Optimal Placement of Energy Storage in Power Networks Christos Thrampoulidis Subhonmesh Bose and Babak Hassibi Joint work with 52 nd IEEE CDC December 13, 2013

Use of energy storage  Fast time scales (seconds, minutes) :Balance intermittency Compensate for supply shortfall Uncertain renewable generation  Slow time scales (hours) : Load Shifting reduce cost of conventional generation minimize (convex) cost of generation 1

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… 2

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 generation subject to network constraints, storage device and generator dynamics 3

Optimization solves for optimal allocation Storage budget 4

Any structural properties? Can we say something about placement of storage before solving the optimization problem ? ? ? ? ? ? ? ? ? ? demand profile ? line-flow limit ? generator characteristics ? storage device characteristics ? 5

Model features 1.neglect reactive power 2.voltage magnitudes at nominal values 3.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 ? ? 6

A structural property Theorem 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

Theorem: Proof idea There exists optimal solution such that Lemma: NO storage at generator 8

Generators with multiple connections? 5 Storage budget May not always have zero storage capacity at optimal. Example: A solution: 2.5 less “flat” not optimal 0 Forcing zero storage at generator: 9

The star network ? Theorem 2 There exists an optimal storage allocation which assigns zero storage at the generator node of a star network when the storage budget is: a) equal to the minimum possible to allow serving the loads, b) big enough. 10

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: NO storage at generators with single links.  Potential to exploit structure of the placement problem. Related recent work: [Castillo&Gayme13] 11

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The Model Finite-time Horizon: Available storage budget: 13

Proof of the Lemma (I) There exists optimal solution such that Lemma: 14

Proof of the Lemma (II) There exists optimal solution such that Lemma: 15

Proof of the Lemma (III) There exists optimal solution such that Lemma: 16

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. 17

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The Model Finite-time Horizon: Line-flow constraints: Power Balance: Available storage budget: Storage Device constraints: Generation constraints: 20

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 1,3,7,6,5,4 Zero storage at generator Non-zero storage at generator Cost of generation 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 21