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Efficient Algorithms for Renewable Energy Allocation to Delay Tolerant Consumers

Michael J. Neely, Arash Saber Tehrani, Alexandros G. Dimakis

arXiv:1006.4649v1math.OC

TL;DR

The paper studies how a renewable supplier can serve delay-tolerant consumers despite variable, unpredictable supply while limiting outside-energy costs or maximizing service profit. It applies Lyapunov optimization to queue-based allocation and pricing without prior statistical knowledge. The resulting algorithms provide cost-delay guarantees, robustness to arbitrary sample paths, and lower tested operating costs than a greedy strategy.

  • Problem

    Renewable suppliers need to serve flexible consumers within deadlines despite variable supply, sometimes purchasing costly outside energy when renewable supply is insufficient.

  • Method

    The paper formulates cost-minimization and dynamic-pricing profit-maximization problems and solves them with Lyapunov optimization using queue-based control.

  • Results

    The algorithms approach minimum stabilizing cost within O(1/V) with worst-case delay O(V), and the tested algorithm reduced operating cost by approximately a factor of 2 versus a greedy strategy.

  • Takeaways & Limitations

    Renewable allocation can be made efficient without prior supply, demand, and price statistics while retaining explicit delay control and robustness to non-i.i.d. or arbitrary sample-path behavior.

Abstract

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We investigate the problem of allocating energy from renewable sources to flexible consumers in electricity markets. We assume there is a renewable energy supplier that provides energy according to a time-varying (and possibly unpredictable) supply process. The plant must serve consumers within a specified delay window, and incurs a cost of drawing energy from other (possibly non-renewable) sources if its own supply is not sufficient to meet the deadlines. We formulate two stochastic optimization problems: The first seeks to minimize the time average cost of using the other sources (and hence strives for the most efficient utilization of the renewable source). The second allows the renewable source to dynamically set a price for its service, and seeks to maximize the resulting time average profit. These problems are solved via the Lyapunov optimization technique. Our resulting algorithms do not require knowledge of the statistics of the time-varying supply and demand processes and are robust to arbitrary sample path variations.

I. INTRODUCTION

The paper addresses renewable-supply variability by serving flexible consumers through queues, using outside energy when needed, and applying Lyapunov optimization without prior process statistics. It develops cost-minimizing and profit-maximizing formulations with explicit cost-delay guarantees and robustness to non-i.i.d. behavior.

  • I. INTRODUCTION: Renewable supply variability and forecast errors increase integration and operating costs, motivating demand-side flexibility and queued service within agreed deadlines.Flexible consumers may accept lower prices in exchange for waiting, allowing suppliers to deliver energy when renewable supply is available.
  • I. INTRODUCTION: A related formulation lets the renewable supplier dynamically price its service to maximize time-average profit.The framework also supports extensions involving multiple queues with different deadlines and other objective functions.
  • I. INTRODUCTION: The algorithms target cost-efficient renewable scheduling and are evaluated on actual CAISO spot-market and wind-production data.The paper reports substantial operating-cost reduction relative to a simple greedy strategy and describes robustness to arbitrary event sample paths.
  • I. INTRODUCTION: The model uses a queue for randomly arriving flexible energy requests, renewable supply s(t), and outside purchases x(t) priced at x(t)γ(t).With no storage, renewable energy must be used or wasted, while outside energy can cover insufficient renewable supply and deadline requirements.
  • I. INTRODUCTION: The first optimization problem minimizes time-average outside-source cost while stabilizing the queue and guaranteeing a maximum delay.Under unknown i.i.d. supply, demand, and price distributions, the algorithm approaches optimal stabilizing cost within O(1/V) with worst-case delay O(V).
  • I. INTRODUCTION: Lyapunov optimization is simpler than dynamic programming, needs no a-priori statistical knowledge, and remains robust to non-i.i.d. and non-ergodic behavior.Its virtual-queue extension converts an O(V) average-delay result into an O(V) worst-case delay guarantee, while avoiding dynamic programming's exponential complexity across users.

