Research on energy storage capacity configuration for PV power plants using uncertainty analysis and its applications

0 Introduction

Photovoltaic (PV) power generation systems are highly random and intermittent and are influenced by weather and environmental conditions and solar irradiance periods [1, 2].PV power generation adversely affects the economic, safe, and reliable operation of power systems [3, 4].Highcapacity energy storage is a key technology in addressing the uncertainty of PV power generation that introduce fluctuations in the grid [5, 6].An energy storage system can respond to dynamic energy changes in a timely manner, effectively absorbing and releasing energy to mitigate grid fluctuations.The capacity configuration of an energy storage system has an important impact on the economy and safety of a PV plant [7].An excessively small capacity cannot ensure a proper function of the energy storage system; an excessively large capacity results in high investment, operation, and maintenance costs.

The energy storage capacity configuration is the one of the hotspots in current study [8, 9, 10].A hybrid wind- photovoltaic energy storage system is proposed to optimize energy storage capacity, and the double-layer decision model of the storage capacity configuration is established [11].In which the target of the outer decision model is the minimum investment of the storage and the contact line penalty, and the target of the inner decision model is to minimize the power fluctuation on system tie line.Lei et al.determined the power value for each cycle through a PV power forecast and battery charging status and controlled the battery using an energy management system.The microgrid load curve for peak load balancing was optimized to increase the economic efficiency of PV power generation [12].Sandhu et al proposed a new approach based on particle swarm method for optimally sizing the storage system employing the battery banks for the suppression of the output power fluctuations generated in the hybrid PV-wind energy system [13].Bludszuweit et al.proposed a method of designing energy storage systems to reduce the uncertainty of short-term wind power forecasts within 48 h [14].Nayak et al.used an improved harmonic search algorithm (IHSA) to optimize the capacity of a battery storage system for grid-connected PV systems to reduce the operating cost as much as possible while satisfying the load demand [15].This study focuses on the energy storage capacity configuration of PV plants considering the uncertainty of PV output and the distribution characteristics of the forecasting error in different weather conditions.

Compensating for PV power forecast errors is an important function of energy storage systems [16, 17].The capacity of an energy storage system is calculated based on the PV power forecast; an energy storage device is used to compensate for the power forecast error [18], effectively reducing the loss caused by the PV power forecast error.PV power has different distribution characteristics in different weather conditions; the traditional single model is significantly limited in practical applications of PV power generation systems.In this study, considering the output characteristics of PV under different weather conditions, a novel optimization method of energy storage capacity has been designed, which can significantly reduce the operating cost of energy storage systems and facilitate operation and maintenance planning.

Considering the uncertainty of PV power generation, a method for configuring energy storage system capacity is proposed to compensate for the power forecasting error.The k-means cluster analysis algorithm is used to reasonably classify the weather; the historical power output and error distribution characteristics of PV power plants are analyzed in different weather conditions.The kernel density estimation (KDE) is used to fit the distributions of the daily maximum power and capacity required for the energy storage system.Power and capacity configurations are calculated at different confidence levels; the degrees of power satisfaction and capacity satisfaction are used to evaluate the energy storage configuration results, and the optimal energy storage system configuration for the PV power station is obtained.The proposed method is validated through calculation and analysis of historical operating data from a PV power station in Qinghai.The results provide a basis for the configuration of an energy storage system for a PV power station.The remainder of the paper is structured as follows: in Section 1, the uncertainty of PV power generation and power forecast errors is analyzed.In Section 2, an energy storage system configuration based on nonparametric estimation is proposed.In Section 3, an example is provided to verify the proposed method using the actual operating data of PV power stations, and Section 4 provides the conclusion.

1 Uncertainty of PV output and forecasting results

1.1 Classification of PV output characteristics and weather types

PV power generation systems have strong uncertainties and are influenced by the external environment, and solar irradiance has the most prominent impact on PV modules [19].Fig.1 shows the output power and solar irradiance variation curves for PV power plants with different solar irradiance conditions on a typical day.The variation in output power is significantly affected by the variation in solar irradiance, and their variation trends are similar.Thus, when analyzing the output power characteristics of PV power plants, the daily operation data must be classified by weather type according to the difference in solar irradiance to fully describe the output power characteristics in different conditions.

