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Optimal energy conversion -- A port-Hamiltonian approach

Guanru Pan, Karl Worthmann, Manuel Schaller, Timm Faulwasser

arXiv:2610.21790v1eess.SYcs.SY

TL;DR

Dynamic constrained operation leaves optimal energy conversion and the bounding force of generalized Carnot efficiencies open questions. This paper formulates finite-horizon optimization for port-Hamiltonian systems and shows that, under suitable reachability conditions, asymptotic optimal efficiencies coincide with cyclic efficiencies; it also establishes structural limits and a generalized Carnot-type bound for a class of systems.

  • Problem

    How to characterize optimal energy conversion under dynamic constraints, and whether generalized Carnot efficiency bounds optimal efficiency, remain open questions.

    Source: §1 · p. 1

  • Method

    The paper formulates finite-horizon optimal-control problems for two energy-conversion efficiencies and studies their relation to cyclic operation and structural limits.

    Source: §1 · p. 2

  • Results

    Under suitable reachability conditions, asymptotic optimal efficiencies coincide with optimal cyclic efficiencies, and an additional scalar structural restriction yields a generalized Carnot-type upper bound for a class of systems.

    Source: §1 · p. 2, §8 · p. 12

  • Takeaways & Limitations

    For the studied structured systems, nontrivial conversion is dynamic, and structural properties connect achievable conversion to thermodynamic efficiency limits.

    Source: §8 · p. 12

    The cyclic-equivalence result relies on reachability from optimal terminal states back to the initial state within a uniformly bounded time.

    Source: §4.2 · p. 6

Abstract

from arXiv · show

This paper studies optimal energy conversion in port-Hamiltonian systems. We analyze state- and control-constrained finite-horizon optimal control problems that maximize energy conversion efficiencies. Under suitable reachability conditions, we show that the asymptotic optimal efficiencies exist and coincide with the corresponding optimal cyclic efficiencies. For a class of port-Hamiltonian systems describing coupled storage-conversion components, we further derive intrinsic limits on achievable energy conversion and establish a generalized Carnot-type efficiency bound. Our results provide an optimal-control framework connecting asymptotic optimal operation, cyclic operation, and fundamental efficiency limits for port-Hamiltonian systems.

1. INTRODUCTION

Dynamic engineering systems make energy conversion an optimal-control problem, while prior work leaves optimal operation and the bounding force of generalized Carnot efficiencies unresolved. This paper connects finite-horizon optimization to cyclic operation and structural efficiency limits in two-port port-Hamiltonian systems.

1. INTRODUCTION

  • The paper addresses constrained dynamic operation, unlike classical cycle analyses that typically prescribe cycles in advance and idealize them as quasi-static.Port-Hamiltonian models represent storage, interconnection, dissipation, and external power exchange.Source: §1 · p. 1
  • Prior pH optimal-control work incorporates energy-, entropy-, and exergy-related objectives, while dissipativity and turnpike studies connect long-horizon operation to steady or periodic behavior.The cited literature also studies passivity-based controller design.Source: §1 · p. 1
  • Whether generalized Carnot efficiency bounds optimal efficiency, and how to formulate optimal conversion beyond the generalized Carnot construction, remain open questions identified in prior work.The cited work also identifies extending Carnot cycles to more general cyclic processes with an efficiency notion as an open problem.Source: §1 · p. 1
  • Optimal finite-horizon efficiencies coincide asymptotically with the corresponding optimal cyclic efficiencies under suitable reachability conditions.The paper formulates two efficiency objectives and characterizes their asymptotic performance.Source: §1 · p. 2
  • For a class of block-diagonal systems, structural restrictions limit achievable conversion, and an additional scalar restriction makes generalized Carnot efficiency a rigorous upper bound.The systems represent coupled storage-conversion components.Source: §1 · p. 2

2. PROBLEM STATEMENT

The problem is dynamic energy conversion between two ports of a nonlinear port-Hamiltonian system. The paper defines two efficiency measures and optimizes them over finite-horizon trajectories subject to input and state constraints.

2. PROBLEM STATEMENT

  • The nonlinear pH model uses skew-symmetric interconnection, positive-semidefinite dissipation, state-dependent inputs, and a continuously differentiable Hamiltonian representing stored energy.The state begins at an initial condition.Source: §2 · p. 2
  • Port 1 is designated for energy injection and Port 2 for extraction, although the passivity supply rate treats both ports symmetrically.The injection/extraction distinction is semantic and chosen according to the intended conversion interpretation.Source: §2 · p. 2, Figure 1 · p. 2
  • The generalized thermal efficiency η+ distinguishes energy supplied through Port 1 from energy returned through that port, extending the input-output distinction beyond thermodynamics.The paper also considers conversion efficiency η.Source: §2 · pp. 2–3
  • For a finite horizon, the optimization maximizes either conversion efficiency subject to input and state constraints.The input is measurable, the state trajectory is absolutely continuous, and both remain within their specified sets.Source: §2 · p. 3

3. OPTIMAL ENERGY CONVERSION PROBLEMS

The section formulates finite-horizon efficiency maximization and uses available storage to bound performance. It explains why these bounds become more informative at long horizons and notes that fixed injection-budget formulations are outside the study's scope.

