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Profit-Maximizing Planning and Control of Battery Energy Storage Systems for Primary Frequency Control

Ying Jun, Zhang, Changhong Zhao, Wanrong Tang, Steven H. Low

arXiv:1604.00952v1eess.SY

TL;DR

BESS participation in primary frequency control requires jointly optimizing control and energy capacity, while prior work largely relied on simulations or experiments. This paper develops a theoretical dynamic-programming framework and proves state-invariant optimal control and decreasing convex operating costs with capacity.

  • Problem

    BESS owners need to jointly optimize energy capacity and operation in PFC markets, where prior analyses were largely simulation- or experiment-based.

  • Method

    The paper formulates optimal BESS control as a stochastic dynamic program and derives planning decisions by analyzing its value as a function of energy capacity.

  • Results

    The optimal target SoC range is invariant to the initial SoC, while optimal operating cost is a decreasing convex function of BESS energy capacity.

  • Takeaways & Limitations

    The results support offline computation of low-complexity BESS control and sizing that balances capital investment against operating cost.

Abstract

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We consider a two-level profit-maximizing strategy, including planning and control, for battery energy storage system (BESS) owners that participate in the primary frequency control (PFC) market. Specifically, the optimal BESS control minimizes the operating cost by keeping the state of charge (SoC) in an optimal range. Through rigorous analysis, we prove that the optimal BESS control is a "state-invariant" strategy in the sense that the optimal SoC range does not vary with the state of the system. As such, the optimal control strategy can be computed offline once and for all with very low complexity. Regarding the BESS planning, we prove that the the minimum operating cost is a decreasing convex function of the BESS energy capacity. This leads to the optimal BESS sizing that strikes a balance between the capital investment and operating cost. Our work here provides a useful theoretical framework for understanding the planning and control strategies that maximize the economic benefits of BESSs in ancillary service markets.

NOMENCLATURE … B. BESS Operation

The paper develops a theoretical profit-maximizing framework for BESS planning and control in primary frequency control markets. It models frequency-excursion operation, SoC management, charging costs, and regulation-failure penalties, and derives state-invariant control and capacity-planning results.

  • NOMENCLATURE: The nomenclature defines variables for electricity prices, regulation penalties, interval lengths, excursion events, PFC power, battery power, efficiency, capacity, power limits, SoC, and operating costs.These symbols support the system timeline and BESS operation model.
  • I. INTRODUCTION: Frequency control maintains nominal system frequency by compensating unforeseen mismatches between generation and load through primary, secondary, and tertiary reserves.Frequency deviations can compromise power quality and security.
  • I. INTRODUCTION: BESSs are attractive PFC providers because of their extremely fast ramp rate, with annual PFC-reserve profits reported as US$236-US$439 per KW in the U.S. market.PFC reserve is identified as the highest-value BESS application in the cited background.
  • I. INTRODUCTION: The theoretical framework models optimal control as a stochastic dynamic program with continuous state and action spaces, while planning analyzes the value function versus BESS energy capacity.This framework complements prior simulation- or experiment-based work.
  • I. INTRODUCTION: The optimal target SoC is state-invariant, can be computed offline with very low complexity, and becomes a fixed point when η approaches 1 or electricity price is much lower than the penalty rate.With slow-varying electricity prices, control reduces to selecting a target SoC whenever frequency lies inside the dead band.
  • II. SYSTEM MODEL: The study formulates BESS planning and control as a profit-maximization problem in which operating cost combines charging/discharging cost with penalties for failing contracted PFC service.The optimal energy capacity balances capital cost against operating cost.
  • A. System Timeline: The system timeline separates I intervals without PFC deployment from J intervals triggered by frequency excursions, whose lengths represent TSO-requested PFC deployment times.Excursions are classified as over-excursions or under-excursions using indicator variables.
  • B. BESS Operation: The BESS has energy capacity Emax, power limit Pmax, and efficiency 0 < η ≤1; it charges or discharges during I intervals to maintain SoC and avoid subsequent regulation penalties.During excursion intervals, requested PFC power is supplied or absorbed, while insufficient SoC can cause penalties proportional to the PFC energy shortage.