Fig.1 Output power and solar irradiance for typical days

The k-means clustering algorithm is an iterative solution-based cluster analysis algorithm [20, 21] that uses distance as a similarity index to classify a given dataset, where each class is described by a cluster center obtained from the mean of all values in the class [22, 23].For a given sample (X) containing n data objects and a number of required classifications (k), k clustering centers Pj (j = 1, 2,...k) are selected to minimize the objective function (F):

The Euclidean distance is selected as the similarity measure, and the error sum-of-squares criterion function (E) is used to evaluate the clustering performance.For sample X containing k clustering subsets Xi (i = 1, 2,...k), the error sum-of-squares criterion function can be expressed as

The basic concept is that if the data are divided into k groups, k objects are randomly selected as the initial cluster centers; the distance between each object and the cluster center is calculated to divide the sample into the cluster centers closest to it, forming k initial clusters [24].New cluster centers are selected based on the existing objects in the clusters, and the data are classified according to the new cluster center.This process is repeated until the maximum number of iterations is reached, or until all cluster centers no longer change.

The output characteristics of PV plants are significantly influenced by solar irradiance.Based on the daily solar irradiance, the historical operating data of PV plants are divided into three types: weather type 1 (best solar irradiance), weather type 2 (good solar irradiance with large fluctuations), and weather type 3 (poor solar irradiance) (Fig.2).

In Fig.2, the solar irradiance curves for weather type 1 exhibit the same trends, all steadily increasing and reaching a peak at 13:00.The solar irradiance curve trends for weather type 2 are similar to those for weather type 1, with more fluctuations in irradiance, possibly due to environmental changes, resulting in insufficient irradiance.The solar irradiance curve for weather type 3 has no apparent trend, with poor irradiance throughout the day.

Fig.3 PV output power distribution for three weather types

Fig.2 Solar irradiance in three weather types

Statistical analysis of a PV plant output power was performed for the three weather types; the results are shown in Fig.3.The PV output power in weather type 1 was concentrated between 30-50 MW, maintaining high output power.In weather type 2, the PV output power was evenly distributed in each power band.In the worst solar irradiation conditions, the PV output power was approximately 25 MW, which was a low power output.

The uncertainty of the PV plant is largely reflected in the significant change in output power as solar irradiance varies.The PV output power distribution characteristics for the three weather types are noticeably different.

1.2 Uncertainty of forecast results

In this study, the historical operation data and power forecasting data from a 60 MW PV plant in Qinghai were selected for error analysis.

Fig.4(a-c) shows the forecasted and actual PV power variation for two typical days in each weather type.Fig.4(a) shows that in weather type 1 with the best irradiance conditions, the actual and forecasted PV power trends were in general agreement, with a smooth curve and high power.Fig.4(b) shows actual power fluctuations due to weather changes; the best irradiance conditions could not be maintained, and the forecast error was occasionally large.Fig.4(c) shows that the forecasted PV power was often slightly greater than the actual power in poor irradiance conditions; the power output was lower and fluctuated frequently.

In this study, the mean absolute error (MAE), rootmean-square error (RMSE), and mean relative error (MRE) were used to evaluate the accuracy of the forecast results [25], and they are expressed as

where Pf is the forecasted power, and Pa is the actual power.

Fig.4 Actual and forecasted PV power generation for two typical days in three weather types

Table 1 Calculated MaxError, MRE, MAE, and RMSE for each weather type

Weather Type MaxError (MW) MRE MAE RMSE Type 1 3 5.034 0.3508 2.1273 4.6708 Type 2 47.251 0.3923 2.5352 5.7682 Type 3 44.928 0.6107 2.1805 4.8545

Table 1 shows the indicator values for the three weather types.The MRE indicates the extent to which the forecasted value deviates from the actual value.The irradiance conditions in type 3 weather were poor and changed frequently, further complicating the power forecasting, resulting in a significantly greater MRE than in other weather types.The MAE reflected the error between the forecasted and actual power.The MAE and RMSE for type 2 were greater than those for type 1 owing to large fluctuations in irradiance that reduced forecast accuracy.With low output power and poor irradiance conditions, the MAE and RMSE for type 3 were lower than those for type 2.

