Adaptive ε-Constraint-Based Scheduling with Three-Network Verification and Closed-Loop Repair for Regional Integrated Energy Systems

Abstract

Low-carbon scheduling of regional integrated energy systems (RIES) based only on energy-balance models may overlook the physical operating limits of distribution, gas, and heating networks, resulting in a gap between scheduling outcomes and actual operating boundaries. To address this issue, this paper proposes a framework integrating bi-objective scheduling, three-network posterior verification, and closed-loop repair. A mixed-integer linear programming model is first formulated with operating cost and carbon emissions as the two objectives, and an adaptive ε-constraint strategy is used to improve the characterization of the compromise region on the Pareto front. Posterior verification models are then established for the distribution, gas, and heating networks to assess the physical feasibility of representative solutions. When infeasibility is detected, a boundary-shrinking repair mechanism is triggered to iteratively update the scheduling boundaries. Case results show that the adaptive refined strategy improves the resolution of the compromise region by 3.2 times with only a 20.4% increase in computational time. Compared with the cost-optimal solution, the carbon-optimal solution reduces carbon emissions but increases peak purchased electricity from 7.333 MW to 11.1 MW, further tightening the lower-voltage margin of the distribution network. The results show that posterior physical verification and closed-loop repair provide additional support for evaluating and improving the engineering feasibility of RIES scheduling solutions.

1. Introduction

In recent years, under the continuous promotion of the “dual-carbon” goals, regional integrated energy systems (RIES) have attracted increasing attention as important carriers for low-carbon energy transition [1]. By coupling electricity, natural gas, heat, and cooling through coordinated conversion, storage, and supply devices, RIES can improve energy-utilization efficiency and provide operational flexibility for industrial parks, commercial districts, and residential communities [2,3].

Substantial progress has been made in the modeling and optimization of RIES. In terms of system representation, existing studies have developed electricity–heat coupling models considering thermal inertia [4], electricity–gas coupling models considering gas dynamic characteristics [5,6], and fully coupled multi-energy flow models integrating multiple energy carriers [7]. Graph-based methods [8], distributed modeling approaches [9], and multi-agent reinforcement learning frameworks [10] have also been introduced to improve representation accuracy, computational scalability, and coordination capability. Some studies have further incorporated physical network constraints into multi-energy flow optimization to better reflect actual operating conditions [11].

On the optimization side, exact approaches such as mixed-integer linear programming, second-order cone programming, and semidefinite programming have been widely used to handle complex operational constraints [12,13,14]. To address uncertainties in renewable generation, load demand, and energy prices, interval optimization, robust optimization, and stochastic programming have also been investigated [15,16,17]. In addition, approximate dynamic programming, deep reinforcement learning, and intelligent optimization algorithms have shown potential in alleviating dimensionality and improving real-time scheduling capability [18,19,20,21,22]. For multi-objective optimization in RIES and related multi-energy systems, ε-constraint formulations have been widely used to characterize trade-offs among economy, carbon emissions, and system security [23,24,25].

More recently, multi-objective optimization for complex energy and infrastructure systems has been extended toward hybrid evolutionary algorithms and learning-based paradigms [26,27,28], surrogate-assisted optimization [29], and quantum-inspired formulations [30,31]. These methods provide useful references for improving adaptivity, search efficiency, and representation flexibility in dynamic, uncertain, and high-dimensional decision spaces. However, the focus of this study is not to replace the optimization paradigm itself, but to improve the physical implementability of RIES scheduling solutions after optimization. When detailed distribution, gas, and heating network constraints are considered, model interpretability, constraint representation, and engineering feasibility remain critical issues. Therefore, the ε-constraint method is adopted in this study as a transparent and interpretable Pareto-front construction tool, rather than as a fundamentally new optimization algorithm. This structure allows representative scheduling solutions to be naturally connected with posterior physical verification and closed-loop repair.

Despite the above advances, the physical implementability of scheduling results still requires further attention. Many scheduling models are mainly established based on energy-balance relationships and device-level operating constraints. Although effective for system-level dispatch, such models do not necessarily guarantee that the obtained solutions remain physically feasible when mapped to distribution, gas, and heating networks. A scheduling-layer feasible solution may still cause voltage violations, pressure-limit tightening, or temperature-related risks at the physical network layer [32,33,34,35,36]. Although some studies have incorporated network constraints, reliability assessment, or physical feasibility checks, many remain at the stage of posterior assessment. Once infeasibility or insufficient margin is identified, a unified mechanism for feeding verification results back to the scheduling layer and forming a closed-loop correction process is still relatively lacking.

This issue is particularly important for RIES operation. As the system shifts among cost-oriented, carbon-oriented, and compromise-oriented dispatch, the underlying supply pattern may change accordingly. For example, a lower-carbon solution may reduce local gas-fired generation and increase dependence on external electricity purchase, thereby transferring part of the operational pressure to the grid side. Variations in gas utilization and heat supply allocation may also alter gas-network pressure and heating-network temperature margins. Therefore, the trade-off among economy, carbon reduction, and operational security cannot be adequately understood only from scheduling-layer objective values; the physical response of coupled networks must also be examined.

For multi-objective scheduling, the compromise region of the Pareto front is particularly important because it contains candidate solutions with balanced economic and environmental performance. However, conventional uniform ε-scanning may not simultaneously achieve satisfactory computational efficiency and sufficient local resolution in this region. Therefore, this paper improves the local characterization of the compromise region within the ε-constraint framework and further connects representative solutions with three-network physical verification and feedback repair.

To address these problems, this paper proposes an integrated framework for RIES that combines bi-objective scheduling, three-network posterior verification, and closed-loop repair. First, a mixed-integer linear programming scheduling model is developed with operating cost and carbon emissions as the two objectives, and a coarse-plus-refined ε-constraint scanning strategy is adopted to improve the characterization of the compromise region. Second, posterior verification models are established for the distribution, gas, and heating networks to assess the network-level feasibility of representative scheduling solutions. Third, when infeasibility or insufficient margin is identified, the verification results are fed back to the scheduling layer through a boundary-shrinking repair mechanism, forming a closed-loop process of “optimal scheduling–three-network verification–feedback repair.” In this way, the physical consistency and engineering implementability of scheduling results can be improved without directly embedding all nonlinear network models into the upper-level scheduling optimization.

Based on a typical-day case study of a northern China industrial park, this paper analyzes the trade-off relationships among operating cost, carbon emissions, and network risk under different scheduling tendencies. In addition, an effective carbon emission intensity (ECI) index is introduced to explain the carbon-attribution characteristics of gas-turbine-based generation under combined cooling, heating, and power operation, helping interpret the mechanism behind the observed cost–carbon trade-off.

The main contributions of this paper are summarized as follows.

(1)

A bi-objective MILP scheduling model for RIES is developed with operating cost and carbon emissions as the optimization objectives. Within the established ε-constraint framework, a normalized compromise metric is used to identify the local refined region, and a coarse-plus-refined scanning strategy is adopted to improve the characterization of the compromise region and provide representative scheduling solutions for subsequent physical verification.

(2)

Posterior verification models are established for the distribution, gas, and heating networks, and a boundary-shrinking closed-loop repair mechanism is designed to improve the physical feasibility and engineering implementability of scheduling solutions.

