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Online Energy Management for a Sustainable Smart Home with an HVAC Load and Random Occupancy

Liang Yu, Tao Jiang, Yulong Zou

arXiv:1706.02831v3eess.SYcs.SY

TL;DR

Smart-home energy management must balance long-term energy and thermal-discomfort costs despite uncertain conditions and unknown future parameters. The paper formulates this problem and proposes an online Lyapunov-based controller; simulations report lower energy cost than a baseline with a small thermal-comfort sacrifice.

  • Problem

    The paper seeks to minimize long-term energy and thermal-discomfort costs despite uncertainty and time-coupled constraints, which make the problem challenging when future system parameters are unknown.

    Source: §I · p. 2

  • Method

    The proposed online Lyapunov-based algorithm stabilizes queues associated with indoor temperature, EV charging, and ESS control without predicting system parameters.

    Source: NOMENCLATURE · p. 1, 1) Constructing virtual queues · p. 5, 3) Minimizing the upper bound · p. 6

  • Results

    Energy cost fell by 36.3% compared with B1, with small sacrifice of ATD (ATD=1.295).

    Source: 2) The impact of · p. 10

  • Takeaways & Limitations

    The paper concludes that its online algorithm is effective on real-world traces and does not require submitting unknown HVAC power demands into an energy queue.

    Source: §VI · p. 11

Abstract

from arXiv · show

In this paper, we investigate the problem of minimizing the sum of energy cost and thermal discomfort cost in a long-term time horizon for a sustainable smart home with a Heating, Ventilation, and Air Conditioning (HVAC) load. Specifically, we first formulate a stochastic program to minimize the time average expected total cost with the consideration of uncertainties in electricity price, outdoor temperature, renewable generation output, electrical demand, the most comfortable temperature level, and home occupancy state. Then, we propose an online energy management algorithm based on the framework of Lyapunov optimization techniques without the need to predict any system parameters. The key idea of the proposed algorithm is to construct and stabilize four queues associated with indoor temperature, electric vehicle charging, and energy storage. Moreover, we theoretically analyze the feasibility and performance guarantee of the proposed algorithm. Extensive simulations based on real-world traces show the effectiveness of the proposed algorithm.

NOMENCLATURE

The paper frames smart-home energy management around uncertain conditions and time-coupled HVAC and EV loads, then develops an online Lyapunov-based approach. Its model and literature review motivate controlling temperature, charging, and storage without forecasting system parameters.

NOMENCLATURE

  • Queue notation maps indoor temperature, EV charging, and ESS control to virtual queues Ht, Qt/Zt, and Kt.The proposed approach stabilizes queues associated with indoor temperature, EV charging, and energy storage.Source: NOMENCLATURE · p. 1, §I · p. 2

I. INTRODUCTION

  • The formulation minimizes long-term expected energy and thermal-discomfort costs under uncertainty, while unknown future parameters and time-coupling make the problem challenging.The cited uncertainties include electricity price, outdoor temperature, renewable output, demand, preferred temperature, and occupancy.Source: §I · p. 2

II. LITERATURE REVIEW

  • Prior Lyapunov methods do not directly fit HVAC loads because their power demand depends on temperature preferences, comfort bounds, occupancy, and outdoor temperature.The paper contrasts this with existing queue-based scheduling for loads with specified energy or power demands.Source: §I · p. 2

I. INTRODUCTION

  • Real-world-trace simulations show the proposed algorithm reduces energy cost with a small sacrifice in thermal comfort.The paper also reports theoretical analysis of feasibility and performance guarantees.Source: §I · p. 2

III. SYSTEM MODEL AND PROBLEM FORMULATION

  • The smart-home model gives HEMS control over generation, storage, and flexible HVAC and EV loads, while other loads are served immediately.The model also includes two-way communication with the utility grid and buying or selling electricity.Source: §III · p. 3

A. Renewable energy model

  • The renewable-energy model estimates photovoltaic output from panel efficiency, radiation area, and solar radiation.The paper focuses on winter heating; it says the algorithm can be extended to summer cooling by changing specified equations.Source: A. Renewable energy model · p. 3, 1) HVAC model · p. 3

1) HVAC model:

  • The HVAC model describes indoor-temperature dynamics using outdoor temperature and system parameters, with inverter input power continuously adjustable within its rating.It models heating and constrains indoor temperature to the comfort range.Source: 1) HVAC model · p. 3

2) EV charging model:

The section models EV charging as a deferrable request managed through an energy queue and delay constraints. It also describes ESS charging, discharging, stored-energy bounds, and the assumed efficiency model.

