a comprehensive model for wind power forecast error and its application … · application in...

J Electr Eng Technol.2018; 13(6): 2168-2177http://doi.org/10.5370/JEET.2018.13.6.2168

2168Copyright ⓒ The Korean Institute of Electrical Engineers

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

A Comprehensive Model for Wind Power Forecast Error and its Application in Economic Analysis of Energy Storage Systems

Yu Huang *, Qingshan Xu†, Xianqiang Jiang*, Tong Zhang** and Jiankun Liu**

Abstract – The unavoidable forecast error of wind power is one of the biggest obstacles for wind farms to participate in day-ahead electricity market. To mitigate the deviation from forecast, installation of energy storage system (ESS) is considered. An accurate model of wind power forecast error is fundamental for ESS sizing. However, previous study shows that the error distribution has variable kurtosis and fat tails, and insufficient measurement data of wind farms would add to the difficulty of modeling. This paper presents a comprehensive way that makes the use of mixed skewness model (MSM) and copula theory to give a better approximation for the distribution of forecast error, and it remains valid even if the dataset is not so well documented. The model is then used to optimize the ESS power and capacity aiming to pay the minimal extra cost. Results show the effectiveness of the new model for finding the optimal size of ESS and increasing the economicbenefit.

Keywords: Forecast error, Mixed skewness model, Copula theory, Energy storage sizing, Wind energy

1. Introduction

Wind power has gained great popularity in recent years as it could reduce pollutant emission and alleviate worldwideenergy crisis. The increasing levels of penetration of wind generated power, however, could pose challenges to system stability [1] and have large impacts on both operational and economic issues, e.g. extra cost for additional operatingreserve or for participation in short-term energy markets. Therefore, forecast systems in wind farms are crucial, but the accuracy is far from satisfactory with current methodologies, from the simplest persistence forecast model [2] to rather sophisticated models that combine statistical methods and numerical weather prediction (NWP) [3]. The stochastic and intermittent nature of windproduction makes it impractical to eliminate forecast errors completely. The error from day-ahead prediction could reach as high as 20%-40% [4] of installed capacity of wind farms and this number can be even higher in extreme weather conditions.

ESS offers to deal with the prediction problems due to its capability of mitigating the effect of forecast errors. The functions of ESS are usually classified by run times as voltage stabilization (seconds), power smoothing (minutes) and ancillary service (hours) [5]. Installation of such utilities near wind farms is advantageous in many aspects

[6-8] and in this context, we mainly concentrate on the reduction of cost with ESS in liberalized electricity market. ESS size is calculated in a deterministic way with fixed proportion of forecast value by using time step simulation [9]. In [10], a spectrum-based analysis of renewable generation and load demand in probabilistic framework was presented considering forecast uncertainties. Bludszuweit et al. [11] optimized ESS power and energy capacity simultaneously with unserved energy by the use of probability distributions of forecast errors and state of charge (SOC). The main obstacle of ESS sizing in the aforementioned literatures is how to better represent forecast errors of wind power.

It is generally assumed that the forecast error follows a Gaussian or near-Gaussian distribution [12,13]. The assumption has proved to work perfectly well for geographically dispersed wind farms that were treated independently due to Central Limit Theorem [14]. However, for an individual site, the normally distributed model has been challenged in [15] since the probability density function (PDF) tends to be asymmetric. It remains an ongoing research area to find an appropriate distribution of forecast errors of wind power, and the difficulty lies in the following three aspects: great diversity of shapes with different forecast methods in multiple timescales [15-17], variable kurtosis and skewness as well as fat tails [5], and insufficient measuring data or information of wind farms [18]. Other types of distributions which are more flexible than Gaussian, including Beta and Gamma-like functions [19], have been proposed to approximate the PDFs of forecast error for different wind power forecast levels. In [20], the uncertainty of wind power forecasts is quantified

† Corresponding Author: School of Electrical Engineering, Southeast University, Nanjing, China. ([email protected])

* School of Electrical Engineering, Southeast University, Nanjing, China. ([email protected])