(3)

Based on a typical-day case study, the trade-off relationships among cost, carbon emissions, and network risk are analyzed, and the dominant network bottleneck and critical periods under different scheduling tendencies are identified. In addition, the ECI index is introduced to support the interpretation of emission-reduction mechanisms under different operating strategies.

The remainder of this paper is organized as follows. Section 2 introduces the system description and scheduling assumptions. Section 3 presents the bi-objective scheduling model and the adaptive ε-constraint strategy. Section 4 describes the three-network posterior verification and closed-loop repair method. Section 5 provides the case study and discussion. Finally, Section 6 concludes the paper.

2. System Description and Scheduling Assumptions

2.1. RIES Architecture and Energy Coupling Relationships

As illustrated in Figure 1, the core coupling chain of the proposed system can be summarized as natural gas–gas turbine–electricity/heat–cooling. In this structure, the energy storage system and the utility grid provide operational flexibility on the electricity side, while the gas boiler serves as a supplementary heat source on the thermal side. This coupling structure indicates that operating cost, carbon emissions, and network security do not evolve independently, but are mutually coupled through the multi-energy conversion chain.

2.2. Scheduling Assumptions

The scheduling horizon is set to 24 h with a time step of 1 h. In this study, a typical-day hourly scheduling framework is adopted to jointly optimize electric, heating, and cooling loads together with renewable energy output.

(1)

Rigid load satisfaction without load shedding

$$\begin{array}{cc}{P}_{shed,t}=H_{shed,t}=L_{shed,t}=G_{shed,t}=0,& \forall t\in T\end{array}$$

(1)

Equation (1) indicates that electricity, heating, and cooling demands are all treated as rigid loads throughout the scheduling horizon, and the model does not permit load shedding as a means of restoring feasibility. Natural gas consumption is determined endogenously by the operating states of the devices, and no interruptible gas load is explicitly considered.

(2)

Renewable curtailment is allowed with penalty costs

Curtailment of wind power and photovoltaic generation is allowed, and the corresponding penalty cost is incorporated into the objective function, as detailed in Section 3.2.

3. Bi-Objective Scheduling Model

This section develops a bi-objective MILP scheduling model for RIES, in which operating cost and carbon emissions are taken as the two optimization objectives. Based on this model, the ε-constraint method is adopted to generate the Pareto front, and the resolution of the compromise region is further improved through adaptive scanning.

3.1. Objective Functions

3.1.1. Operating Cost Objective

The operating cost consists of electricity purchasing and selling, natural gas purchasing, demand charges, and renewable-curtailment penalties, and can be expressed as

$$\min C={\displaystyle \sum _t^T\left[c_{e,t}{P}_{buy,t}+c_{g,t}{V}_{gas,t}+{\lambda }_{curt}{P}_{curt,t}\right]}\cdot ∆t+c_{peak}\cdot \left(P_{peak}\cdot 1000\right)$$

(2)

Equation (2) represents the daily operating cost of the system, including electricity purchasing cost, natural gas purchasing cost, demand charges, and renewable-curtailment penalty cost. Since $P_{peak}$ is measured in MW while the demand charge is charged in kW, the factor 1000 is introduced to ensure unit consistency.

3.1.2. Carbon Emission Objective

The total carbon emissions consist of emissions from electricity purchasing and natural gas consumption, which can be written as

$$\min E={\displaystyle \sum _t^T\left[{\mu }_e{P}_{buy,t}+{\mu }_g{G}_{total,t}\right]}\cdot ∆t$$

(3)

Equation (3) represents the total daily carbon emissions of the system, which are composed of emissions associated with purchased electricity and natural gas consumption. This is also a common accounting approach in integrated energy system studies [36].

3.1.3. Definition of Effective Carbon Emission Intensity (ECI)

In this case study, the power-generation efficiency of the gas turbine is ${\eta }_e$ = 0.26 If the total natural gas consumption of the gas turbine was entirely allocated to electricity generation, the corresponding carbon emission intensity would reach 1.72 tCO₂/MWh, which is much higher than that of the utility grid (0.7035). However, under the combined cooling, heating, and power operating mode, the gas turbine generates electricity while simultaneously providing high-grade waste heat. Therefore, the effective carbon emission intensity of gas-turbine power generation at time period t, denoted by $e_{ECI,t}$, is defined as follows:

$$EC{I}_{GT}={\displaystyle \frac{{\mu }_g\cdot G_{GT,t}-{\mu }_g\cdot \frac{H_{hrb,t}}{{\eta }_{gb}}}{P_{gt,t}}}$$

(4)

Equation (4) defines the effective carbon emission intensity of gas-turbine power generation. The first term in the numerator represents the direct carbon emissions associated with the natural gas consumed by the gas turbine, while the second term represents the equivalent avoided carbon emissions due to waste-heat recovery, which can offset the heat supply otherwise provided by the gas boiler. The denominator is the gas-turbine electricity output. Therefore, ECI reflects the net attributed carbon emissions per unit of electricity generation under the combined heat and power condition, rather than the original combustion-based emission intensity of the gas turbine itself.

Under the present parameter setting, the effective carbon emission intensity of the gas turbine can be reduced from 1.72 kg/kWh to approximately 0.512 kg/kWh. When the recovered waste heat is fully utilized, $e_{ECI,t}$ can further decrease to about 0.48 tCO₂/MWh, indicating that gas-turbine-based generation can exhibit certain emission-reduction potential when the recovered heat is effectively utilized.

It should be noted that ECI is used as an interpretive carbon-attribution index rather than an additional optimization objective or constraint. The carbon-emission objective in Equation (3) still follows the standard purchased-electricity plus natural-gas accounting form, while ECI is introduced to explain the emission-reduction mechanism of gas-turbine-based CCHP operation.

3.2. Constraint Conditions

3.2.1. Grid Interaction Constraints

To avoid physically meaningless arbitrage caused by frequent switching between electricity purchasing and selling under the time-of-use tariff mechanism, binary variables are introduced to enforce mutual exclusiveness between the two actions within the same time period.

(1)

Mutual exclusiveness between electricity purchasing and selling

$$\left\{\begin{array}{l}0\le P_{buy,t}\le P_{grid,\max }\cdot {\mu }_{buy,t}\\ 0\le P_{sell,t}\le P_{grid,\max }\cdot {\mu }_{sell,t}\\ {\mu }_{buy,t}+{\mu }_{sell,t}\le 1\\ {\mu }_{buy,t},{\mu }_{sell,t}\in \left\{0,1\right\}\end{array}\right.$$

(5)

(2)

Capacity limits and peak-period purchasing limit

$$\left\{\begin{array}{l}{P}_{buy,t}\le {\mu }_{grid,t}\cdot P_{grid,\max }\\ P_{sell,t}\le \left(1-{\mu }_{grid,t}\right)\cdot P_{grid,\max }\\ P_{buy,t}\le P_{peak,\max },\forall t\in T_{peak}\end{array}\right.$$

(6)

(3)

Definition of peak demand

$$\left\{\begin{array}{l}{P}_{peak}\ge P_{buy,t}\\ \forall t\in T,P_{peak}\ge 0\end{array}\right.$$

(7)

Equations (5)–(7) describe the mutual exclusiveness between electricity purchasing and selling, the upper limits of grid-interactive trading power, and the definition of the maximum demand during the scheduling horizon, respectively. Here, ${\mu }_{buy,t}$ and ${\mu }_{sell,t}$ are binary variables indicating the electricity purchasing and selling states, respectively; ${\mu }_{grid,t}$ is a time-period indicator used to distinguish the peak-period purchasing limit; and $P_{grid,\max }$ and $P_{peak,\max }$ denote the overall grid-interaction capacity limit and the peak-period purchasing limit, respectively. The numerical values of the grid-interaction limits are listed in Appendix A.

