Dynamic optimal allocation of energy storage systems integrated within photovoltaic based on a dual timescale dynamics model

0 Introduction

Motivated by the goal of “carbon neutrality”,the primary generation mode in power system is gradually transitioning from traditional power generation to renewable energy sources (RESs).The shift towards low-carbon and sustainable development,driven by the stochastic and intermittent nature of RESs,brings new opportunities and challenges to grids [1].From the perspective of the electricity market,the healthy development of power generation units significantly depends on achieving substantial benefits.Compared with wind power generation,PV generation exhibits a more fluctuating output curve and remains inactive during nighttime.Equipping PV units with energy storage systems (ESSs) can help increase the resource utilization efficiency and profitability,resulting in the reduction of unexpected solar energy curtailment.This type of ESS is called an energy storage system integrated within a PV system (PV-ESS).In addition,this equipment enhances the peak support capability of the grid [2].For PV investors,determining the optimal allocation ratio of PV-ESS is crucial for enhancing profitability by reducing solar energy curtailment and minimizing construction costs.Another type of ESS,called independent energy storage systems (IESSs),directly engage in the electricity market and develop through price differentials.For grid planners,conducting a profitability analysis of PV-ESS and IESS is beneficial for informing relevant policy formulations.Addressing these issues is of urgent importance.

To address these issues,many studies have employed traditional optimization models for one-year simulation periods.Reference [3] proposed an active energy storage operation strategy that utilized an empirical mode decomposition algorithm to obtain the optimal energy storage capacity configuration that minimized the upfront planning investments.Reference [4] introduced a multiobjective capacity configuration optimization algorithm based on an improved NSGA-II algorithm and conducted comparative analyses of capacity configurations for off-grid and grid-connected multi-energy systems.Reference [5]employed a PV energy storage optimization configuration model based on the NSGA-II algorithm.This study analyzed how a PV-ESS in rural microgrids can effectively enhance the PV utilization rate and economic viability of a microgrid system.The configuration capacity and load optimization of shared energy storages in PV communities were investigated in [6].Reference [7] investigated the PV-ESS allocation ratio from the perspective of ESS participation to compensate for forecasting errors in PV generation.Reference [8] proposed an improved carbonneutral energy system to explore the market potential of ESSs.Reference [9] introduced a hybrid ESS capacity configuration optimization method aimed at reducing microgrid power fluctuations,extending ESS lifespans,and lowering economic costs.Reference [10] improved the configuration and economic benefits of PV energy storage capacity through dual-layer optimization.Reference[11] proposed a method for optimizing the scale of ESS in distribution networks by considering a reduction in PV power generation.However,variations in the capacity of power generation units alter the generation mix and RES penetration rate,making this a dynamic process.The optimal PV-ESS allocation ratio and profitability of PV-ESS and IESS fluctuate continuously.Thus,analyzing long-term changes in the complex systems formed by electricity market participants and clearing processes is crucial.Therefore,a more appropriate approach was required for this analysis.

System dynamics (SD) is an analytical modeling method used to study complex systems with interconnected variables and feedback loops.This method is widely applied in the field of electricity market research.The application of SD facilitates an in-depth examination of the quantitative relationships among market entities,revealing trends in the development of various types of power generation.A comprehensive review of studies employing the SD method to establish electricity market models is presented in [12],with a particular focus on deregulated electricity market models.Reference [13] constructed an SD model of a three-party evolutionary game to analyze stakeholder strategy interactions under a renewable energy quota system and simulated the corresponding evolutionary processes.Reference [14] established SD models under different scenarios and simulated and evaluated carbon emission allowance allocation methods in the context of power reformation.Reference [15] utilized SD to model the long-term behavior of natural gas markets,extended the existing electricity market behavior models,and conducted a comparative analysis of four investment scenarios.Reference [16] proposed a hybrid simulation model that combined an agent-based simulation with an SD simulation and employed reinforcement learning algorithms to simulate market conditions.Reference [17] employed an SD approach to establish a bidding module for different participants and two-stage market-clearing module for independent system operators.This study discusses the bidding strategies for energy storage projects in an imperfectly competitive market.Reference [18] used the SD model and compared the demand responses in two different market mechanisms,namely scarcity pricing and capacity auctions.Previous studies primarily simulated the long-term development of power generation units using approximately assumed electricity prices,resulting in inaccuracies in simulation outcomes.Additionally,as an economic modeling method,SD cannot be validated against the operational constraints of a power system [19].Therefore,it is necessary to introduce a more suitable method for electricity price simulations that consider the system constraints.

