Optimal Battery Storage Configuration for High-Proportion Renewable Power Systems Considering Minimum Inertia Requirements

Abstract

With the continuous development of renewable energy worldwide, the issue of frequency stability in power systems has become increasingly serious. Enhancing the inertia level of power systems by configuring battery storage to provide virtual inertia has garnered significant research attention in academia. However, addressing the non-linear characteristics of frequency stability constraints, which complicate model solving, and managing the uncertainties associated with renewable energy and load, are the main challenges in planning energy storage for high-proportion renewable power systems. In this context, this paper proposes a battery storage configuration model for high-proportion renewable power systems that considers minimum inertia requirements and the uncertainties of wind and solar power. First, frequency stability constraints are transformed into minimum inertia constraints, primarily considering the rate of change of frequency (ROCOF) and nadir frequency (NF) indicators during the transformation process. Second, using historical wind and solar data, a time-series probability scenario set is constructed through clustering methods to model the uncertainties of wind and solar power. A stochastic optimization method is then adopted to establish a mixed-integer linear programming (MILP) model for the battery storage configuration of high-proportion renewable power systems, considering minimum inertia requirements and wind-solar uncertainties. Finally, through a modified IEEE-39 bus system, it was verified that the proposed method is more economical in addressing frequency stability issues in power systems with a high proportion of renewable energy compared to traditional scheduling methods.

1. Introduction

In recent years, to meet China’s requirements for low-carbon transformation, the development of new energy sources in China has been rapid [1,2]. According to data from the National Energy Administration of China, the cumulative installed capacity of new energy is expected to exceed 1.4 billion kilowatts by the end of 2025 [3]. However, the characteristics of renewable energy generation, such as intermittency, volatility, and randomness [4], add more uncertainty to power grid planning. On the one hand, as the installed capacity of wind and photovoltaic power continues to increase, the issues of power imbalance and curtailment of wind and solar energy caused by large-scale renewable energy integration are becoming more severe [5,6]. On the other hand, in the context of high penetration of renewable energy, the share of thermal power generation is continuously being replaced by wind and photovoltaic power, leading to a persistent decline in power grid inertia and disturbance resistance [7,8]. This significantly impacts the frequency stability of the power grid. Battery storage plays a crucial role in power grids. It can not only work in conjunction with pumped storage units to achieve peak shaving and valley filling, thereby improving the absorption rate of renewable energy, but also provide virtual inertia to stabilize grid frequency. Additionally, compared to other storage devices such as supercapacitors, battery storage has a higher energy density, allowing it to support long-term grid dispatch and meet the requirements for peak shaving and valley filling. Therefore, in energy storage configuration models for power systems with a high proportion of renewable energy, battery storage is more suitable than supercapacitors. Nevertheless, in future research on power system dispatch that considers frequency stability, the combination of battery storage and supercapacitors will offer significant research potential.

Embedding frequency stability constraints into the optimal operation of power systems, and accurately characterizing the uncertainties of large-scale renewable energy, are key to formulating a battery storage planning scheme for high-proportion renewable power systems that considers frequency stability.

In the research on embedding frequency stability constraints into the operation of power systems, Gu et al. [9] established an economic dispatch algorithm for renewable energy grids considering synchronous inertia constraints to meet the minimum synchronous inertia required for frequency control. The results demonstrate that this algorithm can consistently ensure the safety of frequency control during dynamic processes. Zhang et al. [10] proposed the concept of frequency security margin to quantify the frequency regulation capability of the system under emergencies and established a frequency-constrained unit commitment model considering frequency security margin. Finally, the effectiveness of this model was verified through case analysis. Zhang et al. [11] proposed a new linear frequency security constraint method that takes into account more details of frequency response dynamics. Then, a method was designed to assess the accuracy of these frequency constraints. Sedighizadeh et al. [12] considered the uncertainty of wind power and the energy storage in its modeling, and established a new optimal scheduling model that incorporates power grid frequency dynamic constraints. Through case studies, the proposed model was validated, demonstrating its effectiveness in ensuring frequency security while reducing operational costs. Hu et al. [13] studied the frequency response process of high-penetration renewable energy grids and the key indicators in the frequency response process: the rate of change of frequency and the maximum frequency deviation. Based on these perspectives, it proposed an evaluation method for power system inertia and primary frequency response capability. This method provides practical guidance for the planning, construction, and operation of high-penetration renewable energy grids.

The aforementioned research holds significant reference value but also faces two main issues: (1) The non-linear characteristics of frequency stability-related constraints make model solving challenging. Existing studies often employ methods such as time-domain simulation or optimization algorithms to approximate the NF value. However, these methods tend to have high complexity in theoretical derivation and model solving, requiring substantial technical and computational resources, limiting their widespread use in practical power system applications; (2) most existing literature focuses on solving frequency constraint issues from the perspective of dispatch while overlooking the potential of configuring battery storage to provide virtual inertia. In summary, the current research needs to fully address two key issues: first, how to tackle the challenges of model-solving difficulties caused by the non-linear characteristics of frequency stability-related constraints, and, second, whether the configuration of battery storage can effectively provide virtual inertia to enhance the frequency stability of the power system. Solving these issues is crucial for promoting the stable operation of power systems with a high proportion of renewable energy.

Currently, optimization methods that consider the uncertainty of renewable energy mainly include stochastic optimization, robust optimization, and distributionally robust optimization [14,15,16]. The character of robust optimization is such that it does not require the specific probability distribution of uncertain factors to be determined, but rather it makes decisions based on the worst-case scenarios [17,18,19,20,21]. However, this method often leads to overly conservative results, especially when determining the scale of configurations. Distributionally robust optimization improves upon the conservatism of robust optimization decisions. It does this by creating fuzzy sets of probability distributions and then making decisions based on the worst-case probability distribution of uncertain factors [22,23,24]. However, there are challenges in constructing fuzzy sets due to significant subjective influence, and the resulting models tend to be complex and difficult to solve.

In contrast, the stochastic optimization method assumes that the output of renewable energy follows a specific probability distribution. By selecting numerous typical scenarios from actual operations for optimization [25,26,27], this method, compared to robust optimization and distributionally robust optimization, offers the advantages of being easier to solve and yielding more economically favorable results. Additionally, the frequency stability-related constraints in this study exhibit significant non-linear characteristics, which complicates the problem-solving process. To address this challenge, we employed a method that linearizes these non-linear problems and then used a stochastic optimization approach to solve them. This approach allows us to effectively manage the complexity of the model while maintaining high computational efficiency.

In summary, current research has made significant contributions to the optimization of power system dispatch considering frequency stability constraints. However, most of the studies have focused on addressing frequency stability issues through dispatch methods, overlooking the potential of enhancing system inertia by configuring battery storage to provide virtual inertia. Additionally, the challenges in solving models due to the non-linear characteristics of frequency stability-related constraints have not been fully addressed. Therefore, to better tackle these issues, this paper proposes a battery storage configuration model for power systems with a high proportion of renewable energy, considering minimum inertia constraints. The main contributions of this paper can be summarized as follows:

• The frequency stability constraint is transformed into a minimum inertia constraint, mainly considering ROCOF and NF indicators during the transformation process. In solving NF, a piecewise linear analysis of the frequency response process is adopted, converting the non-linear frequency stability constraint into a linear minimum inertia constraint and embedding it into the optimal operation of the power system.
• Using historical wind, solar, and load data, a temporal probability scenario set is constructed using clustering methods to model the uncertainties of wind, solar, and load. Additionally, a storage configuration model is proposed for high-proportion renewable power systems, considering both frequency stability and the uncertainties of wind and solar power.

