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Optimal Power Sharing for Hybrid Energy Storage Systems Based on Karush-Kuhn-Tucker Conditions

Juan Diego Rios-Peñaloza, Milan Prodanovic, Javier Roldán-Pérez

arXiv:2609.31100v1eess.SYcs.SY

TL;DR

Variable renewable generation makes it harder for power plants to provide grid services, while power-sharing methods often rely on constant efficiencies or computationally demanding optimization. This paper proposes a loss-minimizing HESS controller using detailed power-dependent efficiency curves and analytical KKT verification. In the tested daily operation, it achieved the highest operational efficiency and lowest losses among the compared strategies, with real-time simulation also demonstrating implementation feasibility.

  • Problem

    Renewable plants need flexibility to provide grid services, but prior storage power-sharing approaches may use constant loss factors or optimization whose computation can exceed tight ancillary-service timing.

    Source: §I · p. 1

  • Method

    The controller minimizes HESS losses using power-dependent efficiency curves and analytically verifies KKT conditions, solving with a simple numerical root-finding method.

    Source: §I · p. 2, §III · p. 3

  • Results

    The proposed controller achieved the highest operational efficiency and lowest losses among the compared methods, while its power-sharing algorithm had a 6.7 ms mean resolution time in real-time simulation.

    Source: §V · p. 9, C. Real-Time Simulation · p. 9

  • Takeaways & Limitations

    The paper concludes that detailed storage models should be included in hybrid-plant dispatch while preserving low computational complexity for ancillary-service implementation.

    Source: §V · p. 9

    The controller is myopic and cannot anticipate future variable values or forecast conditions, including the need to recover reserved energy.

    Source: D. Remarks and Practical Considerations · p. 9

Abstract

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Renewable energy power plants are increasingly expected to provide ancillary services to the grid. Yet, the variability of sources such as solar photovoltaics requires additional operational flexibility to deliver these services. In this context, hybrid energy storage systems (HESSs) offer an appropriate solution, because different technologies with complementary characteristics can be applied and leveraged to share the power demand and improve the overall performance. Yet, a proper power sharing requires a detailed model of the storage elements. This aspect has barely been studied in the literature, where most models consider constant efficiencies. Moreover, most formulations are based on optimisation problems that are computationally too demanding to be solved in real time. In this paper, a controller is proposed to minimise the losses of a HESS considering detailed power-dependent efficiency curves of each storage technology. The power sharing method is based on the analytical verification of the Karush-Kuhn-Tucker (KKT) conditions, which makes it computationally efficient and suitable for real-world deployment. The proposed controller is applied to a test case consisting of a 10 MW PV power plant with a HESS based on a lithium-ion battery and a redox-flow battery, each with 2.5 MW power and 5 MWh capacity. The main contributions are verified via numerical simulations performed in MATLAB/Simulink, whereas a real-time implementation deployed in OPAL-RT demonstrates the viability of the algorithm for real-time applications.

I. INTRODUCTION

The introduction motivates loss-aware power sharing for hybrid storage at solar plants providing grid services, then outlines a controller and its market-facing operating context. Its design combines detailed storage models with an analytically verified KKT approach intended for computationally efficient dispatch.

I. INTRODUCTION

  • PV variability complicates ancillary-service delivery, while LIBs offer high efficiency and fast response and RFBs offer cycle durability but power-dependent efficiency.The paper motivates HESSs as a way to combine complementary storage characteristics and improve flexibility and overall storage efficiency.Source: §I · p. 1
  • Prior strategies use fixed operating points, constant loss factors, or short-range optimization that may exceed tight ancillary-service timing.Earlier KKT-based approaches for related settings still require numerical solvers.Source: §I · pp. 1–2
  • The proposed PPC minimizes HESS losses using power-dependent efficiency curves and analytical KKT verification, with a simple root-finding solution intended for computationally efficient operation.The stated test system is a 10 MW PV plant with 2.5 MW LIB and RFB units, each with 5 MWh capacity.Source: §I · p. 2

A. Application Description

  • The plant participates in day-ahead energy and secondary reserve markets, with the PPC tracking the cleared profile modified by the operator’s aFRR signal.PV units, a LIB, and an RFB connect to the point of common coupling through power electronic converters.Source: A. Application Description · p. 2

B. Overview of the Power Plant Controller

  • Reserve enforcement prioritizes balancing-energy capacity over the day-ahead profile by zeroing its reference when the aggregated HESS reserve-of-capacity limits are reached.The reserve is held by storage for 15 minutes of the committed capacity; technical SOC limits are enforced in the subsequent KKT-based sharing stage.Source: B. Overview of the Power Plant… · p. 3

III. HYBRID POWER PLANT CONTROLLER

The controller minimizes HESS losses by allocating power subject to the plant reference and storage-unit power limits. It uses KKT conditions and numerical root-finding, with convexity conditions governing when the analytical solution is globally optimal.

