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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

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Optimal battery charge/discharge strategies for consumers and suppliers Ben Mestel

4 April 2014

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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
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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?

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Model (microeconomic) problem

  • Given a prosumer with a storage battery (perhaps in an

electric/hybrid vehicle) and an exogenous price π‘ž(𝑒) 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

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Battery types

Schematic of lithium-air battery charge and discharge cycles

  • Lead-acid
  • Li Ion e.g.

LiCoO2

  • NaS
  • NiCd
  • ZnBr
  • LiO2
  • LiS
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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

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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
  • n battery temperature and lifetime
  • Each battery has its own attributes and manufacturer-

recommended charging regimes

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Battery models

  • State variable: 𝑇 = state of

charge

  • 𝑇 = 0 battery fully discharged
  • 𝑇 = 1 battery fully charged
  • (Ignore battery temperature πœ„)
  • Applied current and voltage

𝐽 𝑒 , π‘Š(𝑒) – 𝐽 𝑒 > 0 - charging – 𝐽 𝑒 < 0 - discharging – 𝐽(𝑒), π‘Š(𝑒) are not independent

  • Time interval [𝑒𝑑, 𝑒𝑓]
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Single storage supplier fixed market

  • Supplier price taker
  • Prices: exogenous variables, set by power system
  • perator
  • Two prices in, say, Β£ per KWh (i.e. money/energy):

– π‘žπ‘(𝑒) offer price i.e. price the storage supplier sells electricity – π‘žπ‘(𝑒) bid price i.e. price the storage supplier buys electricity –In general π‘žπ‘ 𝑒 β‰₯ π‘žπ‘(𝑒)

  • Forward pricing on T = 𝑒𝑑, 𝑒𝑓
  • Here we consider π‘žπ‘(𝑒) = π‘žπ‘(𝑒), which we write π‘ž(𝑒)
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Charging optimisation

  • 𝑇(𝑒) state of charge, 0 ≀ 𝑇 𝑒 ≀ 1, 𝑅 𝑒 = π‘…π‘π΅π‘Œ 𝑇 𝑒
  • 𝐷 =

π‘ž 𝑒 𝑋 𝑒 𝑒𝑒, 𝑋 𝑒 = 𝐻(𝑇 𝑒 , 𝑇 𝑒 )

𝑒𝑓 𝑒𝑑

  • For a simple-battery

– 𝑋 𝑒 = 𝐽 𝑒 π‘Š 𝑒 – π‘Š 𝑒 = π‘Š

𝑃𝐷 + 𝑆𝑐 𝐽 𝑒

– 𝐽 𝑒 = π‘…π‘π΅π‘Œ 𝑇 𝑒 – 𝐻 = 𝐻(𝑇 𝑒 ) = (π‘Š

𝑃𝐷 + π‘†π‘π‘…π‘π΅π‘Œ 𝑇 𝑒 )π‘…π‘π΅π‘Œ 𝑇 𝑒

– 𝐻 convex for 𝑆𝑐 > 0

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Classical Calculus of Variations Approach

  • Theorem. Let 𝑀(𝑒, 𝑇, π‘Š) be a twice continuously

differentiable function with respect to (𝑒, 𝑇, π‘Š) which is convex with respect to 𝑇, π‘Š . Then the functional 𝑀 𝑒, 𝑇(𝑒), 𝑇 𝑒 𝑒𝑒, 𝑇 𝑒𝑑 = 𝑇𝑑, 𝑇 𝑒𝑓 = 𝑇𝑓

𝑒𝑓 𝑒𝑑

has a minimum path S 𝑒 that satisfies the Euler-Lagrange equation

𝑒 𝑒𝑒 π‘€π‘Š 𝑒, 𝑇 𝑒 , 𝑇 𝑒

= 𝑀𝑇(𝑒, 𝑇 𝑒 , 𝑇 𝑒 ) with boundary conditions 𝑇 𝑒𝑑 = 𝑇𝑑, 𝑇 𝑒𝑓 = 𝑇𝑓

  • Note: The Euler-Lagrange equation may be written as

𝑍 (𝑒) = 𝑀𝑇(𝑒, 𝑇 𝑒 , 𝑇 𝑒 ) , 𝑍 𝑒 = π‘€π‘Š 𝑒, 𝑇 𝑒 , 𝑇 𝑒

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Q1: optimal charging

  • For a given price π‘ž 𝑒 , what is the optimal 𝑇 𝑒 (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 𝑒𝑒 π‘ž 𝑒 𝐻𝑇 βˆ’ π‘ž 𝑒 𝐻𝑇 = 0, 𝑇 𝑒𝑑 = 𝑇𝑑, 𝑇 𝑒𝑓 = 𝑇𝑓 Note: 𝐻𝑇 = πœ–π»/πœ–π‘‡ etc

  • For simple battery

𝑇 𝑒 = 𝑇𝑑 + 𝐿

𝑒𝑑 π‘ž(𝑑) 𝑒 𝑒𝑑

βˆ’

π‘Šπ‘ƒπ· 2 π‘…π‘π΅π‘Œ 𝑆𝑐 t βˆ’ ts , S te = Se

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Q2: Optimal price

  • For a given 𝑇 𝑒 what is the price π‘ž(𝑒) for which S(t) is
  • ptimal? And is π‘ž(𝑒) unique and, if not, can we

characterize the degree of degeneracy?

