Collaborative Optimization Strategy for Virtual Inertia Spatiotemporal Distribution Replenishment under Extreme Weather Events

1  Introduction

1.1 Background and Motivation

In recent years, extreme weather events have become increasingly frequent, characterized by high destructive power and prolonged duration, leading to severe damage to transmission lines. Such events hinder rapid restoration of power system stability, resulting in substantial economic losses across sectors including power generation [1] and agricultural production [2]. For instance, Typhoon Capricorn, which struck Hainan Province, China, in 2024, caused extensive grid outages, with 836 transmission lines rated 10 kV and above taken out of service and over 1.6 million users left without power for several weeks [3]. In 2019, Super Typhoon Lekima severely disrupted the power network, damaging 72 substations rated above 35 kV and 4649 transmission lines above 10 kV, leaving over 7.59 million users without power [4]. Meanwhile, to promote sustainable energy development, the large-scale integration of renewable energy into the power grid has resulted in a progressive decline in system inertia. This transformation has markedly reduced the share of conventional synchronous generators, intensified the spatial imbalance of inertia distribution, and diminished the system’s capability to restore frequency stability effectively [5]. For example, during the 2021 Texas cold wave, extensive wind turbine icing led to long-term outages and a 4.3 GW reduction in generation capacity, causing prolonged regional blackouts [6]. Similarly, the 2016 South Australia typhoon event caused severe damage to transmission facilities and the loss of approximately 1.83 GW of load [7]. These incidents demonstrate that as the penetration of renewable generation continues to increase, the power system’s resilience to extreme weather events declines, resulting in rapid regional frequency drops and, in severe cases, system separation or blackout.

In summary, the widespread integration of renewable energy sources has significantly reduced the overall inertia of modern power systems [8]. Moreover, the increasing frequency of extreme weather events has intensified the spatiotemporal heterogeneity of frequency dynamics, posing new challenges to system stability [9]. Therefore, accurately evaluating the inertia distribution and coordinating differentiated virtual inertia support are of great importance for enhancing system resilience [10] under extreme conditions and ensuring both overall and regional frequency stability of the power grid.

1.2 Literature Review

With the rapid development of renewable energy and power electronic technologies, modern power systems are progressively transitioning to low-inertia configurations. Recent research has extensively explored scheduling and optimization strategies for renewable energy–dominated power systems. For example, reference [11] proposed an intelligent scheduling strategy for renewable energy systems to explore day-ahead scheduling in high renewable penetration scenarios and demonstrated its application in energy storage coordination. Reference [12] introduced a scheduling method for virtual power plants that provides joint probabilistic guarantees for power availability under uncertainty. Reference [13] developed a hierarchical control scheme to coordinate inertial frequency response between wind farms and synchronous machines. Reference [14] presented a two-stage distributed robust optimization framework for day-ahead energy scheduling and real-time power dispatch in virtual power plant energy management systems. Traditional optimization-based scheduling frameworks have demonstrated strong potential to enhance energy efficiency and reduce operating costs. However, most existing studies are constrained by system-level assumptions and therefore fail to accurately capture the local frequency dynamics and uncertainty inherent in renewable-dominated systems.

The above studies propose effective renewable energy dispatch strategies that enhance the frequency stability of new power system. However, the large-scale integration of renewable energy sources also introduces greater uncertainty and reduces overall system inertia. Therefore, investigating the uncertainty of inertia distribution and its influence on frequency stability is essential for ensuring the secure and stable operation of power systems. To address this, reference [15] proposed an inertia evaluation method for multi-machine power systems that simultaneously considers both inertia and disturbance distributions to characterize inertia distribution under various disturbance scenarios, enabling evaluation of nodal frequency stability. Reference [16] proposed a method to quantify the inertia of each busbar using generator current and synchronous power coefficient, addressing the challenge of evaluating busbar inertia distribution. Considering the fluctuations of wind force, solar radiation, and load changes, reference [17] established a model of frequency control using a fractional order fuzzy PID controller. To address the inaccuracy of frequency stability assessments caused by the large-scale integration of wind turbines into the grid, Reference [18] introduced a probabilistic frequency stability analysis method that accounts for the dynamic behavior of wind power generation under different control strategies. These studies focus on the differences in system frequency spatial distribution and inertia demand constraints. However, the integration of renewable energy scheduling with system frequency spatiotemporal distribution remains inadequately explored.

Although the above articles have proposed an effective virtual inertia scheduling strategy for improving the frequency stability of the new power system, they are limited to fixed deployment inertia resource scheduling and lack flexible response capabilities to extreme events. Therefore, it is also important to consider coordinating mobile energy storage resources to improve the frequency stability of the power system. Reference [19] proposed a mobile energy storage system configuration to enhance the emergency power supply capability of the distribution network in response to extreme events. This configuration ensured rapid response and restoration of the distribution network in the event of extreme events. Reference [20] introduced a formula for the operation of bidirectional distribution networks in the presence of fixed and mobile batteries. This formula aggregated their operational problems in bidirectional distribution networks, effectively reducing active power loss and peak apparent power of the network, ensuring stable and safe system frequency. Reference [21] developed a concept based on mobile energy storage and power-saving, combined with the power demand of load nodes and the energy storage characteristics of mobile energy storage vehicles. Although numerous studies have investigated inertia optimization, virtual inertia allocation, and mobile energy storage scheduling, most existing works focus on the fixed deployment of system-wide inertia resources, neglecting the dynamic spatiotemporal variations of frequency and inertia under extreme events such as typhoons. Few studies have developed a collaborative optimization framework that simultaneously integrates spatial inertia distribution, mobile energy storage deployment, and economic dispatch under extreme weather–induced contingencies. Considering the spatiotemporal coupling between virtual inertia support and system operating costs is essential to accurately represent system behavior in such scenarios. Therefore, establishing a unified collaborative optimization model is crucial to enhance system resilience and frequency stability under extreme conditions.

1.3 Contributions

To address the above issues, this paper accounts for the spatial heterogeneity of system frequency under extreme weather events and enhances frequency distribution stability through a collaborative optimization of virtual inertia scheduling. Furthermore, the economic aspects of virtual inertia deployment are considered, and a collaborative optimization strategy for virtual inertia spatiotemporal distribution replenishment is developed. The main contributions of this study are summarized as follows:

(1)   A quantitative inertia evaluation model is developed to characterize the spatiotemporal distribution of nodal equivalent inertia under extreme weather–induced disturbances. This model overcomes the limitations of conventional global inertia assessment and enables accurate identification of inertia-deficient regions.

(2)   A collaborative optimization framework is proposed to coordinate renewable-based virtual inertia and Modular Mobile Energy Storage Systems (MMESS). The framework jointly optimizes inertia allocation across space and time, achieving an average 11.4% reduction in the maximum nodal Rate-of-Change-of-Frequency (RoCoF), while maintaining an overall operating cost reduction of 4.9% relative to conventional inertia scheduling strategies.

