Optimization of wind and solar energy storage system capacity configuration based on the Parzen window estimation method

Wind energy and solar energy, as the most important renewable energy sources, will play a crucial role in the future energy system.¹ Due to the complementary characteristics of wind and solar power generation in terms of timescale,² it improves the power quality of wind–solar hybrid power generation systems to a certain extent. However, the intermittent and fluctuating nature still has an impact on the power quality, which cannot be ignored. Therefore, energy storage is an effective method to improve the power quality of renewable energy.³ 

Compared to other forms of energy storage, compressed air energy storage (CAES) systems have advantages such as long lifespan, environmental friendliness, low cost, and long discharge time.⁴ However, the current utilization of CAES configured with typical daily data models, both domestic and international, can lead to low economic benefits due to inaccurate data, which affects its commercialization and application. Therefore, in order to address the aforementioned issues, it is necessary to employ temporal production simulation for planning.⁵ When conducting temporal simulation production calculations, appropriate typical daily data for user load, solar power generation, and wind power output should be selected to ensure the rationality of power planning and generation scheduling.⁶ 

Data processing in energy storage systems can enhance the visibility, operability, flexibility, profitability, and security of energy systems and can be applied to dynamic modeling and economic planning of energy systems.⁷ The selection of typical day data is mainly done through direct method, clustering method, and fitting method. Tiejiang et al.⁸ directly used the day with the largest difference between peak and valley load data as the typical day, which is subjective and has a large error compared to the original data, greatly impacting the accuracy of the final planning results. Guo et al.⁹ first categorized the load data into working day, peak day, and rest day, and then directly selected typical days. The selected results cannot effectively cover the overall characteristics of the original data. Gao et al.¹⁰ proposed a k-means clustering method to process the original load data using clustering and averaging methods to obtain typical day data, but did not consider that k-means clustering method has its own clustering centers, making the averaging meaningless. Pinto et al.¹¹ designed a novel K-Medoids with optimization method for extracting typical day data, using adaptive factors and probabilistic statistics as feature indicators, but ignoring the temporal nature of the load. Jani et al.¹² fitted the original data using the Weibull probability distribution function, which partly reflects the characteristics of the original data, but did not optimize the shape and scale parameters, making the obtained probability distribution not necessarily suitable for all time periods. Azad et al.¹³ optimized the shape and scale parameters using the method of moments (MOM) based on the Weibull probability distribution function, but assumed a certain distribution for the probability distribution in the typical day data in advance, which is subjective. Yu et al.¹⁴ used the Weibull probability distribution function, MOM, and weighted averaging method to obtain the typical day data for wind power, which to some extent improves the accuracy of the typical day data model, but still cannot overcome the excessive subjectivity of the fitting method.

It is generally unknown what the probability distribution of the samples is, and the above-mentioned literature does not address this factor extensively. Known functions, such as the normal distribution, beta distribution, and Weibull distribution, cannot accurately and realistically describe the probability distribution. Moreover, some classical probability density functions are unimodal, while the probability density functions of wind power, solar power, and load samples are mostly multimodal, which directly affects the accuracy of typical daily data. This paper proposes a method to obtain typical daily data using the Parzen window estimation method and weighted averaging method. The Parzen window estimation method is a nonparametric estimation method that is not constrained by the population probability distribution and does not excessively rely on probability parameters. Furthermore, this model has good robustness.¹⁵ Based on typical daily data, the optimization of the system economy needs to be carried out to achieve commercial operation of the wind–solar-storage system.

The utilization efficiency of energy storage systems and the economic benefits of wind–solar-storage systems can be influenced by the various capacity configurations of wind power, photovoltaic, and energy storage systems. For example, Javed et al.¹⁶ developed an optimization model for capacity configuration of wind–solar-storage hybrid power systems with the objective of minimizing total system cost and evaluating the power loss rate. They analyzed the impact of evaluation metrics on capacity and cost. However, their optimization focused only on energy storage capacity without considering the power of the energy storage system. Cardenas et al.,¹⁷ based on a decade of renewable resource and load data in the UK, studied the impact of different proportions of wind and solar power generation on the required energy storage capacity for achieving 100% renewable energy generation. The results showed that the lowest energy storage capacity was required for the 84% + 16% wind–solar output ratio. Meinrenken et al.¹⁸ introduced time-of-use electricity prices during the energy storage system planning phase and established an optimization model with the objectives of maximizing annual revenue and achieving the highest photovoltaic utilization rate. However, they did not consider environmental benefits. Rezk et al.¹⁹ developed a joint optimization planning model for energy storage systems and load demands with the objective of maximizing economic benefits. They analyzed the impact of different scenarios on energy storage systems and economic feasibility, but environmental benefits were not extensively considered.