3. OPTIMAL ENERGY CONVERSION PROBLEMS

  • The finite-horizon optimal-control problem maximizes either η or η+ over a given horizon, with state and input constraints.The problem also imposes a minimum energy extraction requirement that scales with the horizon.Source: §3 · p. 3
  • A minimum extraction requirement ensures positive optimal efficiency for sufficiently large horizons and helps make the optimization problem well-posed.The requirement reflects applications where a prescribed amount of extracted energy must be achieved.Source: §3 · p. 3
  • Available storage bounds achievable efficiencies by quantifying net energy extractable from the initial state, distinct from sustained conversion enabled by Port 1 injection.The available-storage value is bounded on the state set.Source: §3 · p. 3
  • Short-horizon conversion efficiency can exceed one by drawing on initially stored energy, but this finite-storage effect vanishes as the horizon grows, leaving asymptotic efficiency bounded by one.The paper relates this asymptotic bound to the first law of thermodynamics.Source: §3 · pp. 3–4
  • With exogenous Port 1 input, efficiency optimization becomes constrained by the prescribed input, and achievable performance is generally lower than with fully controlled injection.Uncertain exogenous input leads to robust or stochastic formulations.Source: §3 · p. 4
  • Fixed injection-energy budgets are outside the study's scope because repeating a nontrivial cycle requires supplied energy to grow with the number of cycles.An upper bound on injection alone can permit arbitrarily small injection and degenerate efficiency maximization.Source: §3 · p. 4

4. ASYMPTOTIC OPTIMAL ENERGY CONVERSION EFFICIENCY

Under reachability and feasibility conditions, asymptotic finite-horizon optimal efficiencies are characterized by cyclic benchmarks. The results also extend to two-port dissipative systems with uniformly bounded supply rates.

4.1 Asymptotic lower bound

  • Steady-state operation is a constant-trajectory cyclic case, and feasible steady states imply cyclic feasibility with equal steady-state η+ and η.The steady-state input must satisfy the positive Port 1 supply condition.Source: §4.1 · pp. 4–5
  • If an optimal cyclic orbit is uniformly reachable and satisfies the extraction constraint strictly, cyclic operation lower-bounds asymptotic finite-horizon performance.The bounded start-up transition and final partial cycle become negligible over long horizons.Source: §4.1 · pp. 5–6

4.2 Asymptotic upper bound

  • If optimal terminal states can return to the initial state within uniformly bounded time while preserving feasibility, finite-horizon optima admit a cyclic upper bound.The bounded return cost becomes negligible for sufficiently long horizons.Source: §4.2 · p. 6
  • Under Assumptions 1–3, for any x0 ∈ X0, the asymptotic optimal value of OCP (10) exists; periodic operation also provides an asymptotic lower bound on achievable performance.Source: §4.2 · p. 6, §4.1 · p. 6
  • The asymptotic results extend to two-port dissipative systems when their supply rates are separable, uniformly bounded, and the corresponding assumptions hold.The extension uses the same definitions of E1, E2, and E1,+.Source: §4.2 · pp. 6–7
  • For lossless systems, feasible cyclic operation gives η = 1, while η+ can have an upper bound below one when Port 1 power is negative during part of a cycle.The paper distinguishes the energy-rate efficiency η from the positive-injection-normalized efficiency η+.Source: §4.2 · p. 7

5. INTRINSIC CONVERSION LIMITS IN BLOCK-DIAGONAL FORM

The block-diagonal structure constrains energy conversion through the coupling of storage states and port power flows. It rules out nontrivial steady-state conversion and supports extraction bounds for cyclic and finite-horizon operation.

5. INTRINSIC CONVERSION LIMITS IN BLOCK-DIAGONAL FORM

  • Block-diagonal systems describe a single conversion component whose storage variables couple through a common Hamiltonian, unlike separately interconnected subsystems.Examples include electromechanical actuators, thermo-mechanical systems, and electro-thermal devices.Source: §5 · p. 7
  • At steady state, net Port 2 power is nonnegative, so nontrivial energy conversion requires time-varying operation.The block-diagonal structure imposes this limitation on steady-state operating points.Source: §5 · p. 7
  • For any cyclic operation, the Port 2 extracted energy satisfies an upper bound, with positive semidefinite R2 supplying a nonnegative dissipation term.The cyclic condition makes the net Hamiltonian change over a period zero.Source: §5 · p. 8
  • Proposition 2 extends the cyclic extracted-energy bound to finite-horizon optimal trajectories.Finite-horizon trajectories need not be cyclic, so the corresponding bound accounts for that distinction.Source: §5 · p. 8

6. GENERALIZED CARNOT CYCLE AND CARNOT EFFICIENCY BOUND

Under Assumption 4, the generalized Carnot efficiency bounds feasible cyclic operation, and the same structure gives an asymptotic upper bound for finite-horizon trajectories. The bound is built from a generalized Carnot cycle in the Hc–c plane and recovers the classical Carnot bound in the thermodynamic case.

