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Distributed Real-Time Energy Management in Data Center Microgrids

Liang Yu, Tao Jiang, Yulong Zou

arXiv:1608.02274v3cs.DC

TL;DR

Energy management for multiple data center microgrids must account for uncertain parameters and spatial and temporal couplings. The paper combines stochastic cost minimization with a realtime distributed Lyapunov/ADMM algorithm; simulations report operational-cost reductions against benchmarks.

  • Problem

    Energy management for multiple data center microgrids involves spatial coupling from interactive workload allocation and temporal coupling from batch-delay guarantees and storage stability, alongside uncertain system parameters.

    Source: §I · p. 1, §III · p. 5

  • Method

    The paper formulates stochastic operational-cost minimization and solves it with a realtime distributed algorithm combining Lyapunov optimization and a variant of ADMM.

    Source: §I · p. 2

  • Results

    Proposed-2 (w=10^-12) reduced operational cost by 1.48%, 2.55%, and 15.15% versus B1, B2, and B3, respectively.

    Source: 3) Queue weight w · p. 8

  • Takeaways & Limitations

    Simulations show the proposed algorithm can effectively reduce operational costs for multiple data center microgrids.

    Source: §VI · p. 10

Abstract

from arXiv · show

Data center operators are typically faced with three significant problems when running their data centers, i.e., rising electricity bills, growing carbon footprints and unexpected power outages. To mitigate these issues, running data centers in microgrids is a good choice since microgrids can enhance the energy efficiency, sustainability and reliability of electrical services. Thus, in this paper, we investigate the problem of energy management for multiple data center microgrids. Specifically, we intend to minimize the long-term operational cost of data center microgrids by taking into account the uncertainties in electricity prices, renewable outputs and data center workloads. We first formulate a stochastic programming problem with the considerations of many factors, e.g., providing heterogeneous service delay guarantees for batch workloads, interactive workload allocation, batch workload shedding, electricity buying/selling, battery charging/discharging efficiency, and the ramping constraints of backup generators. Then, we design a realtime and distributed algorithm for the formulated problem based on Lyapunov optimization technique and a variant of alternating direction method of multipliers (ADMM). Moreover, the performance guarantees provided by the proposed algorithm are analyzed. Extensive simulation results indicate the effectiveness of the proposed algorithm in operational cost reduction for data center microgrids.

I. INTRODUCTION

Multiple data center microgrids pose coupled spatial and temporal energy-management challenges, including heterogeneous batch-workload delay guarantees. The paper formulates a stochastic cost-minimization problem and proposes a realtime distributed algorithm; its simulations report lower operational costs than benchmark schemes.

I. INTRODUCTION

  • Realtime distributed management for multiple data center microgrids remained unaddressed, while prior work neglected heterogeneous batch-delay guarantees and distributed implementation.Source: §I · p. 1
  • The stochastic program jointly models workload allocation and shedding, delay guarantees, electricity transactions, storage, generators, and power balancing; Lyapunov optimization and variant ADMM underpin the realtime distributed solution.The paper describes avoiding time-coupling constraints, distributing an implementation without nonlinear constraints, then adjusting the solution to satisfy them.Source: §I · p. 2
  • Simulations show that the proposed algorithm outperforms benchmark schemes in reducing operational costs.Source: §I · p. 2

II. MODEL AND FORMULATION

A. Models Associated with Data Centers and Front-end Servers

B. Models Related to the Generation System and ESS

The model combines generator and storage constraints with grid energy exchange to represent microgrid supply and demand. Its operational cost includes energy transactions, workload-related revenue losses, battery depreciation, and conventional generation costs.

1) Generation Model:

  • Conventional generator output is bounded and subject to a ramping constraint limiting changes between consecutive time slots.Source: 1) Generation Model · pp. 3–4

2) ESS Model:

  • The ESS model limits charging and discharging power, accounts for their efficiencies, prohibits simultaneous charging and discharging, and constrains stored energy and its dynamics.Source: 2) ESS Model · p. 4
  • Microgrids balance power through generation, storage, data-center demand, and grid transactions; buying and selling prices satisfy Xi,t > Wi,t.Source: 2) ESS Model · p. 4

C. Operational Cost Model

  • Total operational cost combines electricity transactions, workload-allocation revenue losses, battery depreciation, and conventional-generator costs.Source: C. Operational Cost Model · p. 4
  • Interactive-request revenue loss is modeled from propagation latency, while dropped batch workloads incur a penalty based on their quantity.Source: C. Operational Cost Model · p. 4

D. Operational Cost Minimization Problem

The paper formulates an expected operational-cost minimization problem under uncertain system parameters, then develops a Lyapunov-based sequence of transformations that handles time-average constraints and batch-workload delay.