III. PROBLEM FORMULATION

With fixed remuneration, profit maximization reduces to minimizing capital and operating costs. The control problem is formulated as stationary infinite-horizon dynamic programming and simplified to selecting a target SoC under a full-rate policy.

  • Problem formulation: Fixed remuneration makes profit maximization equivalent to minimizing capital and operating costs for a given BESS capacity Emax.The optimal planning problem later determines Emax.
  • Problem formulation: The BESS chooses interval control p(t) from the observed system state while future interval, PFC energy, and price realizations remain unknown.The resulting stochastic dynamic program uses state transitions determined by the control and exogenous variables.
  • Problem formulation: A tendering period lasting months versus stage durations of seconds or minutes, together with i.i.d. inputs, yields an infinite-horizon dynamic program with stationary policy.Accordingly, stage subscripts can be removed from the formulation.
  • Full-rate target-SoC control: When electricity price ce is constant within an interval, an optimal policy charges or discharges at full rate Pmax until reaching target SoC π or interval end.Because charging and discharging cost depends only on total energy, continuous-time control reduces to choosing π ∈ [0, 1].
  • Bellman formulation: The stationary Bellman equation minimizes expected one-stage charging and regulation-penalty costs plus discounted expected future value over target SoC π.The expected penalty depends on the SoC when the frequency excursion occurs, while the transition follows the target-SoC control.

IV. OPTIMAL BESS CONTROL

The optimal BESS control uses a target SoC range invariant to the current SoC, enabling a simple stagewise charging policy. This range converges to a single point when η →1 or c_e →0.

  • State-invariant target: The optimal target SoC is a range invariant to the BESS SoC observed at each stage.Thus, the optimal target need not be recalculated as a function of the current SoC.
  • State-invariant target: The BESS charges or discharges when its SoC lies outside the target range and remains idle inside it.This rule applies during each I interval and is fixed across stages regardless of system state.
  • State-invariant target: The target range converges to a single point when η →1 or c_e →0.These conditions collapse the optimal SoC range to a single target point.

˜FEP F C

The analysis establishes convexity in the system state and quasi-convexity in the control variable, making the optimal control state-invariant. Consequently, the optimal decision is characterized by two scalars that remain fixed across system states, greatly simplifying computation.

  • Convexity and optimality: H∗(s) is convex in s, with ∂²H∗(s)/∂s² ≥ 0 for all s.The second derivative is characterized as the fixed point of a contraction-mapping operator.
  • Convexity and optimality: Both r1(π)Emax + u(π) and r2(π)Emax + u(π), as well as r(s, π)Emax + u(π), increase with π.This monotonicity establishes the structure needed for the optimality conditions.
  • Convexity and optimality: H(s, π) is quasi-convex in π, so the necessary conditions for π∗ are also sufficient.The conditions distinguish boundary optima from interior solutions and the case π∗ = s.
  • State-invariant control: π∗ remains fixed for all stages regardless of the system state s because the determining expressions are independent of s.The optimal decision is characterized by two scalars, π∗low and π∗high, that remain constant across system states.

V. OPTIMAL BESS PLANNING

Optimal BESS planning weighs the minimum operating cost, which depends on energy capacity, against capital cost, which increases with capacity.

  • The minimum operating cost H∗(s) is a function of the BESS energy capacity Emax.
  • The capital cost Q(Emax) of acquiring and setting up the BESS is an increasing function of Emax.
  • BESS planning therefore considers the operating-cost dependence on Emax together with the increasing capital cost Q(Emax).

˜FEP F C · VI. NUMERICAL RESULTS · A. Optimal Target SoC

The paper establishes that optimal BESS planning balances capital investment against operating cost, while numerical results examine the convexity of operating cost and how efficiency, penalties, and capacity shape the optimal target SoC.