Understanding electricity prices involves simulating electricity market operations.The electricity market mechanisms include medium-and long-term contracts[20],spot markets [21],ancillary services markets [22],and others.Among these,considering system operational constraints,the spot market holds significant engineering importance due to its role in clearing outcomes [23].Spot market-clearing (SMC) models are based on short-term operational optimizations such as security-constrained unit commitments (UCs).UCs can be employed to calculate the clearing energy and prices on a short timescale.Reference [24] proposed two statistical scenario-based probabilistic price forecasting models for pumped-storage hydropower and conducted stochastic UC optimization.Reference [25] conducted a detailed analysis of the reliability unit compensation associated with UCs,considering both the production and opportunity costs of losses.This helps improve economic efficiency by reducing the market-based payments resulting from UCs.Considering the importance of the demand elasticity of the electricity market and its impact on the revenue of power generation companies,a new profit-based UC model was proposed[26].This model effectively captures the uncertainty in willingness to pay for prices set for elastic demand.Reference [27] discussed a stochastic UC model to explore the ability of an ESS to provide valuable grid services by joining the energy and reserve markets.The aforementioned studies utilized UC optimization to simulate the marketclearing process using an hourly timescale.

However,SD models require timescale uniformity across all variables,typically using years or months in electricity market trend research [12].At this timescale,simulating a specific clearing process is infeasible,making it impossible to derive the clearing electricity price.A SMC model based on UCs can be utilized for price discovery;however,this process typically operates on an hourly timescale.The challenge of integrating shorttimescale market clearing with the long-term profitability development of power generation units remains unresolved.Inspired by other dual-layer structure models [28,29],this study introduces a dual-timescale dynamics model in which the SMC model is packaged into a subsystem.The main contributions of this study are as follows:

1) SD models typically operate on an annual or monthly timescale when simulating the development of power generation unit capacities,whereas identifying clearing electricity prices requires an hourly timescale.This study introduces a dual-timescale dynamics model in which a short-timescale SMC model is encapsulated as a subsystem within a long-timescale SD model.This dual-timescale model simulates the developmental trends of market participants by employing SMC prices,which more closely reflect the actual situation.

2) To compare the long-term profitability variations of the PV-ESS and IESS,this study established a dualtimescale model focusing on both PV-ESS and IESS.Furthermore,a dual-timescale simulation was conducted from 2022 to 2040.For each year,12 typical scenarios were selected,with each including 24 hours.The simulations illustrate the profitability differences between the PV-ESS and IESS models over nearly two decades by modeling market interactions under various boundary conditions.

3) To explore the optimal allocation of PV-ESS,this study added additional scenarios to the dual timescale simulation,wherein the PV-ESS allocation ratio was set to 10%.However,in the other scenarios,the investment sensitivities of the PV-ESS varied.The dynamic optimal range of the PV-ESS allocation was determined and verified through comparisons.

Additionally,a comparison of the method proposed in this study with the methods adopted by previous researchers is summarized in Table 1.

Table 1 Comparison between the method proposed in this paper and existing methods

1 Long timescale system dynamics model

1.1 System boundaries and associated assumptions

The first step in constructing an SD model is to determine the system boundaries and associated assumptions.In this study,the system boundaries and assumptions are as follows:

1) In the studied region,medium-and long-term contracts remain the primary mode of electricity trading.These contracts for differences are settled based on spot market prices.Therefore,spot prices serve as the primary reference for both medium-and long-term contract prices [20].Furthermore,all power generation units are assumed to participate only in the spot market.