III. HYBRID POWER PLANT CONTROLLER

  • The controller minimizes HESS losses by deriving KKT conditions and using numerical root-finding instead of advanced optimization solvers.The approach is described as computationally efficient and suitable for real-time implementation.Source: §III · p. 3

A. Optimisation Problem Definition

  • Each storage unit’s losses depend on its efficiency over a time step, while allocated powers must meet the reference without exceeding unit limits.The formulation treats power as constant during each time step.Source: A. Optimisation Problem Definition · p. 3

B. KKT Conditions for Convex Resource Allocation

  • The analytical KKT solution is globally optimal when losses are convex and marginal losses are continuous and strictly increasing; otherwise, marginal-loss equalization may not yield the global optimum.The paper states that numerical optimization would be required if these conditions are violated.Source: B. KKT Conditions for Convex Resource… · p. 3
  • For interior active units, optimal dispatch equalizes marginal losses; units outside their power limits are fixed at a bound and satisfy the corresponding KKT inequalities.The multipliers represent the total-power balance and lower and upper power constraints.Source: B. KKT Conditions for Convex Resource… · p. 3

C. Hybrid Storage Modelling

The storage models make efficiency depend on power, while simplifying the scope by omitting temperature effects and assuming symmetric charging and discharging curves. The LIB loss curve is strictly increasing, whereas the RFB marginal loss decreases at very low power, where the LIB is preferred because its losses are lower.

C. Hybrid Storage Modelling

  • The models use power-dependent efficiency curves adapted from experimental curves, while excluding temperature effects for simplicity.The paper treats the efficiency curves as symmetric for charging and discharging.Source: C. Hybrid Storage Modelling · p. 4

1) Lithium-ion Battery:

  • The LIB efficiency is modeled as a linear function of absorbed or delivered power, and its loss and marginal-loss curves are continuous and strictly increasing.The LIB loss model is derived from its power-dependent efficiency.Source: 1) Lithium-ion Battery · p. 4

2) Redox Flow Battery:

  • RFB efficiency depends strongly on power because auxiliary-system losses, especially pumping losses, have a significant impact at low power.The curve parameters depend on the maximum and minimum efficiencies and the power at which efficiency is maximum.Source: 2) Redox Flow Battery · p. 4
  • At very low RFB power, marginal loss decreases, but the LIB remains preferred there because its losses are significantly lower than the RFB’s.The figure presents power-loss and marginal-loss curves for the RFB.Source: 2) Redox Flow Battery · p. 4, Figure 3 · p. 4

D. Optimisation Implementation

The all-active-unit formulation cannot directly represent the sudden loss increase when a storage unit is activated. The problem therefore includes commitment decisions, with a small number of unit states permitting full enumeration.

D. Optimisation Implementation

  • The all-active formulation identifies optimal dispatch only when every unit is active, because activating an inactive unit causes a discontinuity in marginal loss.The discontinuity at P = 0 is analogous to a generator no-load cost and requires commitment decisions represented as binary variables.Source: D. Optimisation Implementation · p. 4
  • Because the number of units is small, full enumeration can identify the lowest-loss dispatch without relying on dynamic programming or branch and bound.The reformulation uses battery-state binary variables and a vector of losses.Source: D. Optimisation Implementation · p. 4
  • With two storage units, the considered states are both inactive, both active, or either unit alone supplying the required power.The inactive state is u = [0, 0]; the active cases are u = [1, 1], [1, 0], and [0, 1].Source: D. Optimisation Implementation · p. 4

1) Both Units Active:

For two active units, power balance reduces the KKT search to a single condition, with power bounds handled explicitly. The method also considers dispatches in which only one unit is active.