3.2.2. Renewable Generation and Curtailment Constraints

The available photovoltaic and wind power in each time period is allocated between actual utilization and curtailment, as defined below.

$$\left\{\begin{array}{l}{P}_{pv,t}+P_{pv,curt,t}=P_{pv,ava,t}\\ P_{\omega t,t}+P_{\omega t,curt,t}=P_{\omega t,ava,t}\end{array}\right.$$

(8)

Equation (8) describes the allocation of the forecast available photovoltaic and wind power output between actual utilization and curtailed energy.

$$\left\{\begin{array}{l}0\le P_{pv,t}\le P_{pv,ava,t}\\ 0\le P_{\omega t,t}\le P_{\omega t,ava,t}\\ P_{pv,curt,t},P_{\omega t,curt,t}\ge 0\end{array}\right.$$

(9)

Equation (9) gives the non-negativity and upper-bound constraints for the actual photovoltaic and wind power outputs as well as the curtailed power. Here, $P_{pv,curt,t}$ and $P_{\omega ,curt,t}$ denote the curtailed photovoltaic and wind power at time period $t$, respectively, and their sum corresponds to the total renewable curtailment used in the cost objective. The curtailment penalty coefficient used in the objective function is given in Appendix A.

3.2.3. Device Models and Coupling Constraints

(1)

Natural gas consumption and output limit of the gas turbine

$$\begin{array}{cccc}{G}_{GT,t}={\displaystyle \frac{P_{gt,t}}{{\eta }_{e,gt}}},& 0\le P_{gt,t}\le P_{gt}^{\max }& ,& \forall t\in T\end{array}$$

(10)

(2)

Waste-heat recovery and output limit

$$\begin{array}{ccc}{H}_{hrb,t}=P_{gt,t}\cdot {\eta }_{hrb},& 0\le H_{hrb,t}\le H_{hrb}^{\max },& \forall t\in T\end{array}$$

(11)

(3)

Heat output of the gas boiler and natural-gas conversion

$$\begin{array}{cccc}{H}_{gb,t}=G_{GB,t}\cdot {\eta }_{gb},& G_{total,t}=G_{GT,t}+G_{GB,t},& V_{gas,t}={\displaystyle \frac{G_{total,t}}{LH{V}_{gas}}},& \forall t\in T\end{array}$$

(12)

(4)

Energy conversion relationships and capacity limits of cooling devices

$$\begin{array}{ccc}{L}_{ac,t}=CO{P}_{ac}\cdot H_{ac,in,t},& 0\le L_{ac,t}\le L_{ac}^{\max },& \forall t\in T\end{array}$$

(13)

$$\begin{array}{ccc}{L}_{ec,t}=CO{P}_{ec}\cdot P_{ec,in,t},& 0\le P_{ec,in,t}\le P_{ec,in}^{\max },& \forall t\in T\end{array}$$

(14)

(5)

Ramp-rate constraint of the gas turbine

$$\begin{array}{cc}-Ram{p}_{GT}\le P_{gt,t}-P_{gt,t-1}\le Ram{p}_{GT},& \forall t\in T\end{array}$$

(15)

(6)

Output bounds of the gas boiler

$$\begin{array}{cc}{H}_{gb,\min }\cdot {\mu }_{gb,t}\le H_{gb,t}\le H_{gb,\max }\cdot u_{gb,t},& {\mu }_{gb,t}\in \left\{0,1\right\}\end{array}$$

(16)

Equations (10)–(16) describe the multi-energy coupling relationships, capacity limits, and operating constraints of the gas turbine, waste-heat recovery unit, gas boiler, and cooling devices. Specifically, Equation (10) defines the natural-gas consumption and rated output limit of the gas turbine; Equation (11) describes waste-heat recovery and its upper output limit; Equation (12) gives the gas-boiler heat output, the total natural-gas energy consumption, and the conversion between natural-gas energy and volume; Equations (13) and (14) characterize the energy-conversion relationships and capacity limits of the absorption chiller and electric chiller; Equation (15) imposes the ramp-rate constraint of the gas turbine; and Equation (16) gives the operating-state-dependent output bounds of the gas boiler. Here, $P_{gt}^{\max }$, $H_{hrb}^{\max }$, $L_{ac}^{\max }$, $P_{ec,in}^{\max }$, $H_{gb}^{\min }$, $H_{gb}^{\max }$, and $LH{V}_{gas}$ are the rated capacity and energy-conversion parameters listed in Appendix A. To preserve linearity, the recovered waste heat is approximated as a linear function of gas-turbine power output.

3.2.4. Multi-Energy Balance Constraints

Electricity balance

$$P_{buy,t}+P_{gt,t}+P_{pv,t}+P_{\omega t,t}+P_{dis,t}=P_{load,t}+P_{ch,t}+P_{sell,t}+P_{ec,in,t}$$

(17)

Heat balance

$$H_{hrb,t}+H_{gb,t}=H_{load,t}+H_{ac,in,t}$$

(18)

Cooling balance

$$L_{ac,t}+L_{ec,t}=L_{load,t}$$

(19)

Equations (17)–(19) represent the electricity, heating, and cooling energy balance constraints of the system in each time period, respectively. Here, $H_{ac,in,t}$ denotes the thermal input to the absorption chiller, and $P_{ec,in,t}$ denotes the electric input to the electric chiller.

3.2.5. Energy Storage Constraints

SOC dynamics

$$SO{C}_{t+1}=SO{C}_t+{\displaystyle \frac{{\eta }_{ch}\cdot P_{ch,t}\cdot ∆t}{E_{bat}}}-{\displaystyle \frac{P_{dis,t}\cdot ∆t}{{\eta }_{dis}\cdot E_{bat}}}$$

(20)

SOC bounds and cyclical consistency

$$\left\{\begin{array}{l}\underset{¯}{SOC}\le SO{C}_t\le \overline{SOC}\\ SOC\left(1\right)=SOC\left(T+1\right)\end{array}\right.$$

(21)

Charging/discharging power limits

$$\left\{\begin{array}{l}0\le P_{ch,t}c\le {\overline{P}}_{ch}\cdot {\mu }_{ch,t}\\ 0\le P_{dis,t}\le {\overline{P}}_{dis}\cdot {\mu }_{dis,t}\end{array}\right.$$

(22)

Mutual exclusiveness between charging and discharging

$$\left\{\begin{array}{l}{\mu }_{ch,t}+{\mu }_{dis,t}\le 1\\ {\mu }_{ch,t},{\mu }_{dis,t}\in \left\{0,1\right\}\end{array}\right.$$

(23)

Equations (20)–(23) describe the dynamic evolution of the state of charge, the SOC bounds, the charging/discharging power limits, and the mutual exclusiveness between charging and discharging, respectively. Here, $E_{bat}$ is the rated energy capacity of the energy storage system; ${\eta }_{ch}$ and ${\eta }_{dis}$ are the charging and discharging efficiencies; $P_{ch}^{\max }$ and $P_{dis}^{\max }$ are the charging and discharging power limits; and ${\mu }_{ch,t}$ and ${\mu }_{dis,t}$ are binary variables representing the charging and discharging states of the energy storage system. The corresponding numerical values are listed in Appendix A.

