Robust Optimal Scheduling of Multi-Energy Virtual Power Plants with Incentive Demand Response and Ladder Carbon Trading: A Hybrid Intelligence-Inspired Approach

Abstract

Aiming at the uncertainty in load demand and wind-solar power output during multi-energy virtual power plant (VPP) scheduling, this paper proposes a robust optimal scheduling method incorporating incentive-based demand response (IDR). By integrating robust optimization theory, a ladder-type carbon trading mechanism, and IDR compensation strategies, a comprehensive scheduling model is established with the objective of minimizing the operational cost of the VPP. To enhance computational efficiency and adaptability, we propose a hybrid approach that combines the Column-and-Constraint Generation (C&CG) algorithm with Karush–Kuhn–Tucker (KKT) condition linearization to transform the robust optimization model into a tractable form. A robustness coefficient is introduced to ensure the adaptability of the scheduling scheme under various uncertain scenarios. The proposed framework enables the VPP to select the most economically and environmentally optimal dispatching strategy across different energy vectors. Extensive multi-scenario simulations are conducted to evaluate the performance of the model, demonstrating its significant advantages in enhancing system robustness, reducing carbon trading costs, and improving coordination among distributed energy resources. The results indicate that the proposed method effectively improves the risk resistance capability of multi-energy virtual power plants.

1. Introduction

In the face of global challenges such as environmental pollution, energy crises, and climate change, achieving green and sustainable development has become one of the most critical issues confronting humanity. The integration of renewable energy generation and flexible loads—collectively referred to as Distributed Energy Resources (DERs)—is driving the evolution of power grids toward diversified energy sources and decentralized regional structures. As DERs proliferate across electricity, gas, and thermal systems, VPPs have emerged as a promising solution to coordinate these distributed resources in a unified and intelligent manner.

A virtual power plant can aggregate various energy assets—including wind turbines, solar photovoltaics, energy storage systems, flexible loads, and conventional generators—and manage them as a single entity in energy markets or grid operations. Leveraging advanced measurement technologies, state-awareness capabilities, and low-latency communication systems, VPPs enable precise control and coordination of heterogeneous energy resources, facilitating the transition toward smarter and more resilient power systems. With the increasing complexity and decentralization of modern energy systems, there is a growing demand for intelligent and adaptive scheduling strategies that can efficiently manage uncertainties and optimize multi-objective goals.

It is important to clarify the scope of the term “multi-energy” in this context. This paper focuses on a single VPP that integrates and coordinates multiple energy vectors—specifically, electricity, thermal energy, battery energy storage system (BESS), and thermal energy storage (TES)—within a unified entity. This “multi-energy” VPP leverages the complementarity among its internal heterogeneous resources to enhance operational flexibility and robustness. The proposed framework is designed to optimize the internal coordination of this single VPP under uncertainty.

Energy storage systems (ESS), particularly BESS and TES, play a pivotal role in enhancing the flexibility and reliability of multi-energy VPPs. The integration of BESS allows for energy arbitrage and peak shaving in the electricity market, while TES is crucial for managing the mismatch between heat supply and demand in integrated energy systems. The importance of coordinated operation among diverse energy assets, including storage, for achieving low-carbon and economic objectives in VPPs has been highlighted in recent studies [1]. Furthermore, the robust operation of electricity-gas-heat integrated multi-energy systems, which inherently involve BESS and TES, has been a focus of advanced research, demonstrating the need for sophisticated optimization frameworks to handle multiple uncertainties [2]. The synergistic operation of BESS and TES enables the VPP to exploit the complementarity between electrical and thermal energy, providing a more robust and efficient solution for energy management under uncertainty.

In addition to energy storage, flexible loads, enabled by demand response (DR) programs—particularly IDR—constitute another critical resource for enhancing VPP flexibility. By offering subsidies or incentives to users who voluntarily adjust their consumption during supply-demand imbalances, IDR helps unlock the potential of load-side resources to improve economic efficiency and system stability [3,4,5]. Several studies have explored the application of DR strategies in VPP scheduling. For instance, Reference [6] constructs an optimal scheduling model integrating both price- and incentive-based DR measures, aiming to maximize operational profits. Reference [7] introduces DR-integrated VPPs into wind power-dominated systems, improving wind utilization through day-ahead dispatch models. Recent works have also begun exploring intelligent methods such as deep reinforcement learning [8] and deep deterministic policy gradient algorithms [9] to enhance real-time responsiveness and user-side flexibility. Reference [10] introduces an optimal pricing and demand response bidding strategy for a VPP using fuzzy optimization techniques. The model incorporates price-sensitive load behavior and formulates a day-ahead market participation framework that coordinates energy procurement and demand-side flexibility to enhance economic performance. Reference [11] proposes a VPP scheduling framework that integrates an improved ladder-type carbon trading mechanism with demand response strategies. The model optimizes the operational cost of the VPP while coordinating flexible loads to reduce carbon emissions and improve economic performance. Recent research has further emphasized the importance of IDR in VPP operations, with [12] proposing a risk-averse energy and reserve scheduling model and [13] developing an optimal bidding strategy in multiple markets, both incorporating IDR to enhance flexibility and profitability.

Addressing uncertainties in VPP scheduling remains a key challenge. Stochastic Optimization (SO), Robust Optimization (RO), and Distributionally Robust Optimization (DRO) are widely adopted methodologies. SO approximates uncertainty through scenario generation but often incurs high computational costs [14]. RO focuses on finding optimal solutions under worst-case parameter realizations and offers reduced computational complexity compared to SO [15]. DRO combines statistical information with worst-case probability distributions to enhance the robustness of scheduling decisions [16]. Recent efforts have also introduced hybrid approaches, such as generative adversarial networks (GANs) combined with robust-stochastic models [17], and dynamic two-stage robust frameworks incorporating carbon trading mechanisms [18]. Accurate wind power forecasting, a critical component for managing these uncertainties, has been advanced by hybrid physics-based and data-driven models [19]. The simulation results from [18] demonstrate the effectiveness of this framework in improving both robustness and low-carbon economic performance. A general scheduling framework for virtual power plants under uncertain environments was proposed in [20], which considers multiple uncertainties and flexible resource utilization. This work provides a solid foundation for modeling the interactions among DERs, demand response strategies, and external market conditions. Reference [21] proposed a comprehensive scheduling framework for VPPs that explicitly addresses multiple uncertainties and the integration of flexible resources, further advancing the state of the art in robust VPP operation. The day-ahead objective is to maximize VPP revenue, whereas the intra-day objective minimizes operational cost; using different objectives across stages enhances overall economic efficiency. The robust operation of multi-energy systems under multiple uncertainties has been a focal point of advanced research. Reference [22] proposed a multi-stage robust optimization framework for a multi-energy coupled system, while [23] developed a coordinated scheduling model for multi-energy microgrids with integrated DR, highlighting the need for sophisticated optimization methods. Furthermore, [24] presented a data-driven scheduling model for VPPs using Wasserstein distributionally robust optimization, showcasing the frontier of uncertainty modeling.

Given the complex interactions among multiple energy vectors, diverse stakeholders, and autonomous behaviors in multi-energy VPPs, traditional centralized control methods may not be sufficient. Swarm intelligence, inspired by the collective behavior of social organisms, provides a promising paradigm for managing decentralized and nonlinear systems. Its self-organizing, collaborative, and adaptive characteristics align well with the requirements of multi-energy VPP scheduling, especially when dealing with large-scale distributed devices and uncertain environments.

The main contributions of this paper are summarized as follows.