Fig.5 Comparison of actual and forecasted powers for three weather types

Fig.5 shows the degree of deviation between the predicted and actual power for the three weather types.The distributions of the actual and forecasted power were concentrated above 35 MW for type 1; the deviation from the dashed line y=x was smaller, and the forecast accuracy was higher.The distributions of actual and forecasted power were higher even for type 2, with a greater deviation from the dashed line y=x.For type 3, the actual and predicted power were concentrated near 20 MW; the deviation from the dashed line y=x was the greatest, and the forecast accuracy was the lowest.

The PV power output characteristics and the forecast error distributions for the three weather types have apparent differences; the weather classification method can distinguish historical PV operation data and facilitate subsequent analysis.The configured energy storage system compensates for power differences and tracks the target output of the PV system.The required energy storage system capacity depends on the forecast error; the same configuration for all conditions is likely to increase energy storage system operating costs.With the uncertainty of PV power plant output in different weather conditions, different storage capacity configurations can effectively reduce the operation costs, with reasonable operation and maintenance time.

2 Principle and design of energy storage configuration method

2.1 Kernel density estimation

The errors of the forecasted and actual PV power were analyzed, and a model of energy storage capacity and power distribution was established using non-parametric estimation.The required capacity for the storage system was reasonably configured.KDE is a non-parametric method for estimating the underlying distribution or probability density function of a dataset [26, 27].It can be used to estimate the unknown density function without considering the interference of external factors to achieve better fitting results [28].

The KDE at any point x is

where x1, x2…xi is the sample data, h is the bandwidth, and K is the kernel function.The shape and value domain of the kernel function determine the number of data points and the utilization degree to estimate the value at point x [29].Common kernel functions include the uniform, triangular, Epanechnikov, and Gaussian kernel functions; the fitting effects of different kernel functions are shown in Fig.6.For better kernel density estimation [30], the Gaussian kernel function is selected for probability density estimation, and it is expressed as

The bandwidth (h) of the kernel function has a significant influence on the results of the KDE [31].The fitting of the energy storage power distribution with different bandwidths is shown in Fig.7.If the bandwidth is excessively small (h = 0.5), the obtained probability density curve undulates; if the bandwidth is excessively large(h = 9), the obtained probability density curve is excessively smooth and ignores much of the underlying data structure.Therefore, a suitable bandwidth is important.

In selecting the bandwidth, the mean integrated squared error (MISE) can be used [32].

For the MISE, the asymptotic mean integrated squared error (AMISE) is calculated as

Fig.6 Fitting of different kernel functions

Fig.7 Fitting curves for kernel density estimation with different bandwidths

where R ( k)= ∫K 2(u )du; m ( K)= ∫x 2K ( x )dx; R ( f'')= ∫( f ''( x ))2 dx.The selection of an optimal h is transformed into an extreme value problem that minimizes the MISE; when ∂AMISE( h )/∂h=0, the extreme value is obtained.The bandwidth hAMSE can be obtained as

When the kernel function is determined, R, m, and f'' in the equation can be calculated using and AMISE( h ) =[33].If the Gaussian and density functions are selected for non-parametric estimation, h can be expressed as

where is the standard deviation of the sample variables.

2.2 Proposed method

An energy storage system compensates for the difference between the forecasted and actual PV power, such that the PV plant can deliver power to the grid according to the forecasting results and facilitate execution of the power system dispatching plan.The sampling time interval for a PV plant is generally 15 min; the energy storage system can sufficiently respond within 15 min to ensure that the actual power value reaches the predicted power value.The energy storage system power is expressed as

where Ps(t) is the forecasted PV power of the plant at time t, and Pr(t) is the actual PV power of the plant at time t.When Ps(t)＞ Pr(t), the forecasted PV power of the plant is greater than the actual power, and the energy storage system discharges.When Ps(t)＜ Pr(t), the forecasted PV power of the plant is less than the actual power, and the energy storage system charges.When the forecasted PV power is equal to the actual power, the energy storage system stops operating.

E is the energy of the storage system, obtained by integrating the power of the storage system over a period of time, and it is expressed as

where E(t) is the capacity of the energy storage system at time t; η1 and η2 are the charging and discharging efficiencies of the energy storage system, respectively; and Δt is the time interval.