2) The ESSs discussed in this paper are all electrochemical ESSs with storage durations of 2 h.

3) The generation costs of RES generation are disregarded in the SD model.

1.2 Causal-loop diagram

The second step is to establish causal loops between the main variables of the model.Investors usually make investment decisions based on financial metrics,including net present value (NPV) and internal rate of return (IRR).All investments considered in this study had the same investment period.As NPV more intuitively represents the profitability of different investment types under the same discount rate conditions [30],this study employs NPV for assessment.A causal loop diagram was developed based on the profitability of the generation types in the market,as shown in Fig.1.

Fig.1 Causal-loop diagram

This diagram contains a negative feedback loop:NPV → investment decision → unit capacity → spot market competition → power generation profit → net profit→ NPV.

Specifically,as the NPV increases,investment decisions of power generation units become more significant,leading to greater capacity growth.However,competition intensifies with increased market participation,resulting in a reduction in power generation profits.Consequently,both the net profit and NPV decrease.Conversely,decreases in generation and construction costs increase the net profit and NPV.Overall,the causal loop presented a negative feedback loop.

1.3 Stock-flow diagram

The third step was to build a stock-flow system based on previous studies.A stock flow system was constructed according to the quantitative relationships between the market entities in the model.The system included three subsystems: a power generation subsystem,PV and PV-ESS subsystems,and an SMC subsystem,as illustrated in Fig.2.

Fig.2 Stock-flow diagram

The power generation subsystem and PV and PVESS subsystems share many similarities.They simulate the power generation,profitability,capacity growth,and investment decisions of different types of power generation.The most crucial variables are the stocks,which represent the capacities of various generation types.They are equal to the sum of annual capacity growth minus the sum of annual capacity retirements.

whererepresent the annual growth and retirements of capacities for unit I,respectively;TH,W,IE,PV,and PVE represent the thermal units,wind units,IESS,PV units,and PV-ESS,respectively.

The NPV calculation methods for the different types of power generation were the same.The NPV formula is as follows.

Here,pi represents the net profit of unit i;represents the unit construction cost of unit i;λ represents the investment period (10 in this study);d represents the discount rate (set at 8% in this study).

According to [31],an investment decision can be considered as a linear function of the NPV,determining the capacity growth rate of the generation units.As the NPV in this study describes an NPV of 1 kW,it is necessary to divide it by the construction cost per kW.

Here,IDi represents the investment decision of the generation type i.Considering an investment period of 10 years,α is set at 0.3,meaning that when the NPV discounts over ten years equals the construction cost per kW,the capacity growth rate of generation type i is 30%.For thermal units,wind units,PV units,and IESS,identical profitability implies the same capacity growth rate.In this situation,the proportion of each generation type remains unchanged.For the PV-ESS,the investment decision is made by the PV investors.Therefore,the capacity growth of the PV-ESS is also correlated with the profitability of the PV units.

The generation cost per kWh of thermal units includes fuel and carbon emission costs,as follows [32]:

The capacity growth of the PV-ESS differs because this study assumes that the PV-ESS and PV share common investors.They do not independently invest in PV-ESS.Instead,the capacity allocation of the PV-ESS is determined based on the profitability of both the PV-ESS and PV.

where β represents the sensitivity factor of PV-ESS investment (2 in this study).The larger this value,the more sensitive investors are to the profitability of the PV-ESS.When the profitability of the PV-ESS exceeds that of the PV system,investment in the PV-ESS surpasses investment in the PV system,and vice versa.When the profitability of the PV cells and PV-ESS are equal (i.e.,IDP VE=IDPV),the PVESS capacity increases with the current PV-ESS allocation ratio of the newly increased PV capacity,represented as RTPVE.

Annual utilization hours reflect the yearly performance of the power generation units,which is calculated as the ratio of annual output to rated capacity.