(3)   A resilience-oriented simulation platform based on an improved IEEE 39-bus system is established to evaluate the system’s dynamic performance under extreme events. The results verify that the proposed strategy effectively mitigates regional frequency dips and enhances the overall frequency recovery rate by over 10.9% compared to baseline cases. Its effectiveness and economic benefits are validated through simulation experiments under typhoon disaster scenarios.

The remainder of this paper is organized as follows: Section 2 introduces the virtual inertia optimization configuration based on the inertia distribution evaluation model. Section 3 elaborates on the virtual inertia replenishment and mobile energy storage pre-deployment optimization scheduling model for new energy power plants. Section 4 is a case study, which explains the economy and rationality of the model proposed in the paper. Section 5 provides a detailed conclusion.

2  The Inertia Distribution Evaluation Model for New Power System

2.1 Inertia Analysis and Frequency Response Characteristics of New Power System

2.1.1 Inertia Analysis of New Power System

Power system inertia reflects the system’s resistance to frequency deviations following disturbances and is primarily provided by the rotational kinetic energy stored in grid-connected operating units. In traditional power systems, the main source of inertia is the rotational kinetic energy of synchronous machines. The indicators for measuring the inertia of synchronous machines include the rotational inertia J, the kinetic energy EG generated by rotating at the rated mechanical angular speed ωG, and the inertia time constant TG. The rotational kinetic energy of synchronous machines is commonly used to measure the inertia level of synchronous machines, and its expression is:

EG=12JωG2(1)

The equivalent inertia time constant TS is defined as the ratio of the rotational kinetic energy of a synchronous machine to its rated capacity, and its expression is:

TS=2EGSN=JωG2SN(2)

where, SN is the rated capacity of the synchronous machine.

Synchronous generators inherently exhibit voltage-source characteristics, with the amplitude and phase of their internal voltage adjustable according to system operating conditions to allocate disturbance power promptly. During power disturbances in the grid, the rotational kinetic energy of a synchronous generator is converted into electromagnetic power through its power-angle characteristics, releasing or absorbing energy into the system. During the active power–frequency response to disturbances, the inertia response of the synchronous generator occurs spontaneously, and the system’s inertia effect can be quantitatively described by the rotor motion equation [22]. Ignoring the damping coefficient and magnetic flux changes, it can be expressed as:

TSdΔffNdt=ΔPm−ΔPe=−ΔPL(3)

where, Δf is the system frequency deviation, ΔPm is the mechanical power, ΔPe is the electromagnetic power, ΔPL is the unbalanced power.

With the large-scale development of new power system, power supply is increasingly centered around power electronic converters [23], leading to a continuous decline in system inertia. In these systems, renewable power plants provide virtual inertia support through inverter-based control. According to their grid-connection control methods [24], inverters can be classified as Grid-Following (GFL) or Grid-Forming (GFM), each exhibiting distinct active frequency characteristics in the new power system.

The GFL inverter using virtual inertia control change the active power output of the inverter according to the frequency signal [25]. The phase of the GFL inverter is not obtained by the control method, but by tracking the frequency and phase angle through a phase-locked loop. Therefore, the virtual inertia control of the GFL inverter has response delay and adjustment dead zone in the dynamic process of active power-frequency response. It does not have the ability to respond to system power disturbances instantaneously, which is fundamentally different from the inertia response of synchronous machines.

The GFM inverter using virtual inertia control introduces the oscillation motion equation into the control outer loop of the inverter, making the GFM inverter have synchronous external characteristics [26]. Essentially, it can be equivalent to a Virtual Synchronous Generator (VSG) with controlled amplitude and phase containing internal resistance. The essential difference between GFM inverter and GFL inverter is that GFM inverter can autonomously maintain phase angle and voltage amplitude, rather than passively tracking grid frequency. In the inertia response process, the current reference value generated by the GFM inverter does not exceed the limit, and the time lag caused by the power filter is low and can be ignored. In this operating situation, the inertia response mechanism of the GFM inverter power supply and the synchronous machine can be completely equivalent. The active power reference value of the GFM inverter power supply in this article is determined by the optimal operating point of the wind turbine or photovoltaic system. This inertia time constant is the virtual inertia time constant set in the control loop.

In summary, the inertia of new power system includes the rotational inertia of synchronous machines, the virtual inertia of GFL converters, and the virtual inertia of GFM converters. Due to the significant variability in the number, capacity, and control modes of load-side motors, the inertia contribution from the load side is difficult to evaluate accurately and has a relatively minor impact on overall system. Moreover, this study focuses on power systems dominated by synchronous generators and renewable energy plants, where the proportion of load-side inertia is comparatively small. Therefore, for the sake of model simplification and computational efficiency, the inertia contribution from the load side is neglected in this paper. If the inertia time constant of the synchronous machine, the virtual inertia time constant of the inverter, and the rated capacity of the unit can be obtained, the theoretical equivalent inertia Hsys of the system can be calculated, which is expressed as:

Hsys=∑i=1NGHGiSGi+∑j=1NGFLHGFLjSGFLj+∑k=1NGFMHGFMkSGFMk(4)

where, HGi and SGi are the inertia constant and rated capacity of the synchronous generator i, respectively; HGFLj and SGFLj are the inertia constant and rated capacity of the GFL inverter power supply j, respectively; HGFMk and SGFMk are the inertia constant and rated capacity of the GFM inverter power supply k, respectively.

2.1.2 The Frequency Spatiotemporal Distribution Characteristics of New Power System

The present study defines extreme weather events as typhoons, and the uncertainty in the typhoon’s path and speed primarily affects the timing of line faults. To simplify the typhoon model analysis, it is assumed that the typhoon follows a straight-line path at a constant speed, sequentially causing faults on the transmission lines. Due to successive disconnections of the lines within the path, the typhoon results in a serious shortage of power supply and demand in the power system, an abnormal decrease in system frequency, and the generation of disturbance power. In the initial stage of inertia response, synchronous motors and virtual synchronous machines share the disturbance power according to the electrical parameter matrix, and the distribution of disturbance power is related to the disturbance position and magnitude. The uneven distribution of disturbance power causes differences in the speed and frequency of each unit, resulting in the loss of synchronization of the frequency of each node in the system. The spatial distribution characteristics of the system frequency begin to be reflected [27]. Subsequently, the disturbance power is redistributed according to the inertia level of each unit, and the dynamic frequency of each node in the system oscillates continuously around the inertia center frequency. The spatiotemporal distribution characteristics of the system frequency are reflected in the frequency differences of different nodes at different times.

