Flexibility Resource Planning and Stability Optimization Methods for Power Systems with High Penetration of Renewable Energy

Abstract

With the accelerating global transition toward sustainable energy systems, power grids with a high share of renewable energy face increasing challenges due to volatility and uncertainty, necessitating advanced flexibility resource planning and stability optimization strategies. This paper presents a comprehensive distribution network planning framework that coordinates and integrates multiple types of flexibility resources through joint optimization and network reconfiguration to enhance system adaptability and operational resilience. A novel virtual network coupling modeling approach is proposed to address topological constraints during network reconfiguration, ensuring radial operation while allowing rapid topology adjustments to isolate faults and restore power supply. Furthermore, to mitigate the uncertainty and fault risks associated with extreme weather events, a CVaR-based risk quantification framework is incorporated into a bi-level optimization model, effectively balancing investment costs and operational risks under uncertainty. In this model, the upper-level planning stage optimizes the siting and sizing of flexibility resources, while the lower-level operational stage coordinates real-time dispatch strategies through demand response, energy storage operation, and dynamic network reconfiguration. Finally, a hybrid SA-PSO algorithm combined with conic programming is employed to enhance computational efficiency while ensuring high solution quality for practical system scales. Case study analyses demonstrate that, compared to single-resource configurations, the proposed coordinated planning of multiple flexibility resources can significantly reduce the total system cost and markedly improve system resilience under fault conditions.

1. Introduction

As the global energy crisis and environmental pollution become increasingly severe [1], the pursuit of sustainable energy development models has become more pressing worldwide [2]. With the rapid advancement of smart grid and renewable energy technologies, distributed energy systems are becoming more prevalent [3]. A growing number of consumers are installing distributed generation units, such as PV and wind power systems, which enhance local energy self-sufficiency and enable participation in electricity markets [4,5]. However, coordinating energy production and consumption within a community remains challenging in a volatile environment. Flexibility resource planning has thus emerged as a proactive solution to optimize energy utilization while improving energy efficiency and environmental performance [6].

Furthermore, the Chinese government has made notable progress in advancing sustainable energy development, particularly through the adoption of the “carbon peaking and carbon neutrality” targets [7], aimed at establishing a green and low-carbon society. These targets have driven the development of flexibility resources within distribution networks, making source–grid–load–storage coordination a prominent operational paradigm [8,9]. Under this paradigm, the power system maintains a balance between local generation and consumption through the coordinated deployment of flexibility resources, thereby reducing operational costs and enhancing resource utilization efficiency [10]. This approach not only improves system flexibility but also facilitates renewable energy integration and strengthens grid resilience [11]. Reference [12] demonstrates that coordinated planning of diverse flexibility resources can enhance system operational efficiency and grid resilience, while Reference [13] proposes a three-stage management strategy incorporating network reconfiguration to coordinate various resources under grid constraints for optimal operation.

Ensuring rational and efficient flexibility resource planning in distribution networks is essential. Traditional network reconfiguration modeling methods are not always effective at ensuring accurate topology decisions, often facing operational complexity and low computational efficiency. Although Big-M-based methods [14] improve modeling accuracy through linearization, their high computational complexity can burden resource-constrained distribution networks [15]. The virtual network coupling mechanism provides an alternative by constructing a virtual topology to simplify the enforcement of radiality constraints. Reference [16] introduces a virtual network-based configuration strategy that reduces computational complexity through topological coupling, promotes fairness, and maximizes system benefits. By integrating individual incentives with efficient modeling [17], virtual network coupling enables effective network reconfiguration operations.

Flexibility resource planning in distribution networks still encounters significant challenges, especially beyond fundamental resource allocation mechanisms. Existing research often focuses on planning a single type of resource, neglecting the complexities introduced by integrating large shares of renewable energy and ESSs [18]. These factors can lead to increased system volatility, negatively impacting grid stability and introducing economic risks. Reference [19] proposes a CVaR-based multi-objective distributed scheduling model that accounts for distribution network constraints and uncertainty. Additionally, Reference [20] develops a bi-level energy management framework to coordinate energy transactions among multiple consumers, using CVaR to estimate potential losses. However, relying on a single risk measure may be inadequate for balancing risk control for extreme events with the operational flexibility of the system [21].

Bi-level optimization models are widely used in distribution network planning as they effectively address the coupling between planning and operation. The upper-level planning decisions focus on long-term investment returns, while the lower-level operational decisions target short-term operational optimization; these levels interact and constrain each other [22]. However, existing bi-level optimization models often suffer from low computational efficiency when tackling large-scale problems, especially when complex constraints such as network reconfiguration are incorporated [23]. Intelligent optimization algorithms, including PSO and SA, have shown strong performance in solving complex optimization tasks [24]. Bio-inspired optimization algorithms, such as ant lion optimization, have demonstrated effectiveness in power system applications, particularly for load frequency control in multi-area interconnected systems [25]. Nonetheless, individual algorithms typically face limitations such as slow convergence or susceptibility to local optima [26]. Therefore, developing efficient hybrid optimization algorithms is crucial for addressing large-scale distribution network flexibility resource planning problems.

Furthermore, existing research underscores the need to enhance flexibility resource planning in distribution networks, particularly regarding system resilience, risk management, and multi-resource coordination. This study proposes a coordinated planning approach for diverse flexibility resources in distribution networks, explicitly considering network reconfiguration and fault risks. A CVaR mechanism integrated with a virtual network coupling modeling method is employed to quantify and manage risks. Additionally, a bi-level optimization framework is developed to minimize both investment and operational costs as well as risk exposure, and a hybrid SA-PSO algorithm is applied for distributed computation, ensuring high computational efficiency and robust optimization performance.

To further evaluate the proposed method,

Table 1. Comparison of Related Studies on Flexibility Resource Planning.

Reference	Resource Types	Risk Model	Network Reconfiguration	Optimization	Solution Method
[11]	ESS, DR	No	No	Coordinated Planning	MIP
[12]	ESS, DR, SOP	No	Yes	Three-stage dispatch under constraints	Heuristic
[13]	ESS	No	Big-M linearization	Deterministic Optimization	MILP
[15]	ESS	No	Virtual Network	Graph-based Topology Optimization	MILP
[16]	ESS, DR	No	Yes	Incentive-integrated Modeling	Game-based
[17]	ESS	No	No	Basic Planning	MIP
[18]	ESS	CVaR	No	Multi-objective Scheduling	Stochastic Programming
[19]	DR	CVaR	No	Bi-level Energy Management	Mixed Integer
[20]	ESS	VaR	No	Risk-constrained Optimization	MILP
[21]	ESS, DR	No	No	Classical Bi-level	MILP
[22]	ESS	No	Yes	Bi-level (with topology)	MILP
This paper	MT, ESS, SOP, DR	CVaR	Virtual Network Coupling	Bi-level (Planning + Operation)	SA–PSO + SOCP

presents a comparative analysis of existing studies in terms of resource types, risk modeling, network reconfiguration, optimization frameworks, and solution method.

The main contributions of this study are as follows:

• A virtual network coupling modeling method has been implemented to address topological constraints during network reconfiguration. In the planning stage, this method effectively resolves radiality constraints, facilitates truthful topology optimization, and improves operational efficiency. Moreover, accurate topology modeling contributes to the stable operation of the distribution network.
• A CVaR-based risk quantification framework has been established to manage uncertainties arising from extreme weather and fault risks. By integrating CVaR into the bi-level optimization model, an effective balance between investment costs and operational risks is achieved. Compared to traditional deterministic methods, this approach handles uncertainties more effectively, thereby enhancing system robustness and economic efficiency.
• A coordinated planning model for multiple flexibility resources has been developed to enable the integrated configuration of micro gas turbines, ESSs, intelligent soft switches, and other resources. Through a bi-level optimization framework that coordinates planning and operational decisions, and with the aid of a hybrid SA-PSO algorithm, the solution performance is significantly improved. Compared with single-resource configuration methods, this approach markedly enhances overall system benefits and operational resilience.

The remainder of this paper is structured as follows. Section 2 presents the system model and details the modeling methods for various types of flexibility resources. Section 3 outlines the distribution network flexibility resource planning strategy that accounts for network reconfiguration and fault risks. Section 4 describes the distributed algorithm used to solve the problem. Section 5 evaluates the proposed strategy, and Section 6 concludes the findings of this study.

2. Modeling Methods for Multi-Type Flexibility Resources

Flexibility resources are key means for power systems to address the variability of renewable energy and the uncertainty of loads. Through proper configuration and coordinated optimization, they can effectively improve system flexibility, reliability, and economic performance. This section introduces the flexibility resources present on the generation, network, load, and storage sides of the power system, analyzes their primary regulation mechanisms, and establishes the corresponding mathematical models.

2.1. Microturbine Model

A MT features fast start-up and shutdown capabilities, allowing it to swiftly respond to sudden changes in power demand within a short time frame. Its strong load-following capability enables quick adjustments in output power to meet grid requirements and accommodate load fluctuations. In the event of grid frequency deviations or sudden disturbances, the microturbine can react promptly, effectively maintaining grid stability.

The MT model is described by the followed equations:

$$\begin{array}{c}{y}_{i,t,s}^{\mathrm{MT}}-z_{i,t,s}^{\mathrm{MT}}=u_{i,t,s}^{\mathrm{MT}}-u_{i,t-1,s}^{\mathrm{MT}}(\forall t\in \{2,\dots ,T\})\\ y_{i,t,s}^{\mathrm{MT}}-z_{i,t,s}^{\mathrm{MT}}=u_{i,t,s}^{\mathrm{MT}}-I{S}_i^{\mathrm{MT}}(\forall t\in \{1\})\\ y_{i,t,s}^{\mathrm{MT}}+z_{i,t,s}^{\mathrm{MT}}\le 1\\ P_{i,t,s}^{\mathrm{MT}}\le P_i^{\mathrm{MT},\max }{u}_{i,t,s}^{\mathrm{MT}}\\ P_{i,t,s}^{\mathrm{MT}}\le P_i^{\mathrm{MT},\min }{u}_{i,t,s}^{\mathrm{MT}}\\ P_{i,t,s}^{\mathrm{MT}}\le (P_i^{IS}+R{U}_i)u_{i,t,s}^{\mathrm{MT}}(\forall t\in \{1\})\\ P_{i,t,s}^{\mathrm{MT}}\ge (P_i^{IS}-R{D}_i)u_{i,t,s}^{\mathrm{MT}}(\forall t\in \{1\})\\ P_{i,t,s}^{\mathrm{MT}}-P_{i,t-1,s}^{\mathrm{MT}}\le (2-u_{i,t-1,s}^{\mathrm{MT}}-u_{i,t,s}^{\mathrm{MT}})P_i^{\mathrm{SU},\max }+(1+u_{i,t-1,s}^{\mathrm{MT}}-u_{i,t,s}^{\mathrm{MT}})R{U}_i(\forall t\in \{2,\dots ,T\})\\ P_{i,t-1,s}^{\mathrm{MT}}-P_{i,t,s}^{\mathrm{MT}}\le (2-u_{i,t-1,s}^{\mathrm{MT}}-u_{i,t,s}^{\mathrm{MT}})P_i^{\mathrm{SD},\max }+(1-u_{i,t-1,s}^{\mathrm{MT}}+u_{i,t,s}^{\mathrm{MT}})R{D}_i(\forall t\in \{2,\dots ,T\})\end{array}$$

(1)

where i denotes the distribution network bus number to which the microturbine is connected; $u_{i,t,s}^{\mathrm{MT}}$ is a binary variable indicating whether the microturbine is online (1) or offline (0); $y_{i,t,s}^{\mathrm{MT}}$ and $z_{i,t,s}^{\mathrm{MT}}$ are binary variables representing the start-up and shutdown states, respectively; $I{S}_i^{\mathrm{MT}}$ indicates the initial state of the microturbine; $P_i^{IS}$ denotes its initial output power; $R{U}_i$ and $R{D}_i$ represent its upward and downward ramp rates; $P_i^{\mathrm{SU},\max }$ and $P_i^{\mathrm{SD},\max }$ denote the maximum upward and downward ramping power, respectively.