Where Hi and OPi represents the annual utilization hours and output of generation type I,respectively.

The SMC subsystem is used to simulate the SMC process.The key entity in this subsystem is the UC-based SMC simulation.This subsystem has six input values,including the capacities of five power generation types and electricity demand.It has 11 output values,including the annual output,spot market income of the five power generation types,and average SMC price.Detailed information on the SMC model is presented in Section 2.Based on the output values from the SMC subsystem,the average clearing price for each power generation type is calculated annually.

where pri and PFi represent the average clearing price and spot market profit,respectively,of generation type i.

Based on the developments in previous years and future development plans,the quantitative relationships related to electricity demand are as follows:

Here,D represents the electricity demand,Dgr represents the annual growth in electricity demand,and represents the electricity demand growth rate.

2 Short timescale SMC model

The limitation of the SD lies in its inability to discover electricity prices that reflect the actual situation more closely on a short timescale.During a year,which is the timescale of the SD model,electricity generation and clearing prices are influenced by various factors,including bidding curves,weather conditions,load curves,and operational constraints of power system changes within a shorter timescale.To discover prices more accurately,this study introduces a short-timescale SMC model based on UCs.The model reflects the market dynamics on a shorter timescale.

2.1 UC optimization

UC optimization plays a crucial role in the SMC process by determining the on/off status and load allocation for each power generation unit in a short term.UC optimization has a direct influence on bidding,affecting SMC results,such as clearing energy and prices,and shaping market dynamics.In this study,a multi-scenario UC optimization was utilized to simulate the SMC process.The specific contents are as follows.

2.1.1 Objective function

The objective of UC optimization is to minimize the total cost associated with power generation,including the generation,startup,and shutdown costs,thereby ensuring the most economical operation of the power system.

Here,Ctotal is the total cost of the clearing process;t represents different hours,set as 24;k represents different power generation types;Ct ,k and Wt ,k represent the generation costs and output,respectively,of different generation units at each hour; represents the startup and shutdown costs of thermal power units;and si,t is a binary variable,taking values of 0 or 1,representing the start-up or shutdown status change of unit I at time t.A value of 1 indicates that unit i transitions from an idle state to an operational state at time t,and a value of 0 represents other cases.denote the unsatisfied electricity demand and reserve capacity,respectively,at time t.Cens and Crns represent the penalty factors for unserved energy and reserve capacity,respectively.

2.1.2 Power balance constraint

Here,Pi ,t represents the power output of unit i at time t;represent the discharging and charging power,respectively,of the ESS at time t;and Dt represents the electricity demand at time t.Because the timescale of t is 1 h,Dt equals the total power of the power system at time t.

2.1.3 Reserve capacity constraint

where ui ,t is a binary variable representing the startup and shutdown statuses of unit i at time t,represents the maximum power of unit i,and r represents the system reserve margin,which is set at 10%.Whenis considered as the minimum value to satisfy the inequality,it represents the unserved reserve capacity.

2.1.4 Output limit constraint

whererepresents the minimum power of unit i.

2.1.5 Ramping constraints

whererepresent the upward and downward ramping capabilities,respectively,of unit i.

2.1.6 State of charge constraints

whererepresent the minimum and maximum states of charge of all the electrochemical ESSs,respectively.

2.2 Bidding curve considering negative prices

Different power generation systems exhibit distinct characteristics.RES generation units,such as wind and PV,are stochastic and intermittent.Thus,they adopt a “bid zero”strategy (bidding a near-zero price to maximize clearing energy and minimize curtailment to avoid penalties).Thermal power units,whose main generation costs are fuel expenses,tend to bid in a high price range.Therefore,the typical bidding sequence is as follows: wind and PV units bid near zero prices and thermal units bid high prices.During peak-load periods,the ESS and thermal units with strong regulation capabilities may bid even higher.