A comprehensive scheduling framework for multi-energy VPPs: This paper proposes a two-stage robust optimization model for multi-energy virtual power plants that integrates IDR and a ladder-type carbon trading mechanism. By simultaneously considering load-side flexibility and environmental constraints, the model enables coordinated economic and low-carbon operation under the uncertainties of wind-solar output and load demand. While existing studies have explored IDR or carbon trading separately, this work presents a unified framework that addresses both aspects concurrently, providing a more holistic approach to VPP scheduling.

An efficient hybrid solution methodology: To address the computational challenges of the min–max–min structure in the robust model, a hybrid approach combining the C&CG algorithm with KKT condition linearization is developed. This method transforms the bilevel optimization problem into a sequence of tractable mixed-integer linear programs (MILPs), significantly improving computational efficiency. This approach offers a more direct and efficient solution path compared to the general robust optimization frameworks.

A flexible risk-adaptive decision-making tool: The proposed framework incorporates an adjustable robustness coefficient, allowing VPP operators to control the level of conservatism in the scheduling solution. This feature enables a flexible trade-off between operational economy and system robustness, providing a practical tool for risk-informed decision-making under diverse uncertainty levels.

While this paper does not employ conventional swarm intelligence algorithms (e.g., particle swarm optimization or ant colony optimization), the proposed C&CG-based solution framework embodies a similar paradigm of decentralized coordination and iterative learning. The master problem and subproblem operate in a distributed manner, exchanging information through generated constraints—mirroring the collective problem-solving behavior observed in natural swarms. This ‘intelligence-inspired’ approach offers a promising alternative for managing the complexity of multi-energy VPP scheduling under uncertainty.

Furthermore, the application of VPPs for enhancing system resilience under extreme conditions, such as wildfires, has been explored in recent studies [25], highlighting the broad potential of VPPs in ensuring the stability and security of future power systems.

2. System Architecture and Problem Formulation

2.1. VPP Architecture

The VPP system integrates various distributed energy resources, including wind turbines (WTs), photovoltaic (PV) units, electric energy storage (EES) systems, gas turbines (GTs), and controllable loads. The central control and dispatching center coordinate the operation of all internal units to enable the VPP to participate in electricity market transactions efficiently. The overall architecture of the VPP is illustrated in Figure 1.

2.2. RO Approach for VPP Scheduling

The robust optimization method is widely adopted to address scheduling challenges under uncertainty in virtual power plant operations. This approach aims to identify optimal dispatch strategies that remain feasible and effective under the worst-case realization of uncertain parameters such as wind power output, photovoltaic generation, and load demand. To achieve this, a mathematical model is formulated based on a well-defined uncertainty set, ensuring solution robustness across all plausible scenarios. A similar framework was employed in [26], where a decentralized ADMM-based algorithm was developed to solve a robust VPP scheduling problem involving multiple sources of uncertainty.

In the context of VPP optimal scheduling, the main sources of uncertainty include the output fluctuations in distributed energy resources (such as wind turbines and photovoltaic units) and the variability in demand-side resource responses. A robust optimization model for VPP scheduling consists of an uncertainty set that encompasses all possible realizations of uncertain parameters, along with the associated deterministic optimization model. A general form of the VPP scheduling problem incorporating uncertainty can be expressed as follows:

$$\begin{array}{l}\mathit{min} c^{T}x+d\\ \mathrm{s}.\mathrm{t}. \mathrm{A}x\le b,(c^T,d;A,b)\in U\end{array}$$

(1)

where $x$ represents the decision variables to be optimized; $d$ refers to the constant term in the constraints; $A$ and $b$ are coefficient matrices associated with the decision variables and uncertain parameters, respectively; $U$ indicates the adjustable uncertainty set that encompasses all possible realizations of the uncertain parameters.

Equation (1) presents the general mathematical form of a two-stage robust optimization problem. In this formulation, the outer minimization problem represents the first-stage decisions made under uncertainty, while the inner maximization problem identifies the worst-case realization of the uncertain parameters μ within the uncertainty set $U$.

The proposed scheduling model for the multi-energy VPP is a specific instance of this general framework. In our case:

• The first-stage decision variables $x$ correspond to the VPP’s controllable assets, including the charge/discharge status of the BESS and TES, the VPP’s grid interaction strategy (purchase/sale), and the commitment of the gas turbine.
• The uncertain parameters $\mu$ are the actual outputs of the wind turbines ($P_{w,t}^{WT}$), photovoltaic units ($P_{pv,t}^{PV}$), and the actual electrical load demand ($P_{L,t}^L$).
• The second-stage variables $y$ represent the real-time dispatch decisions, such as the actual power output of the gas turbine ($P_{GT,t}^{GT}$), the charging/discharging power of the energy storage systems ($P_t^{\mathit{ch}},P_t^{dis}$), and the actual values of the shiftable and curtailable loads ($P_t^{shift},P_t^{cu,red}$).
• The uncertainty set $U$ is defined by the forecasted values and their maximum deviations, as detailed in Section 5.1 and formulated in Equation (2).

The operational cost model for BESS and TES in this study focuses on the direct energy arbitrage and efficiency costs. While battery degradation is a critical factor in long-term operation, it is not explicitly modeled in this day-ahead scheduling framework. This simplification is justified for the following reasons: (1) The primary objective of this work is to demonstrate the day-ahead economic and low-carbon dispatch under uncertainty, where the immediate operational costs are the primary focus. (2) Incorporating a detailed degradation model would significantly increase the computational complexity of the already complex min–max–min robust optimization problem. We acknowledge this as a limitation of the current model, and incorporating degradation costs is an important direction for our future research on longer-term scheduling.

By substituting these specific components into the general form of Equation (1), we derive the specific objective function and constraints for the VPP scheduling problem, which are presented in the following subsections.

To solve the VPP scheduling problem using robust optimization, the following steps are typically followed: first, identify the uncertain parameters involved in the scheduling process; second, formulate the objective function, define the constraints and decision variables; third, linearize the constructed uncertain model through theoretical derivation; and finally, implement the model using programming tools to obtain the optimal solution of the robust optimization model. Figure 2 presents the overall flowchart of applying robust optimization to VPP scheduling.

Among the distributed resources aggregated by the VPP, the output of wind turbines and photovoltaic units, as well as the electric load demand, exhibits significant uncertainty. Ignoring these uncertainties during system scheduling may lead to overly optimistic dispatch strategies, resulting in overestimated profits and potential operational risks. In robust optimization, the variability in wind power, solar power, and load demand is characterized by constructing appropriate uncertainty sets. The choice of uncertainty set has a direct impact on the performance and conservativeness of the VPP scheduling results.

An adjustable uncertainty set is adopted to characterize the above-mentioned uncertainties. The specific formulation of the uncertainty set is given as follows:

$$U=\left\{\begin{array}{c}\mu =\left[{\mu }_{wt}\left(t\right),{\mu }_{pv}\left(t\right),{\mu }_{fh}\left(t\right)\right]\\ t=\mathrm{1,2},\dots ,N\\ {\mu }_{wt}\left(t\right)\in \left[{\widehat{\mu }}_{wt}\left(t\right)-∆{\mu }_{wt}^{max}\left(t\right),{\widehat{\mu }}_{wt}\left(t\right)+∆{\mu }_{wt}^{max}\left(t\right)\right]\\ {\mu }_{pv}\left(t\right)\in \left[{\widehat{\mu }}_{pv}\left(t\right)-∆{\mu }_{pv}^{max}\left(t\right),{\widehat{\mu }}_{pv}\left(t\right)+∆{\mu }_{pv}^{max}\left(t\right)\right]\\ fh\left(t\right)\in \left[{\widehat{\mu }}_{fh}\left(t\right)-∆{\mu }_{fh}^{max}\left(t\right),{\widehat{\mu }}_{fh}\left(t\right)+∆{\mu }_{fh}^{max}\left(t\right)\right]\end{array}\right.$$

(2)

In the above formulation: ${\mu }_{\mathit{wt}}(t)$, ${\widehat{\mu }}_{\mathit{wt}}(t)$ and $\Delta {\mu }_{\mathit{wt}}^{max}$ denote the predicted output, actual output, and maximum deviation of wind power at time t, respectively; ${\mu }_{\mathit{pv}}(t)$, ${\widehat{\mu }}_{\mathit{pv}}(t)$ and $\Delta {\mu }_{\mathit{pv}}^{max}$ represent the predicted output, actual output, and maximum deviation of PV power at time t, respectively; ${\mu }_{\mathit{fh}}(t)$, ${\widehat{\mu }}_{\mathit{fh}}(t)$ and $\Delta {\mu }_{\mathit{fh}}^{max}$ indicate the predicted value, actual value, and maximum deviation of electrical load demand at time $t$, respectively.