A. The Delay-Aware Virtual Queue

The method introduces a virtual queue that tracks how long unserved requests persist, enabling finite worst-case delay guarantees alongside queue stability. A Lyapunov drift-plus-penalty framework then controls real and virtual congestion using current system information.

  • Virtual-queue construction: The virtual queue Z(t) receives an additional ǫ arrival whenever the real queue Q(t) is nonempty while sharing its service process s(t)+x(t).This makes Z(t) grow when requests remain unserved, so bounded Q(t) and Z(t) imply a worst-case delay bound.
  • Delay guarantee: If Z(t) ≤ Zmax and Q(t) ≤ Qmax for every slot, all requests have maximum delay Dmax = ⌈(Qmax + Zmax)/ǫ⌉.The bound follows by accumulating ǫ over slots during which the real queue remains nonempty.
  • Delay guarantee: With outside-source service delayed by K > 0 slots, the implementation modifies the delay bound to ⌈(Qmax + Zmax)/ǫ⌉ + K.The additional K slots account for requests served through the outside source.
  • Lyapunov control: The concatenated state Θ(t) = (Z(t), Q(t)) is summarized by a quadratic Lyapunov function measuring congestion in both queues.The drift-plus-penalty controller observes queue states and current supply, demand, and outside-energy cost before choosing x(t).
  • Lyapunov control: The drift bound combines expected queue changes with the penalty x(t)γ(t), using a finite constant B determined by bounds on supply, demand, and control.The resulting per-slot optimization is based on minimizing an upper bound on drift plus cost.

C. The Dynamic Algorithm

The dynamic algorithm chooses outside energy purchases each slot by minimizing the Lyapunov drift-plus-penalty bound using current queues and observed system conditions. Its theorem provides deterministic queue and delay bounds and, under i.i.d. conditions, a cost guarantee.

  • Per-slot control: Each slot, the algorithm observes Z(t), Q(t), and (s(t), a(t), γ(t)), then selects x(t) within 0 ≤ x(t) ≤ xmax to minimize the drift bound.The parameter V controls the performance-delay tradeoff in this minimization.
  • Per-slot control: The minimization reduces to a simple threshold rule, followed by updates to the actual and virtual queues.The rule determines the control action from current queue and system states rather than future process values.
  • Implementation: The actual outside-energy purchase ˜x(t) can be smaller than the selected control x(t), because the queue update may not require the full amount.The implementation defines ˜x(t) as the amount actually purchased and guarantees ˜x(t) ≤ x(t).
  • Performance guarantees: The algorithm achieves an [O(1/V), O(V)] cost-delay tradeoff: average cost is within B/V of c∗, while worst-case delay is proportional to V/ǫ.Choosing ǫ = 0 preserves some guarantees but does not provide finite worst-case delay.

D. Proof of Theorem 1

The proof establishes deterministic bounds on both queues by induction and derives the delay guarantee from the virtual-queue lemma. Under i.i.d. system processes, a stationary comparison policy yields the average-cost bound.

  • Deterministic queue bounds: The real queue satisfies Q(t) ≤ Vγmax + amax for all t, proved by considering whether Q(t) is below or above Vγmax.When the queue is high, the algorithm selects x(t) = xmax, preventing the next backlog from exceeding the bound.
  • Deterministic queue bounds: The virtual queue satisfies Z(t) ≤ Vγmax + ǫ for all t by an analogous argument.Together with the real-queue bound, this supplies the boundedness required by the delay lemma.
  • Delay guarantee: The delay result follows immediately by applying Lemma 1 after establishing deterministic bounds for Q(t) and Z(t).Thus the proof connects queue boundedness directly to the worst-case delay guarantee.
  • Average-cost proof: For i.i.d. (s(t), a(t), γ(t)), a stationary randomized policy depending only on the current vector exists and characterizes the optimal average cost c∗.This policy is independent of queue backlogs and provides the comparison point used in the drift analysis.
  • Average-cost proof: The drift inequality is compared against that stationary policy, summed over time, and divided by VM to obtain the cost bound for every M > 0.The argument uses the nonnegative initial and terminal Lyapunov terms and the condition ǫ ≤ max[E{a(t)}, E{s(t)}].