  • ˜FEP F C: The planning objective minimizes λQ(Emax) + Es [H∗(s)], combining capacity-related investment with expected operating cost under optimal charging.λ reflects BESS lifetime, degradation, and the tendering period.
  • ˜FEP F C: H∗(s) is decreasing and convex in Emax, so expected minimum operating cost also decreases convexly as energy capacity increases.The marginal operating-cost reduction diminishes for larger Emax.
  • ˜FEP F C: A unique optimal Emax balances the marginal increase in Q(Emax) against the marginal decrease in expected minimum operating cost.This characterizes the capacity choice in the planning problem.
  • VI. NUMERICAL RESULTS: 10 Hz sampling produced 2,555,377 frequency measurements, representing about 71 hours of realtime data from Sacramento, CA.The data were provided by FNET/GridEye.
  • VI. NUMERICAL RESULTS: Under Emax = 0.1MWh, Pmax = 1MW, PP F C uniformly distributed in [0.5, 1]MW, and α = 0.9, simulations verify H∗(s)'s convexity in s.The verification uses ce = $0.1/kWh and cp = $10/kWh.
  • A. Optimal Target SoC: Lower battery efficiency produces a wider optimal target-SoC interval, which narrows and converges to a single point as η →1.A wider interval reduces the need for costly SoC adjustments caused by power losses.
  • A. Optimal Target SoC: When cp exceeds $35/kWh, the optimal target-SoC bounds overlap; because cp typically exceeds ce, practical designs can treat the target as a single point.The interval is relatively large when cp is comparable with ce and contracts as cp becomes large.
  • A. Optimal Target SoC: Very large Emax makes the optimal target-SoC bounds low because sufficient stored energy can completely avoid regulation failures, leaving charging cost dominant.This occurs when both seEmax and (1 −se)Emax exceed the maximum possible EP F C.

B. Time Response Comparison

The proposed BESS control scheme is compared with three prior recharging benchmarks using real-time frequency data. Simulations show that optimized target SoC reduces operating cost and regulation failure probability relative to heuristic and fixed recharging strategies.

  • Benchmark algorithms: The study compares the proposed control scheme against three benchmark algorithms from previous work.The comparison evaluates operating cost.
  • Benchmark algorithms: “No recharging” adds no charging during I intervals, while “Aggressive recharging” replenishes SoC to 100% during those intervals.These are two of the benchmark strategies used in the figures.
  • Benchmark algorithms: “Heuristic recharging” uses upper and lower target SoCs of 0.92 and 0.73, respectively, instead of optimized targets.This scheme is otherwise similar to the proposed strategy.
  • Simulation results: Time-response simulations use realtime frequency measurements and evaluate regulation failure probability and undiscounted time-aggregate costs at Emax = 0.1MWh and Emax = 1.5MWh.The cost comparisons are reported for both battery energy capacities.
  • Simulation results: With optimal target SoC, the proposed algorithm reduces both operating cost and regulation failure probability compared with “Heuristic recharging”.“No recharging” and “Aggressive recharging” produce higher failure probabilities because their SoC is often too low or too high.

C. Optimal BESS Planning

The minimum operating cost H∗ decreases convexly with BESS energy capacity for every initial SoC, yielding an optimal capacity that balances capital investment and operating cost.

  • C. Optimal BESS Planning: H∗(s) is a decreasing convex function of Emax for all initial SoC s.This behavior is verified as Emax varies from 0.05MWh to 10MWh, with ce = $0.1/kWh, cp = $10/kWh, and η = 0.8.
  • C. Optimal BESS Planning: An optimal BESS energy capacity Emax exists because increasing capacity trades lower operating cost against capital investment.The optimal capacity achieves the balance between capital investment and operating cost described in the passage.
  • C. Optimal BESS Planning: The capacity-effect analysis uses operating cost H∗ and energy capacity Emax as the central planning variables.Figure 13 examines H∗(s) versus Emax under ce = $0.1/kWh, cp = $10/kWh, and η = 0.8.