2.2. Soft Open Point Model

A SOP is a flexible distribution device based on power electronics, widely used in modern distribution networks. It enables flexible control of power flows, precise voltage regulation, and efficient integration of distributed energy resources [27]. By replacing traditional mechanical switches with power electronics, the SOP provides faster and more flexible control capabilities, thereby enhancing the reliability, economic performance, and flexibility of the distribution network. By coordinating multiple types of flexibility resources, the SOP allows flexible control of active power transmission between two feeders, supports flexible switching among various connection modes, enables different topological configurations of bus flexibility resources, and provides reactive power support. The SOP model is described by the following equations:

$$\begin{array}{c}{P}_{i,t,s}^{\mathrm{SOP}}+P_{j,t,s}^{\mathrm{SOP}}+P_{ij,t,s}^{\mathrm{soploss}}=0\\ P_{ij,t,s}^{\mathrm{soploss}}=A_i^{\mathrm{SOP}}\left|P_{i,t,s}^{\mathrm{SOP}}\right|+A_j^{\mathrm{SOP}}\left|P_{j,t,s}^{\mathrm{SOP}}\right|\\ -{\mu }_i{S}_{ij}^{\mathrm{SOP}}\le Q_{i,t,s}^{\mathrm{SOP}}\le {\mu }_i{S}_{ij}^{\mathrm{SOP}}\\ -{\mu }_j{S}_{ij}^{\mathrm{SOP}}\le Q_{j,t,s}^{\mathrm{SOP}}\le {\mu }_j{S}_{ij}^{\mathrm{SOP}}\\ {(P_{i,t,s}^{\mathrm{SOP}})}^2+{(Q_{i,t,s}^{\mathrm{SOP}})}^2\le {(S_{ij}^{\mathrm{SOP}})}^2\\ {(P_{j,t,s}^{\mathrm{SOP}})}^2+{(Q_{j,t,s}^{\mathrm{SOP}})}^2\le {(S_{ij}^{\mathrm{SOP}})}^2\end{array}$$

(2)

where i and j denote the distribution network bus numbers connected at both ends of the SOP; $P_{i,t,s}^{\mathrm{SOP}}$, $P_{j,t,s}^{\mathrm{SOP}}$ and $Q_{i,t,s}^{\mathrm{SOP}}$, $Q_{j,t,s}^{\mathrm{SOP}}$ represent the active and reactive power injected by the two SOP converters, respectively; $S_{ij}^{\mathrm{SOP}}$ and $P_{ij,t,s}^{\mathrm{soploss}}$ denote the converter capacity at both ends of the SOP and the active power transmission loss of the SOP, respectively; $A_i^{\mathrm{SOP}}$, $A_j^{\mathrm{SOP}}$ are the loss coefficients of the SOP converters; ${\mu }_i$, ${\mu }_j$ denote the absolute values of the sine of the power factor angles.

2.3. On-Load Tap Changer Model

An OLTC is a device capable of real-time voltage regulation while operating under load conditions. By adjusting the transformer turns ratio, it stabilizes the voltage level, ensuring that the voltage at the user side remains within an acceptable range [28]. Although OLTCs have the potential to regulate voltage dynamically, traditional devices are limited by their mechanical structure, which restricts their switching frequency. This may hinder their ability to respond effectively to fast renewable generation fluctuations. Therefore, OLTCs are typically more suited to slow voltage variations or are used in conjunction with other faster-acting devices for comprehensive voltage control. The OLTC model can be represented by the following equations:

$$\begin{array}{c}{V}_{i,t,s}^2-V_{m,t,s}^2=2(P_{ij,t,s}{r}_{ij}+Q_{ij,t,s}{x}_{ij})-(r_{ij}^2+x_{ij}^2)I_{ij,t,s}^2\\ V_{m,t,s}^2=k_{ij,t,s}^2{V}_{j,t,s}^2\\ k_{ij,t,s}=1+K_{ij,t,s}\mathsf{\Delta }{k}_{\mathrm{OLTC}}\hspace{1em}{K}_{ij,t,s}\in Z\\ -K_{ij}^{\max }\le K_{ij,t,s}\le K_{ij}^{\max }\hspace{1em}{K}_{ij}\in Z\end{array}$$

(3)

where $r_{ij}$ and $x_{ij}$ denote the resistance and reactance of branch ij; $P_{ij,t,s}$ and $Q_{ij,t,s}$ represent the active and reactive power flows through the branch; $V_{i,t,s}$ denotes the voltage at bus i; $I_{ij,t,s}$ denotes the current flowing through the branch; $k_{ij,t,s}$ denotes the tap ratio of the OLTC; $K_{ij,t,s}$ and $K_{ij}^{\max }$ represent the tap position and its maximum value; $\mathsf{\Delta }{k}_{\mathrm{OLTC}}$ is the unit step size for OLTC adjustment.

2.4. Capacitor Bank Model

A CB provides reactive power by switching capacitor units, thereby regulating voltage and power factor to stabilize grid operation. Its fast response capability makes it an effective flexibility resource that mitigates voltage fluctuations, reduces losses, and supports the integration of renewable energy sources. When coordinated with other flexibility resources, the capacitor bank optimizes grid operation, improving system stability and economic performance. The CB model is described by the following equations:

$$\begin{array}{c}{Q}_{i,t,s}^{\mathrm{CB}}=N_{i,t,s}^{\mathrm{CB}}\times Q^{\mathrm{CB}}\\ N_{i,t,s}^{\mathrm{CB}}\le N_i^{\mathrm{CB},\max }\end{array}$$

(4)

where i denotes the distribution network bus number where the CB is connected; $Q_{i,t,s}^{\mathrm{CB}}$ represents the reactive power compensation capacity of the capacitor bank; $Q^{\mathrm{CB}}$ denotes the reactive power compensation capacity of a single capacitor unit; $N_{i,t,s}^{\mathrm{CB}}$ indicates the number of capacitors switched in; $N_i^{\mathrm{CB},\max }$ represents the total number of capacitors available for switching.

2.5. Demand Response Model

DR is an important strategy by which users, while satisfying their own electricity requirements, actively adjust their consumption behavior in response to price signals or cooperate with power system dispatch under incentive policies [29]. This enhances system flexibility and ensures secure, stable, and economical grid operation. By flexibly regulating loads, demand response effectively addresses supply-demand fluctuations and the uncertainty of renewable energy output, playing a crucial role in promoting renewable energy integration and improving system operational efficiency.

2.5.1. Price-Based Demand Response

Price-based demand response, currently the predominant response mode, regulates loads by establishing a linkage between price signals and user consumption behavior [30]. Its core mechanism is that power companies implement differentiated pricing strategies based on the time-varying characteristics of the load, motivating users to economically optimize their electricity usage periods, thereby promoting peak-valley load shifting. In practice, differentiated pricing schemes such as time-of-use pricing (the main approach at present), real-time pricing, and critical peak pricing are commonly used. Among these, time-of-use pricing is the most widely adopted due to its simplicity in implementation. The price-based demand response is described by the following equations:

$$\begin{gathered} -{\xi }_i={\displaystyle \frac{\mathsf{\Delta }{P}_{i,t,s}}{\mathsf{\Delta }{\rho }_{i,t,s}}}={\displaystyle \frac{P_{i,t,s}^{\mathrm{cur}}-P_{i,t,s}^{\mathrm{load}}}{{\rho }_{i,t,s}^{\mathrm{cur}}-{\rho }_{i,t,s}}} \\ {\rho }_{i,t,s}^{\mathrm{cur}}=\left\{\begin{array}{cc}{\rho }^{\mathrm{peak}}& t\in T^{\mathrm{peak}}\\ {\rho }^{\mathrm{valley}}& t\in T^{\mathrm{valley}}\\ {\rho }^{\mathrm{flat}}& t\in T^{\mathrm{flat}}\end{array}\right. \\ {\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{cur}}=}{\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{load}}} \\ {\rho }_{i,t,s}^{\mathrm{cur},\min }\le {\rho }_{i,t,s}^{\mathrm{cur}}\le {\rho }_{i,t,s}^{\mathrm{cur},\max } \\ {\displaystyle \frac{{\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{cur}}{\rho }_{i,t,s}^{\mathrm{cur}}}}{{\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{cur}}}}}={\displaystyle \frac{{\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{load}}{\rho }_{i,t,s}}}{{\displaystyle \sum _{t=1}^T{P}_{i,t,s}^{\mathrm{load}}}}} \end{gathered}$$

(5)

where i denotes the bus number where demand response is applied; ${\xi }_i$ is the price elasticity coefficient at bus i; $\mathsf{\Delta }{P}_{i,t,s}$ represents the change in electricity demand before and after DR implementation; $\mathsf{\Delta }{\rho }_{i,t,s}$ denotes the change in electricity price before and after DR implementation; $P_{i,t,s}^{\mathrm{load}}$ and $P_{i,t,s}^{\mathrm{cur}}$ represent the load before and after DR implementation; ${\rho }_{i,t,s}$ and ${\rho }_{i,t,s}^{\mathrm{cur}}$ represent the electricity price before and after DR implementation; ${\rho }^{\mathrm{peak}}$, ${\rho }^{\mathrm{valley}}$, ${\rho }^{\mathrm{flat}}$ and $T^{\mathrm{peak}}$, $T^{\mathrm{valley}}$, $T^{\mathrm{flat}}$ denote the time-of-use electricity prices and their corresponding time period sets; ${\rho }_{i,t,s}^{\mathrm{cur},\min }$ and ${\rho }_{i,t,s}^{\mathrm{cur},\max }$ denote the lower and upper limits of the electricity price after DR implementation; T is the total number of divided time intervals.

2.5.2. Incentive-Based Demand Response

Incentive-based demand response achieves load pattern optimization through contractual economic incentives [31]. Its operational framework consists of three key stages: first, establishing rights and obligations via bilateral agreements; second, executing predetermined load control schemes under abnormal operating conditions; and third, implementing differentiated economic settlements based on performance, including positive incentives and penalties for non-compliance. Typical implementation modes include direct load control, interruptible load management, and demand-side bidding. This market-oriented response model significantly increases user participation through diversified incentives, providing essential support for the secure operation of the power system. The incentive-based demand response is described by the following equations:

$$\begin{array}{c}0\le y_{i,t,s}^{\mathrm{cut}}{P}_{i,t,s}^{\mathrm{cut}}\le P_i^{\mathrm{cut},\max }\hspace{1em}t\in [t^{\mathrm{cut},\mathrm{start}},t^{\mathrm{cut},\mathrm{end}}]\\ {\displaystyle \sum _{t=1}^{N_t}}{y}_{i,t,s}^{\mathrm{cut}}=t^{\mathrm{cut},\max }\hspace{1em}t\in [t^{\mathrm{cut},\mathrm{start}},t^{\mathrm{cut},\mathrm{end}}]\\ 0\le y_{i,t,s}^{\mathrm{train}}{P}_{i,t,s}^{\mathrm{train}}\le P_i^{\mathrm{train},\max }\hspace{1em}t\in [t^{\mathrm{train},\mathrm{start}},t^{\mathrm{train},\mathrm{end}}]\\ 0\le y_{i,t,s}^{\mathrm{traout}}{P}_{i,t,s}^{\mathrm{traout}}\le P_i^{\mathrm{traout},\max }\hspace{1em}t\in [t^{\mathrm{traout},\mathrm{start}},t^{\mathrm{traout},\mathrm{end}}]\\ {\displaystyle \sum _{t=1}^{N_t}}{y}_{i,t,s}^{\mathrm{train}}{P}_{i,t,s}^{\mathrm{train}}\mathsf{\Delta }t={\displaystyle \sum _{t=1}^{N_t}}{y}_{i,t,s}^{\mathrm{traout}}{P}_{i,t,s}^{\mathrm{traout}}\mathsf{\Delta }t\\ 0\le y_{i,t,s}^{\mathrm{train}}+y_{i,t,s}^{\mathrm{traout}}\le 1\end{array}$$

(6)

where i denotes the bus number where demand response is applied; $N_t$ denotes one demand response cycle period; $P_{i,t,s}^{\mathrm{cut}}$ and $P_i^{\mathrm{cut},\max }$ represent the reducible load amount and its maximum limit; $P_{i,t,s}^{\mathrm{train}}$, $P_{i,t,s}^{\mathrm{traout}}$, $P_i^{\mathrm{train},\max }$ and $P_i^{\mathrm{traout},\max }$ denote the amounts of shiftable load transferred in and out, and their respective maximum values; $y_{i,t,s}^{\mathrm{cut}}$ is a binary variable indicating whether the reducible load is reduced; $y_{i,t,s}^{\mathrm{train}}$ and $P_{i,t,s}^{\mathrm{traout}}$ are binary variables indicating whether the shiftable load is transferred in or out; $t^{\mathrm{cut},\mathrm{start}}$ and $t^{\mathrm{cut},\mathrm{end}}$ represent the start and end times of load reduction; $t^{\mathrm{train},\mathrm{start}}$, $t^{\mathrm{train},\mathrm{end}}$, $t^{\mathrm{traout},\mathrm{start}}$ and $t^{\mathrm{traout},\mathrm{end}}$ represent the start and end times for load shifting in and out, respectively; $t^{\mathrm{cut},\max }$ denotes the maximum duration of load reduction.