Taking 10 machines and 39 nodes as an example, when a power shortage disturbance occurs, the frequency response curves of some nodes in the system are shown in Fig. 1. From a temporal perspective, the system frequency response process includes three stages: inertia response, primary frequency modulation and secondary frequency modulation. From a spatial perspective, there are differences in the frequency changes of each node during the inertia response phase, and there are also differences in the lowest point of frequency for different nodes. In Fig. 1, the t0~t2 stage is the inertia response stage, during which the system undergoes disturbance power allocation and primary frequency modulation has not yet taken effect. At this time, each node of the system lowers its frequency to different lowest points using different Rate-of-Change-of-Frequency (RoCoF). As a frequency modulation takes effect, the frequency changes of each node tend to be similar. The secondary response (typically achieved through Automatic Generation Control, AGC) gradually restores the frequency to its nominal value after the primary control has arrested the frequency decline. This stage is now illustrated in Fig. 1 to provide a complete depiction of the system’s dynamic frequency regulation process.

Figure 1: Nodal frequency response curve

In the frequency response process, the initial RoCoF of nodes reflect their abilities to resist disturbances from unbalanced power, and these abilities to suppress frequency changes caused by external disturbances is known as inertia. With the integration of a large number of low inertia new energy units, traditional units gradually decrease, and new power system gradually exhibit low inertia level characteristics [28]. When disturbances occur, the initial RoCoF of nodes will be much higher than that of traditional power system, exacerbating the frequency imbalance of nodes and further highlighting the spatiotemporal distribution characteristics of system frequency.

Due to the spatiotemporal distribution characteristics of the dynamic frequency response of the new power system, it will have adverse effects on the analysis and control of the frequency dynamic performance of the new power system [29]. For example, when nodes in the low inertia level region are disturbed, the RoCoF will be much higher than that at Centra of Inertia (COI) frequency, which may trigger the action of protective devices and cause system collapse. In addition, when only the disturbance position changes, although the COI frequency response curve of the system is not affected by the disturbance position, the disturbance power allocated to each unit will change, and the RoCoF of each unit node will also change. At this time, the frequency curves of each node in the system will also change accordingly. In order to avoid greater harm to the frequency stability of the system caused by typhoon disasters, this paper conducts research on the frequency response characteristics of nodes under typhoon disasters.

2.2 Virtual Inertia Optimization Replenishment Based on Frequency Space Characteristics

2.2.1 Evaluation of Nodal Equivalent Inertia Based on Frequency Space Distribution

To study the frequency characteristics of nodes in the new power system, a composite admittance matrix equation [30] is used to investigate this process. The classical second-order model of the generator is extended to the power grid, where only transient reactance exists and the internal potential remains constant. Meanwhile, it assumes that the GFM inverter is used to control the grid voltage of the power supply to be constant. The internal nodes of the generator are treated as equivalent, the system nodes are connected through transmission lines, and it can be written as:

[I˙G0]=[YGG  YGLYLGYLL+YG0+YL0][E˙GejδU˙Lejθ](5)

where, E˙G is the vector composed of the amplitude of the electric potential inside the generator, U˙L is the vector composed of the amplitude of the bus node voltage, I˙G is the vector composed of the generator current value, YLL is the standard node admittance matrix composed of the bus node, YLG is the mutual admittance matrix composed of the generator node and the bus node, YG0 is the diagonal matrix composed of the internal impedance of the generator, YL0 is the diagonal matrix composed of the equivalent admittance of the load, δ is the phase angle of the electric potential inside the generator, where δ=2π∫ΔfGdt+δ0, δ0 is the initial phase angle of the electric potential inside the generator, and ΔfG is the frequency deviation matrix of the generator; θ is the phase angle of the bus node voltage, where θ=2π∫ΔfLdt+θ0, θ0 is the initial phase angle of the bus node voltage and ΔfL is the frequency deviation matrix of the bus node.

Since the amplitude of the potential inside the generator remains constant, but the phase angle between the potential inside the generator and the nodal voltage are time-varying phasors, taking the derivative of the two variables in Eq. (5) over time yields:

ΔfL=−[YLL+YG0+YL0]−1YLGE˙GULcos⁡(δ−θ)ΔfG=BLGΔfG(6)

where, BLG is the composite frequency matrix formed by the network topology relationship between the node and the generator frequency. By taking the derivative of Eq. (6) over time, the initial RoCoF of all nodes in the system can be obtained. The relationship is as follows:

dfLdt=BLGdfGdt(7)

In order to quantitatively describe the spatial distribution of inertia and achieve online evaluation of inertia, this paper conducts research on the equivalent inertia of different nodes. As nodes themselves do not have inertia, the equivalent inertia is the coupling effect of various inertia sources in the system at that node. Its magnitude characterizes the initial frequency response characteristics of the node. Referring to the rotor motion equation of the generator set and the relationship between disturbance power and frequency change rate, the equivalent inertia [31] of node L is defined as the ratio of the per-unit disturbance power (p.u.) to the per-unit nodal RoCoF (p.u.). Its expression is:

HL=−ΔPk2dfLdt=−ΔPk2BLGdfGdt(8)

where, ΔPk is the per-unit value of the disturbance power generated at the disturbance node k.

2.2.2 Virtual Inertia Replenishment Based on Unit Frequency Response Evaluation

In order to further obtain the relationship between the spatial distribution characteristics of frequency and the unbalanced power changes of each generator unit, we summarize the distribution of unbalanced power between units after disturbance into two stages: the initial unbalanced power distribution according to the electrical matrix and the power oscillation stage between units. At the initial moment, the unbalanced power flows only from the disturbance point to the generator set, which can be considered as an instantaneous process. Later, due to the different moments of inertia and disturbance power borne by the units, the rotor accelerations of each generator are different, resulting in frequency differences between the units. The various generator nodes continue to oscillate until the system frequency stabilizes. While the rotor motion equation can be conveniently solved when the unbalanced power is constant, during the power oscillation stage the unbalanced power varies continuously, making it impossible to obtain the frequency dynamic response analytically. Therefore, this paper mainly studies the frequency spatial distribution characteristics of the system at the initial moment, and obtains the frequency dynamic response process of the unit under disturbance power by solving the rotor motion equation.

This paper prepares for the most severe situation of typhoon damage and analyzes the maximum disturbance power generated by the system at the disturbance node at that moment. At the initial moment of disturbance, the phase angle of the internal busbar of the generator will not undergo a sudden change. The disturbance power is distributed to each generator according to the electrical matrix, and the calculation formula [32] for the disturbance power allocated to each generator is as follows:

ΔPi0+=EiUkBikcos⁡δik0+∑i=1NGEiUkBikcos⁡δik0+ΔPk(9)

where, ΔPi0+ is the per-unit disturbance power (p.u.) allocated to the generator i at the initial moment, ΔPk is the per-unit disturbance power (p.u.) generated at the disturbance node k, Ei is the internal potential amplitude of the generator i, Uk is the voltage amplitude at the disturbance node k, Bik is the admittance amplitude between the generator i and the disturbance node k, and cosδGk0+ is the initial phase angle difference of the voltage between the generator i and the disturbance node k.