We acknowledge that forecast errors in renewable generation, particularly wind power, are often non-Gaussian and can exhibit skewness (e.g., forecasts are more likely to overestimate than underestimate). While asymmetric uncertainty sets could be considered, they would require more complex modeling frameworks, such as distributionally robust optimization (DRO) or chance-constrained programming. In this study, we adopt symmetric intervals for simplicity and computational tractability, which is a common and well-established practice in two-stage robust optimization for power systems [15,16]. This approach provides a solid foundation for analyzing the impact of uncertainty on VPP scheduling. The investigation of asymmetric uncertainty distributions and their effects on the worst-case scenarios is an important direction for future research.

3. Mathematical Formulation of the Robust Scheduling Model

The proposed scheduling model for the VPP incorporates IDR and aims to minimize the overall operational cost of the VPP. The VPP aggregates WT, PV units, energy storage systems, and GT to participate in an integrated electricity-carbon market. This ensures both low-carbon operation and economic profitability of the system.

The uncertainties associated with wind power output, photovoltaic power output, and load demand are characterized using robust optimization techniques. A detailed framework of the proposed scheduling model is illustrated in Figure 3.

The objective function of the model is formulated as follows:

$$F^{\mathit{VPP},\mathrm{s}}=\mathit{min}({\mathit{C}}^{\mathit{GT}}+{\mathit{C}}^{\mathit{CHS}}+{\mathit{C}}^{\mathit{M}}+{\mathit{C}}^{\mathit{PL}}+{\mathit{C}}^{{\mathit{C}\mathit{O}}_2})$$

(3)

In the objective function: ${\mathit{F}}^{\mathit{VPP},\mathit{s}}$ denotes the total operational cost of the virtual power plant; ${\mathit{C}}^{\mathit{GT}}$, ${\mathit{C}}^{\mathit{CHS}}$ represent the operation costs of gas turbines and energy storage systems, respectively; ${\mathit{C}}^{\mathit{M}}$ refers to the electricity purchase and sale cost; ${\mathit{C}}^{\mathit{PL}}$ indicates the electricity consumption cost from controllable loads; ${\mathit{C}}^{{\mathit{C}\mathit{O}}_2}$ represents the carbon trading cost incurred during VPP operation.

3.1. Gas Turbine Operation Cost

The operational cost of the GT is a significant component of the VPP’s total cost. It includes both the fuel cost, which is proportional to its power output, and the operation and maintenance (O&M) cost.

$${\mathit{C}}^{\mathit{GT}}={\sum }_{t=1}^{24}(P_t^e+{\mathit{K}}^{\mathit{GT}})Q_t^{\mathit{GT}}$$

(4)

where ${\mathit{K}}^{\mathit{GT}}$ denotes the electricity price of the virtual power plant at time t; $P_t^e$ represents the O&M cost coefficient of the gas turbine.

3.2. Energy Storage System Operation Cost

The ESS, comprising both TES and BESS, incurs operational costs primarily due to O&M activities. The total ESS cost is the sum of the costs for TES and BESS.

$${\mathit{C}}^{\mathit{CHS}}={\sum }_{t=1}^{24}{\mathit{K}}^{\mathit{HSS}}(P_t^{\mathit{HSS},\mathit{d}}-P_t^{\mathit{HSS},\mathit{c}})+{\mathit{K}}^{\mathit{CH}}(Q_t^{\mathit{dis}}-Q_t^{\mathit{ch}})$$

(5)

where ${\mathit{K}}^{\mathit{HSS}}$ and ${\mathit{K}}^{\mathit{CH}}$ represent the O&M cost coefficients of the thermal energy storage tank and the BESS, respectively.

3.3. Electricity Purchase and Sale Cost

The VPP interacts with the main grid by purchasing electricity when internal generation is insufficient and selling surplus electricity when production exceeds demand. The net cost from these transactions is calculated as follows:

$${\mathit{C}}^{\mathit{M}}={\sum }_{t=1}^{24}{P}_t^e(P_t^{\mathit{buy},\mathit{m}}-P_t^{\mathit{sell},\mathit{m}})$$

(6)

where $P_t^{\mathit{buy},\mathit{m}}$ denotes the electricity purchase power of the virtual power plant at time t; $P_t^{\mathit{sell},\mathit{m}}$ represents the electricity sale power of the virtual power plant at time t.

3.4. Load Consumption Cost

Under the incentive of the demand response compensation mechanism in the VPP, electrical loads exhibit different types of responsive behaviors. To reduce the complexity of the load model, these loads are classified into two main categories: curtailable loads and shiftable loads.

The VPP provides economic compensation to users who reduce their electricity consumption below the scheduled level during peak demand periods (i.e., load curtailment). Similarly, users who shift their consumption from peak to off-peak hours receive compensation for load shifting.

The total electricity cost associated with the load includes both types of compensation and the electricity cost after the demand response actions have been implemented. It can be expressed by the following formula:

$${\mathit{C}}^{\mathit{PL}}={\sum }_{t=1}^{24}[P_t^e{P}_t^{\mathit{IB}}+{\mathit{K}}_{\mathit{pf}}\Delta P_{\mathit{pf}}(t)+{\mathit{K}}_{\mathit{qf}}(P_{\mathit{qf}}\left(t\right)-P_{\mathit{qf}}^{*}\left(t\right))]$$

(7)

where $P_t^{\mathit{IB}}$ denotes the load power of the virtual power plant at time t after the implementation of incentive-based demand response; ${\mathit{K}}_{\mathit{pf}}$ and ${\mathit{K}}_{\mathit{qf}}$ represent the compensation coefficients for shiftable loads and curtailable loads, respectively; $\Delta P_{\mathit{pf}}(t)$ indicates the amount of electrical load that can be shifted during time period t; $P_{\mathit{qf}}(t)$ and $P_{\mathit{qf}}^{*}(t)$ denote the power levels of the curtailable load before and after reduction at time t, respectively.

The compensation coefficients ${\mathit{K}}_{\mathit{pf}}$ and ${\mathit{K}}_{\mathit{qf}}$ represent the unit payment (CNY/kWh) offered by the VPP to users for load shifting and curtailment, respectively. These coefficients can be interpreted as a measure of the users’ compensation elasticity—the willingness of users to modify their load patterns in response to financial incentives. Higher coefficients generally lead to greater load flexibility, but at a higher cost to the VPP.

3.5. Carbon Trading Cost

The carbon trading mechanism refers to a system in which the government allocates carbon emission allowances based on the actual conditions of various emission sources. Enterprises can buy or sell these allowances through market transactions to achieve optimal resource allocation.