The remaining sections are organized as follows: Section 2 establishes the minimum inertia requirement evaluation model for power systems. Section 3 presents the battery storage configuration model for high-penetration renewable power systems considering the minimum inertia requirement, and introduces the linearization method of the proposed model. Section 4 conducts case studies, and Section 5 provides conclusions.

2. Assessment of Minimum Inertia Requirements for Power Systems

Inertia in power systems is reflected in the system’s ability to maintain frequency fluctuations within a safe and permissible range when faced with external disturbances or internal equipment failures. Inertia resources within the power system play a crucial role in maintaining frequency stability and reducing frequency deviations. When there is an imbalance between generation and load power, the frequency response of the power system involves four distinct time-scale processes: the inertial response immediately following a frequency disturbance, primary frequency regulation, secondary frequency regulation with adjustments in unit output characteristics, and tertiary frequency regulation. Among these, the inertia response process is supported by synchronous inertia and virtual inertia, while the primary frequency regulation process can be achieved through the co-ordinated operation of reheaters, prime movers, and governors. This paper primarily considers the inertia response process and the primary frequency regulation process in the frequency response sequence.

The purpose of assessing minimum inertia requirements is to ensure power system frequency stability. The frequency response characteristics of the power system can be represented by the rotor motion equation [28]:

$${\displaystyle \frac{2H_{\mathrm{s}}}{f_{\mathrm{N}}}}{\displaystyle \frac{d\Delta f_t}{dt}}+D_{\mathrm{s}}\Delta f_t=\Delta P_{\sum }-\Delta P_{\mathrm{l}}$$

(1)

where $D_{\mathrm{s}}$ is the damping coefficient during the power system frequency response process; $H_{\mathrm{s}}$ is the inertia level of the power system; $f_{\mathrm{N}}$ is the power system rated frequency; $\Delta f$ is the real-time frequency variation of the power system; $\Delta P_{\sum }$ is the frequency regulation power of all dispatchable resources; $\Delta P_{\mathrm{l}}$ is the power deficit of the power system.

The inertia response spans the entire frequency response process and is a crucial support for maintaining power system frequency stability. Factors that hinder frequency changes during this stage can be broadly defined as power system inertia. Generalized power system inertia includes synchronous machine inertia, static load voltage characteristics, and virtual inertia generated by virtual synchronous machine technology. These factors collectively act during this stage, releasing rotor kinetic energy to suppress frequency changes.

ROCOF and the minimum frequency point of the power system (taking load increase as an example) are key indicators of frequency stability during disturbances. Under a certain disturbance power, the factors affecting the instantaneous ROCOF are inertia and load voltage characteristics. The factors influencing the power system’s minimum frequency point include inertia, primary frequency regulation of generators, load frequency regulation, and load voltage characteristics. The minimum inertia requirement of the power system can be assessed based on these two aspects of frequency constraints and influencing factors.

2.1. Minimum Inertia Based on ROCOF Constraints

The extreme value of the power system frequency rate of change usually occurs at the moment of a power system fault. According to the rotor motion equation, the calculation expression for the power system’s maximum rate of frequency change can be obtained, as shown in Equation (2). The constraint expression for the power system’s maximum rate of frequency change is shown in (3).

$$Roco{f}_{\mathrm{M}}={\displaystyle \frac{d\Delta f_t}{dt}}{|}_{t=0_{+}}=-{\displaystyle \frac{f_{\mathrm{N}}\Delta P_{\mathrm{l}}}{2H_{\mathrm{s}}}}$$

(2)

$$\left|Roco{f}_{\mathrm{M}}\right|\le Roco{f}_{\max }$$

(3)

where $Roco{f}_{\mathrm{M}}$ and $Roco{f}_{\max }$ are, respectively, the maximum rate of frequency change of the power system and the allowable upper limit of the power system’s rate of frequency change. Furthermore, the minimum inertia based on ROCOF constraints at this time can be obtained, as shown in Equation (4).

$$H_{\mathrm{R}}={\displaystyle \frac{f_{\mathrm{N}}\Delta P_{\mathrm{l}}}{2Roco{f}_{\max }}}$$

(4)

2.2. Minimum Inertia Based on NF Constraints

Considering the frequency response modes of various dispatchable resources in a high-penetration renewable energy power system and the rotor motion equation, the maximum power system frequency deviation level after a fault can be calculated. This needs to satisfy the power system’s transient frequency deviation extreme value constraint, as shown in (5).

$$\Delta f_{\mathrm{NF}}\le \Delta f_{\max }$$

(5)

where $\Delta f_{\mathrm{NF}}$ is the maximum frequency deviation, and $\Delta f_{\max }$ is the allowable maximum frequency deviation of the power system.

The above expression represents a non-linear constraint. In previous studies, the System Frequency Response (SFR) model was often used for time-domain simulation or simplified by model order reduction to solve this issue. However, as the number of units in the power system increases and the control strategies of renewable energy units vary, the order of the SFR model rises, making model establishment very complex and limiting the efficiency of model solution.

This paper adopts piecewise linearization techniques to simulate the power system frequency response process [29]. The calculation process is shown in Equations (6)–(9).

By integrating Equation (1) from $0$ to $t_{db}$ and solving it, Equation (6) is obtained.

$$t_{\mathrm{db}}={\displaystyle \frac{4H_{\mathrm{s}}{f}_{\mathrm{db}}}{f_{\mathrm{N}}(2\Delta P_{\mathrm{l}}-D{f}_{\mathrm{db}})}}$$

(6)

where $f_{\mathrm{db}}$ and $t_{\mathrm{db}}$ are, respectively, the dead zone and activation time of primary frequency control.

Additionally, by using the power system’s primary frequency control rate to represent the time from the activation of stabilization measures to the frequency reaching the transient extreme value, the time for the power system frequency to reach its extreme value can be expressed by Equation (7).

$$t_{\mathrm{NF}}=t_{\mathrm{db}}+{\displaystyle \frac{\Delta P_{\mathrm{l}}}{R_{\mathrm{s}}}}$$

(7)

By integrating Equation (1) from $0$ to $t_{\mathrm{NF}}$, and using linearization methods for solving it, the result is shown in Equation (8).

$$f_{\mathrm{NF}}-f_{\mathrm{N}}={\displaystyle \frac{f_{\mathrm{N}}}{2H_{\mathrm{s}}}}(-\Delta P_{\mathrm{l}}{t}_{\mathrm{db}}-{\displaystyle \frac{\Delta P_{\mathrm{l}}^2}{2R_{\mathrm{s}}}})-{\displaystyle \frac{f_{\mathrm{N}}D(f_{\mathrm{NF}}-f_{\mathrm{N}})t_{\mathrm{NF}}}{4H_{\mathrm{s}}}}$$

(8)

Combining Equations (6)–(8), we can derive Equation (9).

$$\begin{array}{l}{H}_{\mathrm{NF}}=-{\displaystyle \frac{p_1+p_2+p_3+p_4}{q}}\\ p_1=2t_{\mathrm{db}}{f}_{\mathrm{N}}{R}_{\mathrm{s}}\Delta P_{\mathrm{l}}\\ p_2=f_{\mathrm{N}}\Delta P_{\mathrm{l}}^2\\ p_3=t_{\mathrm{db}}D{f}_{\mathrm{N}}{R}_{\mathrm{s}}(f_{\min }-f_{\mathrm{N}})\\ p_4=t_{\mathrm{db}}D{f}_{\mathrm{N}}\Delta P_{\mathrm{l}}(f_{\min }-f_{\mathrm{N}})\\ q=4R_{\mathrm{s}}(f_{\min }-f_{\mathrm{N}})\end{array}$$

(9)

where $H_{\mathrm{NF}}$ is the minimum inertia requirement based on NF constraints.