By substituting Eq. (9) into the rotor motion formula, the expression for RoCoF of the generator i at the initial disturbance moment can be obtained as follows:

dfidt=−ΔPi0+2Hi(10)

where, Hi is the per-unit (p.u.) value of the rotational inertia of the generator i.

By substituting Eq. (10) into Eq. (8), the expression for the calculated equivalent inertia of node j can be obtained as follows:

Hj=−ΔPk2BLGdfjdt=ΔPkBLGΔPi0+Hi(11)

In summary, the equivalent inertia of nodes is mainly determined by the disturbance position, disturbance quantity, network topology, and moment of inertia. This can be clearly seen from the schematic diagram in Fig. 2. During typhoon disasters, improving the inertia level of weak nodes primarily involves adjusting the generator’s inertia [33]. In traditional power systems, the inertia response is relatively limited, relying mainly on synchronous generators with fixed unit inertia. The rotational inertia of these generators is determined by their start-up mode and cannot be adjusted in real time according to system demand. In contrast, in modern power systems, the virtual inertia parameters of grid-connected renewable units can be dynamically adjusted to meet varying inertia requirements under different operating conditions.

Figure 2: Schematic diagram of generators, nodes, and disturbance

When the frequency risk of the unit is high, the inertia constant can be increased to enhance the frequency support for the node. However, at the same time, a larger inertia constant means that the grid type power supply needs to meet a larger reserve capacity requirement, which means that parameter tuning needs to consider both economy and stability comprehensively. This paper constrains the economic costs required for increasing the reserve capacity and inertia of new energy units without affecting their normal operation. The specific constraints are as follows:

{CminVi=λspΔHspVi0≤ΔHspVi≤0.1HNVi(12)

In the formula, CminVi represents the economic cost target for increasing virtual inertia, λsp represents the economic cost per-unit of inertia increase [34], ΔHspVi represents the virtual inertia increase value of the renewable energy units, and HNVi represents the rated virtual inertia of renewable energy units. The RoCoF expression of node j after virtual inertia enhancement is:

dfjVidt=−(∑i=1NViBjiΔPVi0+2(HNVi+ΔHspVi)+∑i=1NGBjiΔPi0+2Hi)(13)

By reserving reserve capacity for grid type power units, increasing their inertia level, or installing grid type energy storage devices, the frequency safety of the system and nodes is ensured, but it is still affected by the location, quantity, and network topology of disturbances. In order to avoid excessive disturbance power in fault concentration, which may result in the inability to effectively improve weak inertia nodes under the influence of network topology, this paper also proposes the use of Module Mobile Energy Storage System (MMESS) pre-deployment to improve the frequency response of the system and nodes [35]. The specific constraints are as follows:

{0≤|dfjVidt|≤|dfjmindt|0≤ΔPk−ΔPMMESSdch,k≤ΔPkmax(14)

where, ΔPMMESSdch,k is the discharge power provided by the assembled mobile energy storage to the disturbance node k.

In traditional power systems, network topology is difficult to modify once planned and constructed. With a high penetration of renewable energy and complex disturbance propagation paths, relying solely on virtual inertia adjustment is no longer sufficient to ensure frequency stability. In particular, for specific disturbance source nodes, the resulting power imbalances can lead to significant frequency drops and system oscillations under conditions of weak inertia and limited control response. To address this challenge, this paper introduces Modular Mobile Energy Storage Systems (MMESS) as an efficient dynamic control method. MMESS units feature disassembly, mobility, and rapid deployment, allowing them to be prioritized for deployment in areas where line faults or load shedding cause power imbalances, thereby providing a backup solution to compensate for insufficient virtual inertia.

3  Virtual Inertia Replenishment Optimization Scheduling Model

The main sources of inertia in the new power system include thermal power units, gas units, and other equipment mainly composed of synchronous generators, as well as wind turbines, photovoltaic units, and energy storage equipment that adopt virtual inertia control technology. Sufficient system inertia enhances the system’s resistance to frequency disturbances and improves overall frequency stability. However, under typhoon disasters, the frequency response of the system exhibits pronounced spatiotemporal characteristics, leading to heterogeneous nodal inertia levels across the network and reducing the system’s ability to withstand disturbances. This paper focuses on the coupling between nodal inertia requirements and the economic efficiency of virtual inertia scheduling. Considering the dynamic characteristics of frequency response and frequency security constraints, a collaborative optimization strategy for virtual inertia resources—including wind, solar, and energy storage—is developed to achieve precise inertia allocation and system-level economic optimization. The proposed approach enhances the frequency response capability and overall operational resilience of the system under extreme weather events such as typhoons.

3.1 Inertia Scheduling Model

3.1.1 Objective Function

The inertia scheduling model Csysmin aims to minimize the operating cost of the system while meeting the node inertia requirements. It mainly includes thermal power units’ operating cost CrunG, start-stop cost Con−offG, wind and solar power curtailment penalty cost CabW,PV, load shedding penalty cost CshedLoad, and virtual inertia enhancement cost CextraVi. The objective functions are as follows:

Csysmin=min(CrunG+Con−offG+CabW,PV+CshedLoad+CextraVi)(15)

{CrunG=∑t=1T∑k=1NG[akG(Pk,tG)2+bkGPk,tG+ckG]Con−offG=∑t=1T∑k=1NG[Fon,kGuk,tG(1−uk,tG)+Foff,kGuk,t−1G(1−uk,t−1G)]CabW,PV=∑t=1T∑a=1NWFaW(Pa,tYW−Pa,tW)+∑t=1T∑b=1NPVFbPV(Pb,tYPV−Pb,tPV)CshedLoad=∑t=1T∑n=1NLoadFnLoadPn,tLoadCextraVi=∑t=1T∑i=1NViλspiViΔHspiVi(16)

where, T is the scheduling period set to 24 h; NG represents the number of thermal power units; akG, bkG, and ckG are the cost coefficients of the thermal power unit k, respectively; Pk,tG is the actual output of the thermal power unit k at time t; Fon,kG and Foff,kG are the starting and stopping costs of the thermal power unit k, respectively; uk,tG is the operating state of the thermal power unit k at time t; FaW is the wind curtailment penalty coefficient; PatYW is the expected output of the wind farm b at time t; Pa,tW is the actual output of the wind farm b at time t; FbPV is the penalty coefficient for abandoning light; Pb,tYPV is the expected output of the photovoltaic power station b at time t; Pb,tPV is the actual output of the photovoltaic power station b at time t; NLoad is a set of load nodes; FnLoad is the penalty cost for load shedding at the load node n; Pn,tLoad is the load shedding amount of the load node n.