3.2.6. ε-Constraint Formulation

In the ε-constraint method, the carbon-emission objective is converted into a constraint [25]:

$$\begin{array}{l}\begin{array}{cc}\min & C\end{array}\\ \begin{array}{cc}s.t.& E\le \epsilon \end{array}\end{array}$$

(24)

Equation (24) transforms the original bi-objective optimization problem into a single-objective problem that minimizes operating cost subject to an upper bound on carbon emissions. By scanning different values of ε, the Pareto front between operating cost and carbon emissions can be obtained.

3.3. Adaptive ε-Constraint Scanning and Compromise-Solution Identification

An adaptive ε-constraint scanning strategy is adopted in this study to identify the compromise solution on the Pareto front. The procedure is summarized as follows.

(1)

Extreme-point solution

The cost-optimal solution (Opt-Cost) and the carbon-optimal solution (Opt-Carbon) are first obtained, yielding the carbon-emission range $\left[E_{\min },E_{\max }\right]$.

(2)

Coarse scan

A set of $N_c$ uniformly distributed ε values is selected over $\left[E_{\min },E_{\max }\right]$, and the MILP model is solved point by point to obtain the coarse Pareto front.

(3)

Compromise metric and compromise-region identification

The operating cost and carbon emissions are first normalized as

$$\widehat{C}={\displaystyle \frac{C-C_{\min }}{C_{\max }-C_{\min }}}\begin{array}{cc},& \widehat{E}={\displaystyle \frac{E-E_{\min }}{E_{\max }-E_{\min }}}\end{array}$$

(25)

Equation (25) performs dimensionless normalization of operating cost and carbon emissions, thereby eliminating the influence of differences in units and magnitudes between the two objectives.

The compromise metric, defined as the Euclidean distance to the ideal point, is given by

$$J=\sqrt{{\widehat{C}}^2+{\widehat{E}}^2}$$

(26)

Equation (26) defines the Euclidean distance to the ideal point and uses it as the compromise indicator $J$. The coarse-scan solution with the minimum value of $J$ is selected as the compromise-solution candidate. If the index of this candidate is denoted by $i$, the refined interval is selected from the adjacent coarse ε levels around this point, i.e., $[{\epsilon }_{i-1},{\epsilon }_{i+1}]$ for an interior candidate. If the candidate lies at the boundary of the coarse scan, the nearest available adjacent interval is used instead.

Therefore, the refined region is not manually specified, but is automatically determined by the normalized compromise metric.

(4)

Refined scan and merging

A refined scan is then performed by densely sampling $N_f$ ε values within the selected local interval. The resulting refined Pareto points are merged with the coarse-scan results, and duplicate points are removed to form the final Pareto front.

This coarse-plus-refined procedure is not specific to the case study. It only requires the objective values obtained from an initial coarse ε-constraint scan and a normalized compromise metric. Therefore, it can be transferred to other bi-objective scheduling problems where the local compromise region needs higher-resolution characterization. If decision-maker preferences differ, the Euclidean distance in Equation (26) can also be replaced by a weighted or preference-based metric.

As illustrated in Figure 2, the overall workflow of the proposed method proceeds from scheduling optimization to physical verification and then to feedback repair, forming a closed-loop analysis framework of optimization–verification–repair.

4. Three-Network Posterior Verification and Closed-Loop Repair Method

Since park-level scheduling models are usually built on energy-balance relationships and simplified device characteristics, their solutions may lead to voltage violations or pipeline pressure distortions when implemented in actual physical networks. To address this issue, this paper establishes a closed-loop framework of scheduling, verification, and repair. Candidate scheduling solutions are first generated by the MILP model in the scheduling layer and then mapped to the adopted standard benchmark networks as boundary conditions for posterior verification. Specifically, the candidate solutions are examined using the nonlinear verification models of the IEEE 33-bus distribution network, the 14-node gas network, and the 17-node heating network. If physical bottlenecks are identified, the Auto-Repair mechanism shrinks the electricity-purchasing and gas-purchasing boundaries and re-solves the scheduling model until the voltage, pressure, and temperature margins all satisfy the required safety limits. As illustrated in Figure 2, this process forms a closed-loop workflow of optimization, verification, and feedback repair.

4.1. Energy Hub Interface Mapping and Spatiotemporal Alignment

To connect the scheduling layer in Section 3 with the physical-network verification layer in Section 4.2, Section 4.3 and Section 4.4, an energy-hub interface mapping model based on a coupling matrix is established. This model converts the system-level energy decision variables obtained from the scheduling layer into boundary injection conditions at the access nodes of the distribution, gas, and heating networks, as expressed by

$$\left[\begin{array}{c}{P}_{25,t}^{Inj}\\ G_{3,t}^{Load}\\ G_{13,t}^{Load}\\ T_{15,t}^s\end{array}\right]=\left[\begin{array}{ccccc}{\eta }_{tr}& -1/{\eta }_{tr}& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}{P}_{t,E}^{buy}\\ P_{t,E}^{sell}\\ G_{t,GT}\\ G_{t,GB}\\ T_{set,t}\end{array}\right]$$

(27)

In Equation (27), the left-hand side represents the nodal injection or equivalent load quantities in the distribution, gas, and heating networks, while the right-hand side contains the scheduling variables obtained from the optimization model in Section 3. The intermediate matrix describes the cross-network coupling relationships associated with EH1 and EH2. Here, ${\eta }_{tr}$ denotes transformer efficiency.

Through this mapping, the scheduling-layer results are converted into unified nodal boundary injections for the subsequent DistFlow, Weymouth, and thermo-hydraulic verification models. The hierarchical coupling topology of the energy system is illustrated in Figure 3.

4.2. Posterior Verification Model of the Distribution Network

The posterior verification of the distribution network is based on the DistFlow power-flow model, which has been widely applied in distribution network operation analysis and coupled energy-network modeling [37,38].

$$\left\{\begin{array}{l}{p}_j=P_i-r_{ij}{l}_{ij}-P_{load,j}+P_{gen,j}\\ Q_j=Q_i-x_{ij}{l}_{ij}-Q_{load,j}+Q_{gen,j}\\ V_j^2=V_i^2-2\left(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij}\right)+\left(r_{ij}^2+x_{ij}^2\right)\cdot l_{ij}\\ l_{ij}={\displaystyle \frac{P_{ij}^2+Q_{ij}^2}{V_{ij}^2}}\end{array}\right.$$

(28)

Equation (28) gives the DistFlow relationships of the distribution network, including nodal active/reactive power balance, branch voltage drop, and the coupling relationship associated with the squared branch current term.