2.6. Energy Storage System Model

An ESS can perform peak shaving, valley filling, and load smoothing. It also enhances system operational stability, adjusts frequency, and compensates for load fluctuations [32]. In particular, the integration of storage technology and renewable energy can significantly increase the utilization efficiency of renewables. With the continuous rise in renewable energy penetration, the growing complexity of power systems, and the increasing demand for flexibility, the flexibility contribution of storage will become even more prominent, providing crucial support for developing an efficient, low-carbon, and intelligent modern power system. The ESS model is described by the following equations:

$$\begin{array}{c}0\le P_{i,t,s}^{\mathrm{ch}}\le {\beta }_{i,t,s}^{\mathrm{ch}}{P}_i^{\mathrm{ch},\max }\\ 0\le P_{i,t,s}^{\mathrm{dis}}\le {\beta }_{i,t,s}^{\mathrm{dis}}{P}_i^{\mathrm{dis},\max }\\ 0\le {\beta }_{i,t,s}^{\mathrm{ch}}+{\beta }_{i,t,s}^{\mathrm{dis}}\le 1\\ E_{i,t,s}^{\mathrm{ESS}}=E_{i,t-1,s}^{\mathrm{ESS}}+{\eta }_i^{\mathrm{ch}}{P}_{i,t,s}^{\mathrm{ch}}-{\displaystyle \frac{P_{i,t,s}^{\mathrm{dis}}}{{\eta }_i^{\mathrm{dis}}}}\\ E_i^{\mathrm{ESS},\min }\le E_{i,t,s}^{\mathrm{ESS}}\le E_i^{\mathrm{ESS},\max }\\ SO{C}_{i,t,s}^{\mathrm{ESS}}=E_{i,t,s}^{\mathrm{ESS}}/E_i^{\mathrm{ESS}}\\ SO{C}_i^{\mathrm{ESS},\min }\le SO{C}_{i,t,s}^{\mathrm{ESS}}\le SO{C}_i^{\mathrm{ESS},\max }\\ E_{i,0,s}^{\mathrm{ESS}}=E_i^{\mathrm{ESS}}SO{C}_{i,0,s}^{\mathrm{ESS}}\end{array}$$

(7)

where i denotes the distribution network bus where the ESS is connected; $P_{i,t,s}^{\mathrm{ch}}$ and $P_{i,t,s}^{\mathrm{dis}}$ represent the charging and discharging power; $P_i^{\mathrm{ch},\max }$ and $P_i^{\mathrm{dis},\max }$ represent the maximum charging and discharging power; ${\beta }_{i,t,s}^{\mathrm{ch}}$ and ${\beta }_{i,t,s}^{\mathrm{dis}}$ are binary variables indicating the charging and discharging states; ${\eta }_i^{\mathrm{ch}}$ and ${\eta }_i^{\mathrm{dis}}$ denote the charging and discharging efficiencies; $E_{i,t,s}^{\mathrm{ESS}}$, $E_i^{\mathrm{ESS},\min }$ and $E_i^{\mathrm{ESS},\max }$ represent the storage energy level and its lower and upper limits; $SO{C}_{i,t,s}^{\mathrm{ESS}}$, $SO{C}_i^{\mathrm{ESS},\min }$ and $SO{C}_i^{\mathrm{ESS},\max }$ denote the state of charge and its lower and upper limits; $E_{i,0,t}^{\mathrm{ESS}}$ and $SO{C}_{i,0,t}^{\mathrm{ESS}}$ represent the initial energy level and the initial state of charge.

3. Planning Method for Power System Flexibility Resources Considering Network Reconfiguration and Fault Risk

Distribution network reconfiguration optimizes power system operation by adjusting the network topology, modifying switch states, and redistributing loads. Its primary objectives are to reduce network losses, enhance supply reliability, improve voltage quality, and balance load distribution, thereby prolonging equipment lifespan and boosting overall system efficiency.

In terms of flexibility, network reconfiguration enables rapid adaptation to load fluctuations and the uncertainties inherent in renewable energy sources. By optimizing the topological configuration, it mitigates the impact of distributed energy resources on the grid. Particularly during faults, it can rapidly isolate faulted sections and restore power supply, significantly enhancing the system’s disturbance resistance and recovery capability. Additionally, reconfiguration helps balance load distribution, prevent local overloading, and optimize resource allocation, thereby further improving the grid’s operational flexibility and stability. Consequently, distribution network reconfiguration is not only an essential means to enhance system economy and reliability but also a key technology for strengthening grid flexibility and meeting the diverse demands of future power systems. While the unit models developed in Section 2 focus on flexibility resource characteristics, this chapter integrates network reconfiguration to optimize system operation under both normal and fault conditions, achieving a more effective allocation of flexibility resources.

3.1. Virtual Network Coupling-Based Reconfiguration

Based on graph theory, the topology of a distribution network can be represented as an undirected graph G = (V,E), where the bus set V corresponds to electrical buses and the edge set E defines the branch connections [33]. A branch fault can cause the undirected graph to split into multiple independent connected subgraphs, known as “islands.” An island is an independent connected region formed by mutually connected buses, and the number of islands indicates the extent of network partitioning. Specifically, an “islanded microgrid” refers to a subnet that remains internally connected after being disconnected from the main grid. When distributed generation exists within an islanded region, it can sustain the supply of critical loads through an off-grid operating mode. In extreme cases, if a bus loses all its adjacent connections, it becomes an isolated bus, representing the minimal connected unit in a separated power network topology. Figure 1 illustrates the undirected graph of a distribution network and its subgraphs.

A fundamental constraint in distribution network topology reconfiguration optimization is maintaining the radial operational characteristic. Common radial constraint models for distribution networks, such as the “L-1” condition, the spanning tree model, and the virtual network model, must satisfy the following topological conditions:

• The network topology must not contain closed loops.
• All buses must remain connected to the power source.

These constraints ensure that when the network is divided into multiple connected subgraphs, each subgraph still forms a tree structure connected to the power source, thereby providing independent power supply capability and eliminating the risk of unintended islanding.

This study employs a virtual network coupling modeling method, wherein a virtual topological network is constructed to be isomorphic to the actual power grid. Virtual load parameters (source bus = 0, load bus = 1) are used to define the network connectivity criteria, and the Big-M method is applied to transform nonlinear constraints into a mixed-integer linear form. Synchronization between the virtual and actual network topologies is ensured through binary branch state variables, which embed the actual network connectivity requirements within the optimization model while reducing the computational dimension and complexity compared to traditional approaches. The connectivity constraint of the virtual network model is expressed as follows:

$$-y_{i,t,s}\le {\displaystyle \sum _{\forall j\in {\mathrm{\Theta }}_i}{H}_{ji,t,s}-{\displaystyle \sum _{\forall j\in {\psi }_i}{H}_{ij,t,s}\le M(1-y_{i,t,s})}}-y_{i,t,s}$$

(8)

$$-M{\alpha }_{ij,t,s}\le H_{ij,t,s}\le M{\alpha }_{ij,t,s}$$

(9)

where $y_{i,t,s}$ is a 0/1 variable representing the bus type: when $y_{i,t,s}=0$, bus i is a source bus at time t; otherwise, $y_{i,t,s}=1$ indicates a load bus. $H_{ij,t,s}$ denotes the power transferred from bus; ${\psi }_i$ and ${\mathrm{\Theta }}_i$ are the sets of upstream and downstream branch buses associated with bus, respectively; M is a sufficiently large number, equal to the maximum branch transmission capacity in the network. Furthermore, since each load bus outputs a unit value, the maximum output of a source bus in a system with N buses can be set to N, thus $M=N$. (9) defines the branch transmission capacity constraint in the virtual network model: when ${\alpha }_{ij,t,s}=1$, indicating that branch ij is connected, the power transferred from bus i to bus j, $H_{ij,t,s}$ is bounded within ±N. Conversely, when ${\alpha }_{ij,t,s}=0$, meaning that branch ij is disconnected, the power transfer $H_{ij,t,s}$ is zero. Via employing the large number M to limit the branch transmission capacity within the virtual network while coupling the virtual and actual network topologies through the branch connection state ${\alpha }_{ij,t,s}$.

To elaborate on the implementation of the virtual network coupling method, we clarify that the virtual power flow variable $H_{ij,t,s}$ represents the directional flow of a unit virtual load, which propagates from source buses (assigned value 0) to load buses (assigned value 1) in the virtual network. The binary variable ${\alpha }_{ij,t,s}=0$ indicates whether branch ij is active. When ${\alpha }_{ij,t,s}=1$, the corresponding virtual power flow $H_{ij,t,s}$ is allowed within the upper and lower bounds defined by the Big-M formulation. This coupling ensures that power can only propagate through connected branches, thereby enforcing topological connectivity.

Furthermore, the radial operation constraint is guaranteed by ensuring that each connected island contains exactly one source bus and forms a tree-like structure. This is implemented by constraining the number of connected components (islands) to equal the number of source buses, and by avoiding the formation of cycles in the virtual network. These logical conditions, embedded in the virtual network topology, enable a tractable and linearizable way to enforce radial operation without the need for complex combinatorial cycle elimination constraints.

In a radial distribution network topology, when the system operates under steady-state conditions with a single-source supply configuration, the number of islands is strictly equal to the number of source buses in the system.

$${\displaystyle \sum _{ij=1}^{N_{\mathrm{B}}}{\alpha }_{ij,t,s}=N-N_S}$$

(10)

where $N$, $N_{\mathrm{B}}$ and $N_S$ represent the number of buses, branches, and source buses in the distribution network, respectively.

