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Extended Version: Non-Preemptive Scheduling of Flexible Loads in Smart Grids via Convex Optimization

Mehdi Davoudi, Mingyu Chen, Junjie Qin

TL;DR

This paper studies the scheduling of a large population of non-preemptive flexible electric loads, each of which has a flexible starting time but once started will follow a fixed load shape until completion, and proposes an efficient polynomial time relaxation-adjustment-rounding algorithm for solving the problem.

Abstract

This paper studies the scheduling of a large population of non-preemptive flexible electric loads, each of which has a flexible starting time but once started will follow a fixed load shape until completion. We first formulate the scheduling problem as a mixed-integer convex program (MICP), then propose an efficient polynomial time relaxation-adjustment-rounding algorithm for solving the problem. The key novelty of the proposed method lies in its adjustment step, which uses a graph-based algorithm to navigate within the set of optimal points of the convex relaxation while reducing the number of fractional entries in the solution. We establish mathematically that our algorithm yields solutions that are near optimal for a finite number of loads and with its sub-optimality independent of the number of loads. Consequently, the proposed method is asymptotically optimal in a per-load cost sense when the number of loads increases. Despite the gap between the MICP and its convex relaxation, we establish that the solution of the proposed algorithm can be decentralized by marginal prices of the convex relaxation. We also develop and analyze variants of the proposed algorithm for settings with uncertainty and with time-varying realistic load shapes. Finally, we numerically evaluate the proposed algorithm in a case study for the non-preemptive scheduling of electric vehicles charging loads.

Extended Version: Non-Preemptive Scheduling of Flexible Loads in Smart Grids via Convex Optimization

TL;DR

This paper studies the scheduling of a large population of non-preemptive flexible electric loads, each of which has a flexible starting time but once started will follow a fixed load shape until completion, and proposes an efficient polynomial time relaxation-adjustment-rounding algorithm for solving the problem.

Abstract

This paper studies the scheduling of a large population of non-preemptive flexible electric loads, each of which has a flexible starting time but once started will follow a fixed load shape until completion. We first formulate the scheduling problem as a mixed-integer convex program (MICP), then propose an efficient polynomial time relaxation-adjustment-rounding algorithm for solving the problem. The key novelty of the proposed method lies in its adjustment step, which uses a graph-based algorithm to navigate within the set of optimal points of the convex relaxation while reducing the number of fractional entries in the solution. We establish mathematically that our algorithm yields solutions that are near optimal for a finite number of loads and with its sub-optimality independent of the number of loads. Consequently, the proposed method is asymptotically optimal in a per-load cost sense when the number of loads increases. Despite the gap between the MICP and its convex relaxation, we establish that the solution of the proposed algorithm can be decentralized by marginal prices of the convex relaxation. We also develop and analyze variants of the proposed algorithm for settings with uncertainty and with time-varying realistic load shapes. Finally, we numerically evaluate the proposed algorithm in a case study for the non-preemptive scheduling of electric vehicles charging loads.

Paper Structure

This paper contains 31 sections, 14 theorems, 71 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. Organization and contributions
  4. Model
  5. Non-preemptive jobs
  6. Cost model
  7. Non-convex scheduling problem
  8. Relaxation-Adjustment-Rounding Algorithm
  9. Relaxation
  10. Adjustment
  11. Rounding
  12. Performance of the RAR algorithm
  13. Analysis of the RAR algorithm
  14. Necessity of the adjustment step
  15. Extensions
  16. ...and 16 more sections

Key Result

Lemma 1

The while loop in Algorithm alg:adj terminates in $O(J T)$ iterations across all $d$ values, with the complexity of each iteration being $O(T)$.

Figures (6)

  • Figure 1: Illustration of a non-preemptive shiftable load (job)
  • Figure 2: Forming and updating the multigraph for the adjustment step: A toy example
  • Figure 3: Solar irradiance and number of plugged-in EVs for different time of day
  • Figure 4: Relative computational time
  • Figure 5: Scheduled aggregate load for $J=2000$
  • ...and 1 more figures

Theorems & Definitions (20)

  • Example 1: Adjustment for rectangular load shape
  • Lemma 1: Complexity of the adjustment step
  • Theorem 1: Sub-optimality bound
  • Corollary 1: Asymptotic optimality for per-job cost
  • Lemma 2: Feasibility of the adjusted solution
  • Lemma 3: Lossless adjustment
  • Lemma 4: Bound on number of fractional entries
  • Lemma 5: Necessity of adjustment step
  • Example 2: Load serving with renewables
  • Theorem 2: Sub-optimality bound, modified RAR
  • ...and 10 more