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Control and Optimization Meet the Smart Power Grid - Scheduling of Power Demands for Optimal Energy Management

Iordanis Koutsopoulos, Leandros Tassiulas

arXiv:1008.3614v1cs.NIcs.SYeess.SY

TL;DR

The paper formulates demand scheduling to smooth power loads and manage grid operational cost. It analyzes threshold-based control policies and shows that Controlled Release approaches the all-policy lower bound as deadlines increase.

  • Problem

    The grid operator must smooth peak demand to reduce overload periods and support reliability, even though shifting demand does not significantly reduce total energy consumption.

    Source: §I · p. 1

  • Method

    The paper formulates demand scheduling with real-time operator-consumer communication and an increasing convex power-cost model, then studies offline scheduling and two online threshold-control variants.

    Source: §I · p. 1, A. Cost Model · p. 4, 2) Enhanced control space · p. 8

  • Results

    As deadlines increase, the Controlled Release policy with an optimized threshold converges in performance to the lower bound over all policies.

    Source: B. Our contribution · p. 3, C. An Asymptotically Optimal Policy:… · p. 7

  • Takeaways & Limitations

    Demand load management smooths the power profile and avoids overload periods, but does not significantly reduce total energy consumption.

    Source: §I · p. 1

    The analysis assumes the provider has full control over consumer appliances and that consumers always comply with the dictated schedule.

    Source: §V · p. 9

Abstract

from arXiv · show

The smart power grid aims at harnessing information and communication technologies to enhance reliability and enforce sensible use of energy. Its realization is geared by the fundamental goal of effective management of demand load. In this work, we envision a scenario with real-time communication between the operator and consumers. The grid operator controller receives requests for power demands from consumers, with different power requirement, duration, and a deadline by which it is to be completed. The objective is to devise a power demand task scheduling policy that minimizes the grid operational cost over a time horizon. The operational cost is a convex function of instantaneous power consumption and reflects the fact that each additional unit of power needed to serve demands is more expensive as demand load increases.First, we study the off-line demand scheduling problem, where parameters are fixed and known. Next, we devise a stochastic model for the case when demands are generated continually and scheduling decisions are taken online and focus on long-term average cost. We present two instances of power consumption control based on observing current consumption. First, the controller may choose to serve a new demand request upon arrival or to postpone it to the end of its deadline. Second, the additional option exists to activate one of the postponed demands when an active demand terminates. For both instances, the optimal policies are threshold based. We derive a lower performance bound over all policies, which is asymptotically tight as deadlines increase. We propose the Controlled Release threshold policy and prove it is asymptotically optimal. The policy activates a new demand request if the current power consumption is less than a threshold, otherwise it is queued. Queued demands are scheduled when their deadline expires or when the consumption drops below the threshold.

I. INTRODUCTION

Smart-grid demand management shifts flexible demand to smooth the load profile, helping avoid overloads and reduce operating costs. The paper formulates deadline-constrained scheduling under convex power costs and develops offline and online approaches for the operator.

I. INTRODUCTION

  • Demand load management shifts non-emergency demand from peak to off-peak periods, smoothing the load profile and helping avoid overloads and costly supplementary generation.The stated benefits include greater grid reliability and lower operating costs.Source: §I · p. 1

A. State-of-the-art

  • Prior work spans demand-response automation and deadline scheduling, but the paper identifies its contribution as characterizing scheduling structure and solutions for smart-grid demand management.Related work includes convex-cost deadline scheduling in wireless networks.Source: A. State-of-the-art · p. 2, B. Our contribution · p. 2

B. Our contribution

  • The offline formulation treats known task parameters, with an optimal iterative algorithm for elastic, preemptive demands and an NP-hard bin-packing formulation for inelastic, non-preemptive demands.Task parameters include generation time, duration, power requirement, and deadline.Source: B. Our contribution · p. 2, Figure 1 · p. 3
  • For continual stochastic arrivals, the paper studies online scheduling for long-term average cost, including the default policy of serving each task upon arrival.Source: B. Our contribution · p. 2
  • Two consumption-observing control instances yield threshold-based optimal policies; the enhanced instance can activate a postponed demand when an active one ends.The first instance chooses between immediate service and postponement until the deadline.Source: B. Our contribution · p. 3
  • The Controlled Release policy serves arrivals below a consumption threshold and otherwise queues them for deadline expiry or a later drop below the threshold; its performance is asymptotically optimal as deadlines increase.The paper derives a lower bound over all policies that is asymptotically tight as deadlines increase.Source: B. Our contribution · p. 3

II. THE MODEL

The model represents each consumer request as a power task with timing flexibility and requires all tasks to be scheduled by their deadlines. It considers both non-causal offline scheduling and stochastic online scheduling under convex power costs.