Given that the unit models in this study focus on flexibility resources and network reconfiguration, multi-resource integration leads to dynamic changes in the number of islands due to network topology reconfiguration. Therefore, the partitioning mechanism of the source bus set must be redefined to satisfy the radial constraint. Specifically, the system’s source bus set ${\mathrm{\Omega }}_{\mathrm{S}}$ is divided into a fixed source bus set ${\mathrm{\Omega }}_{\mathrm{SUB}}$ (main grid substation buses) and a variable source bus set ${\mathrm{\Omega }}_{\mathrm{DG}}$ (distributed generation buses). Substation buses serve as the system’s fixed sources, while distributed generation buses switch their status between grid-connected and islanded operation modes. Through the optimization process, these buses are modeled either as source buses or as equivalent load buses. It is assumed that each island contains exactly one source bus, ensuring that the number of islands matches the number of source buses. Therefore, the number of system islands is strictly equivalent to the number of activated source buses. Accordingly, the radial logic constraint of the virtual network model can be expressed as:

$${\displaystyle \sum _{ij=1}^{N_{\mathrm{B}}}{\alpha }_{ij,t,s}=N-{\displaystyle \sum _{\forall i\in {\mathrm{\Omega }}_{\mathrm{S}}}(1-y_{i,t,s})}}$$

(11)

$$\left\{\begin{array}{l}{y}_{i,t,s}=0,\forall i\in {\mathrm{\Omega }}_{\mathrm{SUB}}\\ y_{i,t,s}\ge 0,\forall i\in {\mathrm{\Omega }}_{\mathrm{DG}}\\ y_{i,t,s}=1,\forall i\notin {\mathrm{\Omega }}_{\mathrm{S}}\end{array}\right.$$

(12)

In addition, to mitigate the switching losses caused by excessively frequent reconfiguration, Equation (13) imposes a limit on the number of times a switch can be reconfigured:

$${\displaystyle \sum _{t=2}^T\left|{\alpha }_{ij,t,s}-{\alpha }_{ij,t-1,s}\right|}+\left|{\alpha }_{ij,1,s}-{\alpha }_{ij,0,s}\right|\le N_{ij}^{\mathrm{cg}}$$

(13)

where ${\alpha }_{ij,0,s}$ represents the initial state of branch ij, and $N_{ij}^{\mathrm{cg}}$ denotes the maximum allowable number of reconfigurations for branch ij.

Furthermore, for branches equipped with a SOP, it is necessary to ensure that network reconfiguration does not compromise the SOP’s power flow regulation function. Accordingly, the following additional constraint is imposed:

$${\alpha }_{ij}^{\mathrm{SOP}}+{\alpha }_{ij,t,s}\le 1$$

(14)

where ${\alpha }_{ij}^{\mathrm{SOP}}$ is a binary variable indicating whether branch ij is equipped with an SOP.

3.2. CVaR-Based Fault Risk Quantification and Planning

To quantitatively analyze the impact of faults induced by extreme weather on the investment decisions of distribution networks, it is necessary to adopt an appropriate method to measure the associated risks [34]. CVaR, as a risk measure for evaluating potential losses under extreme conditions, has gradually become an important paradigm in risk measurement due to its advantages and practical value in risk management. Compared with the traditional VaR, CVaR not only characterizes the maximum potential loss threshold at a specified confidence level but also quantifies the conditional expected loss exceeding this threshold [35], thereby providing a comprehensive characterization of the tail risk distribution. Therefore, it is more suitable for assessing extreme risks.

$$C_{\mathrm{CVaR},\beta }={\displaystyle \frac{1}{1-\beta }}{\displaystyle \underset{f(x,y)\ge C_{\mathrm{VaR},\beta }(x)}{\int }f(x,y)\rho (y)dy}$$

(15)

where, $\beta$ denotes the confidence level; $C_{\mathrm{CVaR},\beta }$ is the Conditional Value at Risk; ${\mathrm{C}}_{\mathrm{VaR},\beta }(x)$ is the Value at Risk; $f(x,y)$ represents the loss function; x denotes the decision variables and y the random variables; and $\rho (y)$ is the probability density function of y.

Since the analytical expression of $C_{\mathrm{VaR},\beta }(x)$ is difficult to obtain directly, a transformation function $F_{\beta }(x,\alpha )$ derived to compute CVaR as follows [36]:

$$F_{\beta }(x,\alpha )=\alpha +{\displaystyle \frac{1}{1-\beta }}{\displaystyle \int (\max \left\{f(x,y)-\alpha ,0\right\})\rho (y)dy}$$

(16)

where, $\alpha$ represents the VaR value.

Furthermore, the transformation function $F_{\beta }(x,\alpha )$ can be estimated using the following estimator:

$${\tilde{F}}_{\beta }(x,\alpha )=\alpha +{\displaystyle \frac{1}{q(1-\beta )}}{\displaystyle \sum _{k=1}^q(\max \left\{f(x,y_k)-\alpha ,0\right\})}$$

(17)

$$C_{\mathrm{CVaR},\beta }=\min {\tilde{F}}_{\beta }(x,\alpha )$$

(18)

where $y_k$ denotes the k-th realization of the random failure variable y, and $C_{\mathrm{CVaR},\beta }$ represents the final estimated value.

In this chapter, the CVaR theory is incorporated into a coordinated planning framework for diverse flexibility resources. The upper-level problem addresses the planning of multiple flexibility resources in a distribution network, aiming to minimize the annual investment and operation costs by optimizing the siting and sizing of resources such as ESSs, SOPs, and MTs. The configuration scheme determined at the planning stage is then provided as input to the lower-level problem within the framework. The lower-level problem formulates the optimal operation of the distribution network. It coordinates demand response, ESS charging and discharging, and the dispatch of MTs and SOPs to derive an optimal operation strategy across multiple representative scenarios. This strategy enhances economic efficiency, ensures system security, reduces wind and photovoltaic curtailment, and lowers both carbon emissions and network losses. The resulting optimal operation strategy is subsequently fed back to the upper-level problem. This iterative process continues until an equilibrium solution is reached.

The framework of the bi-level planning model is illustrated in Figure 2. Specifically, the planning layer quantifies the operational scheduling risk posed by extreme weather by minimizing the total investment and operation costs of MTs, ESSs, and SOPs along with the CVaR of the operational costs, thereby fully accounting for its impact on planning outcomes. At the operation layer, multiple uncertainties—such as source-load output fluctuations, line faults, and equipment failures—are addressed by coordinating measures including network reconfiguration, demand-side response, and ESS charge-discharge operations, ensuring alignment with the upper-level planning decisions.

• Planning Layer

The planning layer addresses the optimal siting and sizing of MTs, SOPs, and ESSs within the distribution network. The objective is to minimize the sum of their equivalent annualized planning costs and the CVaR of the operation and maintenance costs, as expressed below:

$$C_1=\min \{C_{\mathrm{MT},\mathrm{inv}}+C_{\mathrm{ESS},\mathrm{inv}}+C_{\mathrm{SOP},\mathrm{inv}}+N_{\mathrm{D}}(C_2+L{C}_{\mathrm{CVaR}})\}$$

(19)

where $C_1$ denotes the annual overall economic index, comprising the equivalent annual planning costs of MTs ($C_{\mathrm{MT},\mathrm{inv}}$), ESSs ($C_{\mathrm{ESS},\mathrm{inv}}$) and SOP ($C_{\mathrm{SOP},\mathrm{inv}}$), together with the objective function value of the operation layer ($C_2$)

$$\begin{array}{c}{C}_{\mathrm{MT},\mathrm{inv}}={\displaystyle \frac{r{(1+r)}^{y_{\mathrm{MT}}}}{{(1+r)}^{y_{\mathrm{MT}}}-1}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{MT}}}{c}_{\mathrm{MT},\mathrm{inv}}{P}_i^{\mathrm{MT}}}\\ C_{\mathrm{ESS},\mathrm{inv}}={\displaystyle \frac{r{(1+r)}^{y_{\mathrm{ESS}}}}{{(1+r)}^{y_{\mathrm{ESS}}}-1}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{ESS}}}{c}_{\mathrm{ESS},\mathrm{inv}}{E}_i^{\mathrm{ESS}}}\\ C_{\mathrm{SOP},\mathrm{inv}}={\displaystyle \frac{r{(1+r)}^{y_{\mathrm{SOP}}}}{{(1+r)}^{y_{\mathrm{SOP}}}-1}}{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{SOP}}}{c}_{\mathrm{SOP},\mathrm{inv}}{S}_{ij}^{\mathrm{SOP}}}\end{array}$$

(20)

where r represents the discount rate; $y_{\mathrm{MT}}$, $y_{\mathrm{ESS}}$ and $y_{\mathrm{SOP}}$ denote the service lifespans of the MTs, ESSs, and SOPs, respectively. ${\mathrm{\Omega }}_{\mathrm{MT}}$, ${\mathrm{\Omega }}_{\mathrm{ESS}}$ and ${\mathrm{\Omega }}_{\mathrm{SOP}}$ denote the sets of installation sites for each type of resource. $c_{\mathrm{MT},\mathrm{inv}}$, $c_{\mathrm{ESS},\mathrm{inv}}$, $c_{\mathrm{SOP},\mathrm{inv}}$ along with $P_i^{\mathrm{MT}}$, $E_i^{\mathrm{ESS}}$ and $S_{ij}^{\mathrm{SOP}}$ indicate the unit capacity investment costs and the installed capacities of MTs, ESSs, and SOPs, respectively.

In Equation (19) $C_{\mathrm{CVaR}}$ represents the CVaR of the daily integrated operation and maintenance costs of the distribution network. The risk preference coefficient L reflects the investor’s degree of risk aversion: a higher L indicates stronger risk aversion, implying that the investor is more likely to increase investments to enhance the resilience of the distribution network and mitigate risks caused by extreme weather, thereby avoiding potential losses. In contrast, a lower L suggests greater risk tolerance, indicating a willingness to accept higher risks in exchange for reduced investment. In this paper, the confidence level β is set to 0.95, which is a commonly adopted standard in risk management for power systems. This value reflects a balanced trade-off between risk conservatism and investment economy, as it captures the most critical 5% of tail events while maintaining computational tractability. The calculation of $C_{\mathrm{CVaR}}$ is defined as follows:

$$\begin{array}{c}{C}_{\mathrm{CVaR}}=\min (C_{\mathrm{VaR}}+{\displaystyle \frac{1}{1-\beta }}{\displaystyle \sum _{s=1}^{S^{*}}{p}_s{Z}_s})\\ s.t.\left\{\begin{array}{l}{Z}_s\ge 0\\ Z_s\ge C_{\mathrm{O}\&\mathrm{M}}^{\mathrm{s}}-C_{\mathrm{VaR}}\\ C_{\mathrm{O}\&\mathrm{M}}^{\mathrm{s}}=C_{\mathrm{MT},\mathrm{ope}}^{\mathrm{s}}+C_{\mathrm{ESS},\mathrm{ope}}^{\mathrm{s}}+C_{\mathrm{SOP},\mathrm{ope}}^{\mathrm{s}}+C_{\mathrm{netloss}}^{\mathrm{s}}\\ +C_{\mathrm{soploss}}^{\mathrm{s}}+C_{\mathrm{cur}}^{\mathrm{s}}+C_{\mathrm{buy}}^{\mathrm{s}}+C_{\mathrm{C}{\mathrm{O}}_2}^{\mathrm{s}}+C_{\mathrm{loadloss}}^{\mathrm{s}}+C_{\mathrm{switch}}^{\mathrm{s}}\end{array}\right.\end{array}$$

(21)

Resource allocation must comply with capacity and quantity constraints for the planned units, formulated as follows:

$$\left\{\begin{array}{l}0\le P_i^{\mathrm{M}\mathrm{T}}\le P_i^{\mathrm{M}\mathrm{T},\max }\\ 0\le E_i^{\mathrm{E}\mathrm{S}\mathrm{S}}\le E_i^{\mathrm{E}\mathrm{S}\mathrm{S},\max }\\ 0\le S_{ij}^{\mathrm{S}\mathrm{O}\mathrm{P}}\le S_{ij}^{\mathrm{S}\mathrm{O}\mathrm{P},\max }\end{array}\right.$$

(22)

$$\left\{\begin{array}{l}{P}_i^{\mathrm{M}\mathrm{T}}=m_i^{\mathrm{M}\mathrm{T}}{p}_i^{\mathrm{M}\mathrm{T}}\\ E_i^{\mathrm{E}\mathrm{S}\mathrm{S}}=m_i^{\mathrm{E}\mathrm{S}\mathrm{S}}{e}_i^{\mathrm{E}\mathrm{S}\mathrm{S}}\\ S_{ij}^{\mathrm{S}\mathrm{O}\mathrm{P}}=m_i^{\mathrm{S}\mathrm{O}\mathrm{P}}{s}_{ij}^{\mathrm{S}\mathrm{O}\mathrm{P}}\end{array}\right.$$

(23)

where $P_i^{\mathrm{MT},\max }$, $E_i^{\mathrm{ESS},\max }$ and $S_{ij}^{\mathrm{SOP},\max }$ denote the maximum installable capacities of MTs, ESSs, and SOPs at the selected sites, respectively. $m_i^{\mathrm{M}\mathrm{T}}$, $m_i^{\mathrm{E}\mathrm{S}\mathrm{S}}$ and $m_i^{\mathrm{S}\mathrm{O}\mathrm{P}}$ indicate the maximum allowable number of MTs, ESSs, and SOPs at the selected sites, which must be non-negative integers. $p_i^{\mathrm{M}\mathrm{T}}$, $e_i^{\mathrm{E}\mathrm{S}\mathrm{S}}$ and $s_{ij}^{\mathrm{S}\mathrm{O}\mathrm{P}}$ represent the unit installation capacities of MTs, ESSs, and SOPs at the corresponding sites.