This paper utilizes the Parzen window estimation method to establish a typical weekly data model for wind power, solar power, and user load. Second, an optimization model for a wind–solar-storage system with the objective of maximizing economic benefits is developed. Finally, the optimization results under different scenarios are compared to validate the feasibility of the proposed model.

The objectives of this study are as follows:

1. Compare the accuracy of the typical weekly data models established using the Parzen window estimation method and parametric estimation methods.

2. In islanded/grid-connected mode, an optimization model for wind–solar-storage system capacity configuration is established, taking into account factors such as renewable energy utilization rate, system purchasing rate, and environmental benefits, with the objective of maximizing economic benefits. The optimization results under different scenarios are compared based on the evaluation indicators to ensure the accuracy and objectivity of the proposed model's optimization results.

In Sec. II, the Parzen window estimation method was used to extract characteristic data features of wind power, photovoltaics, and user load. Section III mainly describes the objective function, constraint conditions, evaluation metrics, and optimization algorithms. Section IV focuses on case studies. Section V presents the conclusions.

To create a suitable weekly data model for wind power, this study collected real-time wind speed data from a wind farm in a certain area of Inner Mongolia Autonomous Region, China (N 39°61′, E 109°78′) during the year 2018, as shown in Fig. 1. From Fig. 1, it is evident that the wind speed exhibits characteristics of randomness and fluctuation. Therefore, assuming that the probability density distribution form of the sample is known using parameter estimation, the resulting probability density function obtained from solving it is not accurate.

Nonparametric estimation methods can be used for probability distribution models with unknown probability density functions. These methods do not require assumptions about the data probability distribution and instead derive data distribution characteristics directly from the data itself. The Parzen window estimation method is a typical, efficient, and accurate nonparametric estimation method,²⁰ which can estimate the distribution of any irregular wind power, solar power, and load models and provide a more accurate fit to real data distribution.¹⁴ 

The Parzen window estimation method is influenced by the bandwidth and the window function. Different window functions have little impact on the resulting probability density distribution, while different bandwidths have a significant impact. A larger bandwidth leads to smoother probability density distribution results but lower resolution. Conversely, a smaller bandwidth results in sharper probability density distribution but higher resolution. To ensure the selection of the optimal bandwidth, this paper chooses the bandwidth that minimizes the difference between the probability density distribution curve obtained using the Parzen window estimation method and the real data when the confidence interval is set at 90%.

Taking the first 0:00–0:15 min of the first day as an example, the modeling of the Parzen window estimation method is described as follows.

1. Using IEC61400–12-1 standard,²¹ valid wind speed data are summarized from available samples for this time period.

2. Interval division is conducted using the Bin method, with centers at integer multiples of 0.5 m/s, and the probability density is calculated for each interval.

3. All the probability densities are aggregated to establish a probability distribution curve.

The best Weibull probability density distribution curve is obtained using the method described in Ref. 12. Figure 2 shows the probability distribution of wind speeds obtained using different methods for the 0:00–0:15 min period. At this time, the optimal bandwidth is 0.3209, and the optimal shape parameter and scale parameter in the Weibull probability distribution function are 2.1066 and 8.7573.

In Fig. 2, although the Weibull probability distribution can overall reflect the wind speed density distribution, some smaller oscillations in the probability histogram are not well represented. The error of the probability histogram in the Weibull method is imperfect because the wind speed is irregular, and the assumption of the Weibull probability distribution is made subjectively before establishing the probability model. The Parzen window estimation method does not subjectively describe the wind speed before modeling but builds the probability distribution curve based on the extracted features from the data model. It is more in line with the true distribution of the data and has higher accuracy, which provides preliminary validation of the reliability of the Parzen probability distribution.

To create a suitable weekly data model for photovoltaics, this study collected the solar radiation data from a photovoltaic power station in a certain region (latitude N 39°61′, longitude E 109°78′) in Inner Mongolia Autonomous Region, China, throughout the year of 2018, as shown in Fig. 3. From the graph, it is evident that the solar irradiance data follow a normal distribution pattern overall, with higher values observed between the months of May and August, coinciding with the period of stronger wind speeds.