To ensure the standardization and numerical stability of the inertia scheduling cost minimization process, all cost components were normalized against a unified reference value. Different unit costs were assigned to various types of inertia resources, and the coupling of these cost components represents an economic trade-off among different inertia providers. The proposed formulation minimizes the total scheduling cost while satisfying safety and stability constraints to meet the system’s critical inertia requirements.

3.1.2 Constraints

When conducting inertia scheduling models, the model should also meet the following constraints on the safe and stable operation of the power system. To ensure the clarity of the symbols used in the following constraints, a variable table has been provided in Appendix A Table A1.

(1)   Constraints on the output of coal-fired and solar power units

{Pk,minG≤Pk,tG≤Pk,maxGPa,minW≤Pa,tW≤Pa,maxW0≤Pb,tPV≤Pb,maxPV(17)

where, Pk,maxG and Pk,minG are the upper and lower limits of the output of the thermal power unit, respectively; Pa,maxW and Pa,minW are the upper and lower limits of wind turbine output, respectively; Pb,maxPV is the upper limit of the output of the photovoltaic power station.

(2)   Climbing and minimum start stop time constraints for thermal power units

{Pk,tG−Pk,t−1G≤SkUGPk,t−1G−Pk,tG≤SkDG(18)

{∑s=tt+TGkon−1uk,sG≥TGkon(uk,sG−uk,s−1G)∑s=tt+TGkoff−1(1−uk,sG)≥TGkoff(uk,s−1G−uk,sG)(19)

where, SkUG and SkDG are the uphill and downhill velocities of the thermal power unit, respectively; TGkon and TGkoff represent the minimum continuous start-up time and minimum continuous shutdown time of the thermal power unit k, respectively.

(3)   Rotation backup constraint

∑n=1NGuk,tG(Pk,maxG−Pk,tG)≥μG∑n=1NLoadPn,tLoad(20)

where, μG is the rotation reserve factor.

(4)   Power balance and line constraints

∑k=1NGuk,tGPk,tG+∑a=1NWua,tWPa,tW+∑b=1NPVub,tPVPb,tPV+Pn,tEnd−Pn,tOrigin=Pn,tLoad−ΔPn,tLoad,n∈NLoad(21)

−(1−ΩLine,t)Ω∞≤PLine,t−θLine,start,t−θLine,end,tRLine≤(1−ΩLine,t)Ω∞(22)

{0≤ΔPn,tLoad≤Pn,tLoadΩLine,tPLine,min≤PLine,t≤ΩLine,tPLine,maxθn,min≤θn,t≤θn,max(23)

where, ΩLine,t is a binary variable representing the state of the line at time t, closed to 1 and open to 0; Ω∞ is an infinite constant set to 99,999; PLine,t is the active output of the line at time t; RLine is the resistance of the line; θLine,start,t and θLine,end,t are the voltage angles of the starting and ending nodes in the line at time t, respectively; θn,t is the voltage angle of node n at time t, and the phase angle of the balanced node is 0; θn,max and θn,min are the upper and lower limits of the voltage angle of node n, respectively; PLine,max and PLine,min are the upper and lower limits of the output power for the line, respectively.

(5)   Backup virtual inertia enhancement constraint

0≤ΔHsp,t+1Vi−ΔHsp,tVi≤ΔHsp,maxVi(24)

where, ΔHsp,maxVi is the maximum increment of the virtual inertia of the renewable energy units per unit time.

(6)   Inertia demand constraint of new power system

Hsys,t=∑t=1T∑k=1NGPNkGHkGuk,tG+∑t=1T∑a=1NWPNaWHaWua,tW+∑t=1T∑b=1NPVPNbPVHbPVub.tPV(25)

where, Hsys,t is the overall inertia of the system at time t; PNkG, PNaW and PNbPV are the rated power of the thermal power unit k, the wind power airport a, and the photovoltaic power station b, respectively; HkG, HaW and HbPV are the inertia constants of the thermal power unit k, the wind farm a, and the photovoltaic power station b, respectively.

According to GB 38755-2019 “Guidelines for Safety and Stability of Power Systems”, the criterion for frequency stability of power systems refers to the ability of the system frequency to quickly recover to near the rated frequency. Therefore, this article takes the Rate-of-Change-of-Frequency (RoCoF) and Maximum Frequency Deviation (MFD) as the minimum inertia requirements for the new power system for frequency safety constraints.

Hsys,t≥max{HRoCoF,HMFD}(26)

(7)   RoCoF and MFD constraints

Analogous to the rotor motion equation of a synchronous generator, it can characterize the relationship between the inertia Hsys and frequency fsys of a new power system. Since mechanical power cannot undergo sudden changes, the mechanical power required for frequency regulation can be considered as 0. The specific formula is:

2HsysdfsysfNdt=−ΔP(27)

where, fN is the rated frequency of the system, ΔP is the system disturbance unbalanced power, and the maximum value of RoCoF usually occurs at the initial moment of disturbance when disturbance unbalanced power occurs. Therefore, the minimum inertia requirement expression of the new power system under RoCoF constraint is as follows:

{dfsys0+dt=−ΔP2Hsys≥dfsysmaxdtHsys≥−ΔP2dfsysmaxdt=HRoCoF(28)

Maximum Frequency Deviation (MFD) generally refers to the decrease in frequency value to the lowest point after the system experiences an active power shortage, following the inertia response and primary frequency modulation stage. This article uses a linear model to approximate the maximum frequency deviation of a new power system. Before the primary frequency modulation of the system, it is assumed that the active power deficit of the system remains constant, and this period is recorded as the frequency modulation dead time. The specific expression is as follows:

td=2HsysfdfNΔP(29)

where, fd is the frequency corresponding to the dead time of the primary frequency regulation of the new power system.

The simplified expression for the frequency deviation Δf of the new power system is as follows:

Δf=fd+fNΔP24Hsysvsys(30)

where, vsys is the primary frequency regulation rate of the new power system.

The frequency deviation of the new power system should be within the allowable range of the maximum frequency deviation Δfmax specified in the “Guidelines for Power System Safety and Stability”, with the following constraints:

Hsys≥fNΔP24vsys(Δfmax−fd)=HMFD(31)

3.2 Pre-Deployment of Module Mobile Energy Storage

To mitigate insufficient virtual inertia supply, this paper adopts a pre-deployment strategy for Modular Mobile Energy Storage Systems (MMESS), planning the quantity, location, and routing of mobile energy storage under fault scenarios. First, the number and locations of energy storage units are determined by forecasting potential fault scenarios along the typhoon path using meteorological predictions, ensuring rapid deployment during emergencies. Second, the scheduling of mobile energy storage ensures that nodal rates of frequency change comply with frequency safety constraints. Finally, the transportation and load-shedding costs of mobile energy storage are optimized to achieve both economic efficiency and system stability under extreme weather events.