2.

Operation Layer

The objective function of the operation layer seeks to minimize the expected total operation and maintenance costs across all representative scenarios while accounting for fault risks. The operation and maintenance costs under each scenario comprise the operation and maintenance expenses of MTs, ESSs, and SOPs; network losses and SOP losses; wind and PV curtailment penalties; carbon emission costs; and, additionally, the loss-of-load penalties and switch operation costs newly incorporated in this study. The operation layer objective function is defined in Equation (24):

$$\begin{array}{c}{C}_2=\min \{C_{\mathrm{MT},\mathrm{ope}}+C_{\mathrm{ESS},\mathrm{ope}}+C_{\mathrm{SOP},\mathrm{ope}}+C_{\mathrm{netloss}}+C_{\mathrm{soploss}}\\ +C_{\mathrm{cur}}+C_{\mathrm{buy}}+C_{\mathrm{C}{\mathrm{O}}_2}+C_{\mathrm{loadloss}}+C_{\mathrm{switch}}\}\end{array}$$

(24)

where $C_2$ denotes the economic performance index of the operation layer, comprising the operation and maintenance costs of MTs ($C_{\mathrm{MT},\mathrm{ope}}$), ESSs ($C_{\mathrm{ESS},\mathrm{ope}}$) and SOPs ($C_{\mathrm{SOP},\mathrm{ope}}$); network loss cost ($C_{\mathrm{netloss}}$), SOP loss cost ($C_{\mathrm{soploss}}$), wind and PV curtailment penalty cost ($C_{\mathrm{cur}}$), power purchase cost ($C_{\mathrm{buy}}$), and carbon trading cost ($C_{\mathrm{C}{\mathrm{O}}_2}$).

$$\begin{array}{c}{C}_{\mathrm{MT},\mathrm{ope}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{MT}}}{c}_{\mathrm{MT},\mathrm{ope}}}}}{P}_{i,t,s}^{\mathrm{MT}}\\ C_{\mathrm{ESS},\mathrm{ope}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{ESS}}}{c}_{\mathrm{ESS},\mathrm{ope}}}}}(P_{i,t,s}^{\mathrm{ch}}+P_{i,t,s}^{\mathrm{dis}})\\ C_{\mathrm{SOP},\mathrm{ope}}={\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{SOP}}}{c}_{\mathrm{SOP},\mathrm{ope}}{S}_{ij}^{\mathrm{SOP}}}\\ C_{\mathrm{netloss}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{L}}}{c}_{\mathrm{netloss}}}}}{r}_{ij}{I}_{ij,t,s}^2\\ C_{\mathrm{soploss}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{SOP}}}{c}_{\mathrm{soploss}}{P}_{ij,t,s}^{\mathrm{soploss}}}}}\\ C_{\mathrm{cur}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T({\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{PV}}}{c}_{\mathrm{cur},\mathrm{pv}}{P}_{i,t,s}^{\mathrm{pv},\mathrm{cur}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{WIND}}}{c}_{\mathrm{cur},\mathrm{wind}}{P}_{i,t,s}^{\mathrm{wind},\mathrm{cur}})}}}}\\ C_{\mathrm{cur}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T({\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{PV}}}{c}_{\mathrm{cur},\mathrm{pv}}{P}_{i,t,s}^{\mathrm{pv},\mathrm{cur}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{WIND}}}{c}_{\mathrm{cur},\mathrm{wind}}{P}_{i,t,s}^{\mathrm{wind},\mathrm{cur}})}}}}\\ C_{\mathrm{buy}}={\displaystyle \sum _{s=1}^S{p}_s{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{SUB}}}{\rho }_t{P}_{i,t,s}^{\mathrm{buy}}}}}\\ C_{\mathrm{C}{\mathrm{O}}_2}={\displaystyle \sum _{s=1}^S{p}_s}{\displaystyle \sum _{t=1}^T{c}_{C{O}_2}({\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{SUB}}}{\kappa }_1{P}_{i,t,s}^{\mathrm{buy}}+}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{MT}}}{\kappa }_2{P}_{i,t,s}^{\mathrm{MT}})}\\ C_{\mathrm{loadloss}}={\displaystyle \sum _{s=1}^{S^{*}}{p}_s{\displaystyle \sum _{t=1}^T(c_{\mathrm{loadloss}}^{\mathrm{I}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{I}}}}{P}_{i,t,s}^{\mathrm{loadloss}}+c_{\mathrm{loadloss}}^{\mathrm{II}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{II}}}}{P}_{i,t,s}^{\mathrm{loadloss}}}+}{c}_{\mathrm{loadloss}}^{\mathrm{III}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{III}}}}{P}_{i,t,s}^{\mathrm{loadloss}}})}}\\ C_{\mathrm{switch}}={\displaystyle \sum _{s=1}^{S^{*}}{p}_s(({\displaystyle \sum _{t=2}^T{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{L}}}{c}_{\mathrm{switch}}(\left|{\alpha }_{ij,t,s}-{\alpha }_{ij,t-1,s}\right|)+{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{L}}}\left|{\alpha }_{ij,1,s}-{\alpha }_{ij,0,s}\right|})}}}\end{array}$$

(25)

where, S and T denote the sets of scenarios and time periods; $p_s$ is the probability of scenario s; $c_{\mathrm{MT},\mathrm{ope}}$ and $c_{\mathrm{ESS},\mathrm{ope}}$ are the unit operation and maintenance cost coefficients for MTs and ESSs, while the investment cost of ESS is annualized based on its service lifespan, the degradation-related operational costs are not explicitly modeled in this study. Future work will incorporate lifecycle-aware cost modeling, including battery degradation and replacement; $c_{\mathrm{SOP},\mathrm{ope}}$ is the annual operation and maintenance cost coefficient for SOPs; $c_{\mathrm{netloss}}$ and $c_{\mathrm{soploss}}$ are the unit cost coefficients for network losses and SOP losses; $c_{\mathrm{cur},\mathrm{pv}}$ and $c_{\mathrm{cur},\mathrm{wind}}$ are the unit penalty coefficients for wind and PV curtailment; ${\rho }_t$ is the time-of-use electricity price; and $c_{\mathrm{C}{\mathrm{O}}_2}$ is the unit cost for carbon emission treatment. ${\kappa }_1$ and ${\kappa }_2$ represent the indirect unit carbon emission factor for purchased electricity and the direct unit carbon emission factor for MTs; $C_{\mathrm{loadloss}}$ denotes the loss-of-load penalty under scenario s and $C_{\mathrm{switch}}$ denotes the switch operation cost under scenario s; ${\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{I}}}$, ${\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{II}}}$ and ${\mathrm{\Omega }}_{{\mathrm{N}}_{\mathrm{III}}}$ are the sets of buses for first-class, second-class and third-class loads, respectively; $c_{\mathrm{loadloss}}^{\mathrm{I}}$, $c_{\mathrm{loadloss}}^{\mathrm{II}}$ and $c_{\mathrm{loadloss}}^{\mathrm{III}}$ are the corresponding unit loss-of-load penalty coefficients; $P_{i,t,s}^{\mathrm{loadloss}}$ represents the amount of load loss; and $c_{\mathrm{switch}}$ is the unit cost for a single switch operation.

In the operation of a distribution network system, fundamental power flow constraints must be satisfied. To ensure the rationality and effectiveness of coordinated optimal scheduling, the operational states of various active and reactive power control devices must also be comprehensively considered. This study primarily accounts for the distribution network’s power flow constraints, the operational constraints of PV and wind power units, MTs, SOPs, and ESSs, as well as demand response constraints and constraints related to active and reactive control devices such as OLTCs and CBs. In addition, carbon emissions and wind and PV curtailment are thoroughly addressed. By optimizing system operation strategies to reduce carbon emissions and curtailment, the system’s economic performance can be improved while significantly enhancing its environmental benefits.

Within the operation layer constraints, it is further necessary to incorporate branch power flow constraints resulting from network reconfiguration. By introducing binary variables to indicate whether each branch is connected or disconnected, an extended DistFlow model suitable for dynamic reconfiguration is formulated, as shown in Equations (26)–(29):

$$P_{i,t,s}={\displaystyle \sum _{j=F(i)}{\alpha }_{ij,t,s}{P}_{ij,t,s}-{\displaystyle \sum _{k=T(i)}{\alpha }_{ki,t,s}(P_{ki,t,s}-l_{ki,t,s}{r}_{ki})}}$$

(26)

$$Q_{i,t,s}={\displaystyle \sum _{j=F(i)}{\alpha }_{ij,t,s}{Q}_{ij,t,s}-{\displaystyle \sum _{k=T(i)}{\alpha }_{ij,t,s}(Q_{ki,t,s}-l_{ki,t,s}{x}_{ki})}}$$

(27)

$$v_{i,t,s}-v_{j,t,s}=2(P_{ij,t,s}{r}_{ij}+Q_{ij,t,s}{x}_{ij})-(r_{ij}^2+x_{ij}^2)l_{ij,t,s},{\alpha }_{ij,t,s}=1$$

(28)

$$P_{ij,t,s}^2+Q_{ij,t,s}^2=l_{ij,t,s}{v}_{i,t,s},{\alpha }_{ij,t,s}=1$$

(29)

where, $P_{i,t,s}$ and $Q_{i,t,s}$ denote the active and reactive power injections at bus i; F(i) represents the set of downstream buses connected by branches originating from bus i; T(i) represents the set of upstream buses connected by branches ending at bus i; $P_{ij,t,s}$ and $Q_{ij,t,s}$ denote the active and reactive power flows on branch i-j from bus i to bus j; $l_{ki,t,s}$ is the square of the current magnitude on branch k-i; $r_{ij}$ and $x_{ij}$ denote the resistance and reactance of branch k-i; and $v_{i,t,s}$ represents the square of the voltage magnitude at bus i.

Furthermore, bus voltages in the distribution network must satisfy specified upper and lower bounds, and, due to line transmission capacity limits, branch currents must also remain within allowable ranges:

$$v_i^{\min }\le v_{i,t,s}\le v_i^{\max }$$

(30)

$$l_{ij,t,s}\le l_{ij}^{\max }$$

(31)

where, $v_i^{\max }$ and $v_i^{\min }$ denote the maximum and minimum voltage magnitudes at bus i, and $l_{ij}^{\max }$ indicates the maximum allowable current through branch ij.

