𝑀 𝑢, 𝑇, 𝑊 ≥ 𝑏 𝑢, 𝑇, 𝑊 a mild growth condition • 𝑀(𝑢, 𝑇, 𝑊) is convex in S, 𝑊 • The path 𝑇(𝑢) is absolutely continuous, 𝑇 𝑢 exists a.e. and 𝑇 ∈ ℒ 1
Modern theory • Theorem. Suppose 𝑇(𝑢) is a path with 𝐾 𝑇 < ∞ and suppose 𝑍 𝑢 is a path satisfying the generalised Euler- Lagrange condition 𝑍 𝑢 , 𝑍 𝑢 ∈ 𝜖 𝑇,𝑊 𝑀(𝑢, 𝑇 𝑢 , 𝑇 (𝑢) ) for a.a. 𝑢 and also the generalised transversality condition 𝑍 𝑢 𝑡 , −𝑧𝑍 𝑢 𝑓 ∈ 𝜖ℓ 𝑇 𝑢 𝑡 , 𝑇 𝑢 𝑓 then 𝑦(𝑢) is optimal. • Note 𝜖 𝑌,𝑊 𝑀(𝑢, 𝑇 𝑢 , 𝑇 (𝑢) ) and 𝜖ℓ 𝑇 𝑢 𝑡 , 𝑇 𝑢 𝑓 are subgradients
Extensions • Develop theory for charging/discharging 2 𝑢 2 𝑒𝑋 𝑆 𝑒𝑢 • Include e.g. ramp-up penalty say - 𝑢 1 𝑒𝑢 • Include more complicated pricing structures e.g. price for providing power 𝑄 𝑋 • Consider ensembles of prosumers/battery models • Design of pricing policy to control system and to incentivize development of storage • Incomplete information, stochastic pricing
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