Generalized Carnot cycle

  • The generalized Carnot cycle alternates phases that hold Hc constant with phases that hold the scalar storage coordinate c constant.These are described as isothermal-like and adiabatic-like phases, respectively.Source: §6 · p. 8
  • In the absence of dissipation, the cycle’s extracted energy is expressed using the differences in c and Hc, and its energy ratio yields the generalized Carnot efficiency.The paper identifies this efficiency as the reference performance under Assumption 4.Source: §6 · pp. 8–9

Carnot efficiency bound

  • Under Assumption 4, no feasible cyclic trajectory exceeds the generalized Carnot efficiency.The same structural argument gives an asymptotic upper bound for finite-horizon trajectories, which need not be periodic.Source: §6 · p. 9

Scope of the bound

  • The proof of the cyclic bound relies on a scalar storage coordinate; the argument does not extend directly to genuinely multidimensional coordinates.For multidimensional c, components of the derivative can have different signs and cancel in the inner product.Source: §6 · p. 9

Relation to the classical Carnot cycle

  • The generalized bound reduces to the classical Carnot bound when c is entropy and Hc is temperature.The generalized cycle is a dynamic finite-period process; its isothermal-like phases require constant system temperature, unlike classical reservoir heat exchange.Source: §6 · p. 10

7. SIMULATION EXAMPLES

The examples contrast a reversible lossless gas-piston system, where cyclic optimization recovers the Carnot cycle, with a dissipative linear system that illustrates the long-horizon relation to cyclic efficiency. The linear example also shows that the Carnot-type bound is not applicable when Assumption 4 fails.

7.1 A reversible lossless gas-piston system

  • In the lossless gas-piston system, η+ reaches the Carnot bound of 0.25 for Tp ≥ 3.0 s, while η equals one for every cycle.On the considered period grid, the bound is first attained at Tp = 3.0 s.Source: §7.1 · p. 10, Figure 5 · p. 11
  • At Tp = 3.0 s, the gas-piston optimizer recovers the generalized Carnot cycle, with constant-temperature-like and constant-entropy-like phases.The phases correspond to constant temperatures and constant entropies, respectively.Source: §7.1 · p. 11, Figure 6 · p. 11

7.2 Linear port-Hamiltonian system in block-diagonal form

  • Because the linear example does not satisfy Assumption 4, the Carnot-type bound does not apply, although its best cyclic efficiency tightly characterizes observed long-horizon performance.This behavior is consistent with the asymptotic results.Source: §7.2 · pp. 11–12
  • For the linear system, both finite-horizon efficiencies approach the best cyclic efficiency, 0.9436, as the horizon increases.The best cyclic efficiency is attained at Tp = 8 s.Source: §7.2 · p. 11, Figure 8 · p. 11
  • Along the linear system’s finite-horizon optimal trajectory, the efficiency measures coincide, although Port 2 alternates between extraction and injection while net extraction remains positive.The net integral satisfies the prescribed minimum extraction constraint.Source: §7.2 · pp. 11–12

8. CONCLUSION

The paper links long-horizon optimal behavior to cyclic operation under suitable controllability and reachability conditions. For block-diagonal port-Hamiltonian systems, it also identifies a dynamic requirement for nontrivial conversion and a generalized Carnot bound under an additional scalar structural restriction.

8. CONCLUSION

  • Under suitable controllability and reachability conditions, cyclic operation emerges asymptotically as an optimal mechanism for nontrivial energy conversion in structured systems.The paper presents this as a control-theoretic perspective connecting long-horizon behavior with cyclic operating regimes.Source: §8 · p. 12
  • For the specified block-diagonal systems, steady-state operation cannot sustain net Port 2 extraction, and an additional scalar restriction makes generalized Carnot efficiency an upper bound.The bound applies to achievable positive-injected conversion efficiency under that restriction.Source: §8 · p. 12
  • Future work includes studying turnpike properties around optimal cyclic orbits and incorporating uncertainty and disturbances.These directions are proposed to better understand recurring cyclic operation and extend the framework to more realistic uncertain settings.Source: §8 · p. 12

APPENDIX

The appendix establishes existence of optimal finite-horizon and cyclic controls for both efficiency objectives under stated assumptions. For the positive-injected efficiency objective, it also justifies an equivalent auxiliary-control formulation used in the existence argument.

APPENDIX

  • Under the stated compactness, continuity, and feasibility assumptions, optimal controls exist for both finite-horizon and cyclic problems for each efficiency objective.The existence argument uses continuous terminal costs on the feasible terminal sets.Source: APPENDIX · pp. 12–13
  • For η+, an auxiliary control q yields an equivalent formulation, and an optimal solution can be chosen with q equal to positive injected power.Replacing q this way preserves the trajectory and constraints and does not decrease the objective.Source: APPENDIX · pp. 12–13
  • The available-storage inequality and extraction constraint bound the fractional objectives’ denominators away from zero, supporting continuity of terminal costs.This continuity is used to apply the existence theorem for finite-horizon optimal controls.Source: APPENDIX · p. 13
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