Since Equations (28) and (29) are applicable only to closed branches, this paper introduces large constants $M_1$, $M_2$ and $M_3$ to derive Equation (32), which ensures that when a branch is open, the branch current, active power, and reactive power are all zero. Accordingly, Equations (33)–(36) are obtained:

$$\left\{\begin{array}{l}-{\alpha }_{ij,t,s}{M}_1\le P_{ij,t,s}\le {\alpha }_{ij,t,s}{M}_1\\ -{\alpha }_{ij,t,s}{M}_2\le Q_{ij,t,s}\le {\alpha }_{ij,t,s}{M}_2\\ -{\alpha }_{ij,t,s}{M}_3\le l_{ij,t,s}\le {\alpha }_{ij,t,s}{M}_3\end{array}\right.$$

(32)

$$P_{i,t,s}={\displaystyle \sum _{j=F(i)}{P}_{ij,t,s}-{\displaystyle \sum _{k=T(i)}(P_{ki,t,s}-l_{ki,t,s}{r}_{ki})}}$$

(33)

$$Q_{i,t,s}={\displaystyle \sum _{j=F(i)}{Q}_{ij,t,s}-{\displaystyle \sum _{k=T(i)}(Q_{ki,t,s}-l_{ki,t,s}{x}_{ki})}}$$

(34)

$$v_{i,t,s}-v_{j,t,s}=2(P_{ij,t,s}{r}_{ij}+Q_{ij,t,s}{x}_{ij})-(r_{ij}^2+x_{ij}^2)l_{ij,t,s},{\alpha }_{ij,t,s}=1$$

(35)

$$P_{ij,t,s}^2+Q_{ij,t,s}^2=l_{ij,t,s}{v}_{i,t,s}$$

(36)

When ${\alpha }_{ij,t,s}=0$, both sides of Equation (36) become zero, so the constraint remains valid. Therefore, Equation (36) is no longer restricted to closed branches and can be rewritten as (37):

$${‖\begin{array}{c}2P_{ij,t,s}\\ 2Q_{ij,t,s}\\ l_{ij,t,s}-v_{i,t,s}\end{array}‖}_2\le l_{ij,t,s}+v_{i,t,s}$$

(37)

However, Equation (35) still applies only when the branch is closed and cannot be satisfied for an open branch. Since ${\alpha }_{ij,t,s}$ is a binary variable, Equation (35) remains nonlinear. To overcome this limitation, a virtual voltage $v_{i,t,s}^{ij}$ is introduced for branch ij connected to bus i, defined as follows:

$$0\le v_{i,t,s}^{ij}\le {\alpha }_{ij,t,s}{M}_4$$

(38)

$$-(1-{\alpha }_{ij,t,s})+v_{i,t,s}\le v_{i,t,s}^{ij}\le v_{i,t,s}$$

(39)

where, $M_4$ is a large constant. Equations (38) and (39) indicate that when the branch is closed (${\alpha }_{ij,t,s}=1$), the virtual voltage on branch ij equals the terminal voltage; when the branch is open (${\alpha }_{ij,t,s}=0$), the virtual voltage on branch ij equals zero. Consequently, Equation (35) can be linearized and expressed as:

$$v_{i,t,s}^{ij}-v_{j,t,s}^{ij}=2(P_{ij,t,s}{r}_{ij}+Q_{ij,t,s}{x}_{ij})-(r_{ij}^2+x_{ij}^2)l_{ij,t,s}$$

(40)

In a distribution network with high renewable energy penetration, the operational constraints for PV and wind power generation are formulated as follows:

$$0\le P_{i,t,s}^{\mathrm{pv}}\le {\tau }_{i,t,s}^{\mathrm{pv}}{P}_i^{\mathrm{pv}}(i\in {\mathrm{\Omega }}_{\mathrm{PV}})$$

(41)

$$0\le P_{i,t,s}^{\mathrm{wind}}\le {\tau }_{i,t,s}^{\mathrm{wind}}{P}_i^{\mathrm{wind}}(i\in {\mathrm{\Omega }}_{\mathrm{WIND}})$$

(42)

$$Q_{i,t,s}^{\mathrm{pv}}=P_{i,t,s}^{\mathrm{pv}}\tan {\phi }_i^{\mathrm{pv}}(i\in {\mathrm{\Omega }}_{\mathrm{PV}})$$

(43)

$$Q_{i,t,s}^{\mathrm{wind}}=P_{i,t,s}^{\mathrm{wind}}\tan {\phi }_i^{\mathrm{wind}}(i\in {\mathrm{\Omega }}_{\mathrm{WIND}})$$

(44)

$$P_{i,t,s}^{\mathrm{pv},\mathrm{cur}}={\tau }_{i,t,s}^{\mathrm{pv}}{P}_i^{\mathrm{pv}}-P_{i,t,s}^{\mathrm{pv}}(i\in {\mathrm{\Omega }}_{\mathrm{PV}})$$

(45)

$$P_{i,t,s}^{\mathrm{wind},\mathrm{cur}}={\tau }_{i,t,s}^{\mathrm{wind}}{P}_i^{\mathrm{wind}}-P_{i,t,s}^{\mathrm{wind}}(i\in {\mathrm{\Omega }}_{\mathrm{WIND}})$$

(46)

where ${\tau }_{i,t,s}^{\mathrm{pv}}$ and ${\tau }_{i,t,s}^{\mathrm{wind}}$ denote the output coefficients for PV and wind power, respectively; $P_i^{\mathrm{pv}}$ and $P_i^{\mathrm{wind}}$ are the installed capacities of PV and wind power; ${\phi }_i^{\mathrm{pv}}$ and ${\phi }_i^{\mathrm{wind}}$ are the power factor angles of PV and wind power; $P_{i,t,s}^{\mathrm{pv},\mathrm{cur}}$ and $P_{i,t,s}^{\mathrm{wind},\mathrm{cur}}$ represent the curtailed power of PV and wind power. The operational constraints for various flexible resources in the distribution network, including MTs, ESSs, SOP, demand response, OLTC, and CB, remain consistent with the descriptions provided in Section 2. Finally, the power balance constraint is given by:

$$\begin{array}{c}{P}_{i,t,s}={\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{PV}}}{P}_{i,t,s}^{\mathrm{pv}}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{WIND}}}{P}_{i,t,s}^{\mathrm{wind}}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{MT}}}{P}_{i,t,s}^{\mathrm{MT}}}+\\ {\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{SOP}}}(P_{i,t,s}^{\mathrm{SOP}}+P_{i,t,s}^{\mathrm{soploss}})+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{ESS}}}(P_{i,t,s}^{\mathrm{dis}}-P_{i,t,s}^{\mathrm{ch}})+}}{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{SUB}}}{P}_{i,t,s}^{\mathrm{buy}}}-P_{i,t,s}^{\mathrm{cur}}\end{array}$$

(47)

$$Q_{i,t,s}={\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{PV}}}{Q}_{i,t,s}^{\mathrm{pv}}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{WIND}}}{Q}_{i,t,s}^{\mathrm{wind}}}+{\displaystyle \sum _{ij\in {\mathrm{\Omega }}_{\mathrm{SOP}}}{Q}_{i,t,s}^{\mathrm{SOP}}}+{\displaystyle \sum _{i\in {\mathrm{\Omega }}_{\mathrm{CB}}}{Q}_{i,t,s}^{\mathrm{CB}}}-Q_{i,t,s}^{\mathrm{cur}}$$

(48)

where, $P_{i,t,s}^{\mathrm{cur}}$ denotes the load after demand response, and $Q_{i,t,s}^{\mathrm{CB}}$ denotes the reactive power compensation capacity provided by CBs.

4. Model Solution

The bi-level planning model developed in this study employs an iterative solution strategy. Although exact solution methods such as Benders decomposition are commonly used for bi-level optimization problems [37], their applicability in this study is limited by the complex combinatorial nature of the upper-level problem, which includes multiple integer decision variables related to the siting, sizing, and topology configuration of flexibility resources. These characteristics introduce significant non-convexities and hinder the convergence performance of decomposition-based algorithms, especially for large-scale distribution networks [38]. In contrast, the proposed hybrid SA-PSO algorithm strikes a balance between computational efficiency and solution quality. It effectively handles high-dimensional, non-convex, and mixed-integer search spaces while maintaining robust global search capabilities. Therefore, SA-PSO is selected for the upper-level planning problem, while the lower-level convex operational subproblem is solved exactly via conic programming to ensure optimality within each scenario.

In the upper-level model, the siting and sizing of MTs, ESSs, and SOP are determined using a SA-PSO hybrid algorithm, which provides an optimized configuration scheme and verifies its convergence. Based on the configuration scheme obtained from the upper level, the lower-level model performs multi-scenario operational optimization through conic programming, and the operating strategies for each scenario are fed back to the upper-level model to achieve coordinated optimization. By means of iterative interaction between the two levels, coordinated decision-making for both planning and operation is realized. The iterative process of the bi-level planning model is described in detail below, and its flowchart is presented in Figure 3.

Step 1: Initialize the parameters by inputting the distribution network data and distributed generation parameters. Initialize the parameters of the SA-PSO algorithm by generating the initial positions and velocities of the particle swarm and setting the initial temperature and cooling rate. Set the iteration count to k = 0.

Step 2: Generate representative source-load joint time-series scenarios, these scenarios serve as inputs for the lower-level model.

Step 3: At the current temperature, input the particles into the lower-level model, which is relaxed using second-order cone programming (SOCP). According to the objective function and constraints, solve for the optimal operational strategy for each scenario.

Step 4: Based on the output data from the lower-level model and the current temperature, calculate the fitness value of each particle.

Step 5: Update each particle’s individual best solution and the current global best solution.

Step 6: Update the velocity and position of each particle according to the rules of the particle swarm optimization algorithm.

Step 7: Calculate the fitness of the particle swarm at the new positions $f\left(x_i^{k+1},y_i^{k+1}\right)$ and determine the acceptance probability P under the current temperature. If the fitness difference $f\left(x_i^{k+1},y_i^{k+1}\right)<f\left(x_i^k,y_i^k\right)$, the new position is accepted; if $f\left(x_i^{k+1},y_i^{k+1}\right)\le f\left(x_i^k,y_i^k\right)$, the new position is accepted with probability P.

Step 8: Reduce the temperature according to the cooling schedule and increment the iteration count, k = k + 1.

Step 9: If the convergence criterion is met, output the final optimal result; otherwise, repeat Steps 3 to 8 until convergence is achieved.

Algorithm 1 illustrates the SA–PSO hybrid algorithm designed to solve the bi-level flexibility resource planning problem. It initializes particle positions and velocities, evaluates fitness by embedding the lower-level SOCP model, and updates positions using PSO rules. A simulated annealing mechanism is integrated to enhance global search capability. The process iterates until convergence, yielding the optimal siting and sizing of MTs, ESSs, and SOPs.

Algorithm 1: SA-PSO for Solving the Bi-Level Flexibility Resource Planning Model

Input:     - Population size N     - Maximum iterations MaxIter     - Initial temperature T0, cooling rate α     - Particle positions and velocities initialization     - Lower-level problem model (solved via conic programming) Output:     - Optimal siting and sizing of MTs, ESSs, SOPs 1: Initialize particle positions $x_i$, velocities $v_i$, personal best $pbes{t}_i$, and global best $gbest$ 2: Set current temperature $T\leftarrow T_0$, iteration $k\leftarrow 0$ 3: while k < MaxIter do 4:    for each particle i = 1 to N do 5:       Input current particle $x_i$ into the lower-level model 6:       Solve lower-level operation problem via SOCP to get cost $f$ 7:       Evaluate fitness of particle i using upper-level objective 8:       Update $pbes{t}_i$ and $gbest$ if needed 9:    end for 10:    for each particle i = 1 to N do 11:      Update velocity $v_i$ using PSO update rule 12:      Update position $x_{i+1}\leftarrow x_i+v_i$ 13:      Calculate $\Delta f=f_{new}-f_{old}$ 14:      if $\Delta f\le 0$ then 15:         Accept new solution 16:      else 17:         Accept with probability $P=\exp (-\Delta F/T)$ 18:      end if 19:    end for 20:    $T\leftarrow \alpha \times T$//Simulated annealing cooling 21:    $k\leftarrow k+1$ 22: end while 23: Return optimal solution $gbest$

5. Case Study

5.1. Parameter Settings

In this chapter, simulation analyses are conducted using an improved IEEE 33-bus distribution network. The topology of the IEEE 33-bus system is illustrated in Figure 4, with a nominal voltage level of 12.66 kV.

To thoroughly evaluate the impacts of distributed generation integration, the system is configured with three wind turbine generators and three PV systems. The basic parameters of these units are provided in

Table 2. The parameters of DGs.