2.3. Minimum Inertia Requirement of the Power Systems

The power system inertia must simultaneously be higher than the minimum inertia obtained from both Equations (4) and (9) to ensure frequency stability during the power system operation. Therefore, the minimum inertia requirement of the power system is given by Equation (10).

$$H_{\min ,t}^{\mathrm{s}}=\max \{H_{\mathrm{R}},H_{\mathrm{NF}}\}$$

(10)

The real-time inertia of the power system can be obtained using Equation (11).

$$H_{\mathrm{s},t}={\displaystyle \sum _{i=1}^{Gen\_n}{P}_{\mathrm{Gen},i}}{H}_{\mathrm{Gen},i}{u}_{i,t,s}+{\displaystyle \sum _{i=1}^{ps\_n}{P}_{\mathrm{PS},i}{H}_{\mathrm{PS},i}{U}_i^t}+P_{\mathrm{ES}}{H}_{\mathrm{ES}}$$

(11)

where $Gen\_n$ and $ps\_n$, respectively, are the total number of conventional thermal power units and pumped storage units; $P_{\mathrm{Gen},i}$ and $H_{\mathrm{Gen},i}$, respectively, are the rated power and inertia time constant of synchronous units; $P_{\mathrm{PS},i}$ and $H_{\mathrm{PS},i}$, respectively, are the rated power and inertia time constant of pumped storage units; $P_{\mathrm{ES}}$ and $H_{\mathrm{ES}}$, respectively, are the rated power and inertia time constant of battery storage.

Although wind turbines can simulate virtual synchronous generator (VSG) inertia to participate in primary frequency control, doing so can result in significant wind curtailment, which underutilizes wind energy resources. Therefore, VSG technology from wind turbines is not considered for providing virtual inertia in this power system. Instead, virtual inertia in this power system is primarily provided by battery storage.

3. Battery Storage Configuration Model for a High Penetration Renewable Power System Considering Minimum Inertia Requirements

3.1. Objective Function

Considering the inertia support capability provided by battery storage, a model is developed with the objective function of minimizing the comprehensive operating cost of the power system while meeting the minimum inertia requirements. Additionally, when considering the uncertainty of renewable energy output in the power system, the maximum available active power output of renewable energy plants, influenced by frequent changes in wind speed and solar irradiance, becomes an uncertain stochastic variable instead of a fixed forecast value. This uncertainty needs to be described using probability distribution functions or uncertainty sets. Consequently, the power system optimization scheduling model transforms into an uncertainty optimization model. This paper employs a multi-scenario stochastic programming method to solve the grid-side battery storage optimal configuration problem, considering source-load uncertainties and minimum inertia requirements. The objective function is given by Equation (12).

$$\min F=C_1+C_2+C_3+C_4$$

(12)

$$C_1={\displaystyle \sum _{s=1}^{N\_s}{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_G}[p(f_{i,s}}}(P_{Gi,t,s})u_{i,t,s}+S_{i,t,s}+A_{i,t,s})]}$$

(13)

$$C_2={\displaystyle \sum _{s=1}^{N\_s}{\displaystyle \sum _{t=1}^T[p{C}_{cur}(P_{\mathrm{W},t,s}^{\mathrm{A}}-P_{\mathrm{W},t,s}}}+P_{\mathrm{PV},t,s}^{\mathrm{A}}-P_{\mathrm{PV},t,s})]$$

(14)

$$C_3={\displaystyle \sum _{s=1}^{N\_s}{\displaystyle \sum _{t=1}^T[p{C}_{\mathrm{PS}}}}(P_{\mathrm{PSG},t,s}+P_{\mathrm{PSP},t,s})]$$

(15)

$$C_4=[{\displaystyle \frac{1}{365}}{\displaystyle \frac{i{(i+1)}^{T_{\mathrm{y}}}}{{(i+1)}^{T_{\mathrm{y}}}-1}}](E_{\mathrm{ES}}{C}_{\mathrm{E}}+P_{\mathrm{ES}}{C}_{\mathrm{P}}+C_{\mathrm{H}}{H}_{\mathrm{ES}})$$

(16)

where $f_{i,s}$ is the fuel cost of conventional thermal power units; $P_{Gi,t,s}$ is the output power of conventional thermal power units; $S_{i,t,s}$ is the start–stop cost of conventional thermal power units; $A_{i,t,s}$ is the deep peak-shaving cost of conventional thermal power units; $u_{i,t,s}$ is the operational status of the unit (1 indicates operation, 0 indicates shutdown); $C_{cur}$ is the unit cost of wind and solar curtailment; $P_{\mathrm{W},t,s}^{\mathrm{A}}$ and $P_{\mathrm{PV},t,s}^{\mathrm{A}}$, respectively, are the forecasted outputs of wind and solar power; $E_{\mathrm{ES}}$ is the capacity of the battery storage configuration; $P_{\mathrm{ES}}$ is the maximum power rating of the battery storage configuration; $H_{\mathrm{ES}}$ is the additional virtual inertia of the battery storage configuration; $p$ is the probability of the typical day.

3.2. Constraints

A combined power system with a high proportion of renewable energy, pumped storage, and battery storage needs to satisfy constraints on wind and solar power output, thermal power operation, pumped storage operation, battery storage operation, and other constraints. To address the impacts of uncertainties in wind and solar power output, this paper employs a stochastic optimization method to handle the uncertainty issues.

3.2.1. Battery Storage Constraints

• Constraints on battery storage investment

$$\left\{\begin{array}{l}0\le E_{\mathrm{ES}}\le E_{\mathrm{ES}}^{\max }\\ 0\le P_{\mathrm{ES}}\le P_{\mathrm{ES}}^{\max }\\ H_{\mathrm{ES}}^{\min }\le H_{\mathrm{ES}}\le H_{\mathrm{ES}}^{\max }\end{array}\right.$$

(17)

where $E_{\mathrm{ES}}^{\max }$ and $P_{\mathrm{ES}}^{\max }$, respectively, are the upper limits for the installation of battery storage capacity and power; $H_{\mathrm{ES}}^{\max }$ and $H_{\mathrm{ES}}^{\min }$, respectively, are the upper and lower limits for the virtual inertia time constant of battery storage.

2.