3.2.1 Objective Function

Construct an objective function CMMESSmin to minimize the transportation and operating costs of assembled mobile energy storage, and optimize and adjust the objective function Csysmin to minimize the system operating costs. The specific expression is as follows:

Csys,MMESSmin=Csysmin+CMMESSmin=Csysmin+∑t=1T∑s=1NM∑L=1NL(FsMEs,tMMESSΩs,L,t+RsMDs,L,tΩs,L,t0)(32)

where, NM represents the number of mobile energy storage devices; FsM is the operating cost of mobile energy storage per unit capacity; Es,tMMESS is mobile energy storage capacity for time; Ωs,L,t is binary variable representing the state of mobile energy storage at node L at time t, with a connection of 1 and a disconnection of 0; RsM schedules the cost of mobile energy storage per kilometer; Ds,L,t determines the initial distance between the node L and the mobile energy storage s at time t. This article sets the parameters related to mobile energy storage based on reference [19], as shown in Appendix B Table A2.

3.2.2 Constraints

(1)   Energy storage configuration constraints

Considering the occurrence of multiple fault environments during typhoon disasters, in order to avoid fluctuations in the output of energy storage equipment and residual active reserve, which may affect its operational economy, the constraint expression is as follows:

{∑L∈NLβs,L,tMMESS⩽1∑s∈NMβs,L,tMMESS⩽NLmax(33)

where, βs,L,tMMESS is the number of configurations at node L at time t for MMESS, and NLmax is the maximum number of node L that can access MMESS. Eq. (33) indicates that a certain MMESS can only access one bus node at a certain time.

(2)   Energy storage scheduling constraints

The scheduling constraints for mobile energy storage include charging and discharging constraints (34) and capacity constraints (35), expressed as follows:

{0⩽Ps,L,tMMESS,ch⩽αs,L,tMMESS,chPmaxMMESS,ch0⩽Ps,L,tMMESS,dch⩽αs,L,tMMESS,dchPmaxMMESS,dchαs,L,tMMESS,ch+αs,L,tMMESS,dch⩽βs,L,tMMESS(34)

{Es,t+1MMESS=Es,tMMESS+(Ps,L,tMMESS,chλMMESS,ch−Ps,L,tMMESS,dchλMMESS,dch)ΔtEmaxMMESSδminMMESS⩽Es,L,tMMESS⩽EmaxMMESSδmaxMMESS(35)

where, as,L,tMMESS,ch and as,L,tMMESS,dch represent the charging and discharging states of the mobile energy storage s on the node L at time t in the fault scenario. If the mobile energy storage is in a charging state, then, as,L,tMMESS,ch=1 and if it is in a discharging state, then as,L,tMMESS,dch=1. PmaxMMESS,ch and PmaxMMESS,dch are the maximum power that mobile energy storage can charge and discharge per unit time. Es,tMMESS is capacity of mobile energy storage s at time t in the fault scenario. λMMESS,ch and λMMESS,dch are the charging and discharging efficiencies of mobile energy storage, respectively. δminMMESS and δmaxMMESS the minimum and maximum charging load coefficients for mobile energy storage, respectively.

(3)   Path planning constraints

In typhoon disasters, fault scenarios occur sequentially over time, and the set of path schemes for mobile energy storage vehicles to travel from one fault point to another can be determined using Breadth First Search (BFS) algorithm [36]. At the same time, within the scheduling time interval, the path shall not exceed the maximum operating distance of the mobile energy storage vehicle. The specific expression is as follows:

|Ds,t+1−Ds,t|⩽vmaxMMESSΔt(36)

where, Ds,t is the position of the module mobile energy storage system s at time t. vmaxMMESS represents maximum driving speed for mobile energy storage. Taking the mobile energy storage driving speed proposed in reference [20] as an example, the maximum speed of mobile energy storage is set to 40 km/h.

The strategy proposed in this study consists of three main models, as illustrated in Fig. 3. First, the nodal inertia distribution evaluation model is solved by incorporating the disturbance power resulting from the faulted line together with the relevant grid parameters. The collaborative optimization inertia scheduling model then utilizes inertia demand constraints and frequency safety constraints as boundary conditions to derive the initial unit commitment strategy. Next, the virtual inertia replenishment cost and mobile energy storage deployment cost at different time periods are used as allocation parameters within the particle swarm optimization (PSO) algorithm to calculate the system’s inertia demand and determine the corresponding inertia replenishment scheme. Finally, the results obtained from the collaborative optimization strategy model are re-introduced into the nodal inertia distribution evaluation model to iteratively refine the nodal frequency responses. Through this iterative interaction between the two models, the unit commitment and inertia allocation are continuously optimized to achieve the economically optimal strategy while ensuring system frequency stability.

Figure 3: Collaborative optimization strategy

4  Case Study

4.1 Parameters of System

To verify the economy and stability of the new virtual inertia optimization scheduling model for power systems, this paper studies the simulation experiment of power system under extreme weather events as typhoons. The damage time of the transmission line caused by typhoon disasters was first simulated, with a duration of 10:00–14:00 and a total simulation time of 00:00–24:00. Among them, during the period of 10:00–14:00, typhoons successively damaged the lines L16–L21, L17–L18, L25–L26, and L28–L29. Due to the uncertainty of the repair time of the lines, this article assumes that the damaged lines during the entire simulation period were not successfully repaired [37]. The damage caused by typhoons along their trajectories is modeled as busbar faults set, and the recovery time of affected transmission lines and routes is represented by the duration of the corresponding busbar fault set. The proposed collaborative optimization strategy dynamically optimizes the inertia distribution of nodes based on these busbar fault sets, making it applicable to different fault scenarios. During the undamaged period, the system did not experience any faults but still required inertia support. This article adopts the N-1 principle and sets the system’s power supply capacity loss at 8%–12%, which is not included in the actual load shedding cost. The maximum frequency change rate is set at 0.5 Hz/s, and the maximum frequency deviation is 0.7 Hz.

This paper uses an improved IEEE39 node system to verify the practicality and effectiveness of the scheduling model. The node system network topology diagram is shown in Fig. 4. Wind farms are set up at busbars 8, 30, 34, and 35, photovoltaic power stations are set up at busbars 31, 32, and 33, and thermal power units are set up at busbars 36, 37, 38, and 39. The wind farm at busbar 8 uses GFL inverters to control the power supply to provide virtual inertia, while other new energy power stations use GFM inverters to control the power supply to provide virtual inertia. Considering that the photovoltaic power station studied in this article is located far away from the typhoon affected area, it can be reasonably assumed that it is less affected by typhoon disasters. The total inertia of the system is 18,450 MWs, of which thermal power units can provide 10,500 MWs, and the rest are provided by new energy units. At the same time, 10% of the capacity of new energy units is used as backup capacity. The parameter settings of the units, the expected output curve of the wind farm, and the system load curve are detailed in Appendix C. This model is simulated and solved by calling the Cplex solver in the Yalmip toolbox on the Matlab 2024a platform. The PSO algorithm is configured to run for a maximum of 300 iterations. The optimal replenishment results are determined either when the iteration count reaches 300 or when the convergence error falls below the predefined threshold.