4.3. Posterior Verification Model of the Gas Network

The posterior verification of the gas network is described using the steady-state pipeline-flow relationship and the nodal gas-flow balance equation.

$$F_{mn}\left|F_{mn}\right|=K_{mn}^2\left(f_m^2-f_n^2\right)$$

(29)

$${\displaystyle \sum F_{in}}-{{\displaystyle \sum F}}_{out}=S_n-L_n$$

(30)

Equations (29) and (30) represent the steady-state Weymouth flow relationship of gas pipelines and the nodal flow-balance equation of the gas network, respectively. Owing to the nonconvex nature of this model, its tractability in multi-energy network modeling has attracted considerable attention in the literature [39,40].

4.4. Posterior Verification Model of the Heating Network

4.4.1. Hydraulic Balance Equations

A steady-state hydraulic model is adopted for the heating network to describe the relationship among mass flow rate, nodal pressure, and branch pressure drop.

(1)

Nodal mass conservation

$${\displaystyle \sum _{j\in A_i}{\dot{m}}_{ij}}={\dot{m}}_{load,i}-{\dot{m}}_{source,i}$$

(31)

(2)

Branch pressure-drop equation

$$∆{\beta }_{ij}=K_{h,ij}{\dot{m}}_{ij}\left|{\dot{m}}_{ij}\right|$$

(32)

Equations (31) and (32) represent the nodal mass conservation relationship and the branch pressure-drop equation of the heating network, respectively. Here, $A_i$ denotes the set of branches connected to node $i$, and $K_{h,ij}$ is the pipe resistance coefficient.

4.4.2. Thermal Balance Equations

The thermal model describes heat exchange during circulation and the heat loss along the pipeline.

(1)

Load heat-exchange equation

$$H_{load,i}={\dot{m}}_i{C}_p(T_{s,i}-T_{r,i})$$

(33)

(2)

Temperature attenuation equation

$$T_{end}=T_{amb}+(T_{start}-T_{amb})e^{-{\displaystyle \frac{\lambda L}{\dot{m}{C}_p}}}$$

(34)

Equations (33) and (34) describe the nodal heat-exchange relationship and the temperature attenuation caused by pipeline heat loss, respectively. Here, $C_p$ is the specific heat capacity of water, $T_{amb}$ is the ambient temperature, and $L_{ij}$ is the pipe length.

By combining graph-based topology with physical governing equations, the hydraulic balance, temperature distribution, and heat-loss characteristics of the district heating network can be represented with higher fidelity [41,42]. The hierarchical coupling network topology of the integrated energy system is shown in Figure 3.

4.5. Margin Indices and Bottleneck Identification

To quantify the distance between the operating state and the corresponding physical boundary, three margin indices are defined for voltage, pressure, and temperature constraints.

$$V_{m\arg in,t}=\underset{i\in N}{\min }\left[{\displaystyle \frac{V_{i,t}-V_{\min }}{V_{\max }-V_{\min }}}\right]$$

(35)

$$p_{m\arg in,t}=\underset{j\in M}{\min }\left[{\displaystyle \frac{f_{j,t}-f_{\min }}{f_{\max }-f_{\min }}}\right]$$

(36)

$$T_{m\arg in,t}=\underset{i\in H}{\min }\left[{\displaystyle \frac{T_{s,i}(t)-T_{s,\min }}{T_{s,\max }-T_{s,\min }}},{\displaystyle \frac{T_{s,\max }-T_{s,i}(t)}{T_{r,\max }-T_{r,\min }}}\right]$$

(37)

Equations (35)–(37) define the voltage margin of the distribution network, the pressure margin of the gas network, and the temperature margin of the heating network, respectively. All three indices are constructed from the distance between the operating state and the corresponding safety boundary. A smaller margin indicates that the corresponding network is closer to its physical bottleneck.

4.6. Closed-Loop Repair Strategy

To account for the possible mismatch between the dispatch-layer solution and the benchmark physical networks used for posterior verification, a closed-loop repair strategy is adopted. After the representative scheduling solution is obtained, it is mapped to the electricity, gas, and heating networks for posterior verification. If any network-layer constraint is violated, the corresponding outer-layer boundary is updated and the scheduling model is re-solved until all networks pass verification or the maximum repair iteration number is reached.

For power-network verification, the dispatch-layer electric demand is mapped to the benchmark feeder through a network-scale coefficient $S_{ver}$, as expressed in Equation (38). In this study, $S_{ver}$ = 0.18 is adopted as the calibrated baseline setting, while $S_{ver}$ = 0.16 and $S_{ver}$ = 0.17 are used only for sensitivity analysis in Section 5.6.

$$P_{ver,load,t}=S_{ver}\left(P_{load,t}+P_{ec,in,t}+P_{ch,t}-P_{dis,t}\right)$$

(38)

Equation (38) maps the dispatch-layer electric demand to the benchmark distribution-network load for posterior verification. Here, $P_{\mathrm{v}\mathrm{e}\mathrm{r},\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d},t}$ is the mapped verification load, $P_{\mathrm{e}\mathrm{c},\mathrm{i}\mathrm{n},t}$ is the electric input power of the electric chiller, and $P_{\mathrm{c}\mathrm{h},t}$ and $P_{\mathrm{d}\mathrm{i}\mathrm{s},t}$ are the charging and discharging powers of the energy storage system, respectively.

If posterior verification fails at iteration k, the electricity-purchasing and gas-purchasing boundaries are updated according to Equations (39) and (40), where ${\mathrm{\Omega }}_P^{(k)}$ and ${\mathrm{\Omega }}_G^{(k)}$ denote the violated time sets identified by the power and gas networks, respectively. The shrinking factor ${\eta }_{rep}$ controls the repair intensity. In the main case, ${\eta }_{rep}$ = 0.90 is used as the baseline value, while ${\eta }_{rep}$ = 0.85 and ${\eta }_{rep}$ = 0.95 are compared in Section 5.6.

$${\overline{P}}_{buy,t}^{(k+1)}=\left\{\begin{array}{l}\begin{array}{cc}{\eta }_{rep}\cdot {\overline{P}}_{buy,t}^{(k)}& t\in {\mathrm{\Omega }}_P^{\left(k\right)}\end{array}\\ \begin{array}{cc}{\overline{P}}_{buy,t}^{\left(k\right)}& \begin{array}{c}t\notin {\mathrm{\Omega }}_P^{\left(k\right)}\end{array}\end{array}\end{array}\right.$$

(39)

$${\overline{G}}_{buy,t}^{(k+1)}=\left\{\begin{array}{l}\begin{array}{cc}{\eta }_{rep}\cdot {\overline{G}}_{buy,t}^{(k)}& t\in {\mathrm{\Omega }}_G^{\left(k\right)}\end{array}\\ \begin{array}{cc}{\overline{G}}_{buy,t}^{\left(k\right)}& \begin{array}{c}t\notin {\mathrm{\Omega }}_G^{\left(k\right)}\end{array}\end{array}\end{array}\right.$$

(40)

Equations (39) and (40) update the upper bounds of electricity purchasing and natural gas purchasing, respectively. Here, ${\overline{P}}_{buy,t}^{(k)}$ and ${\overline{G}}_{buy,t}^{(k)}$ denote the purchasing upper bounds at repair iteration $k$, while ${\mathsf{\Omega }}_P^{\left(k\right)}$ and ${\mathsf{\Omega }}_G^{\left(k\right)}$ denote the violated time sets identified by power- and gas-network posterior verification, respectively.