II. THE MODEL

  • Each task has a generation time, duration, fixed power requirement, and deadline; its allowable activation interval depends on whether scheduling is preemptive or non-preemptive.Tasks must eventually be scheduled by their deadlines, and power cannot be adapted by the controller.Source: §II · p. 3
  • The operator communicates schedules to consumer appliances and is assumed to have full control, with appliances complying and communication delays treated as zero.Source: §II · p. 3
  • The offline model assumes task parameters are known non-causally, while the online model treats quantities as stochastic and targets long-term average cost.Source: §II · pp. 3–4
  • Instantaneous cost is an increasing, differentiable convex function of total active-task power, representing higher marginal costs as demand rises.The illustrated piecewise-linear example has three power-consumption classes.Source: Figure 3 · p. 4

A. Cost Model

The cost model makes additional power progressively more expensive, and the off-line problem assumes task parameters are known before scheduling. For preemptive tasks, the paper formulates load balancing as a continuous optimization problem and gives an iterative algorithm that converges to its optimum.

A. Cost Model

  • The increasing, differentiable convex cost C(P(t)) reflects that each additional unit of power becomes more expensive as demand rises.The model also allows a piecewise-linear cost with increasing marginal rates across consumption classes.Source: A. Cost Model · p. 4

III. THE OFF-LINE DEMAND SCHEDULING PROBLEM

  • The off-line problem assumes each task’s generation time, power, duration, and deadline are known non-causally before time zero.The paper notes this setting may apply when task properties are predictable and can provide performance bounds.Source: §III · p. 4

1) Preemptive scheduling of tasks:

  • For preemptive tasks, binary activity variables are relaxed to continuous values, yielding a fluid model that balances load across time.The fluid model represents task power as p_n x_n(t) and uses task-to-time-interval flows over each task’s availability window.Source: 1) Preemptive scheduling of tasks · p. 4
  • The iterative load-balancing algorithm converges to the optimal solution of the continuous-valued preemptive scheduling problem.The paper states that the optimal solution minimizes the maximum power load over time.Source: 1) Preemptive scheduling of tasks · p. 5

2) Non-preemptive scheduling of tasks:

With inelastic tasks sharing a common release time, deadline, and power requirement, minimizing total convex cost is NP-hard. The paper establishes this by relating the scheduling problem to one-dimensional bin packing.

2) Non-preemptive scheduling of tasks:

  • With common deadline D and equal task power p, minimizing peak consumption corresponds to packing task durations into the fewest bins of capacity D.Each bin represents one level of step p in the power-consumption profile.Source: 2) Non-preemptive scheduling of tasks · p. 5
  • Minimizing total convex cost for inelastic demand scheduling is NP-hard, via equivalence to optimization one-dimensional bin packing.The decision version of bin packing is NP-complete.Source: 2) Non-preemptive scheduling of tasks · p. 5
  • Different release times and deadlines can also yield bin-packing instances, while differing power requirements can be represented using a minimum power quantum.A task requiring nΔp is replaced by n tasks of power Δp and the same duration.Source: 2) Non-preemptive scheduling of tasks · p. 5

IV. THE ONLINE DYNAMIC DEMAND SCHEDULING PROBLEM

The online model considers continually arriving requests and minimizes long-run average cost. It assumes stochastic request durations and deadlines, while the controller observes total instantaneous power consumption for its decisions.

IV. THE ONLINE DYNAMIC DEMAND SCHEDULING PROBLEM

  • Requests arrive as a Poisson process of rate λ, and independent durations are exponentially distributed with service rate s.The mean request duration is 1/s time units.Source: §IV · p. 5
  • Request deadlines are independent exponential random variables with expiration rate d and mean 1/d time units.Source: §IV · p. 5
  • The online objective is to minimize long-run average cost, with expectation taken over the stationary distribution of instantaneous power.The controller’s assumed observable state is total instantaneous power consumption.Source: §IV · p. 5

A. Default Policy: No scheduling

The default policy activates every demand immediately, without scheduling regulation. Under unit task power, active demand count follows an M/M/∞ occupation process; with random task power, steady-state mean consumption scales with mean task power.

A. Default Policy: No scheduling

1) Fixed power requirement per task:

  • Under immediate activation of unit-power tasks, active demand count is an M/M/∞ occupation process with Poisson steady-state distribution of parameter λ/s.Its steady-state mean is λ/s.Source: 1) Fixed power requirement per task · p. 6
  • In the unit-power case, arrivals increase active consumption by one at rate λ, while completions decrease it by one at rate P(t)s.Source: 1) Fixed power requirement per task · p. 6
  • For random task power independent of the active-demand process, steady-state average consumption is λE[ P̂ ]/s.The task power is modeled by a discrete distribution, with continuous distributions handled similarly.Source: 1) Fixed power requirement per task · p. 6

2) Variable power requirement per task:

Variable power requirements are represented through equivalent unit-power arrivals. The universal lower bound follows from convexity and fixed average consumption, and is asymptotically tight as deadlines grow.

