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Decentralized Continuous-Time Power Dispatch for Integrated Heat and Power Systems via Bernstein-Galerkin Equivalent Projection

Jie Deng, Zhigang Li, J. H. Zheng, Yue Chen

arXiv:2609.37173v1eess.SYcs.SY

TL;DR

IHPS dispatch must coordinate electric power systems and district heating networks, but VF-VT operation combines thermal flexibility with nonconvex thermal-hydraulic coupling. The paper uses continuous-time Bernstein–Galerkin projection and safeguarded Anderson prediction; case studies report lower operating costs and runtime while maintaining thermal state consistency.

  • Problem

    VF-VT operation can exploit DHN flexibility, but jointly optimizing mass flows and temperatures creates a nonconvex model that challenges decomposition methods relying on convexity.

    Source: §I · p. 2

  • Method

    The framework projects Bernstein-space DHN thermal dynamics onto boundary variables and uses safeguarded Anderson prediction to accelerate alternating EPS–DHN coordination.

    Source: §I · p. 2

  • Results

    On E39–H20, BGEP-D reduces operating cost by 3.59% and total runtime from 69.10 s to 17.32 s versus NMSE-D.

    Source: B. Simulation on a Large-Scale IHPS · p. 9

  • Takeaways & Limitations

    Case studies demonstrate thermal state consistency during decentralized coordination alongside improved economic and computational performance.

    Source: §VI · p. 10

Abstract

from arXiv · show

Decentralized power dispatch for integrated heat and power systems (IHPSs) involves coordinating electric power systems (EPSs) and district heating networks (DHNs) while preserving privacy. The existing methods rely mainly on discretetime DHN models and constant-flow operations, limiting the ability to exploit thermal flexibility. Enabling variable-flow and variable-temperature (VF-VT) operations requires thermal-hydraulic coupling to be addressed together with continuous-time thermal dynamics. However, the simultaneous variations in mass flows and temperature lead to a nonconvex IHPS model, for which conventional decomposition methods are difficult to apply. This paper proposes a decentralized continuous-time dispatch framework using Bernstein-Galerkin equivalent projection. For a prescribed mass flow trajectory, DHN thermal dynamics are formulated in the Bernstein space and projected onto boundary variables, yielding an equivalent feasible-region model without disclosing the internal DHN topology or states. The equivalent model and DHN subproblem are derived from a unified Bernstein-Galerkin formulation, ensuring a consistent thermal state representation throughout the EPS-DHN coordination procedure. A safeguarded Anderson prediction scheme further accelerates the alternating coordination process by using historical updates and fixed-point residuals to predict the next input without additional subproblem solving steps. Numerical results show the proposed method preserves thermal state consistency, exploits VF-VT flexibility, and improves economic and computational performance.

NOMENCLATURE

The nomenclature identifies scheduling and network indices, model parameters and matrices, and the Bernstein-space variables used in the IHPS formulation.

A. Indices and Sets

  • Indices j, k, l, and r identify mass-flow trajectories, scheduling intervals, DHN pipelines, and coordination iterations, respectively.K and L denote the sets of scheduling intervals and DHN pipelines.Source: A. Indices and Sets · p. 1

B. Parameters, Matrices and Functions

  • Pipeline parameters and functions include heat-loss coefficients, lengths, thermal-equation matrices, pressure-loss functions, and ambient temperature.The passages also name Bernstein basis and derivative matrices, Galerkin derivative and mass matrices, and pipeline outlet-temperature equivalent matrices.Source: B. Parameters, Matrices and Functions · p. 1, F ST · p. 1

F ST

  • The listed F_ST and CHP functions and matrices include the ambient-temperature load matrix, CHP operating function, and internal-state reconstruction matrices.The passages also identify constant vectors in the internal-state equivalents.Source: F ST · p. 1

C. Variables

  • Bernstein-space variables include Stage-I slack vectors, EPS and DHN internal and boundary variables, DHN states and mass flows, and pipeline temperature fields.The notation also includes fixed-point residuals and their scaled forms.Source: C. Variables · p. 1

I. INTRODUCTION

IHPS dispatch must coordinate electricity and heat operations while protecting proprietary system data. This paper develops continuous-time Bernstein–Galerkin equivalents and safeguarded Anderson prediction, addressing VF-VT modeling and coordination challenges.

