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A Marginal Cost Consensus Scheme with Reset Mechanism for Distributed Economic Dispatch in BESSs

Yalin Zhang, Zhongxin Liu, Zengqiang Chen

arXiv:2609.19068v1eess.SY

TL;DR

Grid-connected BESS dispatch must account for battery losses, capacity degradation, and electricity trading, motivating an expenditure-minimizing dispatch scheme. The paper designs a distributed PI-based marginal-cost consensus controller with a reset mechanism and analyzes its behavior. The reported scheme has improved transient performance, including no overshoot, and avoids chattering compared with finite-time control.

Abstract

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Battery energy storage systems (BESSs) are often integrated into the smart grid as the key equipment for valley filling and peak suppression. However, the internal power consumption and capacity degradation of battery cells can not be ignored. In addition, there are certain electric trading between the owner of the grid-connected BESSs and the electric company (EC). Therefore, an expenditure function for grid-connected BESSs is constructed in this paper, in which internal power consumption, capacity degradation and power trading are covered while meeting the balance of power supply and demand. On this basis, the Karush-Kuhn-Tucker (KKT) condition of the function is summarized as the consensus problem of marginal cost (MC) converging to time-phased electricity price. Thus, we plan to design a distributed MC consensus scheme for economic dispatch (ED) in BESSs based on multi-agent systems (MASs) on a small-time scale. In view of the low control accuracy and response speed of the existing distributed proportional protocol, this paper proposes a distributed ED (DED) scheme with reset mechanism based on a proportional integral (PI) control. When the proportional term encounters zero crossing, the integral term of the control scheme is reset to 0, which ensures that the signs of the two are aligned, thus accelerating MCs convergence and restraining overshoot. Parameter conditions for consensus, regularity and Zeno-free behavior are given through the relevant theoretical analysis. Several simulation cases are designed to verify the designed DED algorithm.

I. INTRODUCTION

Grid-connected BESS dispatch must account for internal power loss, capacity degradation, and electricity trading while balancing supply and demand. The paper proposes a reset-based distributed PI controller for small-time-scale marginal-cost consensus and analyzes conditions for its behavior.

I. INTRODUCTION

  • Grid-connected BESS dispatch must account for internal power consumption and capacity aging, alongside buying electricity when supply is insufficient and selling excess power.These costs motivate an economic-dispatch algorithm intended to reduce operating costs.Source: §I · p. 2
  • Earlier distributed ED work considered generation cost and other objectives, while BESS studies addressed battery losses, efficiency, or degradation across different time scales.The paper positions its work around a composite objective and small-time-scale dispatch.Source: §I · p. 2
  • A reset-based distributed PI controller targets faster, more accurate marginal-cost consensus for small-time-scale BESS dispatch, addressing slow, low-accuracy proportional control and practical concerns with finite-time controllers.The integral term resets when the proportional term crosses zero; the authors state this improves dynamic and steady-state performance.Source: §I · p. 2
  • The paper analyzes MC consensus, reset regularity, and Zeno-free behavior, and compares the scheme with proportional and finite-time controllers.The authors provide parameter conditions for consensus and regularity and analyze controller progressiveness.Source: §I · p. 2

II. PRELIMINARIES

  • The preliminaries formulate small-time-scale optimal dispatch for BESSs in a resistive network using total expenditure as the objective.The expenditure model covers income, internal power loss in charging and discharging, and capacity deterioration.Source: §II · p. 2

A. BESS and its expenditure model

  • The expenditure model combines grid-trading income, internal battery power loss, and capacity-deterioration cost under a supply-and-demand balance.It is used to develop an output-power strategy for reducing operating cost.Source: A. BESS and its expenditure model · p. 3

B. Analysis for optimal ED

The optimal-dispatch analysis derives KKT conditions from the convex expenditure function. At the optimum, each BESS's marginal cost reaches the electricity price.

B. Analysis for optimal ED

  • The analysis constructs a Lagrangian for the convex expenditure function to derive the BESS optimality conditions.γ is the Lagrange multiplier.Source: B. Analysis for optimal ED · p. 3
  • Total MG expenditure is lowest when each battery's marginal cost reaches the electricity price.The paper identifies this condition from the KKT conditions for BESS dispatch.Source: B. Analysis for optimal ED · p. 3
  • The MC-consensus ED framework is shown for BESSs.Source: Figure 1 · p. 3

C. Control objective

The control objective is to minimize total expenditure by driving BESS marginal costs to the time-phased electricity price. The paper presents a distributed, reset-based PI scheme and the communication-graph assumptions used in its design.