1) Both Units Active:

  • Substituting the power-balance equality reduces the two stationarity conditions to one, simplifying the search over binding constraints for two active units.The unreduced formulation has 16 combinations of binding-variable values.Source: 1) Both Units Active · p. 5

RF B|LIB

  • If the both-active solution violates a power boundary, it is forced to that boundary; the RFB minimum is at least its convexity limit.The convexity limit is required even when the RFB has no technical minimum-power constraint.Source: RF B|LIB · p. 5, 1) Both Units Active · p. 5

2) Single Unit Active:

  • When one storage unit supplies all required power, the loss calculation sets the other unit's power to zero.This covers the single-unit-active cases in which either unit takes the full burden.Source: 2) Single Unit Active · p. 5

3) Optimisation Implementation:

The implementation compares losses across candidate active-unit configurations and returns the least-loss dispatch. It separates parameters computed offline from power references and SOC-dependent power limits supplied online.

3) Optimisation Implementation:

  • Efficiency parameters and minimum technical power limits are set offline, along with the RFB convexity limit used to ensure convexity.The RFB minimum power for the both-active case is set to the larger of its technical minimum and convexity limit.Source: 3) Optimisation Implementation · p. 5
  • Online inputs include the power reference and each unit's maximum power, which changes with SOC and becomes zero when the relevant SOC limit is reached.The implementation uses a numerical root finder for the both-active candidate.Source: 3) Optimisation Implementation · p. 5
  • The algorithm evaluates the both-active and single-unit-active cases, then selects the candidate with the lowest total loss.The three candidates are both units active, LIB only, and RFB only.Source: 3) Optimisation Implementation · p. 5

IV. NUMERICAL VALIDATION

In the numerical validation, the RFB is not worth activating until the power reference exceeds about 0.23 pu. For the studied system, its non-monotonic marginal loss does not create ambiguity in optimal dispatch.

IV. NUMERICAL VALIDATION

  • The validation assumes a 10 MWp PV plant and uses MATLAB/Simulink simulations.The storage parameters are presented in Table I.Source: §IV · p. 5

A. Convexity Limit Power

  • Above about 0.23 pu, RFB activation becomes worthwhile; for the system studied, its non-monotonic marginal loss does not make optimal dispatch ambiguous.The convexity limit power for this RFB is 0.21 pu.Source: A. Convexity Limit Power · pp. 5–6
  • Although the analysis suggests an offline lookup table, the algorithm retains generalisation and flexibility to different parameters.Source: A. Convexity Limit Power · p. 6

B. Numerical Simulations

The simulations examine loss variation when substituting the redox-flow battery (RFB) for the lithium-ion battery (LIB), then test dynamic power allocation as the units move through different state-of-charge conditions.

B. Numerical Simulations

  • The synthetic-reference simulation sets PV generation to zero and drives both storage units to their maximum and minimum SOC limits.The initial RFB and LIB SOC values are 92.5% and 30%, respectively.Source: 1) Dynamic Allocation · p. 6

1) Dynamic Allocation:

  • Fig. 4 compares losses when substituting the LIB with the RFB, using a three-dimensional surface and a cross-sectional view.The simulation varies the RFB power reference to examine loss changes against LIB-only operation.Source: Figure 4 · p. 6
  • When the LIB reaches its minimum state of charge, the RFB takes the entire burden; when both units charge, both contribute to the negative reference.The scenarios also include the RFB reaching maximum SOC while the LIB is at rated capacity, followed by a zero reference.Source: 1) Dynamic Allocation · p. 6
  • The RFB efficiency remains relatively high when both units are active, except when LIB SOC limits force it to supply the entire reference.The simulation also tracks each unit's SOC and efficiency alongside the HESS and unit power references.Source: 1) Dynamic Allocation · p. 6, Figure 5 · p. 6

2) Daily operation:

The daily-operation simulations compare the proposed strategy with equal sharing, reserve-based allocation, and rolling-horizon optimization. The proposed strategy achieves the highest mean and median operational efficiencies in the reported comparison.