I. INTRODUCTION

  • VF-VT operation jointly regulates mass flows and temperatures, increasing DHN flexibility but producing a nonconvex model that challenges decomposition methods relying on convexity.CF-VT simplifies the model but restricts the feasible DHN operation region and thermal flexibility.Source: §I · p. 2
  • Existing DHN equivalents are mainly discrete-time, while separately constructed equivalents and subproblems can represent continuous heat-transport dynamics inaccurately.The flow-dependent equivalent also makes alternating coordination a computationally intensive nonlinear fixed-point iteration.Source: §I · p. 2
  • For prescribed mass flow, Bernstein–Galerkin projection eliminates internal DHN thermal states and expresses the feasible region through boundary variables, retaining continuous-time dynamics without revealing internal information.The framework is developed for decentralized dispatch under VF-VT operation.Source: §I · p. 2
  • A unified continuous-time Bernstein–Galerkin formulation for the DHN equivalent and subproblem maintains consistent thermal trajectories and preserves DHN feasibility during coordination.It avoids inconsistencies associated with separately constructed discrete-time or quasidynamic equivalents and subproblems.Source: §I · p. 2
  • Safeguarded Anderson prediction uses historical alternating updates and fixed-point residuals to predict the next coordination input without additional EPS or DHN subproblem solves.The safeguard rejects unreliable extrapolations.Source: §I · p. 2

A. Representations of Continuous Variables in Bernstein Space

  • Continuous trajectories and spatiotemporal thermal fields are represented by Bernstein coefficients while retaining their continuous domains.The pipeline temperature field is represented by a spatiotemporal coefficient matrix, and temporal differential and integral operators can be evaluated directly from the coefficients.Source: §II · p. 2, A. Representations of Continuous… · p. 3

B. Partitioning of the Subsystem and Boundary Variables

The formulation separates subsystem-local variables from the boundary trajectories that couple the EPS and DHN. Its compact Bernstein-space model represents dispatch costs and operating constraints while using boundary-variable exchange to support decentralized coordination.

B. Partitioning of the Subsystem and Boundary Variables

  • Internal variables remain local to the EPS or DHN, while boundary variables characterize the coupling trajectories between the subsystems.The IHPS couples the EPS and DHN through CHP units and water pumps.Source: B. Partitioning of the Subsystem and… · p. 3

C. Compact Model of the IHPS Power Dispatch Problem

  • The compact Bernstein-space optimization minimizes total IHPS operating cost, including non-CHP generation, CHP operation, and wind-curtailment costs.Its constraints cover EPS operation, flow-dependent thermal conditions, pipeline heat transport, mass balance, pressure-head drops, and CHP-pump coupling.Source: C. Compact Model of the IHPS Power… · p. 3

CHP, T B

  • To coordinate without disclosing proprietary operating data, the DHN’s internal thermal and hydraulic states are represented by boundary variables under a prescribed mass-flow trajectory.The formulation defines Bernstein-space variables for power generation, reserves, temperatures, pressure heads, and mass flows.Source: CHP, T B · p. 3, PUMP, QB · p. 3, C. Compact Model of the IHPS Power… · p. 4

III. BERNSTEIN–GALERKIN EQUIVALENT PROJECTION FOR DISTRICT HEATING NETWORKS

The section represents continuous-time DHN heat transport in Bernstein–Galerkin form and derives outlet-temperature relations for prescribed mass-flow trajectories. It uses the projection to form a DHN feasible region for decentralized coordination.