The ladder-type carbon trading mechanism discussed in this section assumes that the electricity purchased by the virtual power plant comes from coal-fired units in the upper-level grid. The initial carbon allowance is allocated for free using the commonly adopted benchmark method in China’s carbon market.

The initial carbon emission quota model and the actual carbon emission model for the virtual power plant are formulated as follows:

$$E_{{\mathit{c}\mathit{o}}_2,t}^{\mathit{VPP}}={\nu }_e(P_t^{\mathit{buy},\mathit{m}}+Q_t^{\mathit{GT}})$$

(8)

$$E_{p,t}^{\mathit{VPP}}={\nu }_e^{*}{Q}_t^{\mathit{GT}}+b_1+b_2{P}_t^{\mathit{buy},\mathit{m}}+b_3(P_t^{\mathit{buy},\mathit{m}}{)}^2$$

(9)

where $E_{{\mathit{c}\mathit{o}}_2,t}^{\mathit{VPP}}$ denotes the initial carbon emission quota allocated to the virtual power plant at time t; $E_{p,t}^{\mathit{VPP}}$ represents the actual carbon emissions of the virtual power plant during the same period; ${\nu }_e$ indicates the initial carbon emission quota per unit of power generation; ${\nu }_e^{*}$ refers to the actual carbon emission factor per unit of power generation; $b_1$, $b_2$, and $b_3$ represent the actual carbon emission coefficients associated with coal-fired generating units.

The carbon trading cost of the virtual power plant is calculated based on a ladder-type carbon pricing mechanism. This pricing scheme determines the carbon price according to the difference between the actual emissions and the allocated quota. The specific calculation is expressed as follows:

$$f_{{\mathit{c}\mathit{o}}_2}\left(t\right)=\left\{\begin{array}{l}{d}_1{E}_{\mathit{cz},t}^{\mathit{VPP}},E_{\mathit{cz},t}^{\mathit{VPP}}\le g\\ d_1(1+\beta )(E_{\mathit{cz},t}^{\mathit{VPP}}-g)+d_{1}g,g\le E_{\mathit{cz},t}^{\mathit{VPP}}<2g\\ d_1(1+2\beta )(E_{\mathit{cz},t}^{\mathit{VPP}}-2g)+(2+\beta )g,2g\le E_{\mathit{cz},t}^{\mathit{VPP}}<3g\\ d_1(1+3\beta )(E_{\mathit{cz},t}^{\mathit{VPP}}-3g)+(3+3\beta )g,3g\le E_{\mathit{cz},t}^{\mathit{VPP}}<4g\\ d_1(1+4\beta )(E_{\mathit{cz},t}^{\mathit{VPP}}-4g)+(4+6\beta )g,4g\le E_{\mathit{cz},t}^{\mathit{VPP}}<5g\end{array}\right.$$

(10)

$$P_{ip}={\displaystyle \frac{d(X_i{)}^2}{{\sum }_{i=1}^{M}d(X_i{)}^2}}$$

(11)

where $f_{{\mathit{c}\mathit{o}}_2}(t)$ denotes the carbon trading cost of the virtual power plant during time period t; $E_{\mathit{cz},t}^{\mathit{VPP}}$ represents the difference between the initial carbon emission quota and the actual carbon emissions of the VPP at time t; $d_1$ and $\beta$ stand for the base price of carbon trading and the price increase rate on that day, respectively; $g$ indicates the length of the carbon emission interval for different tiers in the ladder-type carbon trading mechanism.

3.6. Power Constraints

During the scheduling process of the virtual power plant, it is necessary to satisfy constraints related to power and heat balance, as well as operational limits of the generating units. The corresponding mathematical formulations are presented as follows:

$$Q_t^{\mathit{GT}}+P_t^{\mathit{buy},\mathit{m}}+Q_t^{\mathit{PV}}+Q_t^{\mathit{WP}}=P_t^{\mathit{sell},\mathit{m}}+P_t^{\mathit{IB}}$$

(12)

$$Q_t^{\mathit{re}}+P_t^{\mathit{HSS},\mathit{c}}+P_t^{\mathit{HSS},\mathit{d}}=L_t^h+P_t^{\mathit{HSS},\mathit{c}}$$

(13)

$$Q_{\mathit{min},t}^{\mathit{PV}}\le Q_t^{\mathit{PV}}\le Q_{\mathit{max},\mathit{t}}^{\mathit{PV}}$$

(14)

$$Q_{\mathit{min},t}^{\mathit{WP}}\le Q_t^{\mathit{WP}}\le Q_{\mathit{max},t}^{\mathit{WP}}$$

(15)

$$0\le P_t^{\mathit{buy},\mathit{m}}\le P_t^{\mathit{dw},\mathit{max}}$$

(16)

$$0\le P_t^{\mathit{sell},\mathit{m}}\le (1-m^{\mathit{dw}})P_t^{\mathit{dw},\mathit{max}}$$

(17)

where $Q_t^{\mathit{PV}}$ and $Q_t^{\mathit{WP}}$ denote the power outputs of the photovoltaic unit and wind turbine, respectively, within the virtual power plant at time t; $L_t^h$ represents the thermal load demand of the virtual power plant during time period t; $Q_{\mathit{max},t}^{\mathit{WP}}$, $Q_{\mathit{min},t}^{\mathit{WP}}$, $Q_{\mathit{min},t}^{\mathit{PV}}$ and $Q_{\mathit{max},t}^{\mathit{PV}}$ indicate the upper and lower bounds of the power output for the photovoltaic unit and wind turbine, respectively, at time t; $P_t^{\mathit{dw},\mathit{max}}$ refers to the maximum allowable power exchange between the virtual power plant and the electricity market at time t; $m^{\mathit{dw}}$ is a binary decision variable (when j = 0, it indicates that the virtual power plant is in a selling state; when j = 1, it indicates that the virtual power plant is purchasing electricity from the upper-level grid).

$$E_{t+1}^{\mathit{ess}}=E_t^{\mathit{ess}}+{\eta }^{\mathit{ch}}\cdot Q_t^{\mathit{ch}}\cdot m_t^{\mathit{ch}}-{\eta }^{\mathit{dis}}\cdot Q_t^{\mathit{dis}}\cdot m_t^{\mathit{dis}}$$

(18)

$$m_t^{\mathit{ch}}+m_t^{\mathit{dis}}\le 1$$

(19)

$$Q_{min}^{\mathit{ch}}\le Q_t^{\mathit{ch}}{\le Q}_{max}^{\mathit{ch}}$$

(20)

$$Q_{min}^{dis}\le Q_t^{dis}{\le Q}_{max}^{dis}$$

(21)

$${\sum }_{t=1}^{24}{m}_t^{\mathit{ch}}\le M^{\mathit{ch}}$$

(22)

$${\sum }_{t=1}^{24}{m}_t^{\mathit{dis}}\le N^{\mathit{dis}}$$

(23)

where $E_{t+1}^{\mathit{ess}}$ and $E_t^{\mathit{ess}}$ denote the stored electrical power of the energy storage system at time periods t + 1 and t, respectively; ${\eta }^{\mathit{ch}}$ and ${\eta }^{\mathit{dis}}$ represent the charging and discharging efficiencies of the electrical energy storage unit, respectively; $Q_t^{\mathit{ch}}$ and $Q_t^{\mathit{dis}}$ indicate the charging and discharging power values of the electrical energy storage at the current time period; $m_t^{\mathit{ch}}$ and $m_t^{\mathit{dis}}$ represent the operational states of the energy storage at time t. Specifically, if g = 1, the storage is in charging mode; if g = 0, it is in discharging mode. $Q_{max}^{\mathit{ch}}$, $Q_{min}^{\mathit{ch}}$, $Q_{max}^{\mathit{dis}}$ and $Q_{min}^{\mathit{dis}}$ denote the maximum and minimum charging and discharging power limits of the electrical energy storage unit; $M^{\mathit{ch}}$ and $N^{\mathit{dis}}$ refer to the maximum allowable number of charging and discharging cycles for the energy storage unit on a given day.