Normalize the data for the period from 10:00 to 10:15 on the first day and then establish a normalized probability distribution of solar irradiance using the steps outlined in Sec. II A for wind speed probability distribution. At this time, the optimal bandwidth for Parzen window estimation is 0.0273.

In Fig. 4, it is evident that the improved Beta probability distribution curve can reflect the overall distribution trend, but it has significant errors and cannot accurately represent the probability density in the presence of multiple peaks. On the other hand, the Parzen probability distribution can effectively fit the variations of photovoltaic data. Compared to the Beta probability distribution, it exhibits minimal overall error.

The user demand load data has a wide range, and it is effective to improve the stability and reliability of the typical weekly data model by normalizing the data samples and then using the Parzen window estimation method to establish the model. The specific method is similar to establishing the typical weekly output power of photovoltaic and will not be further elaborated. Figure 5 shows the real-time load demand power for a year. The sharp fluctuations in short-term load during February, April, May, and October are mainly due to the Chinese New Year, Qingming Festival, Labor Day, and National Day. Normalize the data for the period from 00:00 to 00:15 on the first day, and then establish a normalized probability distribution of load demand using the steps outlined in Sec. II A for wind speed probability distribution, refer to Fig. 6, at this time, the optimal bandwidth for Parzen window estimation is 0.02.

The structural diagram of a wind–solar-coupled hybrid energy storage system is shown in Fig. 7. The system as a whole is powered by the solar and wind farms. The CAES system begins charging when the power demand at the load center is lower than the power supplied by the wind farm and solar power station. The CAES system begins to discharge when the power demand at the load center exceeds the electricity supplied by the wind farm and solar power station. The system will buy electricity from the grid if the power demand at the load center exceeds the total power supplied by the wind farm, solar power plant, and energy storage system.

This study employs an isobaric adiabatic compressed air energy storage (IA-CAES) system, as illustrated in Fig. 8. The main components of the system consist of a compressor, an expander, a heat exchanger, a thermal storage tank, a cold storage tank, and a constant-pressure air storage chamber. The compressor operates in three stages with intercooling, while the expander operates in three stages with inter-stage reheating. Both the hot and cold fluid mediums used in this system are water.

The operational strategy of IA-CAES is shown in Fig. 9. First, the difference between the wind power [P_Wt(t)], photovoltaic power [P_Pv(t)], and the load demand power (P_B(t)) at time t is compared. If P_D(t) is greater than 0, the system enters the charging mode. The remaining capacity of the CAES is calculated, based on which the optimal charging power P2 is determined. If P2 is greater than P_D(t), the surplus power is considered as spilled power, and the compressor is operated at P2 power for charging. If P_D(t) is less than 0, the system enters the discharging mode. The remaining capacity of the CAES is calculated, based on which the optimal discharging power P2 is determined. If P2 is greater than P_D(t), the excess power is regarded as the purchased power, and the expander is operated at P2 power for discharging. In the equation, P_ra, C_ra, S(t), η_ch, and η_dis denote the rated power, rated capacity, energy storage capacity at time t, charging efficiency, and discharging efficiency of IA-CAES, respectively.

The following presumptions form the basis of the system model for the energy storage system: the leakage of gas is not considered during gas circulation at each stage; changes in fluid and heat exchange during flow processes, as well as chemical reactions, are neglected; heat and pressure losses in pipelines are not taken into account; the instantaneous response changes of IA-CAES during the discharging phase are not considered. Table I shows the modeling expressions for different parts of the IA-CAES system.

In Table I, p represents the pressure; λ denotes the compression ratio, which remains constant since the energy storage mode is isobaric; subscripts in and out indicate the inlet and outlet, respectively. The variables i and j represent the ith stage compressor and jth stage expander, respectively; η represents the isentropic efficiency; subscripts c and t refer to the compressor and expander, respectively; γ denotes the adiabatic index; T represents the temperature; W represents the work done; m denotes the air mass flow rate; M_car(0) represents the air mass in the storage tank at the initial state; T_hot represents the temperature of the high-temperature thermal storage tank; and τ represents the time.

P_yh(t), c_gs(t), P_L(t), c_gb(t), P_G(t), and P_du(t), respectively, represent the real-time power for selling electricity, the electricity selling price on the grid, the load power, the grid purchase price, the power for purchasing electricity, and the power for abandoning electricity at time t. c_En represents the environmental benefits.²³  N_Wt, c_Wt, N_Pv, and c_Pv represent the number of wind turbines, unit price of wind turbines, number of photovoltaic systems, and unit price of photovoltaic systems, respectively. Inv_c1 and Inv_p1 represent the initial installation costs of IA-CAES per megawatt-hour and per megawatt, respectively. r represents the investment rate, and L represents the lifespan of the hybrid energy storage system.