III. BERNSTEIN–GALERKIN EQUIVALENT PROJECTION FOR DISTRICT HEATING NETWORKS

  • The projected DHN feasible region is incorporated into the EPS subproblem and updated during decentralized coordination.The projection is derived from the DHN constraints in P0.Source: §III · p. 4

A. Bernstein–Galerkin Reformulation of the DHN Dynamics

  • The Bernstein–Galerkin method converts pipeline heat-transport PDEs into algebraic equations by requiring the normalized PDE residual to be orthogonal to Bernstein testing functions.Substituting Bernstein representations of temperature and mass flow yields the Bernstein–Galerkin equations.Source: A. Bernstein–Galerkin Reformulation of… · p. 4

B. Equivalent Representation of DHN Thermal Dynamics

  • For a prescribed mass-flow trajectory, fixed flow velocities make the Bernstein–Galerkin equations linear in temperature and yield outlet-temperature relations across pipelines and intervals.The paper assembles these relations into a DHN-wide representation.Source: B. Equivalent Representation of DHN… · p. 4

C. Equivalent Projection of the Feasible Region of the DHN

This section expresses internal thermal variables in terms of DHN boundary variables, then projects internal temperature limits onto those boundary variables.

C. Equivalent Projection of the Feasible Region of the DHN

  • Pipeline inlet temperatures are linked to their inlet-node temperatures through topology-determined selection matrices.The relations are collected across pipelines and intervals.Source: 1) Thermal State Equivalent and… · p. 5
  • Under the paper’s radial-DHN assumption, the required matrix is nonsingular, allowing internal temperature bounds to be projected onto boundary variables.The resulting constraints form the thermal portion of the projected feasible region.Source: 1) Thermal State Equivalent and… · p. 5
  • The thermal state equivalent expresses internal thermal variables in terms of DHN boundary variables using the outlet-temperature relation and the network thermal constraints.The remaining internal thermal variables are related to boundary variables through the partitioned state reconstruction equations.Source: 1) Thermal State Equivalent and… · pp. 4–5

2) Hydraulic State Equivalent and Projection:

For a prescribed mass-flow trajectory, the section derives hydraulic state relations and expresses the required supply-to-return pressure margin using DHN boundary variables.

2) Hydraulic State Equivalent and Projection:

  • With mass flow prescribed, pipeline pressure-loss terms are fixed, and pressure losses are accumulated along paths from the reference node to nonreference nodes.The paths are defined separately for the supply and return networks.Source: 2) Hydraulic State Equivalent and… · p. 5
  • Nonreference-node pressure heads are reconstructed using the reference-node pressure head and topology-determined selection matrices.The reconstruction is stated for each network and interval.Source: 2) Hydraulic State Equivalent and… · p. 5
  • The required supply-to-return pressure margin is expressed in terms of DHN boundary variables.The hydraulic state equivalent supplies the relations used in this expression.Source: 2) Hydraulic State Equivalent and… · p. 5

3) Projected Feasible Region of the DHN:

The projected DHN feasible region combines thermal and hydraulic relations and is used in decentralized EPS–DHN coordination. The procedure alternates feasibility and dispatch stages, while updated mass flows require the equivalent model to be refreshed.

3) Projected Feasible Region of the DHN:

  • The projected DHN feasible region combines thermal and hydraulic equalities with projected temperature limits and the pressure-margin constraint.These relations define the equality and inequality components of the projected region.Source: 3) Projected Feasible Region of the DHN · p. 5

IV. DECENTRALIZED POWER DISPATCH WITH ANDERSON PREDICTION

  • Because the DHN subproblem updates mass flow under VF-VT operation, the equivalent model must be updated at each coordination iteration.The paper identifies this repeated update as a computational burden and devises a prediction strategy to address it.Source: §IV · p. 5

A. EPS and DHN Subproblem Formulations

  • The EPS operator receives projected-model coefficients while internal DHN variables remain local to the DHN operator.This preserves the separation of internal DHN variables during coordination.Source: A. EPS and DHN Subproblem Formulations · p. 5
  • The decentralized procedure first alternates feasibility checking with the DHN subproblem, then alternates EPS dispatch with the DHN subproblem to minimize IHPS operating cost.The first stage obtains a feasible initial solution.Source: A. EPS and DHN Subproblem Formulations · p. 6
  • Stage I uses a predefined residual tolerance to determine whether its boundary variables and mass-flow trajectory initialize Stage II.If the condition is not satisfied, the DHN subproblem is solved and Stage I continues to the next iteration.Source: A. EPS and DHN Subproblem Formulations · p. 6

B. EPS–DHN Coordination With Anderson Prediction

EPS–DHN coordination alternates between reconstructing a flow-dependent DHN equivalent and solving the EPS dispatch problem. The paper accelerates this fixed-point process by predicting coordination inputs from historical updates and residuals.