2) Variable power requirement per task:

  • For variable power requirements, a task requiring n units is represented as n unit-power requests, making mean requirement E[P_hat] equivalent to arrival rate λE[P_hat].The default policy activates each task upon arrival without using system-state information.Source: 2) Variable power requirement per task · p. 6

B. A Universal Lower Bound

  • Jensen’s inequality gives a universal cost lower bound because every scheduling policy preserves average power consumption at λE[P_hat]/s.The average consumption is fixed by the arrival rate and service rate, so scheduling shifts demand in time without changing its mean.Source: B. A Universal Lower Bound · p. 6
  • The universal lower bound is asymptotically tight as deadlines become larger.The paper states that a policy achieves the bound asymptotically as deadlines increase.Source: B. A Universal Lower Bound · p. 6

C. An Asymptotically Optimal Policy: Controlled Release

Controlled Release admits new requests only below a power threshold and queues the rest until consumption falls or their deadlines expire. With optimized threshold, its cost converges to the universal lower bound as deadlines increase.

C. An Asymptotically Optimal Policy: Controlled Release

  • Controlled Release activates a new request when power consumption is below P0; otherwise, it queues the request until consumption falls below P0 or its deadline expires.Source: C. An Asymptotically Optimal Policy:… · p. 6
  • With an optimized threshold, Controlled Release converges to the all-policy lower bound as deadlines increase.The paper establishes the result by comparing the deadline system with an auxiliary system without deadline considerations.Source: C. An Asymptotically Optimal Policy:… · p. 7
  • In the auxiliary system, the threshold makes queued requests wait for service capacity, and a sequence of thresholds yields cost equal to the universal lower bound.The auxiliary system is equivalent to an M/M/c queue; its limiting number of busy servers is constant.Source: C. An Asymptotically Optimal Policy:… · p. 7

D. Optimal Threshold Based Control Policies

The section considers two power-control policies that base decisions on instantaneous consumption and differ in their control spaces; proofs of optimality are omitted for space.

D. Optimal Threshold Based Control Policies

1) Bi-modal control space:

For fixed-power demands with exponential, homogeneous durations and deadlines, the bimodal policy either serves arrivals immediately or postpones them until their deadlines. The optimal policy is threshold-based, with a switching curve non-decreasing in the number of postponed demands.

1) Bi-modal control space:

  • The bimodal control space allows each unit-power request to be served immediately or postponed until its deadline.Source: 1) Bi-modal control space · p. 7
  • The model assumes fixed power requirements and homogeneous exponential demand durations and deadlines, with state given by active and postponed demand counts.Source: 1) Bi-modal control space · p. 7
  • The optimal bimodal policy is threshold-based, with a switching curve Pb(Q) that is non-decreasing in the number of postponed demands.Below the threshold the policy serves a new request immediately; above it, the request is postponed until its deadline.Source: 1) Bi-modal control space · pp. 7–8
  • The paper’s intuition is that higher active consumption makes reductions more likely, while more postponed demand makes increases more likely.These transition tendencies motivate a threshold that does not decrease as the postponed queue grows.Source: 1) Bi-modal control space · p. 8

2) Enhanced control space:

The enhanced policy defers arrivals when consumption is high and can activate queued demands when active service ends; with an appropriately selected threshold, it is optimal.

  • ETP serves arrivals immediately when P(t) ≤ Pe, otherwise postpones them until their deadline; active-demand completions also trigger queued-demand activation.Source: 2) Enhanced control space · p. 8
  • The model tracks active-demand count P(t) and postponed-demand count Q(t), with u(t) choosing immediate service or postponement for each arrival.Source: 2) Enhanced control space · p. 8
  • When P(t) ≤ Pe, queued demands leave at rate Q(t)d + P(t)s, reflecting deadline expiry and activation when active demands complete.Source: 2) Enhanced control space · p. 8
  • For an appropriately selected Pe, ETP is an optimal threshold policy over the enhanced control space Ue.Source: 2) Enhanced control space · p. 8

V. DISCUSSION

The paper studies offline scheduling and online threshold control under full operator control of appliances. Its online threshold policy attains a universal lower performance bound asymptotically as deadlines increase.

  • The online threshold policy achieves a universal lower performance bound asymptotically as deadlines increase.Source: §V · p. 8
  • Offline scheduling is solved optimally for elastic, preemptive demands, while the inelastic, non-preemptive problem is NP-hard.Source: §V · p. 8
  • Explicit threshold computation remains a development need: the authors call for fuller policy analysis and real-grid cost and demand data.Source: §V · p. 9
  • Future models could incorporate task classes with differing power and scheduling flexibility, including vehicle charging with energy and completion-window constraints.Source: §V · p. 9
  • Controlling appliance power levels in addition to scheduling would add another dimension to the problem, including for operator-controlled air conditioning.Source: §V · p. 9
  • The studied setting assumes provider control over appliances and compliance with the dictated schedule; consumer choice and pricing incentives are proposed as alternative scenarios.Source: §V · p. 9
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