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Optimal Demand Response with Energy Storage Management
Longbo Huang, Jean Walrand, Kannan Ramchandran
TL;DR
The paper addresses cost-minimizing demand response and energy storage management when finite storage couples decisions across time and system dynamics are uncertain. It develops DR-ESM, a Lyapunov-based low-complexity scheme requiring only a small per-slot convex program and no statistical knowledge. The authors prove near-optimal performance, explicitly compute the required storage size, and report substantial simulated cost reductions.
Problem
Finite-capacity storage couples control actions across time, making demand response and energy management difficult under time-varying prices, stochastic renewable energy, and changing disutility.
Method
DR-ESM combines demand response and storage management using Lyapunov optimization without statistical system knowledge, solving a convex program with 6 variables and 6 linear constraints per decision.
Results
DR-ESM achieves near-optimal performance with explicitly computed storage capacity; simulations reduce average cost by 64%−136% relative to Greedy.
Takeaways & Limitations
The proposed schemes provide lightweight energy management and demand-response control that can be implemented with finite storage and performance guarantees.
Abstract
from arXiv · showhide
In this paper, we consider the problem of optimal demand response and energy storage management for a power consuming entity. The entity's objective is to find an optimal control policy for deciding how much load to consume, how much power to purchase from/sell to the power grid, and how to use the finite capacity energy storage device and renewable energy, to minimize his average cost, being the disutility due to load- shedding and cost for purchasing power. Due to the coupling effect of the finite size energy storage, such problems are challenging and are typically tackled using dynamic programming, which is often complex in computation and requires substantial statistical information of the system dynamics. We instead develop a low-complexity algorithm called Demand Response with Energy Storage Management (DR-ESM). DR-ESM does not require any statistical knowledge of the system dynamics, including the renewable energy and the power prices. It only requires the entity to solve a small convex optimization program with 6 variables and 6 linear constraints every time for decision making. We prove that DR-ESM is able to achieve near-optimal performance and explicitly compute the required energy storage size.
I. INTRODUCTION
The paper studies cost-minimizing energy management for consumers combining renewable energy, grid transactions, demand response, and finite-capacity storage. It develops DR-ESM as a low-complexity alternative that requires little system knowledge and offers near-optimal performance with explicit storage sizing.
- Renewable penetration increases smart-grid uncertainty, while storage smooths fluctuations and reduces supply-demand mismatch.
- The optimization jointly selects consumption, grid purchases and sales, renewable use, and storage charging or discharging for a finite-capacity device.
- Finite storage couples decisions across time, while time-varying prices, stochastic renewables, changing disutility, and demand response complicate control.
- DR-ESM requires no statistical knowledge of system dynamics and solves a convex program with 6 variables and 6 linear constraints per decision.
- The paper explicitly computes required storage size and proves near-optimal performance with the chosen capacity.
B. Energy storage dynamics
The model tracks storage through charging and discharging actions subject to energy availability and physical constraints. The objective is long-term average cost under assumptions about bounded, initially i.i.d. system inputs that are later generalized.
- Storage evolves by subtracting efficiency-adjusted discharge and adding efficiency-adjusted grid and renewable charging.
- The energy-availability constraint requires sufficient stored energy for every discharging action.
- The objective minimizes long-term average disutility, grid-purchase cost, and net selling gains over feasible control policies.
- The baseline model assumes the price, renewable, and system-state quadruple is i.i.d. across slots, while allowing arbitrary component correlation.
- The model bounds renewable energy, residual load, buying prices, and selling prices, and interprets a rate condition as supporting a full load for one slot.
D. Discussion of the model
The paper first isolates non-deferrable load serving to expose its solution approach, then addresses storage sizing and low-complexity control under the energy-availability constraint. Lyapunov optimization replaces statistical dynamic programming requirements with a lightweight adaptive method.
- Prior work uses dynamic programming, receding-horizon control, or convex programs, but related models omit combinations of stored-power selling and demand response.
- The paper first studies fixed-load serving before extending the results to demand response.
- The framework asks how to size storage and control it with low-complexity algorithms that adapt quickly and provide performance guarantees.