As the RES penetration rate increases annually,the power supply stability of the grid decreases.Owing to high start-up and shutdown costs,many thermal units bid for prices below their generation costs to ensure minimum clearing energy and maintain operational status.Simultaneously,when the availability of wind and solar energy is high,the bidding prices for wind and PV will be lower.This is because,with intensive competition,wind and PV units prefer to bid lower prices to avoid significant curtailment penalties,leading to more frequent negative price bidding.Hence,a “negative electricity price”phenomenon occurs in this situation [33],and the bidding sequence undergoes a change as follows: some thermal power units bid negative prices (to maintain operational status at minimum power) → wind and PV units bid nearzero prices → other thermal power units bid high prices.The bidding curves are presented in Figs.3 and 4.

Fig.3 Bidding curve during peak load periods

Fig.4 Bidding curve with negative prices

The marginal electricity price is determined by determining the bid of the marginal unit in each period from the SMC simulation results.

2.3 Framework of the dual timescale simulation

The SD models require timescale uniformity across all variables.When simulating long-term evolution,timescales of years or months are often used,whereas SMCs operate on an hourly timescale.This difference poses a challenge when integrating the two models.The steps to address this problem are as follows [34].

1) Encapsulate the UC-based SMC model into the SD model as a sub-system.

2) Unify the timescales from hours to years by summing up the short-term simulation results of the SMC model,and scale the sums by a conversion ratio.

3) Create interfaces to transfer the processed SMC results back to the SD model for long-term simulation.

Subsequently,a dual-timescale dynamic model is established.The overall simulation workflow is shown in Fig.5.

Fig.5 Flow diagram of the dual timescale dynamics simulation

The algorithm for transferring short-term SMC data into long-term SD data is as follows.

3 Case study

3.1 Parameter configuration

3.1.1 SD model parameter configuration

The simulation period for the SD model was set from 2022 to 2040 with a timescale of one year.The initial values for the stocks in the model were based on data from a provincial power grid for 2022,as shown in Table 2.

Table 2 Initial values of SD stocks

3.1.2 Clearing model parameters configuration

For each year,the number of scenarios was set to 12.Each scenario selected the typical load and RES generation profiles for each month.The thermal power generation was considered using different parameters,as listed in Table 3.The generation ratios of wind and PV,representing the ratio of the maximum output power to the rated power,are illustrated in Figs.6 and 7.

Fig.6 Generation ratio of wind units

Fig.7 Generation ratio of PV units

Table 3 Configuration of thermal units

3.2 Capacity variations of PV-ESS and IESS

The capacity and investment decision variations of the PV-ESS and IESS are shown in Fig.8.The IESS exhibits a more pronounced investment decision than the PV-ESS,meaning that the IESS has superior profitability compared to the PV-ESS.In Fig.9,the IESS has more utilization hours than the PV-ESS,indicating that the IESS demonstrates a better market performance than the PV-ESS.This is because the PV-ESS must prioritize PV generation to avoid curtailment.Consequently,the PV-ESS needs to reserve battery capacity before high-PV generation periods,thus limiting the frequency of charging and discharging cycling.In contrast,the IESS considers only low-cost acquisition and high-priced resale.Besides utilizing PV-ESSs,PV units can also use IESSs for solar energy consumption.Therefore,the role of a PV-ESS is to guarantee minimum consumption.Therefore,allocating a large-scale high-ratio PV-ESS is uneconomical and unnecessary.

Fig.8 Capacity and investment variations of PV-ESS and IESS

Fig.9 Annual utilization hours of power generation units

Figs.10 and 11 illustrate the curtailment reduction achieved by the PV-ESS and IESS,respectively.During the early simulation stage,the decrease in the curtailment reduction rate of the PV-ESS results from the limited pressure on the solar energy consumption,caused by the low RES penetration rate,as shown in Fig.12.At this stage,PV investors prefer to invest directly in PV units,and PVESSs are less economically attractive in comparison.With the decreasing profitability of PV units and increasing RES generation,more PV-ESSs need to be equipped to avoid unexpected curtailment penalties.Low clearing prices significantly affect the profitability of PV units,which explains their slow output growth.The penetration rate of RESs will increase from 24% in 2022 to 48% in 2040.Consequently,the profitability of the PV-ESS becomes apparent,resulting in a sustained increase in the curtailments reduction rate.Notably,the curtailment in the figures differ.Fig.10 compares a PV-ESS with a PV,whereas Fig.11 compares an IESS with a RES,which is equal to wind plus PV.As shown in Fig.11,the RES curtailment rate achieved by the IESS gradually increased by approximately 1%,indicating that the utilization and profitability of the IESS continued to improve.