** Electric Power Research Institute of State Grid Jiangsu Electric Power Company, Nanjing, China.

Received: August 4, 2017; Accepted: June 19, 2018

ISSN(Print) 1975-0102ISSN(Online) 2093-7423

Yu Huang, Qingshan Xu, Xianqiang Jiang, Tong Zhang and Jiankun Liu

http://www.jeet.or.kr │ 2169

by combining wind speed prediction error and power curve of wind turbines. However, the non-linearity of power curve can introduce error propagation via transformation. For ultra-short-term forecasts (within 1 hour), the PDF of forecast error may have extremely steep peaks and fat tails, resulting in large deviations for fitting of single distribution. Mixed distribution-based model taking into account the feature of discreteness [15,21] is a good option. Wu et al. [21] added Gaussian parts to the Laplace distribution for higher accuracy. Though mixture models are attractive for their flexibility and adaptability, it could be time-consuming to cope with discretized parts, and different forms of components would increase the complexity for further studies, e.g. optimal sizing of ESS or reserves [22-24], probabilistic wind power bidding strategies [25], unit commitment (UC) problems [26] etc., which all require post-processing of forecast error distributions.

The paper is organized as follows. Section 2 describes the characteristics of short-term wind power forecast error and its modeling approach using mixed skewness model (MSM). In section 3, Copula theory is employed to formulate conditional forecast error when the measuring data set of wind farms is not well-documented. Section 4 gives the solution to optimal ESS sizing based on cost-benefit analysis. Case study is presented in section 5, followed by conclusions in section 6.

2. Modeling Forecast Error Using MSM

2.1 Characteristics of wind power forecast error

For the universality of our work, the datasets we chose are from several onshore wind farms along the east coast of China. The forecasts (24h) of wind power were based on NWP method at a temporal resolution of 15min, which is most wiely used in short-term wind power prediction.

Fig. 1 shows the histogram of an example of typical wind power forecast error and all the data are set in per unit. It is obvious that the error distribution has two main characteristics, which remains true for other wind farms through statistical analysis.

a) Biased — The mean value of the histogram is non-zero and the entire figure is a little slanted to the right.

b) Fat-tailed — The figure decays rather slow in the right-hand tail region (in the interval [0.4, 0.8]).

The forecast error is also found to be associated with forecast levels [5], showing a highly diversity in the shape and features. Therefore, instead of directly modeling the overall error histogram as in Fig. 1, it is a common practice to represent the conditional forecast errors corresponding to their point forecast levels, respectively. The frequency histograms of forecast errors at different levels of forecasts are shown in Fig. 2. Let us first define an important terminology — forecast bin. Dividing the forecast level [0, 1] p.u. into m non-overlapping intervals of equivalent length (i.e. [0, 1/m], [1/m, 2/m]],L , [(m-1)/m, m/M]), for each interval, we define it as a forecast bin. Thus, the bin mcorresponds to the associated groups of errors with the forecast values located in the interval [(m-1)/M, m/M]. Theerror distributions in four continuous forecast bins (range from [0.4, 0.6] p.u.) have differences in either shape or position. Besides the aforementioned two characteristics of bias and fat-tail, multimodality may occur in some specific bins, which is the case in Fig. 2(d), posing great challenges to any common fitting distributions. It is noted that though the empirical error distribution may vary significantly as the change of forecasting methods and time horizons, the critical characteristics summarized above still hold by statistical tests as in Ref. [5, 15, 16]. Hence, in addition to capturing all of the characteristics, the desired model should be versatile and adaptable to fit the conditional forecast error in different scenarios by just altering the associated parameters.

2.2 Generation of PDF for wind power forecast error

A MSM can be viewed as a weighted sum of several skewed distribution components, whose PDF, for one-dimensional case, can be defined as

( ) ( )2

1

; , ,n

MSM i SN i i ii

f x f xw m s l=

=å (1)

-0.4 -0.2 0 0.2 0.4 0.6 0.80

200

400

600

800

1000

1200

1400

Forecast Error / p.u.

Fre

quency

Fig. 1. Frequency histogram of typical wind power forecasterror

-0.2 0 0.2 0.4 0.60

20

40

60

80

100

120

(a) Forecast Error / p.u.Bin No.8 [0.40,0.45]

Fre

que

ncy

-0.4 -0.2 0 0.2 0.4 0.60

20

40

60

80

(b) Forecast Error / p.u.Bin No.9 [0.45,0.50]

Fre

que

ncy

-0.5 0 0.50

10

20

30

40

50

60

(c) Forecast Error / p.u.Bin No.10 [0.50,0.55]

Fre

que

nc

y

-0.4 -0.2 0 0.2 0.40

10

20

30

40

(d) Forecast Error / p.u.Bin No.11 [0.55,0.60]