Power output limits constraints for battery storage

$$\left\{\begin{array}{l}{m}_{\mathrm{ES},s}^{\mathrm{D}}\le P_{\mathrm{ES},t,s}^{\mathrm{G}}\le u_{\mathrm{G},t,s}(P_{\mathrm{ES},\max }^{\mathrm{G}}-m_{\mathrm{ES},s}^{\mathrm{U}})\\ m_{\mathrm{ES},s}^{\mathrm{D}}\le P_{\mathrm{ES},t,s}^{\mathrm{P}}\le u_{\mathrm{P},t,s}(P_{\mathrm{ES},\max }^{\mathrm{P}}-m_{\mathrm{ES},s}^{\mathrm{U}})\\ u_{\mathrm{G},t,s}+u_{\mathrm{P},t,s}\le 1\end{array}\right.$$

(18)

where $P_{\mathrm{ES},\max }^{\mathrm{G}}$ and $P_{\mathrm{ES},\max }^{\mathrm{P}}$, respectively, are the maximum discharging and charging power of the battery storage; $u_{\mathrm{G},t,s}$ and $u_{\mathrm{P},t,s}$, respectively, are the discharging and charging status variables of the battery storage; $m_{\mathrm{ES}.s}^{\mathrm{U}}$ and $m_{\mathrm{ES},s}^{\mathrm{D}}$, respectively, are the upward and downward reserves of the battery storage.

3.

State of charge (SOC) constraints for battery storage

$$\left\{\begin{array}{l}SO{C}_{\min }\le SO{C}_{t,s}\le SO{C}_{\max }\\ SO{C}_{t+1,s}=SO{C}_{t,s}+({\eta }_{\mathrm{P}}{P}_{\mathrm{ES},t,s}^{\mathrm{P}}-P_{\mathrm{ES},t,s}^{\mathrm{G}}/{\eta }_{\mathrm{G}})\Delta t\end{array}\right.$$

(19)

where $SO{C}_{\max }$ and $SO{C}_{\min }$, respectively, are the maximum and minimum capacity of the battery storage; $SO{C}_{t,s}$ is the real-time remaining capacity of the battery storage; ${\eta }_{\mathrm{P}}$ and ${\eta }_{\mathrm{G}}$, respectively, are the charging and discharging efficiencies of the battery storage.

3.2.2. Constraints on Thermal Power Units

Constraints on thermal power units mainly include output limits, ramp rate limits, and start–stop time limits, as shown in (19) to (21).

1.

Output limit constraints on thermal power units

$$\left\{\begin{array}{l}(P_{\mathrm{Gen},i,mid}+m_{\mathrm{Gen},i,s}^{\mathrm{D}})u_{i,t,s}^{\mathrm{peak}}\le P_{\mathrm{Gen},i,t,s}^{\mathrm{peak}}\le (P_{\mathrm{Gen},i,\max }-m_{\mathrm{Gen},i,s}^{\mathrm{U}})u_{i,t,s}^{\mathrm{peak}}\\ (P_{\mathrm{Gen},i,\min }+m_{\mathrm{Gen},i,s}^{\mathrm{D}})u_{i,t,s}^{\mathrm{deep}}\le P_{\mathrm{Gen},i,t,s}^{\mathrm{deep}}\le (P_{\mathrm{Gen},i,mid}-m_{\mathrm{Gen},i,s}^{\mathrm{U}})u_{i,t,s}^{\mathrm{deep}}\\ u_{i,t,s}^{\mathrm{deep}}+u_{i,t,s}^{\mathrm{peak}}\le 1\end{array}\right.$$

(20)

where $P_{\mathrm{Gen},i,\max }$ and $P_{\mathrm{Gen},i,\min }$, respectively, are the output limits of conventional thermal power units; $P_{\mathrm{Gen},i,t.s}^{\mathrm{peak}}$ and $P_{\mathrm{Gen},i,t.s}^{\mathrm{deep}}$, respectively, are the basic peak power and deep peak power of conventional thermal power units; $u_{i,t,s}^{\mathrm{peak}}$ and $u_{i,t,s}^{\mathrm{deep}}$, respectively, are the basic peak-shaving state and deep peak-shaving state of conventional thermal power units; $m_{Gen,i,s}^{\mathrm{U}}$ and $m_{Gen,i,s}^{\mathrm{D}}$, respectively, are the upward and downward reserves of thermal power units.

2.

Ramp-rate constraints for thermal power units

$$\left\{\begin{array}{l}{P}_{\mathrm{Gen},i,t,s}-P_{\mathrm{Gen},i,t-1,s}\le r_{ui}\Delta t+P_{\mathrm{Gen},i,\max }[1-u_{i,t-1,s}]\\ P_{\mathrm{Gen},i,t-1,s}-P_{\mathrm{Gen},i,t,s}\le r_{di}\Delta t+P_{\mathrm{Gen},i,\max }[1-u_{i,t,s}]\\ P_{\mathrm{Gen},i,t,s}=P_{\mathrm{Gen},i,t.s}^{\mathrm{peak}}+P_{\mathrm{Gen},i,t.s}^{\mathrm{deep}}\\ u_{i,t,s}^{\mathrm{deep}}+u_{i,t,s}^{\mathrm{peak}}=u_{i,t,s}\end{array}\right.$$

(21)

where $r_{ui}$ and $r_{di}$, respectively, are the upward and downward ramp rates of conventional thermal power units; $\Delta t$ is the unit dispatch duration.

3.

Minimum start-up and minimum shutdown time constraints for thermal power units

$$\left\{\begin{array}{l}{T}_{i,on}\ge T_{i,on,\min }\\ T_{i,off}\ge T_{i,off,\min }\end{array}\right.$$

(22)

where $T_{i,on,\min }$ and $T_{i,off,\min }$, respectively, are the minimum allowable continuous running time and minimum allowable continuous shutdown time for conventional thermal power units.

3.2.3. Constraints on Wind and Solar Power Output

$$\left\{\begin{array}{l}{P}_{\mathrm{W},t,s}\le P_{\mathrm{W},t,s}^{\mathrm{A}}\\ P_{\mathrm{PV},t,s}\le P_{\mathrm{PV},t,s}^{\mathrm{A}}\end{array}\right.$$

(23)

where $P_{\mathrm{W},t,s}^{\mathrm{A}}$ and $P_{\mathrm{PV},t,s}^{\mathrm{A}}$, respectively, are the predicted outputs for wind power and photovoltaic power; $P_{\mathrm{W},t,s}$ and $P_{\mathrm{PV},t,s}$, respectively, are the real-time outputs for wind power and photovoltaic power.

3.2.4. Constraints on Pumped Storage Units

1.