2) EV charging model:

  • An EV request specifies start time, completion time, and energy demand, which the HEMS schedules to exploit dynamic prices while meeting the completion deadline.Charging can be deferred when prices are high and performed when they are low.Source: 2) EV charging model · pp. 3–4
  • The EV energy queue tracks arrivals and service for online scheduling, with bounded arrivals and service supporting queue stability.The maximum service rate is at least the maximum arrival rate, and demand beyond the current queue need not be served.Source: 2) EV charging model · p. 4
  • The model limits each slot’s EV charging demand and adds a queue-delay constraint to protect the charging completion time.The allowed service delay is defined relative to the request’s start, completion, and submission duration.Source: 2) EV charging model · p. 4

C. ESS model

  • The ESS model represents charging or discharging power and stored-energy dynamics, with energy constrained between minimum and maximum capacities.The paper assumes perfect charging and discharging efficiency, while noting that the algorithm can be extended to more complex ESS models.Source: C. ESS model · p. 4

D. Power balancing

The paper formulates grid exchange and thermal discomfort costs, then proposes an online Lyapunov-based controller for the coupled HVAC, EV, and storage problem. It uses virtual queues to handle constraints without predicting system parameters or knowing slot-level HVAC demand.

E. Problem formulation

  • Electricity cost uses one signed grid transaction: positive g_t denotes purchasing and negative g_t denotes selling, with buying and selling prices constrained by B_t ≥ S_t.The paper says this assumption prevents profit from buying and reselling electricity at a higher price.Source: D. Power balancing · p. 4, E. Problem formulation · p. 4
  • The objective minimizes energy cost plus thermal discomfort cost, accounting for uncertain prices, outdoor temperature, renewable output, EV demand, comfort temperature, and occupancy.The decision variables are HVAC input, EV service, storage power, and grid transaction.Source: E. Problem formulation · pp. 4–5

A. The proposed online algorithm

  • The algorithm constructs virtual queues for indoor temperature, EV charging delay, and energy storage, then stabilizes them to support feasible operation without forecasts or known HVAC demand.The paper states that queue stability ensures constraints (4), (9), and (12).Source: A. The proposed online algorithm · p. 5

1) Constructing virtual queues:

The method represents indoor temperature, EV delay, and storage energy with virtual queues, alongside the actual EV energy queue. It then combines their stability objective with the one-slot expected total cost through a drift-plus-penalty term.

1) Constructing virtual queues:

  • Virtual queues are shifted versions of indoor temperature and ESS energy, and a delay-aware queue is defined to help enforce the EV charging-delay requirement.The EV delay constraint is guaranteed when the actual energy queue and delay-aware queue have finite upper bounds.Source: 1) Constructing virtual queues · p. 5

2) Obtaining drift-plus-penalty term:

  • The paper includes the actual EV energy queue with the virtual queues in a Lyapunov function, then evaluates one-slot conditional drift over the system’s random inputs and control decisions.The randomness includes prices, outdoor temperature, renewable output, EV demand, comfort level, and occupancy.Source: 2) Obtaining drift-plus-penalty term · p. 5
  • Adding expected one-slot total cost to the drift bound yields a drift-plus-penalty objective weighted by V, which trades queue stability against total-cost reduction.The controller minimizes an upper bound on this term.Source: 2) Obtaining drift-plus-penalty term · p. 6

3) Minimizing the upper bound:

At each slot, the controller solves a convex optimization problem to choose grid exchange, HVAC, EV, and storage actions, with a vacancy rule that can turn HVAC off. Theorems establish feasibility by bounding indoor temperature, EV queues and delay, and storage energy.