From the energy equation, we can calculate the capacity change curve of the energy storage system for one day; the absolute value of the difference between the highest point and the lowest point of the capacity change curve indicates the maximum absolute energy required, which is expressed as Ea.

where d is the number of days with one weather type, and k is the weather type (k = 1, 2, 3).

The KDE is used to fit the distribution of the energy storage capacity (Ea).Based on the fitted probability density curves, the capacity configuration of the energy storage system is calculated at different confidence levels.The degree of capacity satisfaction is calculated to evaluate the energy storage capacity configuration method.The degree of capacity satisfaction (f1) is expressed as

where f 1,p% is the degree of capacity satisfaction at confidence level p%; Dk is the total number of days in weather type k; δd,k is a coefficient indicating whether the storage capacity demand is satisfied on day d in weather type k; Ea,k,p% is the configured storage capacity at confidence level p% in weather type k; and Emax d is the maximum storage capacity on day d.

The maximum absolute value of the peak or trough in the power variation curve for each day is selected as the maximum absolute power required by the energy storage system, expressed as Pa:

where d is the number of days in the dataset; k is the weather type (k = 1, 2, 3).

Similar to the capacity calculation method, the power configuration and degree of power satisfaction are calculated for the energy storage system at different confidence levels.The degree of power satisfaction f2 is expressed as

where f 2,p% is the degree of power satisfaction at confidence level p%; Tk is the number of PV output power samples in weather type k; λj,d,k is a coefficient indicating whether the j-th power sampling point on day d satisfies the storage capacity demand for weather type k; Pa,k,p% is the configured power at confidence level p% in weather type k; and Pj,d is the actual power of the j-th power sampling point on day d.

Fig.8 System diagram of the method presented in this paper

The system diagram of the method presented in this paper is shown in Fig.8 and includes four steps:

Step 1: Historical PV plant operation data are preprocessed to remove abnormal data.

Step 2: The k-means cluster algorithm is used to classify weather types according to different irradiance conditions.The power forecast error for different weather types is analyzed, and the error distribution is established.

Step 3: The maximum daily power and maximum daily capacity of the energy storage system are calculated to compensate the power forecast error.The KDE is used to fit the power and capacity distributions of the energy storage system.

Step 4: The capacity and power configurations are calculated at different confidence levels according to the power and capacity distributions of the energy storage system.The f1 and f2 values at different confidence levels are calculated, and the energy storage configuration is finalized.

3 Verification

Actual operating data from a power plant in Qinghai were selected for analysis and calculation.The k-means clustering algorithm was used to divide the data into the three weather types, occurring in 42%, 37%, and 21% of the days, respectively.

Based on the results of the power forecast, the realtime power and capacity of the energy storage system were calculated for each day.The KDE was used to fit the distribution, and the cumulative distribution function (CDF) was calculated as shown in Figs.9 and 10.

Fig.11 shows the energy storage configurations with different confidence levels.Table 2 shows the f1 and f2 values with different confidence levels.It was observed that when the confidence level was below 90%, the demand for energy storage capacity and the corresponding degree of capacity satisfaction was low; it was difficult for the energy storage system to compensate for the power forecast error.When the confidence level was above 95%, the demand for energy storage capacity increased sharply, and the degree of capacity satisfaction was high with a slight change.Thus, selecting the 95% confidence level to configure energy storage satisfies most of the capacity and power requirements while reducing system costs as much as possible.

Table 2 Power and capacity satisfaction with different weather types

Confidence level Type 1 Type 2 Type 3 f1 f2 f1 f2 f1 f2 80% 0.78 0.993 0.79 0.990 0.68 0.990 85% 0.80 0.995 0.82 0.992 0.76 0.994 87% 0.82 0.996 0.85 0.993 0.79 0.994 89% 0.83 0.996 0.85 0.993 0.85 0.995 91% 0.87 0.997 0.88 0.993 0.87 0.995 93% 0.90 0.997 0.92 0.994 0.89 0.996 95% 0.93 0.998 0.94 0.997 0.94 0.997 97% 0.97 0.998 0.96 0.998 0.96 0.997 99% 0.99 0.999 0.99 0.999 0.98 0.998 100% 1.00 1.000 1.00 1.000 1.00 1.000