Storage capacity constraints for pumped storage units

$$\left\{\begin{array}{l}{V}_{\min }^{\mathrm{up}}\le V_{t,s}^{\mathrm{up}}\le V_{\max }^{\mathrm{up}}\\ V_{\min }^{\mathrm{down}}\le V_{t,s}^{\mathrm{down}}\le V_{\max }^{\mathrm{down}}\\ V_{t,s}^{\mathrm{up}}=V_{t-1,s}^{\mathrm{up}}-{\displaystyle \sum _{n=1}^N{Q}_{n,t,s}^{\mathrm{PSG}}}+{\displaystyle \sum _{n=1}^N{Q}_{n,t,s}^{\mathrm{PSP}}}\\ V_{t,s}^{\mathrm{down}}=V_{t-1,s}^{\mathrm{down}}+{\displaystyle \sum _{n=1}^N{Q}_{n,t,s}^{\mathrm{PSG}}}-{\displaystyle \sum _{n=1}^N{Q}_{n,t,s}^{\mathrm{PSP}}}\\ V_{T,s}^{\mathrm{up}}=V_{\mathrm{pre}}^{\mathrm{up}}\\ V_{T,s}^{\mathrm{down}}=V_{\mathrm{pre}}^{\mathrm{down}}\end{array}\right.$$

(24)

where $V_{t,s}^{\mathrm{up}}$ and $V_{t,s}^{\mathrm{down}}$, respectively, are the water storage volumes of the upper reservoir and the lower reservoir; $V_{\max }^{\mathrm{up}}$ and $V_{\min }^{\mathrm{up}}$, respectively, are the maximum and minimum storage capacities of the upper reservoir; $Q_{n,t,s}^{\mathrm{PSG}}$ and $Q_{n,t,s}^{\mathrm{PSP}}$, respectively, are the flow rates of the pumped storage units during power generation and pumping conditions; $V_{\mathrm{pre}}^{\mathrm{up}}$ and $V_{\mathrm{pre}}^{\mathrm{down}}$, respectively, are the target water level of the upper and lower reservoir at the end time; $T$ is the end time period of the day.

2.

Power output limits constraints for pumped storage units

$$\left\{\begin{array}{l}{U}_{t,s}^{\mathrm{psG}}{P}_{\mathrm{psG},\min }\le P_{t,s}^{\mathrm{psG}}\le U_{t,s}^{\mathrm{psG}}{P}_{\mathrm{psG},\max }\\ U_{t,s}^{\mathrm{psP}}{P}_{\mathrm{psP},\min }\le P_{t,s}^{\mathrm{psP}}\le U_{t,s}^{\mathrm{psP}}{P}_{\mathrm{psP},\max }\\ 0\le U_{t,s}^{\mathrm{psG}}+U_{t,s}^{\mathrm{psP}}\le 1\end{array}\right.$$

(25)

where $U_{t,s}^{\mathrm{psG}}$ is the operational status variable of the pumped storage unit during power generation conditions; $U_{t,s}^{\mathrm{psP}}$ is the operational status variable of the pumped storage unit during pumping conditions; $P_{\mathrm{psG},\max }$ and $P_{\mathrm{psG},\min }$, respectively, are the maximum and minimum power outputs of the pumped storage unit during power generation conditions; $P_{\mathrm{psP},\max }$ and $P_{\mathrm{psP},\min }$, respectively, are the maximum and minimum power outputs of the pumped storage unit during pumping conditions.

3.

Maximum start–stop constraints for pumped storage units

$$\begin{array}{l}{U}_t^{\mathrm{psG}}+U_t^{\mathrm{psP}}\le 1\\ S_t^{\mathrm{psG},\mathrm{u}}-S_t^{\mathrm{psG},\mathrm{d}}=U_t^{\mathrm{psG}}-U_{t-1}^{\mathrm{psG}}\\ S_t^{\mathrm{psG},\mathrm{u}}+S_t^{\mathrm{psG},\mathrm{d}}\le 1\\ S_t^{\mathrm{psP},\mathrm{u}}-S_t^{\mathrm{psP},\mathrm{d}}=U_t^{\mathrm{psP}}-U_{t-1}^{\mathrm{psP}}\\ S_t^{\mathrm{psP},\mathrm{u}}+S_t^{\mathrm{psP},\mathrm{d}}\le 1\\ \max \left\{{\displaystyle \sum _{t=1}^T{S}_t^{\mathrm{psG},\mathrm{u}}},{\displaystyle \sum _{t=1}^T{S}_t^{\mathrm{psG},\mathrm{d}}}\right\}\le N^{\mathrm{psG}}\\ \max \left\{{\displaystyle \sum _{t=1}^T{S}_t^{\mathrm{psP},\mathrm{u}}},{\displaystyle \sum _{t=1}^T{S}_t^{\mathrm{psP},\mathrm{d}}}\right\}\le N^{\mathrm{psP}}\end{array}$$

(26)

where $S_t^{\mathrm{psG},\mathrm{u}}$ and $S_t^{\mathrm{psG},\mathrm{d}}$, respectively, are the start-up and shut-down status variables of the pumped storage unit during power generation conditions; $S_t^{\mathrm{psP},\mathrm{u}}$ and $S_t^{\mathrm{psP},\mathrm{d}}$, respectively, are the start-up and shut-down status variables of the pumped storage unit during pumping conditions; $N^{\mathrm{psG}}$ and $N^{\mathrm{psP}}$, respectively, are the maximum number of daily start-ups and shutdowns of the pumped storage unit during power generation and pumping conditions.

3.2.5. System Constraints

1.

Minimum inertia constraint for the power system

To ensure power system frequency stability requirements, the inertia at each time period should exceed the minimum inertia requirement of the power system, expressed as (18).

$$H_{\mathrm{s},t}\ge H_{\min ,t}^{\mathrm{s}}$$

(27)

where $H_{\mathrm{s},\mathrm{t}}$ is the real-time inertia level of the power system. In practical power systems, this value can be measured by a system inertia monitoring device. In this optimization model, this value can be estimated using Equation (11).

2.

Nodal power balance constraints

$${\displaystyle \sum _{i=1}^{N_{\mathrm{G}}}{P}_{Gi,t,s}}+{\displaystyle \sum _{j=1}^{N_{\mathrm{H},\mathrm{P}}}\left(P_{t,s}^{\mathrm{psG}}-P_{t,s}^{\mathrm{psP}}\right)+}{P}_{\mathrm{W},t,s}+P_{\mathrm{PV},t,s}+P_{\mathrm{ES},t,s}^{\mathrm{G}}-P_{\mathrm{ES},t,s}^{\mathrm{P}}+{\displaystyle \sum _{b\prime \in {\mathrm{\Omega }}_b^B}{P}_{bb\prime ,t,s}^{\mathrm{L}}-P_{b,t,s}^{\mathrm{Load}}}=0$$

(28)

where $P_{bb\prime ,t,s}^{\mathrm{L}}$ is the active power between node $b$ and node $b\prime$; $P_{b,t,s}^{\mathrm{Load}}$ is the active power of the node’s load.

3.

DC power flow and phase angle constraints

$$\left\{\begin{array}{l}{P}_{bb\prime ,t,s}^{\mathrm{L}}=B_{bb\prime }\left({\theta }_{b,t,s}-{\theta }_{b\prime ,t,s}\right),bb\prime \in Lin{e}_{\mathrm{l}}\\ P_{bb\prime }^{\mathrm{L},\min }\le P_{bb\prime ,t,s}^{\mathrm{L}}\le P_{bb\prime }^{\mathrm{L},\max },bb\prime \in Lin{e}_{\mathrm{l}}\\ {\theta }_b^{\min }\le {\theta }_{b,t,s}\le {\theta }_b^{\max }\end{array}\right.$$

(29)

where $B_{bb\prime }$ is the admittance between node $b$ and node $b\prime$; ${\theta }_{b,t,s}$ is the phase angle of the node; $P_{bb\prime }^{\mathrm{L},\max }$ and $P_{bb\prime }^{\mathrm{L},\min }$, respectively, are the upper and lower limits of the allowable active power on the line connecting two nodes.