Parameter	Wind Turbines			PV Systems
Access Location	9	25	32	7	17	22
Capacity (kW)	400	400	400	400	400	400

. To generate representative source–load time-series scenarios, historical wind speed, solar irradiance, and load data are first analyzed. The marginal distributions of wind speed, solar irradiance, and load demand are modeled using the Weibull, Beta, and Gaussian distributions, respectively, with parameters estimated via the maximum likelihood method. A joint probability distribution is then constructed by coupling these marginal distributions using a Copula function, capturing the dependency among the variables. Based on this joint distribution, a large number of correlated source–load time series are generated through Monte Carlo sampling. Subsequently, K-means clustering is applied to group the generated scenarios based on normalized hourly output profiles, and representative scenarios are selected for further analysis. The resulting representative scenarios are illustrated in Figure 5.

The goal of the bi-level planning model is to determine the optimal siting and sizing of three MTs, three ESSs, and two SOPs. The candidate buses for MT placement are {5, 10, 16, 18, 28}, while the candidate buses for ESS installation are {7, 9, 17, 22, 25, 32}. Considering geographical constraints, the candidate positions for SOPs are located at tie switches (marked by red dashed lines). Each MT has a planning capacity of 100 kW, each ESS has a planning capacity of 100 kWh, and each SOP has a planning capacity of 100 kVA. A capacitor bank with switchable units is installed at bus 33, with each unit providing 0.15 MVar of reactive power compensation and up to five units connectable in parallel. An OLTC is installed at bus 1, with a voltage regulation range of ±8 × 1.25%.

The system employs a time-of-use electricity pricing scheme for purchasing electricity from the upstream grid: peak periods are defined as 12:00–15:00 and 19:00–23:00; normal periods as 07:00–11:00 and 16:00–18:00; and off-peak periods as 00:00–06:00.

In the solution process of the bi-level planning model for this case study, the upper-level SA-PSO hybrid optimization algorithm is implemented using MATLAB scripts, while the lower-level model is solved through the cone programming solver Mosek 10.0, accessed via the YALMIP toolbox in MATLAB R2020b.

Furthermore, the loads at the buses are classified into three categories: First-class load buses include buses 13, 15, 21, 24, and 31, with a unit load shedding penalty of 5 CNY/kWh; Second-class load buses include buses 8, 12, 25, 27, and 33, with a unit load shedding penalty of 3 CNY/kWh; and all other buses are designated as Third-class load buses, with a unit load shedding penalty of 2 CNY/kWh. The cost of each switching operation is set at 15 CNY per operation. In this study, a typhoon is selected as the representative extreme weather event affecting the operation of the distribution network. The typhoon’s impact area is illustrated by the blue striped region in Figure 4. Four fault scenarios are considered under typhoon conditions:

Fault Type 1: The typhoon causes the PV unit at bus 17 to disconnect.

Fault Type 2: The typhoon causes the wind turbine at bus 25 to disconnect.

Fault Type 3: The typhoon causes a single line fault on line 24–25.

Fault Type 4: The typhoon causes two simultaneous line faults on lines 12–13 and 6–26.

To ensure the practical solvability of the proposed bi-level planning model, the lower-level operational subproblem is solved using SOCP with the commercial solver MOSEK, accessed through the YALMIP toolbox in MATLAB R2020b. The solver is run with default tolerance settings, and a relative optimality gap of 1 × 10⁻⁴ is used to balance accuracy and computational speed.

For the upper-level SA–PSO algorithm, the population size is set to 50, and the maximum number of iterations is 500. The initial temperature for the simulated annealing process is 1000, with a cooling coefficient of 0.95. In the PSO component, the inertia weight linearly decreases from 0.9 to 0.4, and standard update rules are adopted for velocity and position adjustments.

All simulations are carried out on a desktop computer with an Intel Core i7-12700 CPU, 32 GB RAM, and Windows 11 (64-bit) operating system.

5.2. Impact Analysis of Coordinated Planning of Multiple Flexible Resources

To validate the effectiveness of coordinated planning for multiple flexible resources, four schemes are established for comparative analysis:

Scheme 1: Planning includes only micro gas turbines and energy storage systems; SOP planning and demand response are excluded.

Scheme 2: Planning includes micro gas turbines, energy storage systems, and demand response; SOP planning is excluded.

Scheme 3: Planning includes micro gas turbines, energy storage systems, and SOP; demand response is excluded.

Scheme 4: Planning includes micro gas turbines, energy storage systems, and SOP, along with demand response.

Table 3. Comparison of planning results under different schemes.

Scheme	Configuration Object
	MT/kW (Location)	ESS/kWh (Location)	SOP/kVA (Location)
1	800 (5), 800 (10), 300 (18)	500 (9), 700 (22), 500 (25)	/
2	300 (10), 300 (18), 500 (28)	700 (9), 300 (17), 400 (22)	/
3	700 (5), 800 (10), 400 (28)	600 (17), 600 (22), 600 (25)	300 (9–15), 300 (25–29)
4	500 (5), 300 (16), 500 (28)	200 (7), 800 (17), 400 (22)	200 (9–15), 100 (12–22)

and

Table 4. Comparison of cost distribution results under different schemes.

Scheme	Cost Distribution (CNY)
	MT	ESS	SOP	Power Purchase	Wind & Solar Curtailment	Network Loss	SOP Loss	Carbon Emission	O&M Cost	Annual Total
1	415,192	193,355	/	237,659	12,977	159,748	/	13,433	70,446	1,102,810
2	240,374	165,733	/	203,781	4791	125,236	/	12,512	55,046	807,473
3	415,192	202,563	65,557	170,739	0	31,526	71,178	13,506	68,382	1,038,643
4	284,079	165,733	32,778	167,609	0	32,935	28,602	12,282	54,438	778,456

present the configuration results and cost distributions for the different schemes, respectively. Firstly, a comparison of the annual total system costs reveals that Scheme 4, which comprehensively accounts for flexibility resources across generation, grid, load, and storage sectors, achieves the lowest system configuration cost. In contrast, Scheme 1, which considers flexibility only on the generation and storage sides, results in the highest configuration cost. Specifically, the total costs of Schemes 1–3 are 41.6%, 3.7%, and 33.4% higher than those of Scheme 4, respectively, demonstrating the advantages of coordinated planning of diverse flexibility resources. Furthermore, a comparison between Schemes 4 and Scheme 3 shows that Scheme 4 additionally incorporates demand response. This inclusion enables users to adjust their electricity demand rationally in response to appropriate incentives. When consumption behavior becomes more rational—by reducing demand during periods of insufficient supply capacity and increasing it when supply is adequate, thereby smoothing the load demand curve—the required capacities of resources such as MT, ESS, and SOP, as well as additional electricity purchase costs, are significantly reduced.

Next, a comparison between Scheme 4 and Scheme 2 shows that Scheme 4 additionally incorporates the SOP. The SOP is capable of rapid and continuous adjustment, serving to optimize power flows across distribution lines. The benefits of SOP adjustment manifest in two ways: first, it directly coordinates active power between ports, thereby reducing network loss costs resulting from long-distance power transmission; second, it indirectly contributes to reductions in the required capacity of MT and additional electricity purchase costs. Overall, SOP planning reduces the need for additional generation output and minimizes power transmission losses, effectively regulating power flow both at the source and along the transmission path.

Finally, comparing the total costs of Schemes 2–4 indicates that, relative to SOP, demand response yields a greater reduction in annual total cost. This is because demand response directly targets load demand, producing a more immediate effect: by adjusting load patterns, it fundamentally modifies the system’s electricity consumption intensity across both temporal and spatial dimensions. In contrast, the SOP primarily functions to optimize power flow and mitigate network losses, exerting a comparatively indirect influence; hence, its impact is less pronounced than that of demand response.

In conclusion, fully accounting for the coordinated planning of flexibility resources can significantly lower total system costs, further validating the effectiveness and superiority of multi-faceted flexibility resource coordination.

5.3. Analysis of System Operation Results Under Multiple Flexibility Resource Deployment

Furthermore, several representative scenarios are selected to analyze the system’s operational performance under the optimal configuration of multiple flexibility resources. The overall system load and various output sources are aggregated to evaluate the overall supply–demand balance.

Figure 6 shows the system operation results for Typical Scenario 1. During 01:00–07:00, when there is no photovoltaic generation, the system load is met by wind power and MT output. From 08:00–18:00, with sufficient wind and photovoltaic generation, the MT ceases operation. To mitigate penalties associated with curtailment of wind and solar power, the energy storage system charges during this period, while the SOP also operates to enable large-scale active power flow transfers. Between 19:00–24:00, as renewable output gradually decreases, the MT resumes operation, and the energy storage system discharges the electricity stored during the day to supply the high evening peak load. Owing to the coordinated operation of renewable energy, the MT, and the energy storage system, Typical Scenario 1 does not require additional electricity purchases to maintain the system’s supply–demand balance. As a result, the associated carbon emission costs and network losses are relatively low.

Figure 7 shows the system operation results for Typical Scenario 2. Compared with Typical Scenario 1, the average daily load in Typical Scenario 2 is higher. Although photovoltaic generation remains relatively high during the daytime, the significant reduction in wind power output means that during the nighttime period from 00:00 to 06:00, relying solely on MT output is insufficient to meet demand. Therefore, additional electricity purchases are required to maintain the supply–demand balance. Moreover, because the MT operates at a high output level throughout the entire day, the system’s carbon emission costs and network losses are relatively higher.

Figure 8 shows the system operation results for Typical Scenario 3. Compared with Typical Scenarios 1 and 2, the load in Typical Scenario 3 becomes significantly smoother due to the implementation of demand response. However, owing to low renewable generation levels and the limited ramping capability of the MT, additional electricity still needs to be purchased from external sources at 01:00 during the night to maintain balance. The MT ma in slightly increased carbon emission costs and network losses, although these remain much lower than those in Typical Scenario 2.

The associated carbon emission costs and network losses for all scenarios are summarized in

Table 5. The main operating costs in different scenarios.

Scenario Number	Operating Cost (CNY)
	Network Loss Cost	SOP Loss Cost	Power Purchase Cost	Carbon Emission
Typical Scenario 1	60	187	0	17
Typical Scenario 2	239	30	2975	93
Typical Scenario 3	68	12	86	28

.

To further analyze the peak-shaving and valley-filling effects of demand response and energy storage, Figure 9 illustrates the load curve changes after demand response regulation and energy storage charging/discharging actions. When the energy storage system charges, it acts as a load, shifting the corresponding points on the load curve upward; when discharging, it acts as a power source, shifting these points downward. As observed from the figure, the peak-valley difference significantly decreases under the effect of demand response, resulting in a smoother load curve. Moreover, when incorporating energy storage charging and discharging actions (blue curve) on top of the demand-response adjusted load curve (red curve), the peak-valley difference is further reduced.

The data presented in

Table 6. Analysis of peak-cutting and valley-filling effect.

Category	Peak Load (kW)	Valley Load (kW)	Peak-Valley Difference (kW)	Peak-Valley Ratio
Original Load Curve	2988	763	2225	0.745
After DR	2199	888	1311	0.596
After ESS + DR	1929	888	1041	0.540

provide a more intuitive perspective: the original load curve has a peak-valley difference of 2225 kW, corresponding to a peak-valley difference ratio as high as 74.5%. After considering demand response actions, the peak load decreases from the original 2988 kW to 2199 kW, a reduction of 26.4%. Simultaneously, the valley load increases accordingly, ultimately decreasing the peak-valley difference ratio by 14.9%. This highlights the prominent role of demand response in peak shaving and valley filling. Furthermore, after incorporating the influence of energy storage charging and discharging, the peak-valley difference and its corresponding ratio continue to decrease by an additional 270 kW and 5.6%, respectively. Considering the limited allocation of energy storage resources (only connected at three buses), it is foreseeable that a more widely distributed and abundant deployment of energy storage resources would yield even greater reductions in the system’s peak-valley difference.

5.4. Comparison of Planning Outcomes

The planning models based on risk neutrality and CVaR were solved separately, and the resulting configuration plans and costs for the two methods are presented in

Table 7. The results obtained by solving different models.