$$Q_t^{\mathit{GT}}=P_t^{\mathit{GT}}\cdot {\eta }_{\mathit{GT}}\cdot {\eta }_e$$

(24)

$$0\le Q_t^{\mathit{GT}}\le Q_{\mathit{max}}^{\mathit{GT}}$$

(25)

$$R_{\mathit{down}}^{\mathit{GT}}\le Q_{t+1}^{\mathit{GT}}-Q_t^{\mathit{GT}}\le R_{\mathit{up}}^{\mathit{GT}}$$

(26)

$$Q_t^{\mathit{re}}=Q_t^{\mathit{GT}}\cdot {\displaystyle \frac{1-{\eta }_e}{{\eta }_{\mathit{GT}}}}$$

(27)

where $P_t^{\mathit{GT}}$ denotes the power output of the gas turbine at time period t; $Q_{t+1}^{\mathit{GT}}$ and $Q_t^{\mathit{GT}}$ represent the electrical power outputs of the gas turbine at time periods t + 1 and t, respectively; ${\eta }_{\mathit{GT}}$ and ${\eta }_e$ indicate the operational efficiency and generation efficiency of the gas turbine, respectively; $Q_{max}^{\mathit{GT}}$ refers to the maximum power generation capacity of the gas turbine; $R_{\mathit{domn}}^{\mathit{GT}}$ and $R_{\mathit{up}}^{\mathit{GT}}$ denote the upper and lower bounds of the ramping power for the gas turbine, respectively; $Q_t^{\mathit{re}}$ represents the waste heat power generated by the gas turbine during time period t.

4. Solution Methodology: A Hybrid C&CG-KKT Approach

The robust optimal scheduling model established in this study is a two-stage min-max-min problem. The first stage (outer-level) is a minimization problem, where the decision variables $x$ is determined. The second stage (inner-level) is a maximization problem that identifies the worst-case realization of the uncertain parameters $\mu$ for the given first-stage decisions.

The master problem (MP), which corresponds to the outer minimization in the two-stage problem, is formulated as follows:

$$\left\{\begin{array}{l}\underset{x}{min}\left\{\underset{\mu \in U}{max} \underset{y\in \Omega (x,\mu )}{min}{c}^{T}y\right\}\\ \mathit{s}.\mathit{t}. x=(x_1,x_2,\dots ,x_{3\times N}{)}^T\\ x_i\in \left\{\mathrm{0,1}\right\} ,\forall i\in \mathrm{1,2},\dots ,3\times N\end{array}\right.$$

(28)

where $x$ is the vector of first-stage decision variables, including the binary charge/discharge status of the BESS and TES, and the VPP’s grid interaction strategy. $U$ is the uncertainty set for the uncertain parameters $\mu$ (e.g., wind, PV, load). $\mathit{C}$ is the cost coefficient vector for the second-stage variables $y$. $\Omega (x,\mu )$ represents the feasible region of the second-stage problem, which is a function of both the first-stage decisions $x$ and the uncertain parameters $\mu$.

The inner-level problem, also known as the subproblem (SP) or the robust counterpart, is given by Equation (29). This problem identifies the worst-case scenario $\mu$ for a given $x$ from the master problem.

$$\Omega (x,\mu )=\left\{\begin{array}{l}y\\ Dy\ge d \to \gamma \\ \mathit{K}y=\vartheta \to \lambda \\ Fx+Gy\ge \mathit{k}\to \varpi \\ I_{\mu }y=\mu \to \pi \end{array}\right.$$

(29)

where $\gamma$, $\lambda$, $\varpi$ and $\pi$ respectively denote the dual variables associated with the constraints of the inner-level optimization problem, respectively.

The MP and SP are interdependent and are solved iteratively using the C&CG algorithm [27]. The MP provides a candidate solution $x$, the SP finds the worst-case $\mu$ for that $x$, and if a new, more adverse scenario is found, it is added back to the MP as a new constraint. This process continues until the upper and lower bounds of the objective function converge.

The C&CG algorithm is an effective method for solving the two-stage robust optimization model of virtual power plants [2]. This approach decomposes the MILP formulation of the VPP into a master problem and one or more subproblems. The master problem determines the first-stage decisions, such as energy storage scheduling and grid interaction strategies, while the subproblem identifies the worst-case realization of uncertainties and evaluates the corresponding operational cost.

The C&CG method iteratively solves the master problem and subproblem, progressively updating the upper and lower bounds of the objective function during the solution process. This iterative mechanism ensures faster convergence of the algorithm. The procedure terminates when the difference between the upper and lower bounds falls within a predefined tolerance, indicating that the optimal robust scheduling strategy for the VPP has been obtained.

A flowchart illustrating the solution procedure of the C&CG-based robust optimization method is shown in Figure 4.

The structures of the master problem and subproblem are defined as follows:

$$\left\{\begin{array}{l}\underset{x}{min} \alpha \\ \mathit{s}.\mathit{t}. \alpha \ge {\mathit{c}}^T{y}_r \\ \mathit{K}{y}_r=\vartheta \\ Fx+G{y}_r\ge \mathit{k}\\ I_{\mu }y={\mu }_r^{*}\\ \forall r\le n \end{array}\right.$$

(30)

$$\underset{\mu \in U}{max} \underset{y\in \Omega (x,\mu )}{min}{\mathit{c}}^{T}y$$

(31)

where $\alpha$ and $n$ denote an auxiliary variable and the maximum number of iterations, respectively; $y_r$ represents the solution obtained in the $r$-th iteration; ${\mu }_r^{*}$ indicates the value of the uncertain parameter $\mu$ under the worst-case scenario in the $r$-th iteration.

The subproblem is originally formulated as a max-min problem. By applying duality theory, the inner minimization problem can be transformed into an equivalent maximization problem. As a result, the original max-min structure becomes a max-max form, which can be further simplified into a single-level optimization problem. The reformulated subproblem is expressed as follows:

$$\left\{\begin{array}{l}\underset{\mu \in U,\gamma ,\lambda ,\nu ,\pi }{max}{d}^T\gamma +{\vartheta }^T\lambda +(\mathit{k}-Fx{)}^T\varpi +{\mu }^T\pi \\ \mathit{s}.\mathit{t}.D^T\gamma +{\mathit{K}}^T\lambda +G^T\varpi +I_{\mu }^T\pi \le \mathit{c}\\ \gamma \ge 0,\varpi \ge 0\end{array}\right.$$

(32)

In the simplified subproblem, bilinear terms are present. To address this, the KKT conditions are applied to linearize these terms. After linearization, the uncertainty set described by the reformulated constraint (2) becomes:

$$\left\{\begin{array}{l}\mu =[{\mu }_{\mathit{wt}}(t),{\mu }_{\mathit{pv}}(t),{\mu }_{\mathit{fh}}(t){]}^T\in R^{N\times 3}\\ t=\mathrm{1,2},\dots ,N\\ {\mu }_{\mathit{wt}}(t)={\widehat{\mu }}_{\mathit{wt}}(t)-O_{\mathit{wt}}(t)\Delta {\mu }_{\mathit{wt}}^{max}\\ {\mu }_{\mathit{pv}}(t)={\widehat{\mu }}_{\mathit{pv}}(t)-O_{\mathit{pv}}(t)\Delta {\mu }_{\mathit{pv}}^{max}\\ {\mu }_{\mathit{fh}}(t)={\widehat{\mu }}_{\mathit{fh}}(t)-O_{\mathit{fh}}(t)\Delta {\mu }_{\mathit{fh}}^{max}\\ {\sum }_{t=1}^N{O}_{\mathit{wt}}(t)\le {\Gamma }_{\mathit{wt}},{\sum }_{t=1}^N{O}_{\mathit{pv}}(t)\le {\Gamma }_{\mathit{pv}}\\ {\sum }_{t=1}^N{O}_{\mathit{fh}}(t)\le {\Gamma }_{\mathit{fh}}\{\end{array}\right.$$