P_B(t) represents the load demand power at time t. P_Cdis(t), P_Cch(t), η_dis, and η_ch represent the discharge power, charge power, discharge efficiency, and charge efficiency of the energy storage system, respectively. The capacity constraint of the energy storage system is 10 times the maximum load.²⁵ Except for specific notations, the other constraint conditions apply to any mode.

In the equation, E_wt, E_pv, E_s, and E_g represent the wind power generation, photovoltaic power generation, and total system load, respectively. T_G and T_C denote the purchased electricity time and total time, respectively.

The capacity allocation model for the established energy storage system in this section can be equivalently considered as a single objective problem and solved using the MATLAB Genetic Algorithm Toolbox. Genetic algorithm (GA) has strong global search ability and can distribute the search task to multiple processors or computing nodes, accelerating the optimization process. Moreover, the optimization problem of energy storage system usually involves multiple constraints and objective functions, and often relates to nonlinear and non-convex problems. Genetic algorithm can adaptively adjust based on the characteristics of the problem through fitness function and selection operation, and search for effective solutions in the solution space.²⁶ The algorithm is set with an initial population of 200, a crossover rate of 0.85, and a mutation probability of 0.2. The algorithm flow is illustrated in Fig. 10.

In the expression, v_h(t) represents the wind speed at time t; v_in, v_out, v_r, and P^r_w represent the cut-in wind speed, cutout wind speed, rated wind speed, and rated power of the wind turbine, respectively.

In the equation, f(t, P_Wi) represents the probability of the wind turbine output power being P_Wi at time t.

• A typical weekly data curve for wind turbine output power can be created by repeating steps 1 and 2, as shown in Fig. 11. Using the Parzen distribution probability density function, the average output power for a regular week is 0.86 MW.

In the equation, P_Pv(t) represents the solar irradiance at time t, and f(t, P_Pvi) represents the probability of the solar irradiance being P_Pvi at time t.

• By repeating the above steps, a typical weekly data curve for solar irradiance can be obtained, as shown in Fig. 12. The average irradiance for a typical week, obtained using the Parzen distribution probability density function, is 266.23 W/m².

In summary, the selected photovoltaic system model is CY-TJ 600, with a rated power of 0.60 kW. When the solar irradiance is greater than 0.60 kW, the output power of the photovoltaic system is 0.60 kW. When the solar irradiance is less than 0.60 kW, the output power of the photovoltaic system is equal to the solar irradiance.

To account for the uncertainty in load demand, this paper mainly utilizes the Parzen window estimation method to obtain the typical 15-min load demand curve, as shown in Fig. 13. The specific method is similar to the establishment of a typical weekly output power curve for photovoltaic systems.

Table III represents the basic parameters of the wind–solar energy storage system. IA-CAEA initial capacity is 50%, with upper and lower limits of energy storage capacity at 10% and 90% respectively. The rated energy and ability of the wind turbine, photovoltaic system, and energy storage system are all optimized globally within the constraints, with a step size of 1 and they are all superb integers. The selling price of electricity follows the time-of-use (TOU) pricing of Inner Mongolia Autonomous Region in 2022, as shown in Fig. 14.

This article aims to identify the optimal configuration solution by comparing the technical and economic aspects of different wind and solar energy storage system configurations. The optimal solution can be determined by considering various evaluation criteria comprehensively. The configuration of different scenarios is shown in Table IV.

Economic parameters of wind and solar energy storage systems are shown in Table V.

In grid-connected mode, under the joint action of different configurations of wind power, photovoltaic, and energy storage systems, the economic benefits of the wind–solar-storage system are shown in Table VI. Based on the solution algorithm process of 2.4, the optimal capacity configuration of the wind–solar-storage system that simultaneously satisfies the constraint conditions is obtained as a function of economic optimal in different scenarios. In scenario 3, in the absence of an energy storage system, in order to guarantee the user's load demand, the required wind power and photovoltaic power are both higher than in other scenarios, the REU value is lower than in the other three scenarios, and the LOR value is higher than in the other three scenarios. Additionally, the maximum profit is significantly lower than in the other scenarios, which is unfavorable for commercial promotion. Scenario 4 represents one year of actual data, and the optimization results serve as important references for evaluating scenarios 1 and 2. Among them, the discrepancy between scenario 2 and the actual data is large, while the discrepancy between scenario 1 and the actual data is within 5%, reflecting the accuracy of the typical weekly data model established by the Parzen window estimation method.