D. Operational Cost Minimization Problem

  • The stochastic program minimizes time-average expected operational cost over decisions made under random system parameters and control actions.The cost functions are assumed continuously differentiable and convex.Source: D. Operational Cost Minimization Problem · p. 4

III. ALGORITHM DESIGN

  • The algorithm transforms P1 into P2 with time-average constraints, then into a one-slot Lyapunov problem, and modifies that problem before adjusting the solution for constraint (14).P2 removes constraint (11); the later adjustment restores constraint (14).Source: §III · p. 5
  • Virtual energy, workload, and delay-aware queues support constraint feasibility, while bounded workload queues yield a maximum batch queueing-delay guarantee.The delay-aware queue receives arrivals ε_i,q when batch work remains waiting; the guarantee requires finite queue bounds.Source: §III · p. 5

A. The Proposed Realtime Algorithm

The realtime algorithm minimizes a per-slot Lyapunov-based objective, simplifies a nonlinear constraint and repairs the solution, while its analysis addresses feasibility of constraints omitted during optimization.

A. The Proposed Realtime Algorithm

  • Each slot, the algorithm observes queue and system states, chooses workload, energy, and generator controls, repairs the solution, then updates its queues.The concatenated state includes real workload, virtual workload, and virtual energy queues.Source: A. The Proposed Realtime Algorithm · p. 6
  • The method minimizes a drift-plus-penalty upper bound each slot, using a weighted Lyapunov function that sets workload queues’ importance relative to energy queues.The drift expectation accounts for workload, renewable-output, electricity-price, and control-policy randomness.Source: A. The Proposed Realtime Algorithm · p. 6
  • The algorithm simplifies P4 by removing nonlinear constraint (14), solves P5, then adjusts the solution to satisfy (14).P4 is formed by adding constraint (11) to P3; the algorithm repeats this process each time slot.Source: A. The Proposed Realtime Algorithm · p. 6
  • Although constraints (7), (8), and (15) are not considered when generating the solution, the paper states that the algorithm guarantees their feasibility.The feasibility guarantee is established in Section V.Source: A. The Proposed Realtime Algorithm · p. 6

B. Distributed Implementation

The distributed implementation uses an ADMM-based approach where dual decomposition cannot be applied, then establishes algorithmic feasibility and performance guarantees under stated conditions.

B. Distributed Implementation

  • ADMM is chosen because P5's objective is not strictly convex, making dual decomposition inapplicable, while ADMM does not require strict convexity of the separable objective.The formulation is equivalently transformed to support the ADMM framework.Source: B. Distributed Implementation · p. 7
  • ADM-G provides a distributed implementation of the proposed algorithm, with global convergence provable under mild assumptions and an optimal solution to P6 when one exists.The authors omit the proof for the optimality claim.Source: B. Distributed Implementation · p. 7

IV. ALGORITHMIC PERFORMANCE ANALYSIS

  • The performance analysis gives a sufficient condition for ESS charging and discharging decisions and uses a theorem to establish Algorithm 1's feasibility for P1.The guarantee is stated for the proposed algorithm.Source: §IV · p. 7
  • With initial queues set to zero, the algorithm bounds the workload queues and energy queue and is feasible for the original problem P1.The analysis also bounds the maximum optimality loss from ramping constraints and from online adjustment in Γ_t.Source: §IV · p. 7
  • The stated performance guarantee assumes ε_i,q ≤ E{π_i,q,t} and that λ_f,t, π_i,q,t, r_i,t, X_i,t, and W_i,t are i.i.d. over slots.The supplied passage does not include the guarantee's full expression.Source: §IV · p. 7

V. PERFORMANCE EVALUATION

The evaluation reports that the proposed algorithm is feasible and converges, and that it reduces operational cost against three baselines. Its performance also depends on queue weight, dropping penalty, and tolerant service delay.