C. Control objective

  • The DED controller is designed to make total expenditure optimal when BESS marginal costs reach the time-phased electricity price.The price is set through time-phased contracts between the electric company and users.Source: C. Control objective · p. 3

III. DED ALGORITHM WITH RESET MECHANISM

  • A distributed PI algorithm with reset mechanism targets faster, more accurate MC convergence to the time-phased price without chattering.The design addresses low control accuracy in proportional protocols and chattering in finite-time protocols.Source: §III · p. 3

A. Graph theory and some related conclusions

B. DED algorithm with reset mechanism

The grid-connected BESS dispatch scheme uses a reset PI controller to drive each unit’s marginal cost toward the time-phased electricity price. Its analysis gives parameter conditions for regularity and consensus, with consensus reached after at least one reset.

B. DED algorithm with reset mechanism

  • In grid-connected mode, the dispatch target is for each BESS’s marginal cost to reach the time-phased electricity price.The scheme introduces the reset mechanism to improve control accuracy, avoid overshoot, and accelerate convergence.Source: B. DED algorithm with reset mechanism · p. 4
  • Under Assumption 1, k > 0 and 4h > ηn(H)k2 guarantee regularity of the designed scheme.The proof connects the parameter inequality to complex eigenvalues and at least one zero crossing of ξ.Source: B. DED algorithm with reset mechanism · p. 4
  • Under Assumption 1, the same conditions guarantee consensus with at least one reset, and the marginal costs converge to ρ.The state errors converge to zero as t → ∞, yielding λ1 = · · · = λn = ρ.Source: B. DED algorithm with reset mechanism · pp. 4–5
  • Resetting the integral term to zero when the proportional term crosses zero aligns their signs, allowing a slightly larger gain to speed marginal-cost convergence while avoiding overshoot.The reset is applied when ξ_i = 0; the control coefficients must satisfy the stated conditions for the theoretical guarantees.Source: B. DED algorithm with reset mechanism · pp. 4–5

C. Further discussion on the designed schemes

The discussion links the reset PI controller’s improved transient behavior to its reset rule and analyzes stability, Zeno-free operation, and implementation trade-offs.

C. Further discussion on the designed schemes

  • Resetting each integral term when its proportional term is about to change sign gives shorter settling time and no overshoot versus PC and PI protocols.The reset is applied at the proportional term’s zero crossing.Source: C. Further discussion on the designed… · p. 5
  • Compared with a PC, the reset controller adds ν dynamics and increases closed-loop order; the design can also be represented as a PC with dynamic gain k + h r_i.For a sufficiently small reset interval, r_i is approximately t − t_i^l.Source: C. Further discussion on the designed… · p. 5
  • Aligning the signs of ξ_i and ν_i yields faster convergence than a fixed-gain PC and a compact invariant set unavailable to non-reset PI.The reset scheme’s effective gain is greater than the fixed gain k.Source: C. Further discussion on the designed… · p. 5
  • The reset controller avoids chattering, reduces control input, accelerates convergence, and is described as relatively simple to implement physically.It resets the integral term when the proportional term is zero.Source: C. Further discussion on the designed… · p. 5
  • With the reset mechanism modified as specified, the dynamic DED scheme is asymptotically stable without Zeno behavior for any positive dwell-time constant Δ.The proof concludes that the state norm converges to zero.Source: C. Further discussion on the designed… · pp. 5–6
  • A constant dwell time is used to prevent Zeno behavior, but excessive dwell time may deteriorate system dynamics.For finitely many reset instants, MC stability mainly depends on the non-reset dynamics.Source: C. Further discussion on the designed… · pp. 5–6

IV. CASE STUDY

The case study uses a four-BESS microgrid with an energy router that supplies the electricity-price signal, then tests the algorithm across seven cases. The cases examine validation, controller progressiveness, parameter effects, plug-and-play operation, time-phased pricing, Zeno-free behavior, and scalability.

