Introduction

Owing to the fact of the fast development of renewable energy and the increased concern of the environmental and sustainable impact of fossil fuels, wind farms and photovoltaic plants have been widely increasingly built around the world over past decades [1]. Wind farms and photovoltaic plants are currently not confined in the conventional off-grid generation systems. Those on-grid variable power sources, however, exert an adverse impact on the operation and control of conventional power grid [2]. The power of wind farms and PV plants is normally much more dependent upon the on-site landform, topography and climate. As a result, they inherently contain the congenital defects, including intermittency, fluctuation and undispatchablility. Hence, energy storage systems which nowadays have become less expensive are introduced in such systems to ease the instability tendency of grids, with the ability of compensating for intermitted and fluctuant outputs of wind/PV plants. As the cost gradually falls, energy storage systems have been put into use in grids on a larger scale. Single type energy storage systems cannot meet the demands in real applications, considering the power and energy requirement at different time scale. As a result, hybrid energy storage systems turn out to be the feasible choice. The most common hybrid energy storage system is composed of batteries, such as advanced lead-acid or lithium-ion batteries, and supercapacitors [3].

Refs. [4, 5] present a fundamental frame to analyze the cost in a hybrid energy storage system, pointing out the ramp rate is the principal factor that has effect on the cost. Copula functions are discussed in [6, 7]. The space-time complementarities between wind farms and PV stations are emphasized in [8].

Ref. [9] cautiously explored the wind curtailment phenomenon in nations partly driven by wind power generation, such as America, Spain and Denmark. A new wind-ESS combined control method for surpassing Wind curtailment are in detail discussed in [10].

To further bring down the cost and actually implement the dispatchability of wind/PV plants, hybrid energy storage systems will be increasingly pervasive in the foreseeable future. Therefore, there is a need to penetrate into the major factors that contribute to the cost of the any HESS.

In this study, based on the cost of HESS by considering the ramp rate mentioned in [4], Copulas functions are proposed to analyze the dependence between wind and PV power. Then the cost of HESS is analyzed considering the ramp rate as well as the dependence between wind and PV power. Simulation studies are carried out to verify the above analysis.

Hybrid energy storage system

A hybrid energy storage system often owns the merits of individual single energy storage system. A battery energy storage system, such as advanced lead-acid batteries, has the advantages in pricing and large-scale using. It, however, poorly performs in the situation where power soars dramatically, or on the contrary, drops instantly. So, it is supposed to accumulate massive less fluctuant energy. With the more costly price, a supercapacitor energy storage system is not equipped with possibility of extensive using, but it does well in the moments when power moves fast.

The Fig. 1 depicts the diagram of a common fundamental generation system with a HESS, in which power equipment are omitted. The arrows point out the direction of power in an instant.

Fig. 1
figure 1

The diagram of a HESS-PV/Wind System

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In Fig. 1, P v represents the output power of photovoltaic stations. P w represents the power of wind farms. P is the aggregated power of P v and Pw. Buffered power is represented by P h. P sc is the power of the supercapacitor energy storage system. P b is the power of the battery energy storage system. P d represents the dispatched power into the grid.

Ramp rate

The ramp rate represents the change rate of power. More specifically, ramp rates, including ramp-up rates and rate-down rates, can be approached in two respects. First, they cannot be separated from wind farms or PV stations. For not only protecting wind turbines, photovoltaic cells and girds, but also ensuring that the power fluctuation of wind farms and PV plants can be regulated by other conventional power sources, like fossil-fueled plants, there is a limit to maximum change rate of active power in the wind farms and photovoltaic stations. Tables 1 and 2 bring the standards in China.

Table 1 The limitation of ramp-rates in wind farms
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Table 2 The limitation of ramp-rates in PV stations
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Second, ramp rates also go for the energy storage system. Owing to the chemical properties and specifications by battery manufacturers, a battery energy storage system has the boundaries of maximum ramp rate as well. There is no limit to ramp rates in the supercapacitor energy storage system, which undoubtedly assumes the superiority of supercapacitor energy storage systems.