3.3. Non-Linear Constraint Transformation Based on McCormick Envelopes

For the constraints related to the inertia and operation of battery storage shown in the above model, the power and virtual inertia time constant of the battery storage are generally related to the installed capacity and virtual inertia control coefficients, both of which are variables. When multiplied, they form non-linear constraints. Additionally, the power of the battery storage multiplied by the charging and discharging status also results in non-linear constraints. This paper transforms the non-linear constraints (11) and (18) into linear constraints (30), (31), and (32), based on the McCormick envelopes [30].

$$\left\{\begin{array}{l}{w}_1\ge H_{\mathrm{ES}}^{\min }{P}_{\mathrm{ES}}+H_{\mathrm{ES}}{P}_{\mathrm{ES}}^{\min }-H_{\mathrm{ES}}^{\min }{P}_{\mathrm{ES}}^{\min }\\ w_1\ge H_{\mathrm{ES}}^{\max }{P}_{\mathrm{ES}}+H_{\mathrm{ES}}{P}_{\mathrm{ES}}^{\max }-H_{\mathrm{ES}}^{\max }{P}_{\mathrm{ES}}^{\max }\\ w_1\le H_{\mathrm{ES}}^{\max }{P}_{\mathrm{ES}}+H_{\mathrm{ES}}{P}_{\mathrm{ES}}^{\min }-H_{\mathrm{ES}}^{\max }{P}_{\mathrm{ES}}^{\min }\\ w_1\le H_{\mathrm{ES}}^{\min }{P}_{\mathrm{ES}}+H_{\mathrm{ES}}{P}_{\mathrm{ES}}^{\max }-H_{\mathrm{ES}}^{\min }{P}_{\mathrm{ES}}^{\max }\\ H_{\mathrm{s},t}={\displaystyle \sum _{i=1}^{Gen\_n}{P}_{\mathrm{Gen},i}}{H}_{\mathrm{Gen},i}{u}_{i,t,s}+{\displaystyle \sum _{i=1}^{ps\_n}{P}_{\mathrm{PS},i}{H}_{\mathrm{PS},i}{U}_i^t}+w_1\end{array}\right.$$

(30)

$$\left\{\begin{array}{l}{w}_2\ge u_{\mathrm{G},t,s}{P}_{\mathrm{ES}}^{\min }\\ w_2\ge P_{\mathrm{ES}}+u_{\mathrm{G},t,s}{P}_{\mathrm{ES}}^{\max }-P_{\mathrm{ES}}^{\max }\\ w_2\le P_{\mathrm{ES}}+u_{\mathrm{G},t,s}{P}_{\mathrm{ES}}^{\min }-P_{\mathrm{ES}}^{\min }\\ w_2\le u_{\mathrm{G},t,s}{P}_{\mathrm{ES}}^{\max }\\ 0\le P_{\mathrm{ES},t,s}^{\mathrm{G}}\le w_2\end{array}\right.$$

(31)

$$\left\{\begin{array}{l}{w}_3\ge u_{\mathrm{P},t,s}{P}_{\mathrm{ES}}^{\min }\\ w_3\ge P_{\mathrm{ES}}+u_{\mathrm{P},t,s}{P}_{\mathrm{ES}}^{\max }-P_{\mathrm{ES}}^{\max }\\ w_3\le P_{\mathrm{ES}}+u_{\mathrm{P},t,s}{P}_{\mathrm{ES}}^{\min }-P_{\mathrm{ES}}^{\min }\\ w_3\le u_{\mathrm{P},t,s}{P}_{\mathrm{ES}}^{\max }\\ 0\le P_{\mathrm{ES},t,s}^{\mathrm{P}}\le w_3\end{array}\right.$$

(32)

where $w_1$ is a linear variable of the product of the rated power of the battery storage after conversion and the inertia time constant; $w_2$ and $w_3$, respectively, are the linear variables of the product of the energy storage discharge state and discharge power, and the product of the charging state and charging power.

4. Case Study

4.1. Basic Parameter Settings

In this paper, the modified IEEE-39 bus system is used to verify the accuracy of the model. The network topology of the power system is shown in Figure 1. The modifications to the IEEE-39 bus system are as follows: removing the thermal power units at buses 30 and 39; connecting a 1600 MW wind farm, a 1400 MW photovoltaic plant, and two 160 MW pumped storage units to bus 6; connecting a 1600 MW wind farm, a 1900 MW photovoltaic plant, and two 160 MW pumped storage units to bus 39; setting the battery storage installation bus to bus 39. The typical daily wind and photovoltaic output and load profiles are obtained by clustering the annual wind and photovoltaic output curves and daily load curves of a certain southwestern region using the K-means clustering algorithm, resulting in five typical daily wind and photovoltaic output clustering curves and daily load clustering curves. The typical daily output curves for the stochastic programming are shown in Figure 2 and Figure 3. The output probabilities for each typical day are shown in

Table 1. Probability of different typical days.

Typical Day	1	2	3	4	5
Probability	0.1726	0.2822	0.2959	0.1096	0.1397

. The maximum power system frequency deviation is set to 1 Hz, meaning the lower limit of the power system frequency during dynamic processes is 49 Hz, and the maximum power system rate of change of frequency is 1 Hz/s. The battery storage parameters are shown in

Table 2. Battery storage parameter settings.

Parameters	Numerical Values	Parameters	Numerical Values
Power cost (CNY/KW)	350	Charging and discharging efficiency	0.9
Capacity cost (CNY/KWh)	1800	Maximum SOC	0.9
Virtual inertia cost (CNY/S)	1500	Minimum SOC	0.1
Discount rate	0.05	Design life span	10

, and the power system power deficit is assumed to be 10% of the real-time load level.

4.2. Scenario Comparison and Result Analysis

To verify the cost reduction effect brought by battery storage configuration in a grid containing wind, solar, thermal, and pumped storage units when considering inertia constraints and deep peak regulation of thermal power, we constructed four scenarios for comparison in the case study:

Scenario 1: Without considering inertia constraints and without configuring battery storage;

Scenario 2: Without considering inertia constraints but with configuration of battery storage;

Scenario 3: Considering inertia constraints but without configuring battery storage;

Scenario 4: Considering inertia constraints and configuring battery storage. The comprehensive operating cost results for each scenario are shown in

Table 3. Costs for each scenario.

Scenario	Total Cost (104 CNY)	Renewable Energy Curtailment Cost (104 CNY)	Thermal Power Unit Start-Up and Shutdown Cost (104 CNY)	Daily Average Investment Cost (104 CNY)	Thermal Power Operation Cost (104 CNY)
1	2307	92.64	15.77	0	2163
2	2286.4	29.86	13.72	54.35	2158.7
3	2340.9	93.28	16.62	0	2185
4	2295.6	29.94	13.28	58.79	2163.8

.

