Optimal Placement of Energy Storage in Power Networks

Christos Thrampoulidis

Subhonmesh Bose and Babak HassibiJoint work with

52nd IEEE CDCDecember 13, 2013

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Use of energy storage Fast time scales (seconds, minutes): Balance intermittency

Compensate for supply shortfallUncertain renewable generation

Slow time scales (hours) : Load Shiftingreduce cost of conventional generation

minimize (convex) cost of generation

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Two questions on storage

(2) Where to place, how to size ?

(1) Optimal control policy?Previous work: [Chandy10] ,[Kotsopoulos11], [Zhang11], [Gayme12], [Su13] ...

assume storage resources at each node known a priori.

(1) + (2) Common Framework

Previous work: [Denholm09], [Kraning10], [Harsha13] ...

mainly sizing, purely economic arguments, specific networks.

Previous work: [Sjodin12], [Bose12], [Gayme12]

only simulation results so far.

This work: Find structural properties…

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The optimization framework Slow time scales (Load Shifting) Common Optimization Framework: Placement + Control

given storage budget, demand profiles

find how to place storage devices + how to control

minimize cost of generationsubject to network constraints,

storage device and generator dynamics

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Optimization solves for optimal allocation

Storage budget

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Any structural properties?

Can we say something about placement of storage before solving the optimization problem ?

?

?

? ?

? ?

???demand profile

?

line-flow limit?

generator characteristics

?

storage devicecharacteristics

?

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Model features

1. neglect reactive power2. voltage magnitudes at nominal values3. voltage phase angle differences small

- Demand: known, periodic finite-Horizon

- Storage: finite capacity, ramp rates, losses

- Generators: finite capacity, convex non-decreasing cost

- Network: linearized DC approximation,line-flow limits

??

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A structural property

T h e ore m 1

There always exists optimal storage allocation which assigns zero storage at generator nodes with single-link connections.

Robustness of the result: Holds for arbitrary demand profiles and other network parameters.

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0

0

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Theorem: Proof idea

There exists optimal solution such that L e m m a :

NO storage at generator

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Generators with multiple connections?

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Storage budget

• May not always have zero storage capacity at optimal.• Example:

40.5

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A solution:

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2.5

less “flat”

not optimal

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Forcing zero storage at generator:

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The star network

?

T h e ore m 2

There exists an optimal storage allocation which assigns zero storage at the generator node of a star network when thestorage budget is:

a) equal to the minimum possible to allow serving the loads,b) big enough.

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Summary & future work

? better understanding of generators with multiple connections, role of congested lines.

? cost benefit analysis, performance metrics other than cost of generation.

? fast-time scales, stochastic framework.

Thank You!

• Formulated joint investment decision and control problem.• Identified preliminary structural property of the problem:

N O s t o ra g e a t g e n e ra t o rs w i t h s i n g l e l i n k s .

Potential to exploit structure of the placement problem.Related recent work: [Castillo&Gayme13]

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The Model

Finite-time Horizon:Available storage budget:

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Proof of the Lemma (I)There exists optimal solution such that L e m m a :

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Proof of the Lemma (II)There exists optimal solution such that L e m m a :

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Proof of the Lemma (III)There exists optimal solution such that L e m m a :

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Extensions + Proof of Theorem II

• Same proof technique generalizes to incorporate losses and rate ramp constraints.

• Proof of Theorem II (Star Network)Similar idea. Some more work to show that any storage of

the generator node, can be distributed in a feasible way among load buses.

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The ModelFinite-time Horizon:

Line-flow constraints:

Power Balance:

Available storage budget:

Storage Device constraints:

Generation constraints:

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Model Features - Demand profiles: known, periodic => Finite-Horizon- Storage Units: finite capacity, ramp rates, losses- Generators: finite capacity, convex non-decreasing cost

- Linearized DC approximation, finite line-flow capacities

1,3,7,6,5,4Zero storage at generatorNon-zero storage at generator

Cost of generation900877

Model Features - Demand profiles:

known, periodic Finite-Horizon (storage level same at the end)

- Storage Units: finite capacity, ramp rates, losses

- Generators: finite capacity, convex non-decreasing cost

- Network: Linearized DC approximation, finite line-flow capacities