The proposed Auto-Repair mechanism is a feasibility-restoration layer rather than a globally exact reformulation of the fully coupled nonlinear network-constrained scheduling problem. This decoupled design avoids embedding distribution-power-flow, gas-flow, and thermo-hydraulic constraints directly into the MILP scheduling model, but may introduce a certain loss of optimality because the feasible region is progressively tightened according to posterior verification results. Therefore, the repair process aims to improve physical implementability rather than to guarantee the global optimum of a fully integrated formulation.

$$F_P^{(k)}=F_G^{(k)}=F_H^{(k)}=\begin{array}{ccc}1& or& k\end{array}=K_{rep}^{\max }$$

(41)

Equation (41) gives the stopping criterion of the repair loop $F_p^{(k)}$, $F_G^{(k)}$, and $F_H^{(k)}$ are binary verification flags for the power, gas, and heating networks at iteration $k$, respectively, where a value of 1 indicates that the corresponding network passes posterior verification. The repair process stops when all three networks pass posterior verification or when $k$ reaches $K_{rep}^{\max }$. In this study, $K_{rep}^{\max }$ = 6 is adopted as the formal maximum number of repair iterations.

The method does not claim unconditional convergence under arbitrary stressed conditions; instead, its empirical convergence and parameter sensitivity are examined in Section 5.6. In the studied case, the heating-network temperature margins remain positive and the gas network does not trigger posterior infeasibility; therefore, active repair is mainly driven by the distribution-network voltage violation. For operating conditions with heating-network violations, the same feedback-repair logic can be extended by adjusting heat-source temperature or thermal boundary settings.

5. Case Study and Results Discussion

This study considers a typical-day operating scenario of an industrial park in northern China. The typical daily electricity, heating, and cooling load profiles, together with the electricity and natural gas price curves, are shown in Figure 4 and Figure 5. The detailed device parameters are given in Appendix A, the key settings for posterior verification and feedback repair are summarized in Appendix B, and the complete network data and implementation details are provided in the Supplementary Materials.

5.1. Scenario and Parameter Settings

5.1.1. Physical Network Mapping

To verify the physical feasibility of the scheduling solutions, a scenario-mapping approach is adopted to map the park-level energy-flow configuration onto standard test networks. In this framework, the typical-day data of the industrial park are used for the scheduling layer, while the standard benchmark networks are used for posterior feasibility screening. The network topology and coupling relationships are illustrated in Figure 3, and the interface correspondence is listed in

Table 1. Configuration of system networks and coupling interfaces.

Energy Layer	Distribution Network	Gas Network	Heating Network
Benchmark model	IEEE 33-bus system	14-node gas network	17-node heating network
Core node configuration	EH1 connected to Node 25	EH1 connected to Node 3; EH2 connected to Node 13	EH2 connected to Node 15

. Considering voltage and pressure sensitivity, terminal nodes of the IEEE 33-bus distribution network are selected for distribution-network access verification, while for the gas network, one node close to the main gas source and another pressure-sensitive terminal node are chosen to characterize pressure-gradient variations.

5.1.2. Device and Evaluation Parameters

The parameters related to capacity, efficiency, ramp-rate limits, carbon-emission factors, and energy prices are used consistently in the scheduling, posterior verification, and sensitivity-analysis stages, and their values are listed in Appendix A.

5.2. Comparative Strategies and Evaluation Metrics

5.2.1. Two Baseline Strategies

Two baseline strategies are introduced for comparison under the same modeling framework.

(1)

Baseline-Price (price-driven):

The on/off status and output of each device are determined according to the time-of-use electricity price and device efficiencies, while satisfying the load demand and the same grid-interaction and device constraints as those used in the main model.

(2)

Baseline-Smooth (grid-smoothing):

Under the same operational constraints as the main model, this strategy takes the minimization of the peak purchased electricity power $P_{peak}$ during the scheduling horizon as the primary objective, thereby obtaining a feasible scheduling solution that tends to suppress the peak pressure imposed on the power grid.

5.2.2. Evaluation Metrics

The two baseline strategies are introduced to represent typical operational tendencies rather than competing optimization algorithms. The methodological comparison is conducted between the conventional uniformly coarse ε-constraint scan and the proposed coarse-plus-refined ε-constraint strategy, as further discussed in Section 5.3.2. The evaluation metrics include operating cost, carbon emissions, peak purchased electricity power, and the compromise indicator defined by Equation (26).

5.3. Multi-Objective Optimization Results and Pareto Front

5.3.1. Characteristics of the Pareto Front

The Pareto front is generated using the adaptive ε-constraint scanning strategy. Specifically, the coarse scan contains 20 sampling points and the refined scan contains 25 sampling points, resulting in a total of 45 candidate solutions. After merging and removing duplicate points, 44 distinct Pareto solutions are obtained, denoted as the solution set P. The Pareto front and the locally refined compromise region are shown in Figure 6 and Figure 7, respectively.

Table 2. Summary of representative solutions (Cost/Carbon/Peak Grid).

Solution	Cost (CNY)	Carbon (t)	Peak Grid (MW)	Index_J
Baseline-Price	281,453.78	490.26	10.3	0.9511
Baseline-Smooth	286,149.55	497.73	0.00	1.3095
Opt-Cost	281,211.492	491.320	7.333	1.0000
Opt-Carbon	307,944.084	469.698	11.1	1.0000
Opt-Compromise	289,208.46	477.664	9.216	0.4746

reports the operating cost, carbon emissions, and peak purchased electricity power of the representative solutions. It can be observed that, as the system gradually shifts from the cost-optimal solution toward the carbon-optimal solution, the share of local gas-fired generation decreases while the dependence on external electricity purchasing increases, leading to a higher peak purchased power. This indicates that the emission-reduction pathway is not achieved simply by lowering total energy consumption; instead, part of the operational pressure is transferred to the grid side through energy substitution. Its physical implications will be further discussed in Section 5.5.

5.3.2. Efficiency Analysis of the Adaptive Scanning Strategy

To improve the solution resolution in the compromise region, this paper adopts an adaptive ε-constraint strategy combining a coarse scan with local refined scanning. The coarse scan is first used to identify the overall shape of the Pareto front, and then additional points are inserted around the compromise region to obtain a more stable local boundary.

Table 3. Performance comparison of different scanning strategies.

Scanning Strategy	Number of Samples	Candidate/Effective Solutions	Resolution in Compromise Region	Computational Time Increase
Conventional ε-constraint method (coarse scan)	20	20	Baseline (1.0×)	—
Adaptive ε-constraint method (coarse + refined scan)	45	44	3.2×	+20.4%

compares the performance of this strategy with that of the conventional uniformly coarse ε-constraint strategy.

As shown in Table 3, compared with the conventional coarse ε-constraint method, the proposed coarse-plus-refined strategy improves the resolution of the compromise region by 3.2 times while increasing the computational burden by only 20.4%. This demonstrates that local refinement can significantly improve the characterization quality of the compromise region at a relatively small additional computational cost, thereby enhancing the reliability of compromise-solution identification.

5.3.3. Trade-Off Between Carbon Reduction and Peak Purchased Electricity

Figure 8 shows that peak purchased electricity generally increases as the scheduling solution moves from the cost-optimal end toward the carbon-optimal end of the Pareto set. Compared with the cost-optimal solution, the carbon-optimal solution increases peak purchased electricity from 7.333 MW to 11.1 MW, corresponding to a 51.4% increase. This indicates that aggressive carbon reduction may substantially intensify peak stress on the distribution network.