- Lyapunov optimization temporarily ignores energy availability, then proves the resulting algorithm automatically satisfies it while achieving near-optimal performance.
A. The energy storage management algorithm (ESM)
ESM uses Lyapunov drift minimization to choose storage and grid actions from current system observations. The resulting algorithm is lightweight, needs no statistical model, and supports finite-capacity implementation through deterministic energy bounds.
- The control parameters θ and ϵ determine energy targeting and the distance between algorithmic performance and the optimum.
- ESM uses the min-drift principle to minimize a drift bound over feasible charging, discharging, buying, and selling actions.
- ESM uses storage capacity θ + η_i c_char and automatically prevents energy underflow while keeping stored energy deterministically bounded.
- At each slot, ESM observes energy, residual load, and prices, then solves an optimization over selling, purchasing, charging, discharging, and renewable charging actions.
- ESM requires a linear program with 5 variables and 5 linear constraints per slot and no statistical knowledge of loads, renewables, or prices.
B. Performance analysis of ESM
ESM achieves bounded storage operation while preserving near-optimal average-cost performance. Its storage bound is a sample-path result, enabling finite-capacity implementation under arbitrary system processes.
- The theorem's average-cost performance is summarized for ESM under the stated storage-capacity choice.
- ESM's theorem assumes θ follows (23) and the initial energy satisfies 0 ≤ E(0) ≤ θ + ηicchar.
- The storage level under ESM remains nonnegative and deterministically upper bounded by θ + ηicchar.The bound is established by induction over the storage dynamics.
- The storage bound is a sample-path result that holds under arbitrary processes for consumption, prices, renewable energy, and system conditions.This supports application under more general system dynamics than a specific stochastic model.
- ESM sets rc(t) = 0 whenever E(t) > θ, preventing further charging beyond the threshold and keeping storage finite.The resulting loss of optimality is no more than O(ϵ) when a small fraction of renewable energy is occasionally wasted.
IV. ENERGY MANAGEMENT WITH DEMAND RESPONSE
The demand-response extension lets the user choose consumption dynamically within a maximum level. Its instantaneous cost includes the disutility of the selected consumption level and depends on the storage state.
- With demand response, the user chooses a consumption level 0 ≤ ˜L(t) ≤ Lmax in every time slot.The user may change consumption according to system conditions.
- The instantaneous cost includes D(˜L(t), S(t)), the disutility of consuming power level ˜L(t).
A. ESM with demand response
DR-ESM incorporates demand response by selecting consumption and energy-flow actions through a per-slot optimization. The algorithm uses current system observations and retains a small convex decision problem without statistical system knowledge.
- DR-ESM constructs demand-response actions by minimizing the right-hand side of a drift inequality at each time slot.The construction follows the same argument used for the preceding storage-management algorithm.
- The algorithm uses storage capacity θ + ηicchar and makes decisions after observing E(t), renewable energy r(t), and prices p(t) and q(t).
- The per-slot optimization chooses ˜L(t), hs(t), dl(t), ds(t), dc(t), and rc(t).
- DR-ESM requires no statistical knowledge of system variables and solves a convex program with 6 variables and 6 constraints.The program can be solved efficiently for each decision.
- Theorem 2 states DR-ESM's performance under the prescribed θ and initial-energy condition.
V. ESM WITH MARKOVIAN SYSTEM DYNAMICS
The analysis extends DR-ESM to finite-state irreducible and aperiodic Markovian system dynamics. Under the stated storage and initialization conditions, it achieves performance similar to the i.i.d. case.
- The system state quadruple (p(t), q(t), r(t), S(t)) is modeled as a finite-state irreducible and aperiodic Markov chain.
- Under this Markov model, DR-ESM and ESM achieve performance similar to their performance in the i.i.d. case.
- The Markovian performance theorem assumes θ is chosen according to (23) and 0 ≤ E(0) ≤ θ + ηicchar.
- The result follows from the earlier theorems for one bound as a sample-path result, while the other uses a variable multi-slot drift argument.