Figure 4: Network topology of improved IEEE 39 bus system

To verify the effectiveness of the strategy, this paper designs four cases for simulation analysis. The setting of the virtual inertia cost coefficient in this study considers the hardware, maintenance, and technical costs associated with establishing backup virtual inertia for renewable energy units [38]. Based on these factors, reasonable and representative assumptions were made for the unit virtual inertia cost coefficient. Furthermore, since the actual cost of virtual inertia provision may vary with different technologies, converter types, and system configurations, two distinct cost values were selected for sensitivity analysis to evaluate the robustness and stability of the proposed optimization strategy.

Case 1: This model only considers the inertia requirements of the system for routine unit scheduling, without considering backup virtual inertia replenishment.

Case 2: This model considers the inertia requirements of both the system and nodes for routine unit scheduling, with considering backup virtual inertia replenishment and replenishment cost at $150/MWs.

Case 3: This model considers the inertia requirements of both the system and nodes for routine unit scheduling, with considering backup virtual inertia replenishment and replenishment cost at $150/MWs. And, modular mobile energy storage pre-deployment is added.

Case 4: This model considers the inertia requirements of both the system and nodes for routine unit scheduling, with considering backup virtual inertia replenishment and replenishment cost at $300/MWs. And, modular mobile energy storage pre-deployment is added.

4.2 Simulation Result Analysis

The actual inertia and the minimum inertia requirements of the optimized system are shown in Fig. 5. The RoCoF of each node under different cases is shown in Fig. 6. The backup virtual inertia replenishment for new energy power plants is shown in Table 1 [39]. The disturbance power under load supply imbalance at different time periods is shown in Table 2. The parameters of each unit are shown in Tables A3 and A4 of Appendix C. The start stop status of thermal power units is shown in Table A5 of Appendix C, where a value of 1 represents the operating state and 0 represents the shutdown state.

Figure 5: Comparison of system inertia and inertia requirements under different cases

Figure 6: Comparison of RoCoF across nodes, cases, and time instants

According to the analysis of Figs. A1, A2, 5 and 6, Tables 1, 2 and A5, it can be seen that the line faults caused by typhoons significantly enhance the frequency spatiotemporal imbalance of the system. At 18, 19, and 20 h, the output of the photovoltaic power station weakened and disappeared due to reduced sunlight, resulting in an imbalance in the system’s load supply. In Case 1, the scheduling model does not consider the demand for node frequency stability, and only makes scheduling decisions according to its own system inertia and load demand. Therefore, most thermal power plants participate in power supply to balance the load loss under fault conditions, but the inertia response mode is single, the unit inertia remains constant, and the moment of inertia is only controlled by the synchronous machine startup mode, which cannot change in real time according to the node frequency demand. At this time, the frequency change rate of bus 12 at 18, 19, and 20 h, respectively, reaches −0.5536, −0.6298, and −0.519 Hz/s, which exceeds the frequency change rate limit of 0.5 Hz/s required in this article, and has a significant impact on the stability of the system.

In order to further improve the stability of node frequency, in Case 2, this paper added backup capacity for new energy power plants. By adjusting the virtual synchronous machine inertia constant controlled by the grid type inverter, additional inertia replenishment was provided to nodes with frequency imbalance. Considering the network topology and inertia-weighted profitability, Scheme 2 allocates all available backup virtual inertia to wind farms and photovoltaic power plants at 18 and 19 h. At 20 h, according to the optimization results, Case 2 activates only 77.5 and 120 MW of backup virtual inertia for the wind farms located at busbars 30 and 34, respectively. The results show that the proposed strategy improves the overall frequency recovery rate by 5.989% compared to Case 1, while the maximum Rate of Change of Frequency (RoCoF) at Bus 12 decreases by 8.19%, 7.71%, and 3.65% during the 18–20 h period, respectively. Despite these improvements, the RoCoF at Bus 12 at 18 and 19 h still exceeds the frequency safety limit, indicating that the backup capacity of renewable power plants has been fully utilized. Without additional inertial compensation measures, the frequency security of the system could be severely compromised under such operating conditions.

In order to solve this problem, in Case 3, a modular mobile energy storage pre layout method was added based on the predicted fault scenario of typhoon disasters to cope with the insufficient virtual inertia replenishment. Considering the node load demand and inertia replenishment benefits, modular mobile energy storage was deployed at bus 28 at 19 h to timely supply output to nodes with imbalanced load supply, and combined with backup virtual inertia replenishment. Compared with Case 2, the total backup virtual inertia of the wind farms decreased by 909.7 MWs, while that of the photovoltaic power plants decreased by 91.9 MWs. In Case 3, the coordinated optimization of mobile energy storage deployment effectively addresses the issue of frequency instability at certain nodes caused by the hardware limitations of backup virtual inertia in renewable energy plants. Meanwhile, the backup virtual inertia complements the spatial limitation of mobile energy storage, which cannot always replenish inertia at distant nodes in a timely manner.

Compared with Case 1, the proposed strategy in Case 3 increases the overall frequency recovery rate by 10.9%, and reduces the maximum Rate of Change of Frequency (RoCoF) at Bus 12 to −0.5 Hz/s across all time periods, achieving an average improvement of 11.31% in RoCoF. These results demonstrate that the proposed strategy satisfies both system-wide and nodal frequency safety constraints while significantly enhancing frequency stability.

In Case 4, to highlight the sensitivity of the proposed strategy to virtual inertia costs, the economic cost of backup virtual inertia was increased from $150/MW to $300/MW. Under this condition, the marginal benefit–cost ratio of MMESS becomes more favorable than that of backup virtual inertia provision. Consequently, compared with Case 3, which utilizes only one MMESS, Case 4 employs a second MMESS as the primary inertia supply source, fully replacing the backup virtual inertia from renewable energy power plants. As a result, the maximum Rate of Change of Frequency (RoCoF) at Bus 12 decreased by 14.62%, 25.68%, and 10.46% during the 18–20 h period compared with Case 1, demonstrating a significant improvement in nodal frequency stability. Although the improvement achieved in Case 4 is more significant than in Case 3, the replenishment capacity of each mobile energy storage unit is limited and subject to spatial constraints. Under multiple disturbances, optimizing nodal inertia support often requires activating several mobile energy storage units simultaneously. This can easily lead to over-supply from certain units, while insufficient activation may fail to meet frequency stability requirements. Consequently, the allocation of inertial resources becomes redundant, and precise control of economic costs is difficult to achieve, resulting in inefficient utilization of resources.

