Evaluating the value of storage facilities for buffering and - - PowerPoint PPT Presentation

Evaluating the value of storage facilities for buffering and arbitrage

(Joint work with Lisa Flatley, Richard Gibbens, Stan Zachary)

James Cruise

Maxwell Institute for Mathematical Sciences Edinburgh and Heriot-Watt Universities

26 February 2015

The Problem:

Storage facilities are expensive, high capital cost. To facilitate investment we need to understand storage value. Need to also be able to compare to alternatives, for example demand side management. Also need to understand the effect of multiple competing stores (see Lisa’s talk yesterday). Plan:

1 Simple model for a single store and economic environment for

studying arbitrage value.

2 Consequences of the model. 3 Extensions to include stochastic prices and buffering.

Model

E P

E = size of store — capacity constraint P = max input/output rate — rate constraint

Cost function

At any (discrete) time t,

P P sell buy x ) x (

t

C

Ct(x) = cost of increasing level of store by x (positive or negative) Assume convex (reasonable). This may model market impact efficiency of store rate constraints

Problem

Let St = level of store at time t, 0 ≤ t ≤ T. Policy S = (S0, . . . , ST), S0 = S∗

0 (fixed),

ST = S∗

T (fixed).

Define also xt(S) = St − St−1 (energy “bought” by store at time t – positive or negative) Problem: minimise cost

T

• t=1

Ct(xt(S)) subject to S0 = S∗

0,

ST = S∗

T

and 0 ≤ St ≤ E, 1 ≤ t ≤ T − 1.

Small store

This is a store whose activities are not so great as to impact upon the market, and which thus has linear buy and sell prices. Thus, for all t, Ct(x) =

• c(b)

t

x if 0 ≤ x ≤ P c(s)

t

x if −P ≤ x < 0 where 0 < c(s)

t

≤ c(b)

t

and P is rate constraint. Characteristics Optimal control is bang-bang: at each time buy as much as possible, do nothing, or sell as much as possible. If E = ∞ (no capacity constraint) then optimal solution is global in time. If P = ∞ (no rate constraint) then optimal solution is very local in time.

Example: Periodic Cost functions

Consider sinusoidal prices. Interested in what happens as frequency is varied. Assume P = 1. Then for a given value of E, there exists a pair µb < µs such that we buy if ct < µb and sell if ct > µs. Further µb is increasing in E and µs is decreasing in E. Also µb is increasing and µs is decreasing in frequency. These are bounded by µ∗

b and µ∗ s, the parameters obtained for

E = ∞ which does not depend on frequency. For a given E, as frequency increases profit increases upto the unconstrained case, and there is a frequency beyond which you obtain no further benefit.

Example: Real prices with Dinorwig parameters

E/P = 5 hrs Efficiency = 0.85 (ratio of sell to buy price). Solution is bang-bang: red points buy, blue points sell

2000 4000 6000 8000 10000

Prices (£/MWh)

0.0 2.5 5.0 7.5 10.0 Sun 09−Jan Mon 10−Jan Tue 11−Jan Wed 12−Jan Thu 13−Jan Fri 14−Jan Sat 15−Jan Sun 16−Jan

Storage level

Example: Real prices with Dinorwig parameters

E/P = 5 hrs Efficiency = 0.65 (ratio of sell to buy price). Solution is bang-bang: red points buy, blue points sell

2000 4000 6000 8000 10000

Prices (£/MWh)

0.0 2.5 5.0 7.5 10.0 Sun 09−Jan Mon 10−Jan Tue 11−Jan Wed 12−Jan Thu 13−Jan Fri 14−Jan Sat 15−Jan Sun 16−Jan

Storage level

General Result: Lagrangian sufficiency

Suppose there exists a vector µ∗ = (µ∗

1, . . . , µ∗ T) and a value

S∗ = (S∗

0, . . . , S∗ T) of S such that

S∗ is feasible, xt(S∗) minimises Ct(x) − µ∗

t x

• ver all x,

1 ≤ t ≤ T, the pair (S∗, µ∗) satisfies the complementary slackness conditions, for 1 ≤ t ≤ T − 1,      µ∗

t+1 = µ∗ t

if 0 < S∗

t < E,

µ∗

t+1 ≤ µ∗ t

if S∗

t = 0,

µ∗

t+1 ≥ µ∗ t

if S∗

t = E.

(1) Then S∗ solves the stated problem.

Comment

The above result (essentially an application of the Lagrangian Sufficiency Principle) does not require convexity of the functions Ct. However, convexity the functions Ct is sufficient to guarantee the existence of a pair (S∗, µ∗) as above. The latter result is a standard application of Strong Lagrangian theory (i.e. the Supporting Hyperplane Theorem).

Algorithm

We need to identify the relevant value of µ∗

t at each time t.

It is important to note that the value of µ∗

t only changes at

times when the store is full or empty. Further µ∗

t acts as a reference level since the optimal action at

time t is given by the x which minimizes: Ct(x) − µ∗

t x.

We can use this to carry at a search for µ∗. Further this method is local in time, as we can ignore times after filling or emptying the store.