Fig.10 PV curtailment reduction achieved by PV-ESS

Fig.11 RES curtailment reduction achieved by IESS

Fig.12 RES generation rate

3.3 Clearing prices of PV-ESS and IESS

The average clearing prices of each generation type in this case,which were calculated from the SMC model,are illustrated in Fig.13.In Fig.13,from 2022 to 2040,the SMC prices of all generation types,particularly PV,exhibit a decreasing trend,along with the average market prices.The decline in PV clearing prices is primarily attributed to the competition resulting from the increasing output during sunlight hours.By contrast,the clearing prices of wind units are more stable because wind resources have a more uniform distribution throughout the day.

Fig.13 SMC prices of various power generation types

The SMC prices in the typical scenarios of 2022,2030,and 2040 are illustrated in Figs.14,15,and 16,respectively.Decreasing electricity prices and an increase in the frequency of negative electricity prices are clearly observed.Frequent negative prices can be attributed to increasing PV generation,which result in low charging costs for the IESS.However,PV-ESSs incur no charging costs because they consume solar energy,which cannot be consumed by the grid.After 2026,there will be more annual IESS utilization hours,implying higher charging and discharging frequencies.This will help in achieving considerable profits,even with the decrease in clearing prices.This also explains why the IESS outperforms the PV-ESS in terms of profitability despite lower clearing prices and higher construction costs.The different purposes of IESSs and PVESSs result in profitability differences;PV-ESSs primarily aim to ensure the maximum utilization of solar energy,whereas IESSs focus on profiting from peak-to-valley price differentials.

Fig.14 SMC prices in 2022

Fig.15 SMC prices in 2030

Fig.16 SMC prices in 2040

3.4 PV-ESS allocation ratios in two scenarios

In many regions,a minimum allocation ratio of 10% is required for PV-ESSs when constructing new PV units.This study adds a scenario with a fixed 10% PV-ESS allocation ratio to simulate this situation.The two scenarios are as follows:

S1: The PV-ESS allocation depends entirely on profitability compared to the PV.

S2: The PV-ESS maintains a growth of 10% in the newly increased PV capacity.

All simulations were conducted using S1.The subsequent sections present the comparative experimental results of S1 and S2.

Figs.17 and 18 show the PV-ESS capacity and investment decision changes,respectively,for the two scenarios.Compared to S1,the capacity growth of the PV-ESSs in S2 is faster from 2022 to 2030;then,it slows down.The investment decision for the PV-ESS in S1 is closer to that of the PV unit,indicating that the current PVESS allocation ratio in S1 is a more suitable reference for PV investors.Notably,the investment decision for the PVESS from 2024 to 2031 remains at zero in S2,indicating that the capacity growth of the PV-ESS equals zero during this period.A fixed allocation ratio results in significant investment waste.After 2032,the PV-ESS investment decision in S2 rapidly increases,suggesting that continuing with a 10% allocation ratio construction cannot meet the local consumption needs of the PV units.The fixed allocation ratio in S2 lacks economic feasibility,leading to resource inefficiencies.A more economical choice would be staying at a low level from 2022 to 2026 and gradually increasing to 10%-15% from 2027 to 2040.