Method

By Copulas, the dependences, or correlations between random variables can be set up. On one hand, marginal distributions can easily be obtained with joint probability distribution. On the other hand, it is not easy to get the joint distribution when marginal distributions are known. The emergence and improvement of Copulas functions, to some extent, resolves the problem [6].

Definition

In 1959, Sklar put forward the thinking of Copula function, which divides N-dimension joint probability distribution into N marginal distribution functions and one copula function that describes the correlation degree between variables. Nelsen gave the strict definition of the Copula function in 1999. Copula, denoted as C(u 1, u 2, …, u N), is a connection function that links joint probability distribution F(X 1, X 2, …, X N) of random variables X 1, X 2, …, X N with their individual marginal distribution, F X1(x 1), …, F XN(x N), which can be explicitly formulated as follows.

$$ F\left({X}_1,{X}_2,\dots, {X}_N\right)=C\left[{F_X}_{{}_1}\left({x}_1\right),{F_X}_{{}_2}\left({x}_2\right)\dots {F}_{X_N}\left({x}_N\right)\right] $$
(1)

Common copulas

Supposing Pearson coefficient is, a two-dimensional Gaussian Copula function is as follows,

$$ C\left(u,v;\rho \right)={\displaystyle {\int}_{-\infty}^{\varPhi^{-1}(u)}{\displaystyle {\int}_{-\infty}^{\varPhi^{-1}(v)}\frac{1}{2\pi \sqrt{1-{\rho}^2}} \exp \left[-\frac{s^2-2\rho st+{t}^2}{2\left(1-{\rho}^2\right)}\right] dstdt}} $$
(2)

A two-dimensional t-Copula function with free degree k is as follows,

$$ C\left(u,v;\rho, k\right)={{\displaystyle {\int}_{-\infty}^{{t_k}^{-1}(u)}{\displaystyle {\int}_{-\infty}^{{t_k}^{-1}(v)}\frac{1}{2\pi \sqrt{1-{\rho}^2}}\left[1+\frac{s^2-2\rho st+{t}^2}{k\left(1-{\rho}^2\right)}\right]}}}^{-\left(k+2\right)/2} dsdt $$
(3)

Archimedean copulas are defined as,

$$ \begin{array}{l}C\left({u}_1,{u}_2,\dots, {u}_N\right)=\left\{\begin{array}{l}{\varphi}^{-1}\left[\varphi \left({u}_1\right),\varphi \left({u}_2\right),\dots, \varphi \left({u}_N\right)\right],\hfill \\ {}0\hfill \end{array}\right.{\displaystyle \sum \begin{array}{l}i=1\\ {}{}_{other}\end{array}N\varphi \left({u}_1\right)\le \varphi (0)}\\ {}\end{array} $$
(4)

The measurement of correlation

There are several measurements of correlation, or dependency, between random variables. With regard to different usage, massive of them can be adopted, such as Pearson coefficient ρ, Kendall rank correlation coefficient τ, Spearman rank correlation coefficient ρ and tail dependence λ.

If marginal distributions of continual random variable (X, Y) are F(x) and G(y), the relationships between Copula function C(u, v) and Kendall rank correlation coefficient τ, Spearman rank correlation coefficient ρ, tail dependence λ are as follows, respectively.

$$ \tau =4{\displaystyle {\int}_0^1{\displaystyle {\int}_0^1C\left(u,v\right)dC\left(u,v\right)-1}} $$
(5)
$$ {\rho}_s=12{\displaystyle {\int}_0^1{\displaystyle {\int}_0^1C\left(u,v\right) dudv-3}} $$
(6)
$$ {\displaystyle {\lambda}^{up}}=\underset{{{}_{u\to 1}}^{-}}{ \lim}\frac{1-2u+C\left(u,v\right)}{1-u} $$
(7)
$$ {\displaystyle {\lambda}^{lo}}=\underset{{{}_{u\to 0}}^{+}}{ \lim}\frac{C\left(u,v\right)}{u} $$
(8)

Results

Dispatchability

Generally, a generation plant bids against each other. It is common for bidders in several power trading markets to forecast its power (nowadays, more and more forecast data are purchased from third-party service providers) and to propose generation schedule N hours before the market begins every trading day, which can be exemplified with Nordic power exchange market. The mechanism is called N-hour rule. To simulate the operation of real world, a forecast example is given below, with the hypothesis that dispatched power is derived from hourly average of forecasting power. A forecast algorithm using historical power data can be found in Fig. 2.