From Table 3, the following conclusions can be summarized:

Generation costs account for the largest proportion of long-term scheduling costs. By comparing Scenario 1 and Scenario 3, we find that considering inertia constraints leads to the operation of more thermal power units during periods of low system inertia, increasing their start–stop and operating costs, while also exacerbating wind and solar curtailment. The comparison between Scenario 3 and Scenario 4 shows that integrating inertia constraints into grid operations and investing in battery storage to provide virtual inertia can reduce the frequent start–stop of thermal power units, lower operating costs, and reduce wind and solar curtailment. The comparison between Scenario 2 and Scenario 4 indicates that considering inertia constraints requires configuring higher-capacity battery storage, which not only reduces wind and solar curtailment but also provides the necessary inertia support for grid operation.

From an economic perspective, the method proposed in this paper has several advantages compared to the traditional approach of meeting system inertia requirements by dispatching thermal power units. Firstly, the quick response of battery storage can reduce the need for frequent start–stop operations of thermal power units, thereby lowering the overall operation and maintenance costs of the power system. Secondly, meeting system inertia demands through battery storage can reduce the adjustment requirements of thermal power units, thus decreasing the consumption of fossil fuels, which in turn lowers fuel costs and enhances environmental friendliness. Finally, although the initial investment in storage systems is relatively high, in the long run, battery storage can optimize the integration of renewable energies such as wind and solar, and reduce the reliance on traditional power plants, thereby significantly lowering the total costs of the power system.

However, there are also some disadvantages. Firstly, the initial investment cost of providing virtual inertia through battery storage is relatively high, especially when deployed on a large scale, which may impact economic efficiency in the short term. Secondly, the lifespan of battery storage is shorter compared to traditional thermal power units, so multiple replacements of battery modules are required during the lifecycle of battery storage, increasing the long-term maintenance and replacement costs of battery storage.

Next, we take one of the typical days as an example to observe the operation under each scenario. The start–stop situations of thermal power units for each scenario are shown in Figure 4.

When battery storage is not considered in the power grid configuration, analyzing Figure 4, Figure 5 and Figure 6 reveals the following.

When inertia constraints are not considered, thermal power units will be scheduled to minimize operating costs throughout the day. This means that units with lower operating costs and higher maximum technical output, such as thermal power units 4 and 8, will be prioritized. However, as shown in Figure 5, the system inertia during the entire operation process generally does not meet the minimum inertia requirements. Consequently, if a fault occurs, the system frequency minimum point will exceed limits and fail to meet the system’s frequency requirements.

When inertia constraints are included, analyzing Figure 6 shows that the system frequency will meet the pre-determined requirements at any time of fault occurrence, specifically, the post-fault minimum system frequency will be greater than 49 Hz. By analyzing Figure 5, it is evident that, to meet the system’s minimum inertia requirements, an additional thermal power unit (unit 7) is operated from midnight to 8:00 AM in Scenario 3, along with units 4 and 8 from Scenario 1. From 9:00 AM to noon, due to the higher load, Scenario 1 operates additional thermal power units 3 and 5 to meet load requirements, but still falls short of the minimum inertia requirements. In this situation, Scenario 3 continues to dispatch thermal power units 6 and 7. After noon, unit 7, which has a lower inertia time constant, is shut down, and unit 6, which has a relatively higher inertia time constant, is kept running to meet both load and inertia requirements. Overall, in Scenario 3, the frequency of thermal power units entering deep peak shaving mode increases significantly, which also increases the wear and tear on these units.

When battery storage is considered in the power grid configuration, analyzing the on–off situations of thermal power units in Scenarios 3 and 4 from Figure 4 shows that battery storage reduces the frequent start–stop of thermal power units, lowering their operating costs and the costs of wind and solar curtailment. This is because battery storage can provide virtual inertia, working in conjunction with thermal power units and pumped storage units to meet the minimum inertia requirements. Specifically, in Scenario 4, due to the virtual inertia provided by battery storage, unit 7 can be shut down from midnight to 8:00 AM when the inertia demand is not high, with battery storage compensating for the missing inertia. After 9:00 AM, unit 2, which has a low inertia time constant and poor economic performance, can be shut down. The entire system maintains frequency stability through the combined support of thermal power units, pumped storage units, and the virtual inertia provided by battery storage. Overall, this significantly reduces the wear and tear on thermal power units caused by deep peak shaving, as well as the associated environmental pollution.

Next, taking Scenario 4 as an example, we observe the system’s operating output and the SOC (State of Charge) of the battery storage, as shown in Figure 7 and Figure 8.

According to Figure 7 and Figure 8, it can be seen that by integrating battery storage into a high-proportion renewable power system, two key benefits are achieved. Firstly, battery storage provides the necessary virtual inertia for the system. Secondly, battery storage, in conjunction with pumped storage, enables peak shaving and valley filling for the entire system. For example, during midday when renewable energy generation is high, both battery storage and pumped storage are in a charging state, absorbing excess renewable energy. In the evening, when renewable energy is scarce, battery storage and pumped storage discharge to work with thermal power units to meet the system’s load requirements. Additionally, analysis of the battery storage SOC shows that the SOC remains between 0.1 and 0.9 throughout the day, which meets the safety requirements for battery storage operation. Furthermore, at the end of the day, the SOC of the battery storage returns to the initial 0.5 level, ensuring that the battery storage can meet the dispatching requirements for the next day.

4.3. Price Sensitivity Analysis

The grid-side battery storage configuration model constructed in this paper introduces power system inertia constraints and storage virtual inertia, enabling the grid to minimize costs while meeting inertia constraints through the co-ordinated operation of thermal power, pumped storage, and battery storage. This involves considering the investment capacity, power, and virtual inertia time constant of storage, as well as the operation and shutdown scheduling of thermal and pumped storage units during daily operations. An essential idea of this paper is the trade-off between the decision-maker’s investment cost in storage and the costs associated with the operation and shutdown of thermal and pumped storage units to meet inertia demands through dispatching. Based on this concept, by studying the impact of different units’ virtual inertia costs of battery storage investment on the grid’s investment in storage capacity and power, we can further explore how to optimize the operation scheduling of units. Additionally, low virtual inertia costs may prompt grid operators to adopt more flexible operating strategies to adapt to changes in wind and solar resources. The planning results for different unit virtual inertia costs of battery storage are shown in

Table 4. Impact of virtual inertia costs of battery storage units on planning results.

Virtual Inertia Cost (CNY/S)	Storage Capacity Configuration (MWh)	Storage Power Configuration (MW)	Storage Virtual Inertia Time Constant (S)
1000	753.9	670.1	5.68
1500	754.1	718.9	4.31
1800	754.1	718.9	4.31
2000	753.86	686.4	4.20

.

From the above table, it can be seen that the unit virtual inertia cost of the battery storage has little impact on the configuration of storage capacity in the grid. The main impact lies in the power size and the virtual inertia time constant of the battery storage configuration. When the units’ virtual inertia cost is below 1500 yuan/s, the grid tends to configure a larger virtual inertia time constant for the battery storage, thus reducing the required power size and minimizing the investment cost while meeting the overall inertia demand. When the unit virtual inertia cost is between 1500 and 2000 yuan, the increase in unit virtual inertia cost leads the power system to reduce the virtual inertia time constant by increasing the maximum power investment in battery storage, thus minimizing the total cost while still meeting the inertia requirements. When the unit virtual inertia cost exceeds 2000 yuan/s, the power system gradually tends to rely on scheduling the start and stop of thermal power units to meet the inertia demand.