B. EPS–DHN Coordination With Anderson Prediction

  • Each coordination update evaluates the DHN equivalent at the current boundary variables, embeds it in the EPS problem, and uses the resulting boundary variables in the DHN subproblem.The successive EPS and DHN solutions form one alternating update.Source: 1) Alternating EPS–DHN Coordination · p. 6
  • The alternating scheme is a fixed-point iteration on a coordination vector containing boundary variables and DHN-related variables, with a residual measuring the mismatch from its update.The naive scheme sets the next vector directly to the latest update.Source: 1) Alternating EPS–DHN Coordination · p. 6
  • Anderson prediction uses historical alternating updates and fixed-point residuals to choose the next coordination vector, addressing slow convergence without additional subsystem solves.Residual scaling accounts for different numerical scales; scaled residuals and recent updates are retained for prediction.Source: 1) Alternating EPS–DHN Coordination · p. 6, 2) Safeguarded Anderson Prediction · p. 6

2) Safeguarded Anderson Prediction:

The safeguarded Anderson prediction constructs a candidate coordination displacement from recent alternating updates, then limits its scaled displacement from the latest update.

2) Safeguarded Anderson Prediction:

  • Anderson weights are obtained from a regularized least-squares problem using historical alternating updates and their scaled displacement from the latest update.The memory depth is bounded by a predefined maximum.Source: 2) Safeguarded Anderson Prediction · p. 6
  • The least-squares formulation regularizes the weights and imposes a predefined upper bound on their L1 norm.The regularization parameter is positive, and the maximum memory depth is predefined.Source: 2) Safeguarded Anderson Prediction · p. 6
  • A safeguard scales the predicted displacement so its distance from the latest alternating update does not exceed the prescribed limit.The resulting safeguarded prediction is used as the next coordination vector.Source: 2) Safeguarded Anderson Prediction · pp. 6–7

3) Termination Criterion and Iteration Restart Strategy:

After feasible initialization, Stage II uses safeguarded predictions for operating-cost minimization and retains the latest successful alternating solution when needed. The procedure also covers constant-flow/variable-temperature and variable-flow/constant-temperature modes.

3) Termination Criterion and Iteration Restart Strategy:

  • The predicted vector is used for the next iteration without solving another EPS or DHN subproblem at that vector.The latest alternating update is retained as a restart point if either subproblem fails under the prediction.Source: 3) Termination Criterion and Iteration… · p. 7
  • Stage II stops when the relative operating-cost error meets its predefined tolerance and retains the latest successful alternating solution as the final dispatch.Stage I supplies a feasible starting point for Stage II.Source: 3) Termination Criterion and Iteration… · p. 7
  • Under constant-flow/variable-temperature operation, the DHN model is linear and needs one equivalent projection; variable-flow/constant-temperature operation remains nonlinear and requires iterative updates.The method is designed for variable-flow/variable-temperature operation and also applies to these two modes.Source: 3) Termination Criterion and Iteration… · p. 7

V. CASE STUDIES

On the E6-H6 IHPS, BGEP-D achieved the lowest operating cost under VF-VT while accommodating all available wind power. Case studies also compare its thermal consistency, computational performance, Anderson prediction, and performance across DHN operation modes.