Fre

que

nc

y

Fig. 2. Frequency histogram of forecast error in different bins


2170 │ J Electr Eng Technol.2018; 13(6): 2168-2177

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

MSM

ComponentsMSM

1

X

PD

F

2

3

Fig. 3. Mixed skewness model

where iw is the weight coefficient of the ith component of MSM. m , 2s and l denote the parameters that reflect the location, scale and skewness of the distribution, respectively. ( )SNf × is the skewed distribution, defined as

( )2

SN

x xf x

m mf l

s s s

- -æ ö æ ö= Fç ÷ ç ÷

è ø è ø(2)

where ( )f × and ( )F × are PDF and CDF of the standard normal distribution, respectively. Specifically, when 0m = ,

2 1s = , 0l = , x is a standard normal random variable, denoted as ( )~ 0,1x N .

Fig. 3 shows the example of MSM approximation of a nonstandard random variable with three components. Theoretically, with n approaches infinite, MSM can inerrably fit any atypical distributions. However, the computation time is expected to grow exponentially with the number of components. Usually, n=2 or 3 can meet the requirements of engineering purposes in terms of accuracy. The estimation of parameters iw , im , 2

is and

il of each component are calculated iteratively by applying the expectation maximization (EM) algorithm [27] or nonlinear fitting [28], which works well with large number of measured datasets.

Therefore, the generation procedure of PDF for wind power forecast error is listed as follows: 1) Dividing the whole forecast range into m forecast bins. The number massociated with bin width should be selected according to the scale of measured data. 2) Sorting the forecast values into m forecast bins, and the measured outputs of forecasts at the same time instant are assigned to corresponding forecast bins. 3) Fitting the output dataset of each bin with the proposed MSM. 4) Adding up the PDFs of forecast error for all bins with weighted coefficients. The weight

iv of the ith bin corresponds to its forecast frequency, which can be written as in

1

( ) ( )m

i ii

f fe v e=

=å (3)

i iN Nv = (4)

where ( )f e and ( )if e are the PDFs of total forecast error and the error in bin i. N is the number of total historical samples, and the subscript “ i ” denotes the bin

that these samples belong to.

3. Modeling Forecast Error Using Copulas

3.1 Existing problems of proposed model

A noticeable benefit of MSM is that it is more flexible and could better represent the asymmetry, fat-tail and multimodality of the short-term wind power forecast error, compared with other commonly used distributions. There is no restriction on this statistical data-driven model and the procedure remains valid no matter what forecasting methods or timescales that one may choose. Such a modeling approach needs to be performed for a certain level of wind power in each corresponding forecast bin. However, it may sometimes occur that the historical output data set monitored in wind farms is not so well documented, presenting a relatively high missing rate, that the samples for some bins are few in number. The low occurrence of these output levels could make it hard to determine the distribution parameters and diminish the modeling accuracy. Therefore, an effective solution is to find the intrinsic correlations between the wind power outputs and their forecasts, by which all the measurement data can be utilized rather than those only in specific bins, thus providing a better estimate of forecast uncertainties.

3.2 Stochastic dependence measurement and copulas

The stochastic dependence of variables can be fully demonstrated by multivariate probability distributions. To obtain such multivariate analytical formulas, however, is intractable as the marginals for different variables are likely to be diverse from one another. In [29], concept of dependence modeling is classified as dependence structure and the degree of dependence. Though, versatile correlationcoefficients have been widely used in probabilistic studies, involving linear coefficient, rank correlation coefficient, tail dependence coefficient etc., they only provide quantitative measurement of stochastic dependence and do not contain the complete information, e.g. the dependence structure.

Copula theory is an effective tool for the analysis of dependence between random variables. According to Sklar [30] in 1959, there exists a Copula function ( )C × that can be written as

( ) ( )1 1 2 2 1 2( ), ( ), , ( ) , , ,n n nC F x F x F x F x x x=L L (5)

where 1x , 2x ,L , nx are random variables with invertible cumulative distribution functions (CDF) 1 1( )F x , 2 2( )F x ,

,L ( )n nF x and joint CDF F 1 2( , , , )nx x xL . If we treat( )i iF x ( )1, 2, ,i n= L as random variables, they are

uniformly distributed in [0, 1]. The joint PDF of variables ( )1, , nf x xL can be further derived by partial



differentiation

( )( )

( )