Example

Again consider real price data but incorporate market impact by the store. Use a quadratic cost function: Ct(x) =

• ptx(1 + λx)

if 0 ≤ x ≤ P −ptνx(1 + λνx) if −P ≤ x < 0 pt is the historic price. λ is a measure of market impact by the store. ν is the round trip efficiency of the store. P is the power constraint.

2 4 6 8 10

Store level

E = 10 , P = 1 , η = 0.8 , λ = 0.05

5 10 15 20 01/Dec 08/Dec 15/Dec 22/Dec 29/Dec

Lookahead time (days)

2 4 6 8 10

Store level

E = 10 , P = 1 , η = 0.6 , λ = 0.05

5 10 15 20 01/Dec 08/Dec 15/Dec 22/Dec 29/Dec

Lookahead time (days)

2 4 6 8 10

Store level

E = 10 , P = 1 , η = 0.8 , λ = 0.5

5 10 15 20 01/Dec 08/Dec 15/Dec 22/Dec 29/Dec

Lookahead time (days)

2 4 6 8 10

Store level

E = 10 , P = 0.25 , η = 0.8 , λ = 0.05

5 10 15 20 01/Dec 08/Dec 15/Dec 22/Dec 29/Dec

Lookahead time (days)

Market impact of storage

A large store may impact costs (be a price-maker), and hence the rest of society.

Market impact of storage

A large store may impact costs (be a price-maker), and hence the rest of society. Impact of storage on consumer surplus is in general beneficial, but not necessarily.

Market impact of storage

A large store may impact costs (be a price-maker), and hence the rest of society. Impact of storage on consumer surplus is in general beneficial, but not necessarily. Example (T = 2): Buy x at time 1 (increasing price by p1) and sell at time 2 (decreasing price by p2). Increase in consumer surplus is p2d2 − p1d1 where d1 and d2 are the respective demands at times 1 and 2. In general expect d2 > d1 and p2 to be comparable to p1, so effect

• n consumer surplus is beneficial.

However, the latter need not be the case.

Optimisation of value of storage

Suppose it is desired to maximise store profit plus consumer surplus.

Optimisation of value of storage

Suppose it is desired to maximise store profit plus consumer surplus. Then it is straightforward to modify the cost function at each time t so as to reflect total costs: for each time t, the cost ct(x) is now the cost to the store of buying x units (positive or negative) plus the cost to society (reduction in consumer surplus) of doing so.

Stochastic Prices

Suppose that the cost functions Ct evolve randomly in time, and that Ct = ξt ¯ Ct, 1 ≤ t ≤ T, where (¯ C1, . . . , ¯ CT) is a sequence of deterministic cost functions and where (ξ1, . . . , ξT) is a sequence of strictly positive real-valued random variables such that E(ξt | Ft−1) = ξt−1, 1 ≤ t ≤ T, and where the deterministic functions ¯ Ct are assumed to satisfy the same conditions as the cost functions Ct of the deterministic problem. Then the optimal control (cost minimisation) strategy remains exactly as previously, with the optimal sequence of store levels as given in the case where stochastic cost functions Ct are replaced by their deterministic counterparts ¯ Ct.

Integration of buffering for uncertainty

Suppose that the store is also used to provide buffering against relatively rare unexpected events. Then the previous optimization problem may be rewritten as: Problem: minimise

T

• t=1
• Ct(xt(S)) + At(St)
• subject to

S0 = S∗

0,

ST = S∗

T

and 0 ≤ St ≤ E, 1 ≤ t ≤ T − 1. Here the functions At, such that At(St) is the cost (negative benefit) of having a level of reserve St in the store at time t, are independently calculable.

Result - Lagrangian sufficiency

Assume, for simplicity, we incorporate the capacity constraints into the functions At, and the functions Ct and At are differentiable. Suppose there exists a scalar ν∗ and a value S∗ = (S∗

0, . . . , S∗ T)

(with S∗

0 and S∗ T having their required values) such that

C ′

t(xt) = ν − T

• u=t

A′

u(Su)

=: µt Then S∗ solves the stated problem.

Example

E/P = 5 hrs Efficiency = 0.85 (ratio of sell to buy price). At(S) = ν/S (Black:ν = 0.02,Red:ν = 0.2, Blue: ν = 1)

Time (March 2011) Store level 2 4 6 8 10

Functions At

The function At is represents the extra cost associated with random deviation. In many cases we can either determine this a priori or use a suitable approximation. Examples: Random deviations are corrected immediately. Failure to be able to provide the service, leads to penalty. Cases where the probability of the buffering service being called upon is small.

Demand Side Response

Demand side response can be viewed as also moving energy through time. We now have an energy debt S, such that −E ≤ S ≤ 0. So we have to sell before we can buy, but the problem formulation is the same. Often demand response has further binding constraints:

Frequency at which actions can be taken. Length of time energy debt can be maintained.

But for a first approximation this work provides some insight.

Going Forward

Explore the control of multiple stores:

With different parameters sets, e.g. a slow large store and a fast small store. Integration of network constraints.

Utilize understanding in the study of competitive stores. Consider alternative formulation for stochastic prices. Extend results to better analysis demand side response.