VII. CONCLUSIONS

The paper concludes that optimal BESS control uses a SoC target range invariant to the initial SoC, enabling offline computation, while future work should study combining multiple services with different capacity requirements.

  • Control strategy: Optimal control charges or discharges the BESS during I intervals until its SoC reaches a target range invariant to the beginning-of-interval SoC.This state-invariant target can be calculated offline and remain unchanged over the entire system time.
  • Future research: BESSs can serve multiple purposes beyond PFC, including demand response, energy arbitrage, and peak shaving.The paper identifies combining these services in a single BESS as an interesting future research topic.
  • Future research: PFC reserves require low energy capacity but are sensitive to regulation failures, whereas demand response, energy arbitrage, and peak shaving require high energy capacity.These differing capacity requirements motivate studying their optimal combination in a single BESS.

APPENDIX C PROOF OF THEOREM 2 · ˜FEP F C

Appendix C proves that the optimal cost H∗(s) is convex and decreasing with respect to BESS energy capacity Emax. The proof establishes convexity through a contraction mapping argument and decreasing monotonicity by first showing that the optimal single-stage cost decreases with Emax.

  • APPENDIX C PROOF OF THEOREM 2: The convexity argument relies on showing that ˜a(s, Emax) is nonnegative for every state s and capacity Emax.Its first term is nonnegative, while the remaining terms are nonnegative because PDFs and CCDFs are nonnegative.
  • APPENDIX C PROOF OF THEOREM 2: The proof establishes ∂2H∗(s)/∂Emax^2 ≥ 0, so H∗(s) is convex in Emax.The argument uses the contraction-mapping property of the right-hand side of (39).
  • APPENDIX C PROOF OF THEOREM 2: To prove decreasing monotonicity of H∗(s), the proof first shows that the optimal single-stage cost h∗(s) = minπ h(s, π) decreases with Emax.The monotonicity of H∗(s) then follows from the monotonicity property of contraction mapping.
  • ˜FEP F C: The minimizing policy π1st of h(s, π) yields the decrease of h∗(s) with Emax for all states s.The proof derives this condition from the expression ∂h(s, π)/∂π = r(s, π)Emax ˜FI(Q1).
  • ˜FEP F C: The infinite-horizon Bellman equation is treated as a contraction mapping through the operator in (44).This operator transfers the single-stage cost comparison into a comparison of value functions.
  • ˜FEP F C: Applying contraction-map monotonicity at every iteration gives H∗+(s) ≤ H∗−(s) as k approaches infinity.This completes the proof that the optimal infinite-horizon cost decreases with BESS energy capacity.

APPENDIX D ALGORITHM TO OBTAIN π∗

The appendix reduces computation of the optimal policy π∗ by replacing iterative dynamic-programming updates with a direct optimization based on transition probabilities and cost vectors. The resulting π∗high requires only one optimization problem with two scalar variables.

  • Computational reduction: Discretizing the continuous SoC state space into N levels requires N optimization problems per iteration in traditional value- or policy-iteration methods.The discretized states are i ∈ {0, δ, 2δ, · · ·, 1}, with δ = 1/(N−1).
  • Direct solution: For a policy pair d = (πlow, πhigh), transition probabilities pij(π) are calculated from the distributions of I, J, q, and EPFC.The policy uses πlow below the lower threshold, πhigh above the lower threshold, and π(i) within the interval πlow ≤ i ≤ πhigh.
  • Direct solution: The method forms the matrix Pd and vectors Hd and hd, with hd entries determined by the corresponding policy action in each SoC region.Hd is obtained as the solution of the stated matrix-based system.
  • Computational reduction: No iteration is required, and π∗high is obtained by solving one optimization problem with two scalar variables.The optimization is identified as problem (47).
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