Through Fig. 7, the spatial distribution of system inertia under four different schemes can be directly observed, reflecting the influence of the topological position of the system line on the equivalent inertia of nodes. Due to the disconnection of lines L16–L21, L17–L18, L25–L26, and L28–L29, the equivalent inertia level of nearby nodes is low, and the anti-interference ability is significantly reduced. However, the inertia distribution near bus 5 is more concentrated, and the equivalent inertia value of the node is larger, indicating that it is less affected by line disconnection and has stronger anti-interference ability. By comparing the inertia levels of different schemes, it can be seen that the inertia of the same node changes significantly in different cases and at different times, which conforms to the spatiotemporal distribution characteristics of the system’s inertia. At the same time, the differentiated replenishment of backup virtual inertia and the pre layout configuration of mobile energy storage make the frequency response indicators of each node meet safety constraints, further improving nodes with weak inertia levels in the system and improving system stability.

Figure 7: Nodal inertia at different times in cases 1–4

Table 3 shows the detailed operating costs for four different cases. In this paper, the penalty cost for wind power curtailment is set at $120/MWh, the penalty cost for solar power curtailment is set at $100/MWh, and the reference [40] cost for load shedding is set at $500/MWh.

By comparing and analyzing the data in Table 3, Figs. 5 and 6, it can be seen that the inertia scheduling model generates different costs in the four cases. Case 1 only met the system inertia requirement by adjusting the scheduling of the thermal power plant, with an actual total cost of only $901,384, but ignored the frequency stability requirement of the nodes, resulting in incalculable potential economic losses. Case 2 effectively mitigated the RoCoF exceedance issue by incorporating 1547.5 MWs of additional backup virtual inertia. Compared with Case 1, although the increased inertia investment led to a 25.75% rise in total economic cost, the overall system and nodal frequency stability were significantly enhanced, thereby improving the operational security and robustness of the power system. However, relying solely on this method for supply cannot handle complex system disturbances and poses a safety hazard.

Case 3 enriches the method of virtual inertia replenishment through modular mobile energy storage pre layout, enabling flexible calling of virtual inertia to meet the frequency stability requirements of nodes in all time periods. Compared with Case 2, Case 3 achieved a reduction of $15,000 in load-shedding penalty cost and $150,270 in virtual inertia cost through the coordinated configuration of modular mobile energy storage systems (MMESS). As a result, the total economic cost decreased by 4.9%, effectively mitigating the risk of system frequency imbalance. In Case 4, a sensitivity analysis of the backup virtual inertia cost coefficient was conducted to assess the marginal economic benefits of inertia provision. By increasing the cost coefficient to $300/MW·s, the model reduced the load-shedding penalty cost by $30,000 and fully substituted the renewable-based virtual inertia with backup inertia sources, leading to a 3.5% reduction in total cost compared with Case 2. However, despite meeting frequency stability requirements, Case 4 exhibited a loss of fine economic control, resulting in partial resource inefficiency compared to Case 3. These findings indicate that backup virtual inertia and mobile energy storage play complementary roles in formulating optimal scheduling strategies. Appropriately coordinating virtual inertia allocation with MMESS deployment can enhance system frequency stability while maintaining economic efficiency and operational robustness.

5  Conclusion

This paper proposes a collaborative optimization strategy for virtual inertia spatiotemporal distribution replenishment under extreme weather events. The strategy integrates the reserve capacity of renewable power plants with the pre-deployment of mobile energy storage, aiming to meet system inertia requirements and ensure nodal frequency stability, while simultaneously minimizing the economic cost of inertia provision. It is designed for new power systems with a high penetration of renewable energy sources, aiming to provide reliable inertia support under extreme conditions such as typhoon disasters, thereby improving both the reliability and the economic efficiency of system operation. Finally, the effectiveness of the proposed optimization scheduling strategy is validated through simulations, and the main conclusions are summarized as follows:

(1)   Line faults induced by extreme weather events intensify the spatial distribution characteristics of system frequency. To address this, this paper incorporates nodal frequency differences into inertia evaluation and develops a quantitative model for the spatiotemporal distribution of nodal equivalent inertia. The proposed model enables accurate assessment of the frequency response of renewable generators and synchronous generators with GFM inverter interfaces in new power systems under extreme weather conditions. Simulation results show that the model enhances the maximum nodal frequency deviation by 15.2%, and reduces the maximum nodal Rate-of-Change-of-Frequency (RoCoF) by 11.4% compared with traditional global inertia assessment methods. Furthermore, it quantitatively characterizes the influence of different types of inertia resources on nodal frequency stability within the network topology, thereby ensuring frequency stability at the nodal level while simultaneously optimizing the economic benefits of inertia provision.

(2)   The virtual inertia replenishment strategy proposed in this paper emphasizes the dynamic variation of virtual inertia levels in renewable-based power systems. It enables accurate evaluation of the inertia support capability provided by renewable generators to nodes under multiple fault disturbances across different time periods. Moreover, the strategy integrates differentiated backup inertia replenishment from renewable power plants with the flexible configuration of modular mobile energy storage, thereby establishing a multi-objective economic scheduling framework. To address multiple random disturbances within the expected fault set of extreme weather events, the proposed approach coordinates diverse virtual inertia resources to ensure the stable operation of the power system. Numerical analysis indicates that the proposed strategy enhances the overall frequency recovery rate by over 10%, and achieves a total operating cost reduction of 4.9% compared with the benchmark scheduling scheme.

The collaborative optimization strategy proposed in this study relies on virtual inertia to supplement nodal inertia but does not consider the probabilistic forecasting of virtual inertia from renewable energy sources, thereby limiting its ability to capture inherent randomness. Future research will focus on developing integrated optimization strategies that incorporate machine-learning-assisted inertia prediction, energy storage infrastructure planning, and AI-based decision-making. These efforts aim to further enhance the frequency resilience and operational reliability of high-renewable-penetration power systems under extreme weather events.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Taotao Zhu and Pai Pang; methodology, Taotao Zhu and Pai Pang; software, Taotao Zhu; validation, Yang Wang, Taotao Zhu and Pai Pang; formal analysis, Yang Wang and Pai Pang; investigation, Taotao Zhu and Pai Pang; resources, Yang Wang and Pai Pang; data curation, Taotao Zhu; writing—original draft preparation, Taotao Zhu; writing—review and editing, Yang Wang and Pai Pang; visualization, Taotao Zhu; supervision, Yang Wang; project administration, Yang Wang. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The authors confirm that the data supporting the findings of this study are available within the article.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Appendix A

Appendix B

Appendix C

Figure A1: Predictive output curves of new energy power plant and load

Figure A2: Three-dimensional diagram of the output of power plants

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