4.4. Impact of Deep Peak Regulation of Thermal Power on Battery Storage Configuration Results

In the model described in this paper, we assume by default that thermal power units have the capability for deep peak regulation. However, it is important to consider that some remote thermal power units may not have undergone deep peak regulation modifications. In this case, we need to study the impact on the configuration of battery storage capacity and power in the grid when thermal power units cannot achieve deep peak regulation, which involves issues of grid reliability and stability. When thermal power units cannot provide deep peak regulation support, the battery storage will play a more critical role. By reasonably configuring battery storage capacity and power, it can compensate for the deficiency of thermal power units’ inability to meet deep peak regulation demands.

Table 5. Impact of virtual inertia costs of battery storage on planning results.

Ability to Deep Peak Shave	Storage Capacity Configuration (MWh)	Storage Power Configuration (MW)	Storage Virtual Inertia Time Constant (S)	Total Cost (104 CNY)
Yes	754.1	718.9	4.31	2295.6
No	887.2	814.3	4.65	2352.8

presents the battery storage configuration results when thermal power units can and cannot perform deep peak regulation.

From the results in Table 5, it can be seen that when thermal power units cannot enter deep peak regulation mode to provide inertia and load support in co-ordination with battery storage and pumped storage units, the required battery storage capacity, power, and virtual inertia in the grid will be larger. The main reason is that, when thermal power units cannot enter deep peak regulation mode, issues that could originally be resolved by reducing the output of thermal power units now have to be addressed by changing the startup and shutdown of units or configuring more battery storage. This leads to a greater demand for battery storage in the grid and an increase in overall operating costs. This indirectly confirms the necessity of deep peak regulation modifications for thermal power units.

4.5. The Impact of Different Renewable Energy Penetration Levels on the Results

To ensure the reliability and resilience of the model proposed in this paper, we tested the model under scenarios with different renewable energy penetration levels. The results are shown in

Table 6. The impact of different renewable energy penetration levels on the results.

Renewable Energy Penetration Levels (%)	Proposed Method Cost (104 CNY)	Traditional Method Cost (104 CNY)
33	3218.4	3209.1
41	2942.0	2943.3
51	2491.7	2495.2
55	2331.0	2361.7
59	2340.0	2636.9

.

From the results in Table 6, it can be seen that when the renewable energy penetration level is low, the overall cost of meeting the system inertia requirements using traditional methods is lower than that of the method proposed in this paper. This is because, at low penetration levels, the installed capacity of thermal power units is relatively large, and many units operate simultaneously, providing the necessary inertia support. In contrast, the initial investment cost of battery storage is high, making traditional methods more economical in this scenario. However, as the renewable energy penetration level increases, the share of installed capacity and output from thermal power units decreases, leading to a decline in the economic efficiency of traditional methods. Conversely, the economic efficiency of the method proposed in this paper improves with increasing penetration, gradually demonstrating greater economic benefits. The higher the penetration level, the more apparent the advantages.

It is also important to note that the overall operating costs in this study exhibit a trend of first decreasing and then increasing as the renewable energy penetration level rises. This is because, in this paper, penetration is defined as the proportion of installed renewable energy capacity. Changing the penetration level affects the installed capacity, but the overall operating costs in the model do not include the investment costs of renewable energy installations, only considering the costs of wind and solar curtailment. As a result, an initial increase in penetration leads to a reduction in operating costs, but as penetration continues to rise, limitations in grid transmission capacity lead to increased wind and solar curtailment, causing overall operating costs to increase.

4.6. Comparison between Stochastic Optimization and Deterministic Optimization

The comparison of economic costs between the stochastic optimization method adopted in this paper and the deterministic optimization method, which considers only a single typical scenario of wind and solar output, is shown in Figure 9.

Based on the analysis of the above figure, the following conclusions can be drawn: The deterministic optimization method makes decisions based on a given single wind and solar output scenario without accounting for the uncertainty in wind and solar output throughout the year. Therefore, its planning scheme is greatly affected by the typical daily wind and solar output magnitude and volatility. When the typical daily wind and solar output are high and volatility is low, the configuration cost and overall operating cost are low, whereas, when the typical daily wind and solar output are low and volatility is high, the configuration cost and overall operating cost are high.

In contrast, the stochastic optimization method comprehensively considers the volatility of wind and solar output and load. As a result, its outcomes ensure economic efficiency while meeting operational requirements under the most adverse conditions, thereby reducing overall operating costs. This method balances a degree of conservatism, making the system more robust and resilient.

5. Conclusions

This study focuses on addressing the challenges of power system frequency stability that may arise after large-scale integration of wind and solar energy. By transforming frequency stability considerations into inertia constraints, it establishes a grid-side battery storage configuration model based on minimum inertia constraints. This model can identify the optimal battery storage configuration scheme that meets various constraints through a stochastic optimization method, based on the probabilistic distribution of wind and solar energy output uncertainties. The model and stochastic optimization method’s effectiveness were validated through simulations on a modified IEEE 39-bus system. The study draws the following conclusions:

• The frequency stability issue of the power system becomes increasingly prominent after large-scale wind and solar energy integration. Considering the impact of frequency stability on the unit commitment of the power system, it is necessary to increase the operation of thermal power units or allow them to enter deep peak regulation states at certain times to meet minimum inertia constraints. This significantly increases the start-up and shut-down costs and operating costs of thermal power units, while also leading to the increased curtailment of wind and solar energy due to the minimum technical output of thermal power units.
• By configuring battery storage and virtual inertia, battery storage can co-ordinate with thermal power units and pumped storage units in the grid to meet the power system’s inertia requirements, optimize the operation of thermal power units and pumped storage units, and enhance the absorption capacity of wind and solar energy.
• The unit inertia cost of battery storage has a minimal impact on the battery storage configuration capacity, mainly affecting the power and virtual inertia time constant of the battery storage configuration. This is because the model considers the minimum inertia constraints, and the deep peak regulation capability of thermal power units also significantly influences the battery storage configuration results.
• Compared to deterministic optimization, the stochastic optimization method adopted in this study can more fully consider the uncertainties of wind and solar energy and load, ensuring the economic efficiency and robustness of the configuration results.

At present, the development of renewable energy in China is progressing rapidly. Many wind farms and photovoltaic power stations are gradually being integrated into the power system, forming a new power system with a high proportion of renewable energy. However, the intermittency, volatility, and randomness of renewable energy add more uncertainties to power system planning. Additionally, the increase in renewable energy capacity affects the proportion of thermal power units, thereby reducing the power system’s inertia support. Therefore, with the continuous development of renewable energy in China, it is of great significance to study the optimal configuration of battery storage in power systems with a high proportion of renewable energy. In our future work, we will further investigate the impact of uncertainties on battery storage configuration results. Additionally, we will conduct in-depth research on the optimal dispatch of power systems with a high proportion of renewable energy, considering frequency stability. We will explore the role that the combination of battery storage and other energy storage devices, such as supercapacitors, can play in optimal dispatch and frequency stability control.