III. PRICING FOR MAXIMUM PROFIT

The pricing extension lets the renewable plant decide whether to accept requests, set a price, and purchase outside energy while managing the same real and virtual queues. Its drift-plus-penalty formulation targets time-average profit and can exploit current demand states.

  • Pricing model: The demand state y(t) captures conditions affecting requests in response to the advertised price, such as peak status or the number of willing consumers.The state is assumed known at the beginning of each slot.
  • Pricing model: Each slot, the plant chooses b(t) ∈ {0, 1}, a price p(t) within [0, pmax], and outside energy x(t), with accepted demand influenced by p(t), γ(t), and y(t).Here b(t) = 0 means no new requests are accepted, while b(t) = 1 permits requests at the selected price.
  • Model assumptions: The arrival process remains bounded by amax regardless of price, demand state, or market price, and F(p, y, γ) is assumed continuous in p.The continuity assumption ensures a well-defined minimizing price in the per-slot drift decision.
  • Profit optimization: Instantaneous profit φ(t) is optimized through the limiting time-average profit objective, using −φ(t) as the drift-plus-penalty term.The pricing problem therefore reuses the real queue, virtual queue, and Lyapunov function from the allocation problem.
  • Profit optimization: The joint algorithm observes Q(t), Z(t), s(t), γ(t), and y(t), then chooses price and allocation by minimizing the drift bound.In the special scaling case F(p(t), y(t), γ(t)) = y(t) ˆF(p(t), γ(t)), the pricing policy does not need to know y(t) after the algorithm is stated.

A. Defining Optimality

The joint pricing and allocation policy achieves optimal time-average profit under stated i.i.d. conditions while preserving explicit backlog and delay bounds.

  • The policy jointly selects allocation, service admission, and price decisions using only the current supply, demand state, and market price.
  • The optimal time-average profit φ∗ is achievable by a stationary randomized policy satisfying the energy-balance constraint.The policy class depends only on (s(t), y(t), γ(t)).
  • Theorem 2 applies the joint policy with fixed ǫ and V under the assumption xmax ≥ max[amax, ǫ] and initially empty queues.
  • The resulting worst-case backlog is proportional to V, while worst-case delay is proportional to V/ǫ.
  • Under i.i.d. system states and ǫ ≤ max[a∗, E{s(t)}], the policy is evaluated against the optimal time-average profit φ∗.

IV. NON-I.I.D. MODELS

The paper extends its scheduling analysis to arbitrary time correlations and sample paths, retaining delay bounds and comparing performance with ideal lookahead and real-data experiments. The results show a cost-delay tradeoff and lower cost than a greedy deadline policy.