In scenario 1, the power dispatch of the wind–solar-storage system is shown in Fig. 15. The weekly curtailed energy is 4887.56 MW·h, accounting for 17.41% of the total duration. The purchased energy is 477.62 MW·h. The total time when the energy storage system is at the upper and lower capacity limits is 18.75%. After 7 days of operation, the state of charge (SOC) the CAES system reaches 58.04%, slightly higher than the initial capacity, ensuring the periodicity, accuracy, and reusability of the constructed model.

In island mode, the economic benefits of the wind–solar-storage system are shown in Table VII. Based on the solution algorithm process of 2.4, the optimal capacity configuration of the wind–solar-storage system that simultaneously satisfies the constraint conditions is obtained as a function of economic optimal in different scenarios. In scenario 5, in the absence of an energy storage system, in order to guarantee the user's load demand, the required wind power and photovoltaic power are both higher than in other scenarios. Since there is no curtailment penalty cost, the profit is higher than in scenario 3. However, the REU value is minimal, which is unfavorable for commercial promotion. Scenario 8 represents one year of actual data, and the optimization results serve as important references for evaluating scenarios 6 and 7. Among them, the discrepancy between scenario 6 and the actual data is large, while the discrepancy between scenario 7 and the actual data is within 5%, reflecting the accuracy of the typical weekly data model established by the Parzen window estimation method.

In scenario 5, the power dispatch of the wind–solar-storage system is shown in Fig. 16. The weekly curtailed energy is 4030.95 MW·h, accounting for 14.88% of the total duration. There is no purchased energy (0 MW·h), and the total time when the energy storage system is at the upper and lower capacity limits is 13.83%. After one week of operation, the energy storage system's capacity reaches 63.08%, slightly higher than the initial capacity.

This article aims to improve the accuracy of typical weekly data by comparing the accuracy of Parzen window estimation method and parametric aggregation methods (such as Weibull distribution method, Beta distribution method, etc.) in constructing typical weekly data models. It evaluates the capacity configuration of wind–solar energy storage systems obtained by different methods based on one-year time series data as the evaluation basis. The final results indicate that

1. The typical weekly data model established using the Parzen window estimation method can more accurately reflect the true data characteristics, and the obtained capacity configuration of the wind–solar energy storage system is more in line with reality.

2. To meet the average user hourly load of 699.26 MW in terms of power demand, in grid-connected mode, the rated power of wind and solar PV is 1107 and 346 MW respectively. The rated power of CAES is 290 MW, with a rated capacity of 2320 MW·h. It can save electricity purchase costs by 6.55 million dollars per week, with a maximum weekly profit of 0.61 million dollars. If the penalty costs are not included, the payback period for system investment is 5.6 years. In island mode, the rated power of wind and solar PV is 1041 and 455 MW, respectively. The rated power of CAES is 466 MW, with a rated capacity of 3724 MW·h. It can save electricity purchase costs by 6.74 million dollars per week, with a maximum weekly profit of 0.61 million dollars. If the penalty costs are not included, the payback period for system investment is 6.2 years.

3. In the grid-connected mode, the rated power of CAES is 290 MW, with a rated capacity of 2320 MW·h, which can reduce the rated power of wind and solar PV by 111 and 17 MW, respectively. It can reduce the power shortage rate by 27.24% and increase the wind–solar utilization rate by 11.68%. In the island mode, the rated power of CAES is 386 MW, with a rated capacity of 3085 MW·h, which can reduce the rated power of wind and solar PV by 766 and 201 MW, respectively. It can increase the wind–solar utilization rate by 38.82%.

The research work presented in this paper was financially supported by a Grant (No. 52065054) from the National Natural Science Foundation of China, a Grant (No. 2022LHMS05023) from the Natural Science Foundation of Inner Mongolia, and the Beijing Outstanding Young Scientists Program (No. BJJWZYJH01201910006021).

The authors have no conflicts to disclose.

Qihui Yu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal). Shengyu Gao: Writing – original draft (equal); Writing – review & editing (equal). Guoxin Sun: Project administration (equal); Supervision (equal). Ripeng Qin: Investigation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.