21 1

1

1

1 11

( ), , ( ), ,

( ), , ( ) ( )

n n

n

nn

n n i ii

C F x F xf x x

x x

c F x F x f x=

¶=

¶ ¶

= Õ

LL

L

L

(6)

where ( )1 1( ), , ( )n nc F x F xL is the Copula density function.( )i if x is the marginal PDF of random variable ix .As shown in (6), the feature of Copula theory is that it

transforms the acquisition of joint PDF into the modeling of Copula function c and marginal PDFs separately. Given an arbitrary data set, a suitable Copula function should be selected first, though it may not be a simple task. There are a number of Copula families, among which the most commonly used are elliptical ones including Gaussian Copula, t-Copula, and the Archimedean family (e.g. Gumbel Copula, Clayton Copula etc.). Each Copula is specialized in fitting a specific dependence structure between random variables. More information can be found about Copulas in [29].

3.3 Conditional PDF estimation for forecast error of single wind farm

The dependence between actual wind power and its forecast is stochastic and can be modeled using copula function. Their joint PDF can also be obtained from (6)

( , )( , )

( ( ), ( )) ( ) ( )

a f

a f

a f a f

P P a f

P P a f

a f

P a P f P a P f

F p pf p p

p p

c F p F p f p f p

=¶ ¶

=

(7)

whereaPf ,

fPf are marginal PDFs of the actual wind power

ap and its forecast fp respectively. ( )c × is a 2D copula density function. If a point forecast of wind power ˆ

fp is given, conditional PDF of the actual output

a fP Pf is then as follows

ˆ( , )ˆ( )

ˆ( )

ˆ( ( ), ( )) ( )

a f

a f

f

a f a

P P a f

a f fP PP f

P a P f P a

f p pf p p p

f p

c F p F p f p

= =

=

(8)

Substituting the real power ap in (8) for the sum of forecast error e and point forecast ˆ

fp , we get the conditional forecast error distribution as

ˆ( , )ˆ( )

ˆ( )

ˆ ˆ ˆ( ( ), ( )) ( )

a f

f

f

a f a

P P a f

f fE PP f

P f P f P f

f p pf p p

f p

c F p F p f p

e

e e

= =

= + +

(9)

Eq. (9) is the key for modeling forecast error based on copulas, and it is made up of two components: the

distribution of measured wind power as the base function and the selected copula density function as a variant multiplier. Note that though versatile models are available for the unconditional PDF of wind power output and its forecast, e.g., the GMM, Beta distribution etc., few of them have analytical forms of CDF, for which empirical CDF is adopted to give numerical solutions to

apF and fpF .

Partition point forecast levels ranged from 0% to 100% evenly with a step of 10%., and the total error distribution is approximately a weighed sum of conditional PDF of forecast error in each level.

Above all, the procedure of modeling the total forecast error of wind power using copula, which takes advantage of the dependence between real output and its forecasts can be summarized as in Fig. 4.

4. Economic Analysis of ESS Sizing

4.1 Cost of wind power forecast uncertainty

Wind power is not dispatched like conventional resourcesdue to its intermittent nature. In liberalized electricity market, though wind generation does not necessarily need to bid in the day ahead (DA) market, it should stick to the real time dispatch instructions. Any injection that is out of the tolerance band centered around the instructed value (wind power forecasts), known as uninstructed deviations (UD), would be penalized. Usually, the tolerance band is set within 10%± of the hourly average dispatch instructions. Additional cost of installation of supplemental-up and

Input historical data set of wind power output and its corresponding forecasts

Model the marginal PDF and CDF of wind power output and its forecasts

Transform the data set of real output and forecasts into uniformly distributed numbers

using their own CDFs, respectively

Select the best copula by fitting the joint PDF of all the uniformly distributed numbers and

identify the key parameters

Calculate the conditional PDF of forecast error for different point forecast levels ranged from 0% to 100% with a step of 10% , and the

weight coefficient of each level

Sum up the weighed conditional PDFs for all forecast levels to obtain the total error

distribution

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Fig. 4. Block diagram of modeling the forecast error of wind power using copulas



down reserves should also be included in penalties. For the sake of simplicity, penalties aroused from UD are arbitrarilyset 40% of the real time locational marginal price (RT-LMP), and hence, the loss L associated with UD for nhours can be formulated as

( )1

0.4n

i ii

L D l=

= ´å (10)

where iD refers to the deviation (either positive or negative deviation) in the ith hour, and il is the corresponding real-time locational marginal price. To compensate for the deviations and make wind generation more “dispatchable”, ESS is thought to be the optimal choice, for which the associated penalized cost can be largely reduced as well. Suppose that the rated ESS power equals to ESSP , and TP

is the tolerance band width. Then if forecast error( )ESS TP P P Pe ¢ ¢> = + , the excessive part for penalties

would be calculated according to (10), which could easily be expanded to expected daily loss ( )E L in DA market, written as:

( ) ( )24

1

0.4 i ii

E L E D l=

= ´å (11)

where ( )iE D is an integration with forecast error PDF formed before.