  • IV. NON-I.I.D. MODELS: The non-i.i.d. model allows arbitrary time correlations while retaining bounded sample paths for supply, demand, and market prices.The processes satisfy 0 ≤ s(t) ≤ smax, 0 ≤ a(t) ≤ amax, and 0 ≤ γ(t) ≤ γmax.
  • IV. NON-I.I.D. MODELS: The same algorithm preserves the exact worst-case backlog and delay bounds in the non-i.i.d. setting.The delay bound remains proportional to V/ǫ because its proof is sample-path based.
  • IV. NON-I.I.D. MODELS: The T-slot lookahead metric compares the algorithm with ideal frame-level allocations that know future supply, demand, and prices.Each ideal frame allocation must cover total arrivals and at least ǫT energy.
  • IV. NON-I.I.D. MODELS: For any interval of RT slots, the achieved average cost is at most the average ideal frame cost plus a fudge factor of BT/V.The ideal frame costs use full knowledge of future events within each T-slot frame.
  • V. EXPERIMENTAL EVALUATION: On a six-month CAISO and wind-production dataset, the proposed algorithm reduced renewable-supplier cost by approximately a factor of 2 versus Purchase at deadline.The evaluation used 10-minute slots and varied V, ǫ, and the resulting deadlines.
  • V. EXPERIMENTAL EVALUATION: 14 hours was the maximum experimental delay for ǫ = 0, compared with 9.5 hours for ǫ = E{a(t)}.The ǫ = 0 setting does not provide a finite delay guarantee but maintained delays below Dmax = 2.9 days in the simulations.
  • V. EXPERIMENTAL EVALUATION: 3.5, 5.8, 10.2, and 15.2 hours were the observed maximum waiting times for V = 20, 50, 100, and 200, respectively.Increasing V improved cost while increasing the waiting-time tradeoff.

VI. CONCLUSIONS

The paper presents Lyapunov optimization for efficient renewable-energy allocation with flexible demand, including cost-minimization and pricing objectives. The algorithms require no statistical process knowledge and provide a tunable performance-delay tradeoff.

  • Lyapunov optimization is presented as an approach to efficient use of renewable energy sources.
  • Flexible consumers can improve renewable-energy efficiency by tolerating delayed service.
  • The paper studies minimizing outside-plant cost and maximizing profit through dynamically selected service prices.
  • The algorithms operate without knowing statistical properties and handle i.i.d., non-i.i.d., and non-ergodic sample paths.
  • Cost is within O(1/V) of optimal while worst-case delay is O(V), creating a tunable performance-delay tradeoff.

APPENDIX A – PROOF OF LEMMA 2

The proof derives a drift bound by squaring queue updates and applying a quadratic inequality, then adds the cost penalty and takes conditional expectations.

  • The proof squares the queue update and applies a bound involving Q, µ, and a.
  • The resulting bounded term includes 2[(smax + xmax)^2 + a^2max].
  • After combining inequalities, the proof adds V E{γ(t)x(t)|Θ(t)} to both sides.

APPENDIX B – PROOF OF THEOREM 3

The proof extends drift analysis to arbitrary bounded sample paths using a T-slot sample-path drift, queue-state updates, and frame-based summation.

  • The analysis defines the state as Θ(t) = [Q(t), Z(t)] and uses the corresponding Lyapunov function.
  • Unlike the one-slot i.i.d. drift, the T-slot sample-path drift spans T slots and uses no expectation.
  • The arbitrary supply, demand, and price paths are assumed only to satisfy boundedness constraints for a(t), s(t), γ(t), and x(t).
  • The lemma applies algorithm (12) with queue updates for Q(t) and Z(t), comparing its decisions with any feasible x*(τ).
  • The proof sums the drift inequality over successive frames and divides by RT V to obtain the result.

APPENDIX C – PROOF OF THEOREM 2

The proof compares the dynamic algorithm with alternative feasible actions through conditional drift inequalities, then takes expectations and sums over time to establish the theorem.

  • The proof preserves the queue dynamics and the condition a(t) ≤ amax when extending the preceding theorem's argument.
  • The proof assumes ǫ ≤ max[a*, E{s(t)}] and separately analyzes the case ǫ ≤ a*.
  • The dynamic actions b(t), p(t), and x(t) minimize the drift bound over alternative feasible choices.
  • The comparison uses an (s, y, γ)-only policy independent of queue states, so conditional and unconditional expectations coincide.
  • Taking expectations yields E{L(Θ(t + 1))} − E{L(Θ(t))} − V E{φ(t)} ≤ B − Vφ*.
  • Summing over M slots and using nonnegative Lyapunov values establishes the result for all M > 0; an additional argument covers ǫ ≤ E{s(t)}.
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