Method	Configuration Scheme Capacity (Location)	Total Planning Cost (CNY)	Total Operating Cost (CNY)	Total Cost (CNY)
Risk-neutral	ESS: 400 (9), 1400 (25), 900 (32) MT: 300 (5), 600 (10), 500 (18) SOP: 200 (12–22), 200 (18–33)	483,805	353,386	837,191
CVaR	ESS: 1000 (22), 500 (25), 1900 (32) MT: 700 (5), 800 (10), 900 (28) SOP: 200 (9–15), 300 (12–22)	738,626	274,158	1,012,784

.

A comparative analysis shows that, compared with the conventional risk-neutral planning model, the CVaR-based model yields significantly larger capacities for various flexibility resources. Specifically, the total capacity of the ESS increases by 25.9%, the total MT capacity increases by 71.4%, and the total SOP capacity increases by 25%. This is because, although the operating cost under contingency scenarios is substantially higher than under normal conditions, the low probability of occurrence diminishes its impact on the annual overall planning cost. As a result, the risk-neutral model primarily satisfies normal operating conditions, leading to more risk-taking configurations and consequently larger losses under contingencies. By contrast, the CVaR-based model amplifies the influence of such low-probability, high-impact events, resulting in a more resilient configuration plan with correspondingly higher resource capacities and planning costs. Moreover, the increased capacity of flexibility resources provides greater regulation capability, which reduces excessive external electricity purchases and load shedding penalty costs during operation, thereby lowering the overall operating cost.

Additionally,

Table 8. Scenario comparison of different models.

Method	Risk-Neutral				CVaR
Cost Distribution (CNY)	Load Shedding	External Purchase	Other Costs	Total Cost	Load Shedding	External Purchase	Other Costs	Total Cost
Typical Scenario 1	0	0	895	895	0	0	572	572
Typical Scenario 2	75	0	690	765	0	0	647	647
Typical Scenario 3	2254	2188	2080	6522	1594	526	1890	4010
Fault Scenario 1	4821	2097	1977	8896	1920	627	2074	4621
Fault Scenario 2	5023	2725	1992	9740	2415	763	2199	5377
Fault Scenario 3	2366	2328	2289	6983	1694	585	1960	4239
Fault Scenario 4	5255	2593	2058	9907	2093	823	2276	5192

presents the distribution of system operating costs under several normal and contingency scenarios for both planning models. A comparison indicates that the CVaR-based planning model consistently reduces operating costs across all scenarios, with this effect being particularly pronounced in contingency conditions. Specifically, in typical scenarios, the incorporation of CVaR amplifies the influence of contingency scenarios on flexibility resource allocation. The resulting increase in resource investment lowers the previously high costs associated with electricity purchases and load shedding under the original risk-neutral planning model, reducing total costs by 36.1%, 15.4%, and 38.5%, respectively.

Fault Scenario 3 builds upon the generation and load profiles of Typical Scenario 3 by additionally considering a single line outage between buses 24 and 25. However, timely network reconfiguration optimizes the system topology, significantly mitigating the additional operational costs that could arise from the outage, resulting in similar cost distributions for both scenarios. In Typical Scenario 3, low wind and solar generation throughout the day combined with high load demand inevitably compel the system to rely on external electricity purchases and to shed some non-critical loads. Under the CVaR-based planning approach, however, the enhanced resource allocation further reduces these costs, leading to a lower overall operating cost.

Fault Scenarios 1, 2, and 4 involve, respectively, the disconnection of PV units, wind turbines, and multiple simultaneous line outages. Although these scenarios share the same generation and load profiles as Typical Scenario 3, the impacts of the faults are far more severe. Under the traditional risk-neutral planning model, the system’s limited regulation capability results in a sharp increase in load shedding and external electricity purchases, causing significant economic losses. In contrast, the configuration plan derived from the CVaR-based planning method substantially narrows the cost gap between contingency and normal operating conditions. This demonstrates that the proposed approach effectively mitigates the impact of contingencies on system operations and enhances overall system resilience.

5.5. Parameter Analysis of the CVaR Model

This study further investigates how the decision-maker’s risk attitude influences the planning of multiple flexibility resources by varying the risk preference coefficient L. The specific results are presented in

Table 9. Comparison of results under different risk preference parameters.

L	Investment Cost (CNY)	Annual Total Cost (CNY)	CVaR (CNY)
0.01	619,139	902,948	3,191,488
0.05	625,077	933,782	2,711,396
0.1	639,662	952,633	2,135,347
0.2	648,312	979,901	2,054,545
0.5	697,351	992,768	1,913,240
1	738,626	1,012,784	1,853,854
2	782,195	1,064,974	1,838,959
3	837,459	1,144,030	1,831,561

. The findings show that as the value of L increases, the system’s total investment cost increases accordingly. This is because a higher L indicates a stronger risk-averse tendency. To ensure reliable power supply to critical loads under contingency conditions, risk-averse decision-making leads to larger capacities of flexibility resources such as energy storage systems and microturbines, thereby enhancing the system’s emergency backup capability.

As illustrated in Figure 10, a clear trade-off emerges between investment cost and annual total cost as the risk preference coefficient L increases. Initially, small increments in investment lead to noticeable reductions in annual total cost, reflecting high marginal benefit. However, beyond L = 1, further investment results in diminishing returns, as annual cost reduction plateaus while investment cost rises steeply. This trend highlights the importance of selecting an appropriate risk preference level to achieve cost-effective planning without overinvestment.

Figure 11 presents the efficient frontier between system investment cost and CVaR. When the decision-maker exhibits a risk-seeking tendency (i.e., a low L), the marginal investment cost required to improve system resilience remains relatively low, and the investment cost rises gradually as the risk level decreases. Conversely, when the decision-maker becomes more risk-averse (i.e., a high L), the investment cost required to achieve the same level of resilience improvement increases significantly, which is reflected in the steeper slope of the curve.

5.6. Performance Analysis of the Algorithm

This section compares the proposed method with conventional PSO, SA, and the GA. All algorithms adopt the same population size of 50 and a maximum of 500 iterations. For the SA-PSO algorithm, the initial temperature is set at 1000 with a cooling coefficient of 0.95; for the PSO algorithm, the inertia weight decreases linearly from 0.9 to 0.4; and for the GA, the crossover probability and mutation probability are set at 0.8 and 0.1, respectively.

As shown in

Table 10. Performance Evaluation of Algorithms.

Algorithm	Best Solution (CNY)	Average Solution (CNY)	Standard Deviation	Average Computation Time (s)	Convergence Iterations
SA-PSO	778,456	780,832	2156	45.2	152
PSO	812,347	817,529	4287	38.7	245
SA	785,692	791,276	3825	52.8	298
GA	823,571	832,184	5967	41.3	320

, the SA-PSO algorithm achieves the best performance in terms of solution optimality, producing a minimum objective function value of 778,456 CNY, which represents improvements of 4.2%, 0.9%, and 5.8% over the PSO, SA, and GA algorithms, respectively. In terms of algorithm stability, the SA-PSO algorithm yields the smallest standard deviation (2156), indicating strong robustness. Although the PSO algorithm offers a slight advantage in computation time, the SA-PSO algorithm converges within just 152 generations, exhibiting a convergence rate significantly faster than those of the other algorithms.

As shown in

Table 11. Computational Effects of Different Big-M.

M	Solution Time (S)	Average Solution (CNY)	Constraint Violation
N	45.2	778,456	No
2 N	47.6	778,460	No
0.5 N	Error	-	Yes
0.7 N	46.3	778,457	No

, we tested the impact of different Big-M values on model performance. When M = N, 2 N, or 0.7 N, the model remains feasible with similar objective values and acceptable computation times. In contrast, setting M = 0.5 N leads to constraint violations and solution failure. These results confirm that M = N offers a good balance between feasibility and efficiency, and is thus adopted in this study.

To evaluate the scalability and computational efficiency of the proposed SA-PSO algorithm, additional experiments were conducted on distribution networks with varying bus sizes. As shown in

Table 12. Performance of the SA-PSO Algorithm under Different Network Sizes.

Network Size	Objective Value	Computation Time	Convergence Iterations
33	778,456	45.2	152
69	1,083,920	98.7	214
118	1,626,370	183.5	289
150	1,975,280	265.4	327

, the network sizes range from 33 to 150 buses, and the corresponding objective values, average computation times, and convergence iterations are reported.

The results indicate that the SA-PSO algorithm maintains a good balance between solution quality and computational burden as the network size increases. Specifically, the objective function value grows proportionally with system complexity, while the computation time increases moderately, from 45.2 s for the 33-bus system to 265.4 s for the 150-bus system. Moreover, the number of iterations required for convergence increases steadily but remains within a reasonable range, demonstrating the algorithm’s stability and robustness under larger-scale scenarios.

5.7. Voltage Quality Analysis

The integration of photovoltaic and wind power in distribution networks increases the risk of voltage violations, making it difficult to maintain voltage quality. Therefore, this section analyzes the impact of coordinated operation of multiple flexibility resources on voltage quality. Figure 12 illustrates the voltage profiles under the same operating scenario for the four different configuration schemes.

Through a comparative analysis of the system voltage profiles under the same scenario, it can be observed that although the voltages of all schemes remain within the allowable range, the voltage fluctuations at critical buses and during peak load periods vary significantly in both amplitude and frequency. Scheme 1, which relies solely on the regulation effects of ESS, OLTC, and CB, exhibits relatively stable voltages at most buses, but some buses still show significant voltage fluctuations during periods of large supply-demand imbalance.

Scheme 2 and Scheme 3 further incorporate DR and SOP, respectively. DR adjusts power demand through dynamic load shifting, thereby improving the load balance and smoothing power flows to mitigate voltage variations. SOP enables fast and continuous power flow regulation, providing both active and reactive power support to enhance voltage levels. Compared with Scheme 1, both approaches effectively reduce the magnitude and frequency of voltage oscillations, thus improving overall voltage quality. Scheme 4, which integrates both DR and SOP, achieves the best performance. Most buses maintain voltage within an extremely narrow fluctuation range, indicating superior voltage quality.

6. Conclusions

This paper develops a comprehensive planning and stability enhancement strategy for distribution networks with high renewable energy penetration, emphasizing the coordinated deployment of diverse flexibility resources. By systematically modeling generation-side micro gas turbines, network-side smart soft open points and on-load tap changers, load-side demand response, and distributed energy storage systems, the proposed approach exploits multiple flexibility channels to mitigate renewable variability and enhance network operational resilience. To address uncertainties and extreme weather-induced faults, a risk-coupled network reconfiguration method based on CVaR is integrated within a bi-level planning model, ensuring an economic balance between investment and operational risks. The hybrid SA-PSO and conic programming solution enables tractable optimization for realistic system scales.

The following conclusions can be drawn from the case studies:

• By introducing the virtual network coupling modeling method to handle topological constraints in network reconfiguration, the proposed approach significantly improves computational efficiency while ensuring radial operation requirements. This method effectively resolves the complexity challenges in traditional network reconfiguration modeling and provides rapid topology adjustment capabilities for fault isolation and power restoration, enhancing both the accuracy of topological decision-making and the overall system operational flexibility.
• The proposed CVaR-based risk quantification framework offers a comprehensive approach to managing extreme weather uncertainties and fault risks by incorporating conditional value-at-risk into the bi-level optimization model. By tuning risk preference parameters, system planners can effectively balance investment costs and operational risks under uncertain conditions, improving system robustness and economic performance while enhancing the system’s resilience against extreme events.
• The coordinated multi-type flexibility resource planning model significantly increases system operational efficiency and resilience. By integrating microturbines, energy storage systems, and soft open points through the bi-level optimization framework combined with the SA-PSO hybrid algorithm, the proposed method achieves superior overall benefits compared to single-resource configuration approaches, demonstrating substantial improvements in both system reliability and economic viability while contributing to the sustainable development of modern power systems.

To further improve the applicability of the proposed model, it is worth noting that while the current fault scenarios are representative of typical renewable energy disturbances, future work can extend the framework to incorporate more complex failure mechanisms, such as cascading outages and large-scale fault propagation. Additionally, future research may explore reinforcement learning or mixed-integer SOCP to further improve the model’s scalability and adaptability in complex operational environments.