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
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