2) Daily operation:

  • The proposed algorithm is executed every 200 ms of real operation time during the full-day test.The simulation uses a time scaling in which five seconds correspond to one minute of real operation.Source: 2) Daily operation · p. 6
  • The daily-operation figures show the reserve activation signal and the storage-unit set-points alongside total plant power injection.The plant output is compared with the committed power.Source: Figure 8 · p. 7, Figure 9 · p. 7, 2) Daily operation · p. 7
  • The plant schedule includes PV forecast generation, day-ahead market power commitment, and secondary-reserve commitment, shown at one-minute granularity.Energy and reserve prices are based on historical data; scheduling is quarter-hourly despite prices being constant within each hour.Source: 2) Daily operation · pp. 6–7, Figure 7 · p. 7
  • The comparison includes equal sharing, allocation weighted by available storage reserves, and rolling-horizon power-sharing optimization.The reserve-based weighting factors account for charging and discharging and are calculated from unit SOC.Source: 2) Daily operation · p. 7
  • The proposed strategy achieved the highest operational efficiencies in both mean and median values over a daily operation; Strat. C optimizes power sharing but approximates efficiency curves during operation.Table II reports median efficiency values and daily-operation losses; Figs. 10–11 show cumulative losses and efficiency distributions, including mean and median.Source: 2) Daily operation · p. 7, Table II · p. 7, Figure 10 · p. 7

3) PV Mismatch and Reserve Enforcement:

Under PV forecast mismatch, reserve enforcement stops day-ahead energy delivery when reserve limits are violated while allowing balancing energy to continue. The KKT-based controller generally respects energy limits with small deviations, while daily strategy outcomes are similar.

3) PV Mismatch and Reserve Enforcement:

  • When reserve limits are violated, reserve enforcement stops day-ahead energy delivery but continues balancing energy; between 8h and 12h, plant injection falls below commitment mainly because PV output is below forecast.The comparison in this mismatched-PV scenario is only against Strat. C.Source: 3) PV Mismatch and Reserve Enforcement · p. 8
  • Reserve energy limits are aggregated across storage technologies and adjusted for aFRR energy already deployed within each 15-minute market interval.The cumulative activation terms reset at the start of each interval.Source: 3) PV Mismatch and Reserve Enforcement · p. 8
  • The proposed controller generally respects energy limits with small deviations, while optimization recovers the required levels by each market period at high computational expense.The KKT controller lacks a predictive horizon to anticipate and recover energy when reserve falls below a limit after aFRR activation.Source: 3) PV Mismatch and Reserve Enforcement · p. 8
  • Both strategies perform similarly in daily operation, with a slight improvement from the KKT-based strategy.Table III reports median efficiencies and losses by storage technology, along with deviation of day-ahead energy.Source: 3) PV Mismatch and Reserve Enforcement · p. 8, Table III · p. 8

C. Real-Time Simulation

The paper reports that its controller performed best among the tested dispatch strategies and was feasible in real-time HIL simulation. Its applicability is bounded by efficiency-curve convexity, limited constraint complexity, and myopic dispatch.

C. Real-Time Simulation

  • In HIL testing, algorithm resolution averaged 6.7 ms, and power-plant set-point and output variance was negligible compared with Fig. 12.The power plant ran on OPAL-RT 4510 while the KKT-based algorithm ran on an external computer; the algorithm executed once per second.Source: C. Real-Time Simulation · p. 9, Figure 15 · p. 9

D. Remarks and Practical Considerations

  • The KKT formulation requires convex efficiency curves; the simple formulation also limits constraint complexity, while myopic control cannot anticipate future conditions.The paper notes that non-convex curves may require reformulation or approximation, and that conservative energy limits or intraday trading could address reserve-energy recovery needs.Source: D. Remarks and Practical Considerations · p. 9

V. CONCLUSION

  • A convexity power limit for the RFB is introduced to ensure unambiguous optimal dispatch, and reserve enforcement keeps SRM-reserved energy unavailable for DAM trading.Source: §V · p. 9
  • The proposed controller achieved the highest operational efficiency and lowest losses among three benchmark strategies, with a 6.7 ms mean resolution time in HIL testing.Its performance advantage also held with PV forecast deviations, although its myopic nature could cause small deviations from contracted reserve energy limits.Source: §V · p. 9, C. Real-Time Simulation · p. 9
  • The paper concludes that dispatch should include detailed storage models while retaining low computational complexity; future work will extend the formulation to simultaneous services across multiple markets.Source: §V · p. 9
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