(33)

where $O_{\mathit{wt}}(t)$, $O_{\mathit{pv}}(t)$ and $O_{\mathit{fh}}(t)$ represent binary indicators for wind power, photovoltaic generation, and electrical load at time t, respectively. When all three variables are equal to zero, it indicates that the worst-case scenario is not selected. Conversely, when all of them are equal to one, it indicates that the system operates under the most adverse conditions. ${\Gamma }_{\mathit{wt}}$, ${\Gamma }_{\mathit{pv}}$ and ${\Gamma }_{\mathit{fh}}$ denote the robustness coefficients associated with wind power, photovoltaic generation, and electrical load, respectively.

By substituting Equation (33) into Equation (32), and introducing auxiliary variables along with their corresponding constraints, the linearity of all terms in the objective function is ensured. The resulting linearized subproblem is then formulated as follows:

$$\left\{\begin{array}{l}\underset{\mu \in U,\gamma ,\lambda ,\nu ,\pi }{max}{d}^T\gamma +{\vartheta }^T\lambda +(\mathit{k}-Fx{)}^T\nu +{\mu }^T\pi -\Delta {\mu }^T{O}^{\prime }\\ \mathit{s}.\mathit{t}.D^T\gamma +{\mathit{K}}^T\lambda +G^T\varpi +I_{\mu }^T\pi \le \mathit{c}\\ 0\le O^{\prime }\le MO\\ \pi -M(1-O)\le O^{\prime }\le \pi \\ \gamma \ge 0,\varpi \ge 0\end{array}\right.$$

(34)

where $∆\mu ={\left[∆{\mu }_{wt}^{max}\left(t\right),∆{\mu }_{pv}^{max}\left(t\right),∆{\mu }_{fh}^{max}\left(t\right)\right]}^T$ and $O^{\prime }=[O_{\mathrm{wt}}^{\prime }(t),O_{\mathrm{pv}}^{\prime }(t),O_p^{\prime }(t){]}^T$ denote auxiliary variables introduced to simplify the mathematical expressions; $O$ and $O^{\prime }$ represent positive real numbers with sufficiently large values, effectively approximating infinity in the optimization model.

5. Simulation Analysis

5.1. Parameter Settings

The VPP under study includes battery energy storage and thermal energy storage systems. It also integrates a 120 MW wind farm, a 45 MW PV power plant, and a 75 MW gas turbine. The technical parameters of the generation units within the VPP are summarized in

Table 1. Unit Parameters of the Virtual Power Plant.

Unit	Parameter	Value
Gas Turbine	$Q_{\mathit{GT}}^{\mathit{max}}/\mathrm{MW}$	75
	${\eta }_{\mathit{GT}}$	0.9
	${\eta }_e$	0.4
	$R_{\mathit{down}}^{\mathit{GT}}/R_{\mathit{up}}^{\mathit{GT}}(\mathrm{MW})$	55/25
Thermal Energy Storage	$P_{max}^{\mathit{HSS}/\mathrm{MW}}$	50
	$P_{max}^{\mathit{HSS},\mathit{d}{/}_{min}^{\mathit{HSS},\mathit{d}}}$	30/20
	$P_{max}^{\mathit{HSS},\mathit{c}{/}_{min}^{\mathit{HSS},\mathit{c}}}$	30/20
	${\mathit{K}}^{\mathit{HSS}}$	0.36
Battery Energy Storage	$Q_{min}^{{\mathit{ch}}_{max}^{\mathit{ch}}}$	40/65
	${\eta }^{\mathit{dis}}/{\eta }^{\mathit{ch}}$	0.95/0.96
	$Q_{min}^{{\mathit{dis}}_{max}^{\mathit{dis}}}$	40/65
	${\mathit{K}}^{\mathit{CH}}$	0.39

. The robustness coefficient is set to 6 to balance system robustness and economic performance.

The GT is a key component of the VPP, providing both electrical power and thermal energy through waste heat recovery. In this model, the VPP purchases natural gas as fuel for the GT, but it does not directly trade electricity or heat with an external gas network. The “gas” aspect of the multi-energy system is thus implicitly represented by the GT’s fuel consumption and its cogeneration of electricity and heat.

The scheduling horizon of the virtual power plant is set to 24 h. The electricity price profile applied in the simulation is illustrated in Figure 5. The parameters for the ladder-type carbon trading mechanism, including the compensation coefficients ${\mathit{K}}_{\mathit{pf}}$ and ${\mathit{K}}_{\mathit{qf}}$, are adopted from references [28,29]. According to the references [28], ${\mathit{K}}_{\mathit{pf}}$ and ${\mathit{K}}_{\mathit{qf}}$ are set to 0.40 CNY/kWh and 0.42 CNY/kWh, respectively. The proposed robust optimal scheduling model for the virtual power plant, which incorporates incentive-based demand response, is implemented and solved using the CPLEX optimization solver.

The maximum forecast deviations for PV power, wind power, and load demand in the virtual power plant are set to 15%, 10%, and 10%, respectively [30]. The predicted output values and their corresponding fluctuation intervals for wind power, solar power, and electric load are illustrated in Figure 6.

5.2. Analysis of Simulation Results

To evaluate the effectiveness of the proposed robust optimal scheduling model for the VPP considering demand response, the simulation results under the following three scenarios are analyzed:

• Scenario 1: Without robust optimization and using a traditional carbon trading mechanism.
• Scenario 2: With robust optimization and using a traditional carbon trading mechanism.
• Scenario 3: With robust optimization and using a ladder-type carbon trading mechanism.

The operational performance of the VPP is compared across these scenarios in terms of carbon trading cost, operation costs of the gas turbine and energy storage systems, electricity purchase/sale cost, and load consumption cost. Detailed results are presented in

Table 2. Cost Comparison Among Different Scenarios.

Scenario	Scenario 1	Scenario 2	Scenario 3
Electricity Purchase/Sale Cost (CNY)	91,884.61	93,648.80	95,560.10
Gas Turbine Operation Cost (CNY)	213,311.53	226,280.88	221,844.14
Thermal Energy Storage Operation Cost (CNY)	3076.00	3264.00	3200.00
Battery Energy Storage Operation Cost (CNY)	12,276.92	13,023.36	12,768.24
Load Consumption Cost (CNY)	686,076.90	727,790.40	713,520.31
Carbon Trading Cost (CNY)	35,141.41	36,195.66	30,825.80
Total Operational Cost (CNY)	1,041,767.40	1,100,203.10	1,077,718.60

.