( ) ( ) ( )1

1

P

i PE D f d f de e e e e e

¢-

¢ -= -ò ò (12)

Separating Eq. (12) into two parts: the loss caused by positive deviations ( )iE D+

and negative deviations ( )iE D-.

( ) ( ) ( )i i iE D E D E D+ -= + (13)

In each part, the integral computation is time-consuming and could be solved in a numerical way based on Monte Carlo simulation defined as

( )( )

( )( )

1

1

, 0

, 0

C

C

N

ij ijj

i ij ij

CN

ij ijj

i ij ij

C

P

E D PN

P

E D PN

e

e

e

e

=+

=-

ì¢-ï

ï ¢= £ £ïíï ¢ -ï

¢= £ £ïî

å

å(14)

where ije is sampled from the proposed forecast error distribution, CN is the sampling size.

4.2 Cost of ESS

In [31], various types of ESS, including pumped hydroelectricity storage (PHS), Redox Flow Battery (RFB)

etc., are shown to have different structures and technical characteristics. But in general, these are all composed of two main parts: power conversion system (PCS) and energy storage unit. Take RFB as an example, Fig. 5 illustrates its structure schematic where the two parts are apparently shown.

Since the similarity in structural framework, a unified cost model could be established for different ESS. From ownership perspective, we use daily life cycle cost (DLCC) as an indicator for cost evaluation. DLCC consists of all the daily expenses associated with acquisition, operation and maintenance (O&M), replacement etc. of the ESS during its service time. It can be calculated by applying Eq. (15), subject to three terms: the initial cost or namely, capital cost InitialC , energy loss cost LossC and variable cost

VariableC . ODN is the operating days of the year, and the capital recovery factor CRFm based on the present value of money is written in (16).

( ),CRF

ESS d Initial Loss Variable

OD

C C C CN

m= + + (15)

(1 )

(1 ) 1

y

CRF y

i i

im

+=

+ -(16)

where i is the interest rate and y is the total lifetime. For detailed description of each term, the initial cost associated with rated power and energy is expressed as

Initial e ESS p ESSC c W c P= + (17)

where ec ($/kWh) and pc ($/kW) are associated cost coefficients of ESSW and ESSP . A real ESS will have losses, and the cost of which is dependent on its efficiency. Total variable cost of ESS can be expressed by adding the cost of O&M, replacement, disposal and recycling as shown in (18).

&Variable O M replace disposalC C C C= + + +L (18)

4.3 Optimization model for ESS sizing

The determination of ESS size is an optimization problem considering both the economic benefit of ESS and its cost. The objective is to minimize total daily cost TDC

expressed as

AC

DC Cell

Stack

PositiveElectrolyte

PositiveElectrolyteElectrolyte

Electrolyte

Membrane

Pump

Pump

Storage TanksDC bus

PCSAC system

.

Fig.5. Schematic of redox flow battery system



Min ,( )TD ESS dC E L C= + (19)

where ( )E L explained in (11) can reflect economic benefit associated with the decreased penalties of UD, and ,ESS dC

is found in Eq. (15). The constraints are listed as follows:Power balance constraints:

( ) ( ) ( ) ( )t tdisp a ESS ESSt p p P t D t P te = - + = + (20)

wheretdispp is the dispatched power of conventional

power plants at time t. This value is based on the wind power forecasts and is determined before the realization of the actual wind power t

ap . It is noted that ( )D t is only relevant to the forecast error and ESS power at time t, thusit will not be influenced by the predicted or actual wind power separately. Specifically, if ESSP equals the installed wind power, i.e., 1ESSP = p.u., any power mismatch caused by wind power forecast error could be mitigated by the ESS.