Fig. 2
figure 2

An example of forecast power

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The allocation of P b and P sc

P h is the aggregated power of P b and P sc. P h is the power that buffers from/into hybrid energy storage system. Y is the ramp rate. P b and P sc are confirmed in (9) and (10) [4].

Supposing

$$ \begin{array}{l}{P}_{B,r}\left({t}_{i-1}\right)+Y\cdot \varDelta t\ge {P}_{H,r}\left({t}_i\right)\hfill \\ {}{P}_{B,r}\left({t}_i\right)={P}_{H,r}\left({t}_i\right),{P}_{SC,r}\left({t}_i\right)=0\hfill \end{array} $$
(9)

Otherwise,

$$ \begin{array}{l}{P}_{B,r}\left({t}_i\right)={P}_{B,r}\left({t}_{i-1}\right)+Y\cdot \varDelta t\hfill \\ {}{P}_{SC,r}\left({t}_i\right)={P}_{H,r}\left({t}_i\right)-{P}_{B,r}\left({t}_i\right)\hfill \end{array} $$
(10)

The determination of energy capacity

After integrating P b and P sc with time, E b and E sc can be achieved. E b and E sc display the energy variation, or the energy level, of the energy storage system [4]. In Fig. 3, the zero-interface represents the assumptive initial energy level. The energy storage system is charged when the curve goes up with system being discharged as the curve comes down. Thus, the minimum energy capacity of the BESS or the SC-ESS is based on its own moving range, respectively. This implies that the best energy capacity is no less than these least value.

Fig. 3
figure 3

The determination of the BESS energy capacity

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Cost function

Solving (9) and (10), every ramp rate Y corresponds to a set of P b(t) and P sc(t), from which E b(t) and E sc(t) can be gained. The fit value of P b is the maximum of P b(t); the same for P sc. Note that the fit values of E b and E sc depend primarily on their difference between upper and lower bounds in Fig. 3.

Then, the basic cost function is as follows [4].

$$ f={k}_1\times {E}_b+{k}_2\times {E}_{SC}+{k}_3\times {P}_b+{k}_3\times {P}_{SC} $$
(11)

where k 1, k 2, k 3 and k 4 are obtained in [11].

Statistical observation

To minimize the cost f, the proper ramp rate Y should be selected. Meanwhile, P b (t), P sc (t), E b (t) and E sc (t) are successively determined. It is natural that the Cumulative Distribution Function (CDF) of these capacities is obtained as shown in Fig. 4 to depict the complete spectacle in statistical manners [4].

Fig. 4
figure 4

The CDF of BESS Power in 2013

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Definitions of coefficients

To make the problem more intuitive, here we separately define two types of coefficients, wind curtailment coefficient and energy storage coefficient. This approach will undoubtedly construct the connection between energy storage status and wind curtailment condition. The similar solar power abandonment is ignored for parallel approach.

The key to the problem is that we make a probabilistic modification to wind power output. In other words, P w is replaced by \( \left[1-\xi \left({\mathrm{t}}_{\mathrm{i}}\right)\right]\cdot {P}_w^{\hbox{'}}; \) note that \( {P}_w^{\hbox{'}} \) is the original 100 % of wind power.