  • π‘ž(𝑒) = π‘ž 𝑒𝑑

𝐻𝑇

ts

𝐻𝑇

𝑒 exp [

𝐻𝑇(𝑑) 𝐻𝑇

𝑑 𝑒𝑑]

𝑒 𝑒𝑑

  • For simple battery

π‘ž 𝑒 = π‘ž(𝑒𝑑)

π‘Šπ‘ƒπ·+2 π‘…π‘π΅π‘Œ 𝑆𝑐 𝑇 𝑒𝑑 π‘Šπ‘ƒπ·+2 π‘…π‘π΅π‘Œ 𝑆𝑐 𝑇 𝑒

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Q3: Specifying power 𝑋(𝑒)

  • For a given power 𝑋(𝑒), what is 𝑇(𝑒) (and is it unique and

in a given class of approved charging functions)?

  • 𝐻(𝑇, 𝑇 ) = 𝑋(𝑒) is an implicit differential equation for 𝑇(𝑒),

which can be solved with one boundary condition 𝑇 𝑒𝑑 = 𝑇𝑑. It’s then possible in principle to determine the price function inducing this charging function providing 𝑋(𝑒) is compatible with 𝑇 𝑒𝑓 = 𝑇𝑓.

  • Simple battery:
  • 𝑇 𝑒 = 𝑇𝑑 +

1 2 π‘…π‘π΅π‘Œπ‘†π‘

π‘Š

𝑃𝐷 2 + 4 𝑆𝑐𝑋 𝑑

1 2 βˆ’ π‘Š

𝑃𝐷 𝑒𝑑 𝑒 𝑒𝑑

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Q3 cont’d

  • For a given power 𝑋(𝑒), what is the price π‘ž(t) that

induces 𝑋 𝑒 ?

  • Apply the previous theory.
  • For a simple battery:

π‘ž(𝑒) = π‘ž 𝑒𝑑

π‘Š

𝑃𝐷 2 +4 𝑆𝑐 𝑋 𝑒𝑑

π‘Š

𝑃𝐷 2 +4 𝑆𝑐 𝑋 𝑒 1 2

𝐷 = π‘ž 𝑒𝑑

π‘Š

𝑃𝐷 2 +4 𝑆𝑐 𝑋 𝑒𝑑

π‘Š

𝑃𝐷 2 +4 𝑆𝑐 𝑋 𝑑 1 2 𝑋 𝑑 𝑒𝑑

𝑒𝑓 𝑒𝑑

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Example

𝑋 𝑇 π‘ž 𝑇 π‘Š

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Advantages of the modern theory

  • Relaxation in the smoothness requirements for 𝑀 𝑒, 𝑇, π‘Š

and 𝑇 𝑒 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

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Modern theory of CoV (after Rockafellar)

  • Functional J =

𝑀

𝑒𝑓 𝑒𝑑

𝑒, 𝑇 𝑒 , 𝑇 𝑒 dt + β„“ 𝑇 𝑒𝑑 , 𝑇 𝑒𝑓 where 𝑀 and β„“ may take value ∞ to incorporate constraints

  • Technical assumptions:
  • β„“ is lower semi continuous (lsc), proper
  • 𝑀 is lsc, proper, a normal integrand and
  • 𝑀 𝑒, 𝑇, π‘Š β‰₯ 𝑏 𝑒, 𝑇, π‘Š a mild growth condition
  • 𝑀(𝑒, 𝑇, π‘Š) is convex in S, π‘Š
  • The path 𝑇(𝑒) is absolutely continuous, 𝑇 𝑒 exists a.e.

and 𝑇 ∈ β„’1

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Modern theory

  • Theorem. Suppose 𝑇(𝑒) is a path with 𝐾 𝑇 < ∞ and

suppose 𝑍 𝑒 is a path satisfying the generalised Euler- Lagrange condition 𝑍 𝑒 , 𝑍 𝑒 ∈ πœ–π‘‡,π‘Šπ‘€(𝑒, 𝑇 𝑒 , 𝑇 (𝑒)) for a.a. 𝑒 and also the generalised transversality condition 𝑍 𝑒𝑑 , βˆ’π‘§π‘ 𝑒𝑓 ∈ πœ–β„“ 𝑇 𝑒𝑑 , 𝑇 𝑒𝑓 then 𝑦(𝑒) is optimal.

  • Note πœ–π‘Œ,π‘Šπ‘€(𝑒, 𝑇 𝑒 , 𝑇 (𝑒)) and πœ–β„“ 𝑇 𝑒𝑑 , 𝑇 𝑒𝑓

are subgradients

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Extensions

  • Develop theory for charging/discharging
  • Include e.g. ramp-up penalty say -

𝑒𝑋𝑆 𝑒𝑒 2

𝑒𝑒

𝑒2 𝑒1

  • Include more complicated pricing structures e.g. price for

providing power 𝑄𝑋

  • 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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Thank you for your attention