A. Simulation Setup

  • The simulations cover six months of 4320 one-hour slots, use real-world workload and dynamic electricity-price traces, and compare three baselines; details specify B1 and B2 only.The tolerant-delay cases use Ti,q ∈ {4, 8} and Ti,q ∈ {12, 24}.Source: A. Simulation Setup · pp. 7–8

B. Simulation Results

  • Maximum queue lengths and queueing delays stay within their respective bounds, while ESS energy levels remain in normal ranges, supporting feasibility for the original problem P1.The reported checks cover workload queue constraints, heterogeneous batch-service delay guarantees, and ESS energy constraints.Source: 1) Algorithmic feasibility · p. 8
  • The algorithm converges to the same optimal value as the GAMS commercial solver across different penalty parameters ρ.The evaluation tracks operational cost, primal residual, and feasibility violation during iteration.Source: 2) Convergence results · p. 8
  • 1.462 seconds for 600 iterations: the single-server implementation took this long, while the authors say parallel implementation can solve subproblems concurrently.The authors note that they lacked hardware resources to test a parallel implementation and relate the runtime to time slots that can last minutes or hours.Source: 2) Convergence results · p. 8
  • 1.48%, 2.55%, and 15.15%: Proposed-2 with w = 10−12 reduces operational cost versus B1, B2, and B3, respectively.The authors attribute the result to coordinating batch service, storage charging and discharging, and electricity sales with temporal price diversity.Source: 3) Queue weight w · p. 8
  • Larger queue weight w yields smaller average maximum queueing delay but uses less temporal electricity-price diversity for cost reduction.A very large weight such as w = 10−5 can produce small queueing delay while limiting the benefit of long batch-workload deadlines.Source: 3) Queue weight w · p. 8, 5) Tolerant service delay Ti,q · p. 10
  • Proposed-2 has the lowest operational cost in the tested penalty-factor comparison, and a larger θi leads the algorithm to discard fewer batch workloads.As θi approaches zero, the algorithm reduces to B1, with batch workloads dropped and no energy queue needed.Source: 4) Dropping penalty factor θi · p. 8
  • With w = 10−10, increasing tolerant service delay lowers operational cost and raises electricity-selling profit; with w = 10−5, both are nearly unchanged.The latter queue weight places a very large emphasis on stabilizing workload and virtual queues, limiting use of temporal price diversity.Source: 5) Tolerant service delay Ti,q · pp. 8, 10

VI. CONCLUSIONS

The conclusion presents a distributed realtime algorithm for minimizing long-term operational cost across multiple data center microgrids. Simulations show operational-cost reduction, and the authors report low computational complexity and no need for prior statistical knowledge of system parameters.

VI. CONCLUSIONS

  • The proposed distributed realtime algorithm targets long-term operational cost across multiple data center microgrids while accounting for workload, energy-storage, electricity-trading, and generator constraints.The considered factors include heterogeneous batch-workload delay guarantees, interactive workload allocation, batch shedding, battery charging/discharging efficiency, and generator ramping constraints.Source: §VI · p. 10
  • Extensive simulations show that the proposed algorithm can effectively reduce data center microgrid operational cost.The algorithm is reported to require no prior statistical knowledge of system parameters and to have low computational complexity.Source: §VI · p. 10

APPENDIX A PROOF OF LEMMA 1

The appendix supplies proofs for workload-queue and drift-related claims used in the analysis. The provided passages show a deadline-based argument for serving batch workloads and combine queue bounds in a Lyapunov inequality.

APPENDIX A PROOF OF LEMMA 1

  • The Lemma 1 proof argues that batch workload πi,q,t must be served before t + Rmax_i,q by contradiction using the workload and virtual-queue updates.It assumes the batch remains unserved throughout the interval and uses the queue relations to derive a contradiction with the definition of Rmax_i,q.Source: Appendix A · p. 10

APPENDIX B PROOF OF LEMMA 2

  • The Lemma 2 proof combines upper bounds for the workload, virtual, and energy queues, then adds V E{Γt|Θt} to complete the stated proof.The passage presents the combined inequality as part of the Lyapunov analysis.Source: Appendix B · pp. 10–11

APPENDIX C THE DISTRIBUTED IMPLEMENTATION OF ALGORITHM 1

The appendix describes an iterative distributed implementation with prediction and correction steps, parallel subproblem solves, and a feasibility-aware stopping option. Its subproblems are described as convex and solvable by closed forms or binary search.