ESS power limits:

min max( )ESS ESS ESSP P t P- £ £ (21)

SOC limits:

min ( ) 1SOC SOC t£ £ (22)

Charge/discharge limits:

( ) ( 1) ( )ESSSOC t SOC t P t Th= - + ×D (23)

where h is charge efficiency. TD is time interval.

5. Case Study

The case study is performed from two different onshore wind farms (site A and B in Fig. 6), located in Fujian, east coast of China. The day-ahead forecasts of wind power are obtained from a specialized tool NSF3100 V1.33 designed by State Grid Electric Power Research Institute. In site A (55 MW), the actual outputs and their forecasts were generated in 2014 with 1-h resolution from a well-documented entity (8760 data values). While the available data corresponding again to hourly measurements in site B

(35 MW) for the same period is not so well documented, with only 2920 data values. All the data are standardized in p.u. with the installed capacity as the base value in the modeling process.

Three PDFs are used to fit the error distribution of dataset A: the normal PDF, Laplace PDF and the proposed MSM. Beforehand, the data pairs [measured, forecast] should be sorted and assigned to respective forecast bins, in this case, 20 bins with 0.05 p.u. bin width seem to be suitable. Fig. 7 compares the performance of the three PDFs in specific bins with the actual error histogram. Adding up the weighed PDFs of all bins, the total forecast error PDF can be obtained. The result is shown in Fig. 8 with a reduced number of components.

It is obvious that MSM offers a better fit for the fat-tailed and bimodal error PDFs in the example bins. To verify this numerically, chi-square test is applied to measure the goodness-of –fit for various distributions, as shown in Table 1. The lowest value of MSM confirms it as the best fit of all.

Due to the vast capacity and deep discharge capability of RFB, we choose it as an appropriate ESS for site A and B in this case study. Suppose the RT-LMP is set fixedly at 25$, and average width of tolerance band is kept 0.02p.u. Table 2 provides the basic parameters of selected ESS. Detailed ESS data can be found in Ref [37].

-0.2 0 0.2 0.4 0.6 0.80

2

4

6


PD

F

Normal

Laplace

MSM

Actual

(a)

-0.4 -0.2 0 0.2 0.40

1

2

3

4


PD

F

Normal

Laplace

MSM

Actual

(b)

Fig. 7. Comparison of curve fitting with various distri-butions for forecast error in example bins. (a) forecast bin 6 (b) forecast bin 11

Table 1. Chi-square goodness-of-fit for various distributions

Chi-square testBin No.

Normal Laplace MSM6 3719.47 2986.02 72.5911 5306.83 4541.65 88.92

.

.

A

B

Fig. 6. Sites used for wind power data in this study



Table 2. Associated parameters for selected ESS (RFB)

ce/kWh cp/kW η/% y/years interest rate/%

120 476 75 14 5

Table 3. Optimal size of ESS and associated daily cost by using different error distributions

PESS/MW WESS/MWh Daily Penalties/$ Daily cost/$

Actual 3.75 45.8 564.5 1587.9

Normal 4.03 51.4 614.6 1711.5

Laplace 3.98 50.9 608.2 1698.7

MSM 3.78 46.3 565.1 1594.6

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

Wind Forecast Error / p.u.

Pro

bability

MSM

Actual

Fig. 8. Total error PDF of wind forecast (dataset A) with MSM approximation

02

46

810

0

50

1001400

1600

1800

2000

2200

PESS

/MWWESS

/MWh

CT

D /$

Fig. 9. 3-D diagram of total daily cost with reference to ESS power and capacity in site A

Wind Power Forecast

Win

d P

ow

er O

utp

ut

Fig. 10. Scatter plot of actual wind power and its forecast after CDF transformation

0 0.5 10

0.5

1

(a) Gaussian Copula

0 0.5 10

0.5

1

(b) t Copula

0 0.5 10

0.5

1

(c) Clayton Copula

0 0.5 10

0.5

1

(d) Gumbel Copula

Fig.11 Scatter plot of random numbers generated by different Copulas

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

Wind power output / p.u.

Pro

babili

ty

10% Point forecast

30% Point forecast

60% Point forecast

90% Point forecast

Fig. 12. Conditional PDF of wind power output for different percentage of point forecast in site B

The total daily cost estimated with reference to rated power and capacity is represented in Fig. 9. Table 3 compares different distributions in terms of optimal ESS size, daily penalties and total cost. It can be seen that the results of MSM are quite close to the actual one, while using normal and Laplace distributions could overestimate the size of ESS and decrease the economic benefit.