A. Simulation on a Small-Scale IHPS

  • BGEP-D’s operating cost was 1.47% lower than BGM-C’s while accommodating all available wind power under VF-VT.The comparison concerns the E6-H6 IHPS, where both mass flow and temperature trajectories are updated.Source: A. Simulation on a Small-Scale IHPS · p. 7, Table I · p. 8
  • During peak demand from 9–15 h, BGEP-D coordinated higher CHP power output with greater mass flow, whereas NMSE-D’s later mass-flow increase mismatched heat transport and CHP output.The paper links BGEP-D’s coordination to faster heat transport while meeting heat demand; NMSE-D’s mismatch limited CHP utilization and resulted in wind curtailment.Source: A. Simulation on a Small-Scale IHPS · p. 8, Figure 1 · p. 8
  • BGEP-D maintained zero average closure gap, while NMSE-D’s average gap reached 3.53 °C and its componentwise maximum gaps reached approximately 16 °C.The comparison evaluates internal DHN-subproblem states against states reconstructed from the DHN equivalent; BGEP-D derives both representations from the same Bernstein–Galerkin relations.Source: A. Simulation on a Small-Scale IHPS · p. 8, Figure 2 · p. 8
  • BGEP-D completed Stage II in 14 iterations and ran in 8.43 s, compared with 23 iterations and 16.57 s for NMSE-D.BGEP-D’s runtime was 1.65 s longer than BGM-C’s 6.78 s and 8.14 s shorter than NM-C’s 42.88 s; Anderson prediction added 0.01 s.Source: Figure 2 · p. 8, A. Simulation on a Small-Scale IHPS · p. 8, Table II · p. 8
  • Anderson prediction reduced BGEP-D’s Stage-II iterations from 31 to 14 and total runtime from 15.23 s to 8.43 s, while yielding a lower operating cost.Both variants satisfied the termination tolerance.Source: Figure 3 · p. 9, Table III · p. 9
  • Under CF-VT, BGEP-D reduced operating cost by 3.0% and wind curtailment from 12.84 to 3.11 MWh; under VF-CT, it reduced cost by 3.65% and accommodated all available wind.Under VF-CT, BGEP-D completed Stage II in two iterations versus ten for NMSE-D, which curtailed 25.64 MWh; BGEP-D cost was $27,595.75 under CF-VT and $30,773.51 under VF-CT.Source: A. Simulation on a Small-Scale IHPS · p. 9

B. Simulation on a Large-Scale IHPS

On the E39–H20 IHPS, BGEP-D outperforms NMSE-D on operating cost and runtime while maintaining thermal consistency at the reported precision. The conclusion describes a unified Bernstein–Galerkin, boundary-variable approach for decentralized VF-VT dispatch and identifies uncertainty-aware dispatch as future work.

B. Simulation on a Large-Scale IHPS

  • With five Stage-II iterations in both cases, Anderson prediction reduces DHN-subproblem time from 50.87 s to 16.65 s and total runtime from 51.73 s to 17.32 s.Operating costs differ only marginally between the prediction and no-prediction cases.Source: B. Simulation on a Large-Scale IHPS · p. 10
  • 3.59% lower operating cost and total runtime reduced from 69.10 s to 17.32 s versus NMSE-D.Among the methods considered, BGEP-D achieves the lowest operating cost and shortest total runtime on the E39–H20 IHPS.Source: B. Simulation on a Large-Scale IHPS · p. 9, Table V · p. 10
  • Zero average closure gap at the reported precision accompanies BGEP-D’s lower Stage-II operating cost; NMSE-D’s gap remains above 1.4 °C after the initial iteration.The paper attributes the contrast to BGEP-D’s shared Bernstein–Galerkin thermal relations, which maintain consistency between its equivalent and subproblem as mass flow changes.Source: B. Simulation on a Large-Scale IHPS · p. 9, Figure 5 · p. 10

VI. CONCLUSION

  • The framework uses a unified Bernstein-space formulation for the DHN equivalent and subproblem, coordinating through boundary variables without disclosing internal information.The conclusion states that this representation preserves continuous spatiotemporal thermal dynamics and thermal-state consistency during decentralized coordination.Source: §VI · p. 10
  • Future work will address decentralized IHPS dispatch under electricity and heat uncertainties, which can affect thermal-dynamics modeling and decentralized-optimization performance.Source: §VI · p. 10
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