Fig.9 Storage power for three weather types

Fig.10 Fitting of energy storage capacity for three weather types

Fig.11 Three energy storage configuration scenarios

Fig.13 shows the power curve of the energy storage system for seven consecutive days with the three weather types, including the actual power demand of the energy storage system and power of the energy storage system configured with 85% and 95% confidence levels.The power requirement could not always be satisfied when the storage power was configured with 85% or 95% confidence levels, as indicated by the circles in Fig.13.However, the power requirement was satisfied significantly more often at the 95% confidence level than at the 85% confidence level.Fig.12 shows the variation in the maximum demand for energy storage capacity for each day with three weather types; the red line represents the 95% confidence level capacity configuration, and the blue line represents the 85% confidence level capacity configuration; the 95% confidence level capacity configuration can better satisfy the capacity demand of the storage system.

Table 3 Capacity and power configurations for 85% and 95% confidence and full satisfaction (capacity: MWh, power: MW)

Weather type 85% confidence level 95% confidence level Full satisfaction Capacity Power f1 f2 Capacity Power f1 f2 Capacity Power f1 f2 1 36.8 13.2 0.80 0.995 49.7 15.9 0.93 0.998 70.4 23.8 1.00 1.000 2 41.4 16.0 0.82 0.992 53.1 19.0 0.94 0.997 74.2 28.1 1.00 1.000 3 18.7 8.8 0.76 0.994 25.8 12.3 0.94 0.997 43.1 18.0 1.00 1.000 unclassified 47.8 15.8 0.85 0.995 61.7 19.9 0.95 0.998 74.2 28.1 1.00 1.000

Fig.12 Maximum daily capacity required for energy storage (From top to bottom: type 1, type 2, and type 3)

Fig.13 Power change for seven consecutive days (From top to bottom: type 1, type 2, and type 3)

Fig.14 Capacity and power configurations for 85% confidence, 95% confidence, and full satisfaction

Fig.14 and Table 3 show the capacity and power configurations for the 85% and 95% confidence levels and full satisfaction.The 95% confidence level capacity configuration provided 49.7, 53.1, and 25.8 MWh for the three weather types, respectively; f1 was approximately 0.94 for the 95% confidence level.Compared with full satisfaction, the capacity decreased by 29.4%, 28.4%, and 40.1%, respectively.

The energy storage system capacity required for all three weather types was lower than that for unclassified weather; the energy storage capacity satisfaction was similar with or without weather classification.Configuring different energy storage system capacities for different weather types can effectively reduce the required capacity.For PV plants, reasonable configuration of the energy storage system capacity according to the forecasted weather can effectively reduce the operation cost and provide a reference for energy storage system operation and maintenance plans.

4 Conclusion

In this paper, a method of energy storage capacity allocation is proposed based on the distribution characteristic of PV output power forecasting errors.The objective is to use energy storage systems to smooth the fluctuation of PV power plant outputs and improve the reliability of PV power generation.We analyze the uncertainty of PV plant output power, and use the energy storage system to compensate for PV power differences and track the target PV output.Depending on the differences in irradiance conditions, the k-means clustering algorithm is used to classify the historical operating data of the PV plant into three types; we analyze the output characteristics and power forecast error distributions for the three types of PV plants.An energy storage system is used to compensate for the difference between the actual and forecasted PV powers.We calculated the power of the energy storage system based on the forecast error and used the KDE to analyze the capacity distribution characteristics of the energy storage system.The required capacity and power of the energy storage system was calculated for different confidence levels.The following conclusions were drawn:

1.Solar irradiance is the main external factor affecting the output of a PV plant.Using the k-means clustering algorithm to divide the historical operating data of the PV plant into three categories based on irradiance can effectively describe the output characteristics of PV plants in different conditions and the distribution characteristics of power forecast errors.

2.The KDE can describe the energy storage system capacity and power distributions.The energy storage capacity configuration with a 95% confidence level can reduce the cost of energy storage and satisfy the energy storage requirements in most conditions.

3.A method of configuring the energy storage capacity based on the uncertainty of PV power generation is proposed.Configuring different energy storage capacities for different weather conditions can significantly reduce the required capacity, reducing the operating cost of energy storage equipment and providing a reference for energy storage system operation and maintenance plans.

Acknowledgements

This work was supported by Nation Key R&D Program of China (2021YFE0102400).

Declaration of Competing Interest

We declare that we have no conflict of interest.