IV. CASE STUDY

  • The test system contains four BESSs and an energy router; only agent 1 receives the leader’s electricity-price information, and ηn(H) = 4.7913.Seven cases are arranged to verify the analysis in Section III.Source: §IV · p. 6
  • Case 1 verifies the proposed reset-control scheme under k = 1 and h = 2, while Case 2 tests the algorithm’s progressiveness under the same settings.The Case 1 test is described as satisfying inequality (12).Source: §IV · p. 6
  • Case 3 examines how k and h affect marginal costs, and Case 4 tests plug-and-play operation with k = 1 and h = 2.Source: §IV · p. 6
  • Case 5 tests the designed algorithm in a time-phased electricity-price environment; Case 6 verifies the DED algorithm under a Zone-free condition with a reset mechanism for Δ = 1 and Δ = 1.5.Source: §IV · p. 6
  • Case 7 increases the number of BESSs and their managing agents to 12 to test scalability for a large-scale power system.Source: §IV · p. 6

A. Case 1: Validation of the Proposed DED algorithm

In Case 1, the designed algorithm drives marginal costs to the constant electricity price while the control input and state difference converge to zero. Its reset behavior also aligns the integral and proportional terms.

A. Case 1: Validation of the Proposed DED algorithm

B. Case 2: Discussion on the progressiveness

The Case 2 simulations compare the designed controller with a proportional controller and a finite-time controller. The designed controller reaches the electricity price faster without overshoot, while the finite-time controller exhibits chattering and practical implementation difficulty.

B. Case 2: Discussion on the progressiveness

  • The designed algorithm drives marginal costs to electricity price ρ faster and without overshoot than the proportional controller.Fig. 6 compares the designed controller, proportional controller, and finite-time controller.Source: B. Case 2: Discussion on the… · p. 7
  • The finite-time controller accelerates convergence but produces chattering in control input and marginal costs because of its fractional integral term.Source: B. Case 2: Discussion on the… · p. 7
  • Fractional integration is currently physically difficult to achieve in projects, whereas the designed controller uses a PI controller described as widely used in industry.The paper therefore considers its controller more convenient for engineering application.Source: B. Case 2: Discussion on the… · p. 7

C. Case 3: Algorithm Test on different parameters

Case 3 varies the controller parameters k and h in simulations. Increasing them can speed convergence, but large k can violate regularity, while larger h increases BESS 1’s control input without breaking regularity.

C. Case 3: Algorithm Test on different parameters

D. Case 4: The play-and-plug function

The simulations test plug-and-play operation, response to changing electricity prices, and a zone-free reset scheme. MCs ultimately converge after BESS connection changes and price switches, while the zone-free scheme exhibits performance trade-offs.

D. Case 4: The play-and-plug function

E. Case 5: Test in time-phased electricity price environment

  • At electricity-price switches, BESS 1’s MC fluctuates more but still converges to the time-slot price, with regularity and consensus maintained.The tested prices are 0.8 for 0–15 s, 1 for 15–35 s, and 0.6 for 35–50 s.Source: E. Case 5: Test in time-phased… · p. 8, Figure 13 · p. 9

F. Case 6: Test on a Zone-free distributed scheme

  • The zone-free scheme converges more slowly than Case 1, and increasing ∆ increases both convergence time and MC overshoot.The comparison covers ∆=1 and ∆=1.5; Fig. 14 shows simulation results for ∆=1.Source: F. Case 6: Test on a Zone-free… · pp. 8–9, Figure 14 · p. 9
  • Larger ∆ delays resets and can leave dynamic gains negative; the authors identify a trade-off between system performance and Zeno-free behavior.Ignored reset operations mean ξi and νi do not remain aligned at all times.Source: F. Case 6: Test on a Zone-free… · pp. 8–9

G. Case 7: Test on a large-scale system

The reset mechanism is tested on a 12-BESS system, where MCs converge to the electricity price. The conclusion describes its design, analyzed guarantees, and potential extension to other distributed energy dispatch settings.

G. Case 7: Test on a large-scale system

V. CONCLUSION

  • The method constructs a small-time-scale ED objective covering capacity degradation, internal power loss, and electricity trading, then designs MC consensus control with a reset mechanism.The reset aligns integral and proportional term signs; the conclusion reports faster response and convergence and higher control accuracy than a proportional controller, with no overshoot or finite-time-controller chattering.Source: §V · p. 10
  • Consensus and regularity conditions support the stated control properties, while a linear reset interval based on minimum dwell time is used to avoid Zeno behavior.Source: §V · p. 10
  • The authors state that the MC consensus scheme can be transferred to ED problems in microgrids and distributed integrated energy networks with convex objective functions.Source: §V · p. 10
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