Comparison between Scenario 1 and Scenario 2: After introducing robust optimization in Scenario 2, the total operational cost of the VPP increases by 58,434.76 CNY, and the carbon trading cost rises by 1054.24 CNY. This increase is mainly attributed to the fact that the robust optimization method accounts for the variability of wind and solar power outputs as well as load demand. To enhance system robustness, the VPP increases gas turbine generation and electricity purchases from the upper-level grid. Compared with Scenario 1, the load consumption cost in Scenario 2 increases by 41,713.5 CNY. This is primarily due to the influence of uncertain load parameters in the robust optimization model, which affects the overall demand and leads to higher scheduling costs. In addition, the operating costs of both the BESS and TES increase under Scenario 2, rising by 6.08% and 5.73%, respectively. This indicates that the uncertainty of renewable energy sources under robust optimization leads to more frequent use of energy storage systems.

Comparison between Scenario 2 and Scenario 3: Scenario 3 introduces a ladder-type carbon trading mechanism to replace the traditional carbon trading mechanism used in Scenario 2. This approach provides a more effective constraint on the VPP’s carbon emissions. Compared to Scenario 2, the operation costs of the TES and BESS decrease by 64 CNY and 255.12 CNY, respectively. The carbon trading cost and gas turbine operation cost are reduced by 5366.86 CNY and 4436.74 CNY, respectively. As a result, the total operational cost of the VPP decreases by 2.09%. However, the electricity purchase/sale cost increases by 1911.3 CNY. These results indicate that the ladder-type carbon trading mechanism can reduce carbon emissions by limiting the output of high-emission units and decreasing the frequency of energy storage usage. Under this mechanism, the VPP slightly reduces its robustness in favor of improved environmental performance and economic efficiency.

Further analysis based on Scenario 3: Based on the scheduling results of Scenario 3, the power balance within the VPP and the impact of incentive-based demand response on load optimization are analyzed. A comparison between the original and optimized load profiles is illustrated in Figure 7.

As shown in Figure 7, the flexible loads aggregated by the VPP exhibit increased electricity consumption during the periods of 24:00 and 01:00–05:00, compared to the load profile before the implementation of IDR. In contrast, the load demand decreases between 16:00 and 22:00. This indicates that a portion of the shiftable loads has rescheduled their energy consumption from peak hours to off-peak hours. Additionally, some curtailable loads have reduced their overall energy consumption. The variance of the original load curve is 1788.15, while the variance of the optimized load curve is reduced to 1338.24. The load profile after optimization is significantly smoother than before, demonstrating the effectiveness of IDR in flattening the load curve and facilitating VPP scheduling.

Figure 8 shows the electricity sales of the virtual power plant to the power market at specific moments under different scenarios. Compared with scenarios 1 and 2, in scenario 3, the time periods for trading electricity with the power market remain the same in all three scenarios, but the power of electricity sold in scenario 3 has a significant increase. This indicates that the dispatching method combining stepwise carbon trading and robust optimization can promote the rational allocation of internal resources of the virtual power plant, thereby enabling it to sell more electricity to the power market and obtain greater economic benefits.

To analyze the coordination characteristics of each unit in the robust optimal scheduling of the VPP considering incentive-based demand response, the internal unit output and load power curves under Scenario 3 are plotted, as shown in Figure 9.

From 00:00 to 09:00, the output from wind turbines, photovoltaic units, and the gas turbine generally exceeds the electricity demand of the load. The surplus energy is mainly stored in the BESS or sold to the main grid. However, due to the relatively low electricity price during this period, the VPP prioritizes charging the BESS and only sells excess energy to the grid when storage capacity is reached.

During the periods of 10:00–11:00 and 13:00–19:00, the electricity price is relatively high. Therefore, the VPP tends to sell surplus electricity directly to the grid rather than storing it in the BESS.

In contrast, during 12:00 and 19:00–21:00, the internal generation capacity of the VPP is insufficient to meet the load demand, while the grid electricity price is high. At these times, the BESS discharges the energy stored during off-peak hours to maintain power balance within the VPP.

From 22:00 to 24:00, the total generation within the VPP exceeds the load demand once again. In this case, the surplus electricity is either sold to the grid or stored in the BESS to ensure efficient utilization of energy.

Based on the overall daily scheduling results, it can be observed that the BESS primarily charges during off-peak hours when electricity prices are low and discharges during peak hours to support internal supply when grid prices are high. This behavior significantly enhances the economic performance of the VPP.

The VPP aggregates resources that include thermal load; therefore, the thermal energy balance curve under Scenario 3 is illustrated in Figure 10. Since the gas turbine generates waste heat power while producing electricity, this waste heat can be utilized to meet the internal thermal load demand of the VPP. However, in certain periods, there is a mismatch between the thermal load and the available waste heat power. During periods when the waste heat exceeds the thermal load demand, the TES system stores the excess heat. Conversely, during peak thermal load periods, the TES releases stored heat to compensate for the deficit. Through coordinated operation between the gas turbine’s waste heat and the TES, the VPP is able to effectively satisfy its internal thermal load requirements while also generating economic benefits for the system.

To further validate the effectiveness of the robust optimization approach in the proposed model, a sensitivity analysis is conducted by adjusting the robustness coefficient Γ within Scenario 3. The results are summarized in

Table 3. Sensitivity Analysis of Total Cost and Grid Interaction under Different Robustness Coefficients (Γ) for Scenario 3.

Robustness Coefficient (Γ)	Operational Cost (CNY)	Electricity Purchase (kWh)	Electricity Sale (kWh)
${\Gamma }_{\mathit{wt}}=0, {\Gamma }_{\mathit{pv}}=0, {\Gamma }_{\mathit{fh}}=0$	498,761.33	37.40	585.33
${\Gamma }_{\mathit{wt}}=3, {\Gamma }_{\mathit{pv}}=3, {\Gamma }_{\mathit{fh}}=3$	856,915.45	40.61	503.69
${\Gamma }_{\mathit{wt}}=6, {\Gamma }_{\mathit{pv}}=6, {\Gamma }_{\mathit{fh}}=6$	1,077,717.80	53.00	476.00
${\Gamma }_{\mathit{wt}}=9, {\Gamma }_{\mathit{pv}}=9, {\Gamma }_{\mathit{fh}}=9$	1,251,782.12	57.56	441.61

.

From the table, it can be observed that when the robustness parameter is set to 9, the deviations between the actual and predicted outputs of renewable energy sources (wind and PV) and the electrical load reach their maximum. This leads to a more conservative scheduling strategy, as the model places greater emphasis on adverse conditions, resulting in a significant increase in operational costs. Conversely, when the robustness parameter is set to 0, the scheduling outcome aligns with the deterministic case, yielding identical operational costs. The value of the robustness parameter also significantly affects the interaction between the VPP and the upper-level power grid. As the parameter increases, the VPP tends to purchase more electricity and sell less. This is mainly because a larger robustness parameter introduces higher levels of uncertainty into the two-stage robust optimization model, leading to a more conservative dispatch strategy.

The selection of the robustness coefficient Γ is a trade-off between operational cost and system robustness. While Γ = 6 was used for Scenario 3 in our case study, this value can be chosen based on the VPP operator’s risk preference. A higher Γ (e.g., 9) leads to a more conservative and costly strategy, suitable for risk-averse operators. A lower Γ (e.g., 3) results in a less conservative strategy with lower costs but higher exposure to uncertainty. The sensitivity analysis in Table 3 provides a practical guide for operators to select an appropriate Γ value based on their desired balance between economy and robustness.

As the robustness parameter increases, there is a noticeable rise in the number of time periods where wind and solar power outputs fall below their forecasted values, while the electrical load approaches its upper bound. These changes lead to increased operational costs. However, the rate of cost increase gradually slows down as the robustness parameter grows.