The step-like variation of peak purchased electricity is mainly caused by the definition of the indicator itself, since it is determined by the maximum hourly electricity purchase within the day. When the operating state crosses certain carbon-emission thresholds, the peak purchased electricity may change abruptly rather than continuously. From the ECI perspective in Section 3.1.3, when the recovered waste heat of the gas turbine is not sufficiently utilized, the marginal emission-reduction advantage of local gas-fired generation weakens, and the model becomes more inclined to use external grid electricity.

The compromise solution lies between the two extreme solutions and restrains the expansion of peak purchased electricity while still achieving considerable emission reduction. Therefore, it provides a more balanced trade-off among economic performance, carbon reduction, and grid-side security.

5.4. Strategy Comparison and Discussion

5.4.1. Comparison Between Baseline and Optimized Strategies

Under the unified assumption of rigid load satisfaction (Shed = 0), two baseline strategies and three representative optimized solutions are compared. As shown in Figure 9, Baseline-Smooth achieves the smoothest purchased-electricity profile but incurs a pronounced carbon penalty, whereas Opt-Compromise provides a more balanced performance in the compromise region of the Pareto front.

Compared with Baseline-Price, Opt-Compromise reduces carbon emissions by approximately 2.57% (490.26 → 477.66 t), increases operating cost by approximately 2.76% (281,454 → 289,208 CNY), and decreases peak purchased electricity by approximately 10.5% (10.3 → 9.216 MW), indicating a more balanced trade-off among economy, carbon reduction, and grid-side security.

5.4.2. Operating Characteristics of the Baseline-Smooth Strategy

The Baseline-Smooth strategy represents an extreme grid-smoothing operating mode. Under this strategy, peak purchased electricity is close to 0.00 MW because the objective strongly suppresses grid exchange. However, maintaining internal energy balance requires sustained gas-turbine output, resulting in the highest daily carbon emissions among all compared strategies. This result indicates that grid-side smoothing alone may reduce external electricity dependence but can sacrifice economic and environmental performance. By contrast, the proposed multi-objective framework provides a more balanced use of grid interaction and local energy conversion.

5.4.3. 24-H Multi-Energy Balance Under the Compromise Solution

As shown in Figure 10, the compromise solution satisfies the electricity, gas, and heat demands throughout the 24 h horizon under the rigid-load assumption. The electricity supply is jointly provided by grid purchase, renewable generation, gas-turbine output, and battery discharge, while battery charging and electric cooling consume part of the available power. The gas demand is mainly allocated to the gas turbine and gas boiler, with the external base gas load acting as a background component. The heat demand is primarily met by recovered waste heat and gas-boiler output. Overall, the compromise solution exhibits a balanced multi-energy dispatch pattern and provides a time-domain interpretation of the posterior network-margin variation discussed in Section 5.5.

5.5. Three-Network Posterior Verification Results and Bottleneck Analysis

The representative solutions are further examined by calibrated posterior verification over the electricity, gas, and heating networks. Because the scheduling layer does not explicitly embed the physical network constraints, the initial solutions may violate the benchmark-network limits in some periods. Closed-loop repair is then applied to restore implementable solutions whenever possible.

5.5.1. Overall Calibrated Posterior-Verification Results

Under the calibrated posterior-verification setting with $S_{ver}$ = 0.18, the three test scenarios all trigger power-network violations in the initial verification, whereas the gas and heating networks remain non-binding. The initial critical-hour minimum voltage is 0.946210 p.u. for all three scenarios, and the corresponding initial cumulative voltage violation is 0.003790. After the closed-loop repair with the baseline shrinking factor ${\eta }_{rep}$ = 0.90, the nominal and moderate scenarios are restored to full feasibility within six iterations, while the severe scenario still retains a residual voltage violation. These results indicate that the distribution network is the dominant vulnerable layer in the present case, whereas the gas and heating networks mainly provide auxiliary physical screening. The calibrated posterior-verification and repair outcomes under $S_{ver}$ = 0.18 are summarized in

Table 4. Calibrated posterior-verification and repair outcomes under $S_{ver}$ = 0.18.

Scenario	Initial All-Pass	Dominant Violated Network	Initial Minimum Voltage (p.u.)	Initial Cumulative Voltage Violation	Violation Duration (h)	Repair All-Pass $({\mathit{\eta }}_{\mathit{r}\mathit{e}\mathit{p}} = 0.90$)	Repair Iterations	Final Minimum Voltage (p.u.)	Final Cumulative Voltage Violation
Nominal	FAIL	Power	0.946210	0.003790	1	PASS	6	0.951428	0.000000
Moderate	FAIL	Power	0.946210	0.003790	1	PASS	6	0.950401	0.000000
Severe	FAIL	Power	0.946210	0.003790	1	FAIL	6	0.947306	0.002694

.

5.5.2. Analysis of Distribution-Network Verification Results

The distribution network is the dominant bottleneck under the calibrated baseline setting. As shown in Figure 11, the lower-voltage boundary is the main source of infeasibility, whereas the upper-voltage boundary remains non-binding. This indicates that the physical risk introduced by the compromise scheduling solution is mainly reflected in the tightening of the low-voltage margin of the benchmark feeder.

Under $S_{ver}$ = 0.18 and ${\eta }_{rep}$ = 0.90, the nominal scenario is repaired from an initial minimum voltage of 0.946210 p.u. to a final minimum voltage of 0.951428 p.u., with the cumulative voltage violation reduced from 0.003790 to zero. The moderate scenario is also restored to full feasibility, with the final minimum voltage reaching 0.950401 p.u. By contrast, the severe scenario remains infeasible after six repair iterations, with a final minimum voltage of 0.947306 p.u. and a residual cumulative voltage violation of 0.002694.

These results indicate that the proposed repair strategy is effective for mild-to-moderate voltage violations under the calibrated baseline setting, while a clear recovery boundary still exists under more stressed conditions. Therefore, the severe scenario is interpreted as a stress-testing case rather than as the baseline operating condition.

5.5.3. Analysis of Gas- and Heating-Network Verification Results

Compared with the distribution network, the gas and heating networks do not trigger posterior infeasibility in the present case. For the gas network, the pressure margins remain positive under all tested scenarios, although the minimum upper-pressure margin is relatively small, reaching 2.539398 kPa. This indicates that the gas network is feasible but operates relatively closer to its upper-pressure boundary.

The heating network also remains non-binding. The minimum supply-side temperature margin is 14.230307 °C, and the minimum return-side temperature margin is 9.997410 °C for all three scenarios. Both margins remain positive with negligible scenario-to-scenario variation, indicating that the thermal side retains sufficient physical redundancy throughout the day.

Taken together, the posterior-verification results show a clear hierarchy among the three physical layers: the distribution network is the dominant bottleneck, whereas the gas and heating networks mainly provide complementary physical screening. Therefore, the closed-loop repair process is primarily triggered by power-network voltage violations. The detailed margin profiles of the gas and heating networks are provided in the Supplementary Materials.