The insufficiency of dataset B can be settled via the proposed copula modeling. Fig. 10 reveals the inherent relationship between the real output and forecast of site B, using the scatter plot of data pairs which have been transformed to the uniform domain. Four typical copula functions are then selected for the fitting of the same data (in Fig. 11). Comparing Fig. 11 with Fig. 10, we can intuitively tell the performance of fitting from best to worst: Gumbel, Gaussian, t and Clayton. The same conclusion could be made by the 2c test shown in Table 4. Therefore, we use Gumbel Copula for dependence modeling in the case study.

Fig. 12 partitions the total distribution of wind power output by different point forecast levels. The peak of conditional PDF is shifted from left to right as the percentage of forecast increases.



The optimal size of ESS (PESS=2.23MW, WESS=24.2MWh) as well as the associated daily cost (CTD=963.3$) is captured from the 3-D diagram shown in Fig. 13. In table 5, comparative results using different types of copulas are also calculated to validate the effectiveness of the proposed model.

6. Conclusion

A new comprehensive model has been developed to calculate the forecast error of wind power in this paper. Provided that the historical measurement of wind power output is well documented, MSM can be flexible enough to give an accurate fit for the biased, fat-tailed error PDF compared to other distributions. However, when there is a lack of data set, a copula-based approach that builds the dependence structure between the real output and its forecast, is adopted as an alternative to model the conditional PDF of forecast error effectively. The model is further applied to the economic analysis of ESS, to determine its optimal rated power and capacity with the consideration of both the economic benefit and cost. The case study results show that the size of ESS can be largely reduced with the application of the proposed model, and

thus could save a large sum of money and increase the economic profits.

Acknowledgements

This work was supported by National Natural Science Foundation of China (51377021), the Fundamental ResearchFunds for the Central Universities (2242016K41064) and Postgraduate Research & Practice Innovation Program of Jiangsu Province.

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Table 4. Chi-square goodness-of-fit of dependence modelingusing different Copulas

4 Gaussian t Clayton Gumbel2 statisticsc - 46.28 48.17 210.43 35.81

Table 5. Optimal size of ESS and associated daily cost by using different type of copulas

PESS/MW WESS/MWh Daily Penalties/$ Daily cost/$

Actual 2.22 23.8 235.6 955.7Gaussian 2.25 25.8 264.4 992.6

t 2.27 25.9 271.4 1006.2

Clayton 2.35 28.7 309.2 1105.4Gumbel 2.23 24.2 238.1 963.3

02

46

8

0

20

40

60

80900

1000

1100

1200

1300

1400

1500

PESS

/MWWESS

/MWh

CT

D /$

Fig. 13. 3-D diagram of total daily cost with reference to ESS power and capacity in site B



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Yu Huang Received the B.S degree in electrical engineering from Nanjing University of Aeronautics and Astro-nautics, China, in 2015. He is now a visiting research scholar at Purdue University, the USA and is pursuing the Ph.D. degree in the School of Electrical Engineering, Southeast Uni-

versity, China. His research interests include probabilistic load flow calculation, stochastic analysis and evaluation of Renewable energy, new energy power generation forecasting.

Qingshan Xu (IEEE Member) He was born in China, on May 25, 1979. He received B.S., M.S. and D.E. inelectrical engineering from Southeast University, Hohai University and Southeast University, in 2000, 2003, and 2006, respectively. Now he is a professor in the School of Electrical

Engineering, Southeast University, China. He did Visiting Scholar and cooperates with Aichi Institute of Technology



in Japan from 2007 to 2008. His research interests include renewable energy, power system operation and control of and etc.

XianQiang Jiang Received the B.S degree in electrical engineering from Shandong University, Jinan, China, in 2015. He is now pursuing the M.S. degree in electrical engineering in Southeast University, Nanjing, China. His research interests include renewableenergy and power system reliability.

Tong Zhang Received the B.S. and M.S. degree in electrical engineering from Nanjing University of Aeron-autics and Astronautics and State Grid Electric Power Research Institute, in 2015 and 2018, respectively. She is nowa researcher in Electric Power ResearchInstitute of State Grid Jiangsu Electric

Power Company, China. Her research interests include power system monitoring and control.

Jiankun Liu He is currently a senior engineer and director in the power grid control center, Electric Power Research Institute of State Grid Jiangsu Electric Power Company, China. His research interests include power system opera-tion and control.

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