3) Minimizing the upper bound:

  • The proposed controller minimizes an upper bound on drift-plus-penalty through a four-variable convex problem, and turns HVAC off under specified vacancy and temperature conditions.The problem can be solved with available convex methods or tools.Source: 3) Minimizing the upper bound · p. 6

B. Algorithm feasibility

  • With an initial indoor temperature in the comfort range and prescribed V and Γ intervals, Theorem 1 guarantees indoor temperature remains within its bounds at every slot.The allowable parameter intervals are specified in the theorem.Source: B. Algorithm feasibility · p. 6
  • If maximum EV charging power is at least the maximum arrival rate and ξ, and both initial EV queues are zero, Theorem 2 bounds the energy and delay queues and gives a maximum-delay bound.The passages provide queue bounds Q_max = V B_max + a_max and Z_max = V B_max + ξ.Source: B. Algorithm feasibility · p. 6
  • With initial storage energy in range and prescribed V and α intervals, Theorem 3 guarantees ESS energy remains within its bounds for all slots.The theorem gives parameter intervals for V and α.Source: B. Algorithm feasibility · p. 7
  • The authors conclude that the algorithm is feasible for P1 because the theorems ensure the remaining constraints and the algorithm explicitly incorporates the others.The paper tests effectiveness using real-world traces of outdoor temperature, electricity price, and PV generation.Source: B. Algorithm feasibility · p. 7

C. Performance guarantee

The performance guarantee is stated for system parameters that are i.i.d. over slots and bounds the algorithm’s expected time-average cost relative to the optimal objective of P1. The authors caution that the optimality gap need not decrease monotonically with V.

C. Performance guarantee

  • Under i.i.d. prices, outdoor temperatures, renewable output, EV demand, comfort levels, and occupancy, Theorem 4 gives an expected time-average cost guarantee relative to P1’s optimal objective.The bound includes a Θ term whose dependence on V is complex.Source: C. Performance guarantee · p. 7
  • The optimality gap need not decrease monotonically as V increases, and performance analysis for non-i.i.d. parameters is left to future work.When ε = 1, Θ becomes constant and performance improves with a greater V.Source: C. Performance guarantee · p. 7

V. PERFORMANCE EVALUATION

In simulations using real-world traces, the proposed algorithm satisfies the reported feasibility checks: indoor temperature and ESS energy stay within their normal ranges, and maximum EV charging delay is below R=5.

A. Simulation setup

  • Simulations use January 2017 hourly outdoor-temperature and retail-price data from Nanjing, solar-radiation data from Golden, and step counts as an occupancy proxy.A threshold of 1800 hourly steps is used to classify the home as unoccupied.Source: A. Simulation setup · p. 7

1) Algorithm feasibility:

  • B1 turns off HVAC during vacancy when next-slot temperature would stay above T min; B2 preheats when Bt < εBt+1 and otherwise lets indoor temperature decrease.B1 aims to maintain the most comfortable temperature when occupied; B2's control depends on consecutive Bt values.Source: 1) Algorithm feasibility · pp. 7, 10
  • Step-count thresholding can mistake cases such as office occupancy for home occupancy, though the method remains effective for sleep hours (22pm-7am).The threshold is 1800 steps per hour.Source: 1) Algorithm feasibility · p. 7, A. Simulation setup · p. 7
  • Indoor temperature and ESS energy remain within their normal ranges, and maximum EV charging delay stays below R=5, verifying feasibility for P1.The figure reports this check for ε = 0.98, T min = 15oC, and γ = 0.Source: 1) Algorithm feasibility · p. 7, Figure 3 · p. 8

2) The impact of

The impact analysis shows that the proposed algorithm’s cost and comfort outcomes depend on comfort-range and control settings. The conclusion returns to online HVAC management under random occupancy and reports effectiveness in real-world-trace simulations.