VI. SIMULATION
The simulations evaluate DR-ESM for a 10-user-equivalent consumer using renewable generation, finite storage, and time-varying conditions. DR-ESM substantially lowers average cost and maintains a bounded, nonnegative storage trajectory.
- Simulation setup: The simulation models an apartment or small commercial building representing 10 residential users, with one-hour time slots.The entity is assigned a 9kW wind turbine and the consumption level is treated in kW and kWh equivalently because each slot lasts one hour.
- Simulation setup: The simulations use wind data averaged and normalized from 2006 records for 100 turbines near the California west coast, with mean wind power of 8kW.The model assumes equal-probability high and low environmental states and specifies charging and discharging efficiencies of 80%.
- Simulation setup: The experiments simulate storage capacities defined by 18V + 24.6kWh for V = {2, 5, 10, 20, 50} over 10^4 slots, comparing DR-ESM with a no-storage Greedy policy.The Greedy baseline minimizes instantaneous cost at each time slot.
- Cost results: 64%−136%: DR-ESM reduces average cost relative to Greedy without storage.For V = 5, average cost falls from 8.24¢ to −1.61¢, a 120% saving; negative cost reflects profit from buying and selling power.
- Storage behavior: 118.35kWh: the provisioned storage capacity when V = 5, equivalent to about 12kWh per individual user.The reported sample path remains below capacity and never drops below 0; the paper also reports implementation with 75kWh storage.
- Conclusion: The paper concludes that DR-ESM uses a small convex optimization program and achieves near-optimal performance while allowing explicit storage sizing.This conclusion summarizes the proposed energy-management schemes for finite storage and renewable energy.
APPENDIX A - PROOF OF LEMMA 1
The appendix proves a drift-based performance result by comparing ESM with an existing stationary randomized policy. The argument establishes bounded long-run performance while highlighting why the comparison policy may be impractical to implement.
- Proof of Lemma 1: The proof derives a one-step quadratic energy-level bound involving the storage deviation E(t) − θ and the net charging and discharging actions.The bound is obtained by squaring the storage relation, applying feasibility constraints, and taking conditional expectations.
- Proof of Lemma 1: The resulting inequality is converted into an expected drift relation by conditioning on E(t) and using the definition of Δ(t).This completes the stated proof of Lemma 1.
- Benchmark policy: Theorem 4 asserts the existence of a stationary randomized energy-management policy achieving the benchmark used in the performance proof.The expectation includes random system dynamics and possible randomness in charging, discharging, purchasing, and selling actions.
- Practical limitation: The benchmark policy may be impractical because it requires complete statistical knowledge and its required storage size may be difficult to compute or infinite.The paper presents ESM as a low-complexity alternative with similar performance.
- Performance comparison: ESM minimizes the right-hand side of the drift inequality, so substituting the benchmark policy yields the comparison needed for the theorem bound.The proof then takes expectations, sums over t = 0, ..., T − 1, rearranges terms, and divides by TV.
- Performance comparison: The proof concludes by taking a lim sup as T → ∞, using finite expected energy, and establishing the target long-run result.The theorem applies to both load-serving and demand-response cases.
APPENDIX - PROOF OF THEOREM 2
Theorem 2 is proved by showing that the proposed control actions preserve the storage lower bound through separate cases. The induction uses the maximum possible discharge and the signs of the optimization weights.
- Inductive setup: The proof treats the optimized load as a given load because it does not directly affect the other actions.This isolates the load variable before proving the storage lower bound by induction.
- Inductive lower bound: The lower-bound proof assumes the storage bound at time t and analyzes the next energy level under separate cases.The cases distinguish whether current energy is above or below ηe min[Lmax, cdis].
- Inductive lower bound: When E(t) ≥ ηe min[Lmax, cdis], the next energy level remains nonnegative because this is the maximum amount that can be discharged.This directly establishes the lower bound for the first case.
- Inductive lower bound: For c(t) = cgrid or c(t) = cchar, the chosen actions imply E(t + 1) ≥ E(t) ≥ 0, after which the proof follows the earlier theorem argument.The charging case uses that Wl(t) < 0, so the optimal load is chosen as large as possible.