5.6. Sensitivity Analysis of Calibration and Repair Parameters

To examine the influence of posterior-verification calibration and repair-parameter selection, this section compares different values of the network-scale coefficient $S_{ver}$ and the shrinking factor ${\eta }_{rep}$. The main-text baseline setting is $S_{ver}$ = 0.18 and ${\eta }_{rep}$ = 0.90, while $S_{ver}$ = 0.16 and 0.17, as well as ${\eta }_{rep}$ = 0.85 and 0.95, are used only for sensitivity analysis.

5.6.1. Network-Scale Sensitivity

As shown in Figure 12, increasing $S_{\mathrm{v}\mathrm{e}\mathrm{r}}$ strengthens the mapped loading level of the benchmark distribution network. The critical-hour minimum voltage decreases from 0.952475 p.u. at $S_{ver}$ = 0.16 to 0.949352 p.u. at $S_{ver}$ = 0.17, and further to 0.946210 p.u. at $S_{ver}$ = 0.18. Meanwhile, the initial cumulative voltage violation increases from 0 to 0.000648 and then to 0.003790.

At $S_{ver}$ = 0.16, all scenarios pass posterior verification directly, indicating a relatively mild verification condition. At $S_{ver}$ = 0.17, the voltage violation is detectable but can be repaired within two iterations. At $S_{ver}$ = 0.18, the nominal and moderate scenarios remain repairable, whereas the severe scenario cannot be fully recovered within the default repair budget. Therefore, $S_{ver}$ = 0.18 is retained as the baseline setting because it reveals the voltage bottleneck while preserving repairability under mild-to-moderate stress. The red line denotes the lower-voltage safety boundary used for posterior verification.

5.6.2. Shrinking-Factor Sensitivity Under the Default Repair Limit

With $S_{ver}$ fixed at 0.18, the influence of ${\eta }_{rep}$ is compared in

Table 5. Shrinking-factor sensitivity under the default repair limit ($S_{ver}=0.18$).

${\mathit{\eta }}_{\mathit{r}\mathit{e}\mathit{p}}$	Scenario	Initial Cumulative Voltage Violation	Final Cumulative Voltage Violation	Repair Iterations	Final All-Pass
0.95	Nominal	0.003790	0.002313	6	FAIL
0.90	Nominal	0.003790	0.000000	6	PASS
0.85	Nominal	0.003790	0.000000	4	PASS
0.95	Moderate	0.003790	0.002313	6	FAIL
0.90	Moderate	0.003790	0.000000	6	PASS
0.85	Moderate	0.003790	0.000000	4	PASS
0.95	Severe	0.003790	0.002313	6	FAIL
0.90	Severe	0.003790	0.002694	6	FAIL
0.85	Severe	0.003790	0.002694	6	FAIL

. A smaller ${\eta }_{rep}$ gives a stronger boundary contraction and faster repair, while a larger ${\eta }_{rep}$ results in a milder but slower update. Specifically, ${\eta }_{rep}$ = 0.85 restores the nominal and moderate scenarios within four iterations, whereas ${\eta }_{rep}$ = 0.95 cannot eliminate the voltage violation within the default six iterations. The baseline value ${\eta }_{rep}$ = 0.90 provides an intermediate behavior and restores the nominal and moderate scenarios within $K_{rep}^{\max }$ = 6.

The severe scenario remains infeasible within six iterations for all tested shrinking factors, indicating that the repair outcome also depends on the severity of the mapped network stress. Under more stressed conditions, additional operational flexibility, a larger repair-iteration limit, or a modified repair policy may be required.

It should be emphasized that the severe scenario is not used as the baseline operating condition, but as a stress-testing case for identifying the applicability boundary of the default repair setting. Under this condition, simply adjusting ${\eta }_{rep}$ within the tested range cannot fully restore feasibility within $K_{rep}^{\max }$ = 6. For more stressed operating conditions, alternative repair strategies may be required, such as increasing the repair-iteration budget, applying stronger multi-boundary contraction, coordinating storage dispatch and renewable curtailment, introducing demand-side flexibility, or embedding selected voltage-support constraints into the scheduling model. These strategies are beyond the default repair setting adopted in this study and will be further investigated in future work.

5.6.3. Extended Convergence Behavior

To illustrate the convergence trend, the moderate scenario under $S_{ver}$ = 0.18 is extended beyond the default repair limit for visualization only. As shown in Figure 13, the cumulative voltage violation decreases monotonically for all three shrinking factors. The first feasible iteration occurs at k = 4 for ${\eta }_{rep}$ = 0.85, at k = 6 for ${\eta }_{rep}$ = 0.90, and at k = 11 for ${\eta }_{rep}$ = 0.95. The corresponding minimum voltages at the feasible iteration are 0.950401, 0.950401, and 0.950746 p.u., respectively.

These results show that the closed-loop repair process exhibits a clear empirical convergence trend under the calibrated moderate scenario, while the shrinking factor mainly affects the repair speed. This extended test is used only to visualize the convergence tendency; the formal maximum repair iteration number in the manuscript remains $K_{rep}^{\max }$ = 6.

Although the case study is based on a typical operating day of one industrial park, the proposed scheduling–verification–repair framework is modular and can be extended to other RIES configurations by updating device parameters, load profiles, price or emission settings, and physical network models. The sensitivity analyses under different network-scale coefficients, shrinking factors, and stress scenarios further show that the method can reflect changes in operating conditions and identify the applicability boundary of the default repair setting. Nevertheless, the quantitative results remain case-dependent, and further validation under multi-season, multi-day, and uncertain operating conditions will be considered in future work.

6. Conclusions

This paper proposes an integrated scheduling–verification–repair framework for regional integrated energy systems. A bi-objective MILP model is first established to characterize the trade-off between operating cost and carbon emissions, and an adaptive ε-constraint strategy is adopted to improve the local resolution of the compromise region. The case results show that the refined scanning strategy improves the resolution of the compromise region by 3.2 times with only a 20.4% increase in computational time, thereby improving the reliability of compromise-solution identification.

The posterior verification results show that scheduling-layer feasibility based on energy-balance constraints does not necessarily guarantee physical-network feasibility. Under the calibrated baseline setting, the dominant infeasibility is associated with the lower-voltage boundary of the distribution network, while the gas and heating networks remain feasible and mainly provide complementary physical screening. The closed-loop repair mechanism further improves the physical implementability of the representative scheduling solution. Under $S_{ver}$ = 0.18 and ${\eta }_{rep}$ = 0.90, the nominal and moderate scenarios are restored to feasibility within six iterations, whereas the severe scenario still retains a residual voltage violation after the default repair budget is exhausted. This indicates the applicability boundary of the current repair setting under more stressed operating conditions.

Sensitivity analysis shows that $S_{ver}$ affects the triggering intensity of posterior verification, while ${\eta }_{rep}$ influences the repair speed and boundary-contraction intensity. The extended convergence results further show a monotonic reduction in cumulative voltage violation under the calibrated moderate scenario, supporting the empirical stability of the proposed repair procedure. Although the present validation is based on a typical-day case study, the modular scheduling–verification–repair framework can be extended to other RIES configurations by updating device parameters, operating scenarios, and network models. Future work will further validate the method under multi-season, multi-day, and uncertain operating conditions, and develop advanced repair strategies, such as multi-boundary repair, demand-side flexibility, and selected network-constrained re-optimization, to better balance feasibility restoration and scheduling optimality.