Now we choose ξ(ti) to describe the level of wind curtailment and the definition of ξ(ti) is as follows,

$$ \xi \left({\mathrm{t}}_{\mathrm{i}}\right)=\left\{\begin{array}{l}0,\hfill \\ {}0\sim 1,\hfill \\ {}1,\hfill \end{array}\begin{array}{c}\hfill zero\; wind\; curtailment\hfill \\ {}\hfill partly\; wind\; curtailment\hfill \\ {}\hfill complete\; wind\; curtailment\hfill \end{array}\right. $$

Then, the wind curtailment coefficient is denoted by,

$$ C=\frac{{\displaystyle \sum_i^{i+k}\xi \left({\mathrm{t}}_{\mathrm{i}}\right)}}{T/\varDelta t}\times 100\% $$

Finally, the energy storage compensation coefficient is represented,

$$ \lambda =\frac{{\displaystyle \sum_i^{i+k}\left|\left[1-\xi \left({\mathrm{t}}_{\mathrm{i}}\right)\right]{P}_w+{P}_v-{P}_{H,r}\right.\left|\cdot \varDelta t\right.}}{{\displaystyle \sum_i^{i+k}{P}_h\cdot \varDelta t}} $$

As a result, we obtain that

  • λ > 1, the cost of hybrid energy storage system goes up

  • λ = 1, the cost of hybrid energy storage system remains unchanged

  • λ < 1, the cost of hybrid energy storage system goes down

Procedure to determine the optimum BESS ramp-rates

The MATLAB program is developed following the methods mentioned above. The flowchart is shown in Fig. 5.

Fig. 5
figure 5

The flowchart of calculation

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Discussion

Here, the paper exhibits a numerical example. All the data are based on real historical scene. The installed capacities of wind farms and PV stations are 143.45 MW and 431.17 MW, respectively, which are located at Belgium (http://www.elia.be).

The dependence using copulas

The paper adopts Copulas to readily calculate the dependences between wind farms and photovoltaic stations, which are shown in Fig. 6.

Fig. 6
figure 6

The dependence between wind and PV in 2013

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From the above dependences, three typical sets of data, representing three different common dependences, are picked up. The copula PDF of one of the three sets of data is given in Fig. 7.

Fig. 7
figure 7

The PDF of Frank Copula between wind/PV

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The buffered power

The paper utilizes the method in [12] to forecast the dispatchable power P d into the grid in next N hours. Note that the following procedures are based merely on one set of data. Then, P h is obtained after P deducting P d. An example of P h is shown in Fig. 8.

Fig. 8
figure 8

An example of buffered power P h

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Power and energy capacities

Once ramp rate Y is decided, a set of corresponding data is determined as well, including P b, P sc, E b and E sc, where energy level E b and E sc are from the integral of output powers. Here, the designed MATLAB Program endeavors to traverse Y from 0 through 10. Then, a great number of sets of {P b , P sc , E b , E sc} are acquired by the method mentioned in Subsection Cost function. The allocation of P b and P sc is revealed in Fig. 9. It is not difficult to find out that majority of P h is supported by the BESS, while the minority is assumed by SC-ESS in this case.

Fig. 9
figure 9

An allocation of P b and P sc

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The essence of cost function

As seen in (11), the cost of the HESS can be obtained by traversing ramp rate Y. By the input of three sets of data, the Fig. 10 intuitively demonstrates the relation between cost and dependence.

Fig. 10
figure 10

The cost curves with different dependences

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Hence, the essence of the HESS cost is as follows.

$$ f=f\left(Y,\alpha, \lambda; t\right) $$
(12)

In (12), f 1, f 2 and f 3 correspond the dependence -0.1, -0.2 and -0.3, respectively. As the dependence α between wind farms and PV stations goes up, the cost f diminishes. On determining the best ramp rate Y, the minimum energy capacities of the BESS and the SC-ESS are also confirmed. The cost curves with different lambda are showed in Fig. 11.

Fig. 11
figure 11

The cost curves with different lambda

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Conclusion

The paper concluded that the cost of a hybrid energy storage system is greatly affected by ramp-rate and dependence between the power of wind farms and photovoltaic stations, in which dependence can easily be determined by Copulas. A numerical example shows as the dependence between wind farms and PV stations goes up, the cost decreases. Moreover, the cost is also influenced by energy storage compensation coefficient.