2) The impact of

  • The proposed algorithm reduces energy cost by 36.3% versus B1 with ATD of 1.295; its energy cost is lower than B3’s when V ≥ 13, while B3 is lower otherwise.The discussion attributes the latter pattern to small V limiting use of temporal price diversity.Source: 2) The impact of · p. 10
  • Larger ε generally lowers the proposed algorithm’s energy cost, but cost begins increasing at ε ≥ 0.98; versus B3, it lowers energy cost without sacrificing ATD under the given configuration.B2 has the lowest energy cost but the largest ATD, so it is not necessarily best when both costs matter.Source: 2) The impact of · p. 10
  • The proposed algorithm achieves the lowest total cost for γ in [0.002, 0.016] without vacancy and [0.002, 0.02] with vacancy; below 2°C ATD, it reduces energy cost versus B1 by 40.41% and 38.95%, respectively.B1 is best when γ > 0.02, while B2 performs best only in [0, 0.001].Source: 2) The impact of · p. 10

VI. CONCLUSIONS AND FUTURE WORK

  • The paper concludes that its online Lyapunov-based algorithm manages HVAC energy and thermal discomfort under random occupancy without predicting system parameters.Real-world-trace simulations showed effectiveness; future work proposed online HVAC control in commercial buildings and HVAC-load aggregation in residential demand response.Source: §VI · p. 11

APPENDIX A PROOF OF LEMMA 1

The appendix proofs establish threshold-dependent HVAC decisions and use induction to verify the stated temperature inequalities across time slots.

APPENDIX A PROOF OF LEMMA 1

  • The Lemma 1 proof concludes that HVAC input is at maximum or zero under the specified threshold conditions.The displayed conditions compare V and price-related terms with the indoor-temperature virtual queue term.Source: Appendix A · p. 11

APPENDIX B PROOF OF THEOREM 1

  • The Theorem 1 proof uses mathematical induction, starting from the initial comfort-range condition and checking the inequalities for the next slot.Its case analysis uses zero or maximum HVAC input to verify the temperature bounds.Source: Appendix B · p. 11

APPENDIX C PROOF OF LEMMA 2

The Lemma 2 proof establishes threshold rules for the EV queue’s service rate: zero service below one threshold and maximum feasible service above another.

APPENDIX C PROOF OF LEMMA 2

  • When Qt + Zt < V Smin, the proof concludes that the optimal EV service rate x*t is zero.Source: Appendix C · p. 12
  • When Qt + Zt > V Bmax, the proof concludes that x*t equals min{xmax, Qt}.Source: Appendix C · p. 12

APPENDIX D PROOF OF THEOREM 2

The appendix bounds the EV queues to establish a charging-delay guarantee and proves directional conditions for ESS control.

APPENDIX D PROOF OF THEOREM 2

  • The proof bounds Qt and Zt, then uses their bounds to derive a maximum queueing-delay expression and satisfy the tolerant delay R.The expression is continued across the supplied passages.Source: Appendix D · p. 12

APPENDIX E PROOF OF LEMMA 3

  • The Lemma 3 proof concludes y*t ≤ 0 when Kt > −V Smin and y*t ≥ 0 when Kt < −V Bmax.Source: Appendix E · p. 12

APPENDIX F PROOF OF THEOREM 3

Theorem 3’s proof uses induction and case-by-case ESS decisions to keep the stated bounds on G_t satisfied. Theorem 4’s proof derives a drift-plus-penalty bound and compares feasible control policies under stated i.i.d. assumptions.

APPENDIX F PROOF OF THEOREM 3

  • Induction starts from G_0 within the bounds and uses case-specific optimal ESS decisions to show G_{t+1} remains within them.The proof treats three ranges of G_t and selects α_min or α_max as needed.Source: Appendix F · p. 12

APPENDIX G PROOF OF THEOREM 4

  • When P3 is feasible and the listed system parameters are i.i.d. over slots, a stationary randomized policy depending on current observations is available.The proof also notes that any feasible P1 solution is feasible for P3, so y_2 ≤ y_1.Source: Appendix G · p. 13
  • The drift-plus-penalty bound is Θ + V y_1, with Θ defined from the Ω_l terms and temperature bounds.The proof says the proposed algorithm minimizes this upper bound over control strategies, including an optimal stationary randomized strategy.Source: Appendix G · p. 13
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