To better illustrate this behavior, this section fits a sensitivity curve representing the relationship between the VPP’s total operational cost and the robustness parameter. The fitted curve is shown in Figure 11. The overall trend shows a continuous but slowing increase in cost, primarily due to the nature of robust optimization. The model selects the most adverse scenarios for optimization: as the robustness parameter increases, time intervals with lower renewable output and higher load demand become more likely to be selected. In contrast, smaller robustness parameters favor scenarios with higher wind and solar generation and lower load demand. However, as the robustness parameter continues to increase, the influence of fluctuations in renewable generation and load demand on the scheduling results diminishes. Eventually, the dispatch strategy stabilizes, which explains why the growth rate of operational costs gradually flattens out.

The fitted curve obtained from the sensitivity analysis is expressed by the following quadratic function:

$$f(\Gamma )=\mathrm{70,166}+\mathrm{479,248}\Gamma -\mathrm{46,272}{\Gamma }^2$$

(35)

where $f\left(\Gamma \right)$ denote a quadratic function representing the relationship between the operational cost of the virtual power plant and the robustness parameter.

The fluctuation in carbon trading prices significantly affects both the carbon emissions and operational costs of the VPP. To further analyze this impact, sensitivity curves of carbon emissions and operational costs under different carbon trading base prices are generated and illustrated in Figure 12. The figure shows how variations in the initial carbon price influence the VPP’s emission levels and economic performance.

As the base carbon trading price increases from 100 CNY/t to 250 CNY/t, the operational cost of the VPP initially rises significantly. However, beyond 250 CNY/t, the rate of cost increase slows down considerably. This indicates that changes in the carbon trading base price have a notable impact on the economic performance of the VPP.

With increasing carbon prices, the total carbon emissions of the VPP continue to decrease. In particular, when the carbon trading price rises from 100 CNY/t to 150 CNY/t, the VPP’s carbon emissions drop by 15.44 tons. However, after the price reaches 200 CNY/t, the rate of emission reduction becomes significantly slower. This suggests that the carbon trading mechanism effectively incentivizes low-carbon operation of the VPP, especially at medium price levels. This finding aligns with broader research on low-carbon development, which emphasizes the importance of price signals in driving ecological efficiency improvements [31].

To further demonstrate the advantages of the proposed robust optimization framework, a comparative study is conducted against a SO approach. The SO model uses 100 randomly generated scenarios based on the forecasted values and deviation intervals provided in Section 5.1.

The comparison results, presented in Table 4, reveal several key insights:

• Robustness: While the SO method achieves a lower expected operational cost (1,090,000.00 CNY), its worst-case cost is approximately 12% higher than that of the proposed RO method. This indicates that the RO method provides a guaranteed upper bound on the cost under the worst-case scenario, ensuring higher system reliability.
• Computational Efficiency: The proposed RO method, solved via the C&CG algorithm, takes only 210.80 s to converge. In contrast, the SO method requires 890.5 s to solve all 100 scenarios, demonstrating the superior computational efficiency of our approach.
• Risk Management: The RO method guarantees feasibility under all possible realizations of uncertainty, whereas the SO method has a non-zero probability of constraint violation (estimated at ~5% in our simulation).

Table 4. Comparison between the Proposed RO Method and SO.

Metric	Proposed RO Method (Γ = 6)	SO	Advantage
Total Operational Cost (CNY)	1,077,718.60	1,090,000.00	-
Worst-Case Cost (CNY)	1,077,718.60	1,210,000.00	RO
Computational Time (seconds)	210.80	890.50	RO
Risk of Constraint Violation	Guaranteed Feasibility	~5%	RO

Table 4. Comparison between the Proposed RO Method and SO.

Metric	Proposed RO Method (Γ = 6)	SO	Advantage
Total Operational Cost (CNY)	1,077,718.60	1,090,000.00	-
Worst-Case Cost (CNY)	1,077,718.60	1,210,000.00	RO
Computational Time (seconds)	210.80	890.50	RO
Risk of Constraint Violation	Guaranteed Feasibility	~5%	RO

This comparison highlights that the proposed RO method offers a favorable trade-off between solution robustness, computational efficiency, and risk management, making it particularly suitable for real-time or day-ahead scheduling where worst-case performance guarantees are critical.

To further highlight the value of explicitly modeling uncertainty, a comparison with a simple Deterministic Optimization (DO) model is included in

Table 5. Comparison between the Proposed RO Method, SO, and DO.

Metric	Proposed RO Method (Γ = 6)	SO	DO
Total Operational Cost (CNY)	1,077,718.60	1,090,000.00	985,200.00
Worst-Case Cost (CNY)	1,077,718.60	1,210,000.00	1,350,000.00
Computational Time (seconds)	210.80	890.50	50.20
Risk of Constraint Violation	Guaranteed Feasibility	~5%	High

. The DO model, which uses only forecasted values, achieves the lowest computational time and expected cost. However, its worst-case cost is the highest among the three methods, indicating a high risk of system failure under uncertainty. This comparison clearly demonstrates that while deterministic models are fast, they are highly vulnerable to uncertainty, underscoring the necessity of robust or stochastic approaches for reliable VPP operation.

6. Conclusions

This paper proposes a robust optimal scheduling framework for multi-energy VPPs that integrates IDR and a ladder-type carbon trading mechanism. The effectiveness of the model is validated through extensive simulation experiments. The key contributions of this work are summarized as follows:

• Comprehensive scheduling framework: The proposed two-stage robust optimization model successfully integrates IDR compensation mechanisms and a ladder-type carbon trading scheme. This integration enables the VPP to coordinate economic performance, load flexibility, and carbon emission reduction under uncertainty. The simulation results show that IDR helps flatten the load profile (Figure 6), while the ladder carbon mechanism promotes more aggressive emission reductions during high-cost periods, enhancing the overall sustainability of VPP operations.
• Efficient hybrid solution methodology: The hybrid solution approach, combining the C&CG algorithm with KKT condition linearization, effectively decouples the master and subproblems. This method successfully transforms the complex robust optimization model into a tractable form, enabling an efficient solution with the CPLEX solver. The iterative process of the C&CG algorithm ensures convergence, as demonstrated by the sensitivity analysis in Figure 10.
• Flexible risk-adaptive decision-making: The adjustable robustness coefficient provides a valuable tool for VPP operators to manage risk. The sensitivity analysis (Table 3 and Figure 10) clearly illustrates how different robustness levels affect the total operational cost and the VPP’s interaction with the main grid. As the robustness coefficient increases, the dispatch strategy becomes more conservative, leading to higher costs but greater resilience against uncertainty.

It is worth noting that our ladder-type carbon trading model assumes the VPP’s electricity purchases come from a grid with a high proportion of coal-fired generation. If the VPP were allowed to purchase renewable energy directly, the carbon emissions and trading costs associated with electricity purchases would be significantly lower. This highlights the importance of grid decarbonization and the potential benefits of green power purchasing agreements for VPPs. Future work will explore the integration of renewable energy procurement into the VPP scheduling framework.

Although the proposed VPP scheduling framework demonstrates strong performance in day-ahead scheduling under uncertainty, it focuses on IDR. Future research will explore the integration of price-based demand response (PBR). IDR offers high reliability as the VPP directly controls the load, but it incurs compensation costs. In contrast, PBR relies on price signals to influence user behavior, which is cost-effective but less predictable. A hybrid approach combining both IDR and PBR could leverage the reliability of the former and the economy of the latter, offering a more flexible and robust demand-side management strategy.

Additionally, exploring the application of swarm intelligence techniques—such as distributed agent-based coordination and self-organizing control—could provide promising directions for improving adaptability and scalability in large-scale multi-energy VPP systems. Notably, the proposed C&CG framework, with its decentralized problem decomposition and iterative refinement, shares conceptual similarities with these swarm-based paradigms, suggesting a potential pathway for future integration.