Fig.17 Capacity variations of a PV-ESS in S1 and S2

Fig.18 Comparison of the investment decisions of a PV-ESS and PV system in S1 and S2

The variation in the PV-ESS allocation ratio in S1 is shown in Fig.19.In the early stage simulation,owing to the large profit margin of the PV system and high construction cost of the PV-ESS,the construction of PV-ESSs stopped.The PV-ESS allocation ratio gradually decreased to～5%.After 2026,the increase in the PV capacity and proportion of power generation increased the risk of curtailment and decreased clearing prices.Meanwhile,the construction costs of the PV-ESS decreased gradually.Profitability gradually became apparent,and the profitability of the PV system gradually lagged that of the PV-ESS.The PV-ESS allocation ratio gradually increased to～15%.After 2026,the investment decision of PV-ESSs remained slightly ahead of that of PV systems,implying that the profitability of PV-ESSs remains superior to that of PV systems.This can be attributed to the low investment sensitivity factor in Equation (6).Additional sensitivity scenarios are incorporated to address this issue.

Fig.19 PV-ESS allocation ratio in S1

Table 4 provides a clearer depiction of the optimal PVESS allocation ratio for each year of the simulation,along with its confidence probability.The confidence probability is derived based on the alignment of profitability between the PV-ESS and PV system.

Table 4 Optimal PV-ESS allocation ratio and its confidence probability

3.5 PV-ESS allocation ratios under different investment sensitivities

Figs.20 and 21 depict the investment decisions and allocation ratios,respectively,of PV-ESSs under different sensitivity scenarios,where a higher β value indicates greater sensitivity to PV-ESS investment.As shown in Fig.20,higher β values lead to more significant fluctuations in the PV-ESS investment decisions.In scenarios of β =3,β =3.5,and β =4,investment decisions drop to zero in some years,indicating inefficient investments in excessive PV-ESS construction due to high sensitivity.However,despite temporary excessive investments in PV-ESSs,the investment decision could be rectified by profitability changes within three years.This rectification can bring the PV-ESS allocation ratio back to an appropriate range,as illustrated in Fig.21.

Fig.20 Investment decisions of PV-ESS under different sensitivities

Fig.21 PV-ESS allocation ratios under different sensitivities

The PV-ESS allocation ratio fluctuates by year but shows similar values under different sensitivities.This indicates a dynamic optimal range for PV-ESS allocation.Therefore,from 2022 to 2026,PV investors need to allocate 5% of the PV-ESS when investing in PV units.After 2026,the allocation ratio gradually increases each year,reaching 15% by 2040.A sensible allocation ratio helps reduce the output rate of the PV units,enhancing their profitability during periods of low or zero electricity generation,thereby avoiding unexpected curtailment penalties.

4 Conclusions

This study proposes a dual-timescale dynamic model that combines a long-term SD model with a short-term UCbased SMC model.Based on this dual-timescale model,a profitability analysis was conducted between two different ESSs,and the optimal allocation ratio of PV-ESSs was studied.According to the simulation results,both the IESS and PV-ESS achieved sufficient capacity growth motivated by the price signal.The IESS demonstrated stronger profitability than the PV-ESS,owing to the higher number of utilization hours.The allocation of PV-ESSs is essential for reducing the simultaneous PV output rate and solar energy curtailment during peak periods.The optimal allocation ratio from 2022 to 2026 is approximately 5%,and thereafter,it increases by approximately 0.7% per year,reaching 15% by 2040.

Grid planners recommend that subsidies for ESSs be enhanced before 2026.The utilization of renewable energy sources can be increased by incentivizing the earlystage development of ESSs.PV investors are suggested to allocate a certain proportion of the investment to PVESSs,with this proportion significantly changing with the investment year.Constructing a PV-ESS within the optimal ratio range can assist PV units in clearing more energy and maximizing profits,while avoiding penalties for solar energy curtailment.

In future research,we will introduce more market types,such as the capacity and ancillary services markets,to comprehensively analyze the profitability of ESSs and optimal allocation of PV-ESSs.

Acknowledgments

This project is supported by National Natural Science Foundation of China (U2066209).

Declaration of Competing Interest

We declare that we have no conflict of interest.