APPENDIX C THE DISTRIBUTED IMPLEMENTATION OF ALGORITHM 1

  • The implementation initializes decision variables at zero and repeats prediction and correction steps until convergence or an acceptable feasible solution is obtained.The authors describe early termination when the primal residual is small enough and the solution is feasible; a feasibility metric is used to assess the solution.Source: Appendix C · p. 11, §4 · p. 12

2. ADMM step (prediction step)

  • In the ADMM prediction step, front-end servers, queue controllers, generators, ESSs, and EMSs solve their assigned subproblems in parallel.The appendix states that P7–P17 are convex and can be solved using closed-form expressions or binary search.Source: §2 · pp. 11–12

2.12 Dual update

  • The correction step obtains the next iteration's input parameters using Gaussian back substitution, with α set to one based on practical experience.The stopping rule can end the algorithm before convergence once an acceptable feasible solution is found.Source: §2.12 · p. 12, §4 · p. 12

4. Stopping criterion

Each iteration begins with independent local decisions by front-end servers, queue controllers, and conventional generators. Decisions are then exchanged among microgrid components so entities can update their decisions for the next iteration.

4. Stopping criterion

  • Front-end servers, queue controllers, and conventional generators independently make their initial local decisions in parallel.These decisions are made at the initial iteration.Source: §4 · p. 12
  • The components broadcast decisions to other components, after which ESSs and EMSs make local decisions.The sequence includes decisions sent to the EMS and decisions obtained by the EMS.Source: §4 · p. 12
  • The EMS broadcasts the obtained decisions for iteration k+1 so entities can update their respective decisions through Gaussian back substitution.The entities include front-end servers, queue controllers, conventional generators, and ESSs.Source: §4 · p. 12

APPENDIX D PROOF OF LEMMA 3

The proof examines the optimality of the battery discharging decision under a threshold condition by comparing it with an alternative decision for the same energy demand.

APPENDIX D PROOF OF LEMMA 3

  • The nonoptimality argument compares objective values for alternative decision vectors under the same energy demand pi,t.One alternative vector is given as (0, 0, x*i,t).Source: Appendix D · p. 12
  • When Zi,t < −V ηd,iγi,max, the optimal discharging decision u∗d,i,t is greater than zero, as stated in the summary of the proof for this condition.Source: Appendix D · p. 13
  • The proof states that its second part follows by an argument similar to the first and omits the details.Source: Appendix D · p. 13

APPENDIX E PROOF OF THEOREM 1

Theorem 1’s proof establishes feasibility of the proposed algorithm and derives performance and operating-constraint guarantees. It also accounts for generator ramping and the resulting adjustment between generator output and grid energy exchange.

APPENDIX E PROOF OF THEOREM 1

  • The proof uses queue thresholds and induction to establish the queue constraint and bound charging-request delay.When Qi,q,t exceeds V βiXmax, the algorithm chooses the maximum possible xi,q,t.Source: Appendix E · p. 13
  • The algorithm can ensure heterogeneous service delays for charging requests, including a one-slot delay for requests in the stated case.The proof also relates the guarantee to Rmax_i,q.Source: Appendix E · p. 13
  • The algorithm satisfies constraints (7), (8), and (15), and its solution is feasible for the original problem P1.The proof states that the other constraints in P1 are also guaranteed.Source: Appendix E · p. 13
  • Generator ramping can require more grid purchases when generator output is constrained downward, or more grid sales when it is constrained upward.The proof analyzes both cases as adjustments to balance power.Source: Appendix E · pp. 13–14
  • The proposed algorithm’s long-term objective is bounded by a comparison involving the relaxed problems and the drift-based cost terms.The proof combines the comparison with a stationary randomized control strategy to obtain an average-cost bound, but the supplied passage does not expose the complete bound.Source: Appendix E · p. 14
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