Evaluation Method of Wind Speed Time-Shifting ...

Research ArticleEvaluation Method of Wind Speed Time-ShiftingCharacteristics at Multiple Scales and Its Application inWind Power System

Han Wang Shuang Han Yongqian Liu and Aimei Lin

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources School of Renewable EnergyNorth China Electric Power University (NCEPU) Beijing 102206 China

Correspondence should be addressed to Shuang Han hanshuang1008sinacom

Received 31 July 2020 Accepted 23 October 2020 Published 21 November 2020

Academic Editor Haoran Zhang

Copyright copy 2020 Han Wang et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(e wind speed sequences at different spatial positions have a certain spatiotemporal coupling relationship It is of greatsignificance to analyze the clustering effect of the wind farm(s) and reduce the adverse impact of large-scale wind powerintegration if we can grasp this relationship at multiple scales At present the physical method cannot optimize the time-shifting characteristics in real time and the research scope is concentrated on the wind farm (e statistical method cannotquantitatively describe the temporal relationship and the speed variation among wind speed sequences at different spatialpositions To solve the above problems a quantification method of wind speed time-shifting characteristics based on windprocess is proposed in this paper Two evaluation indexes the delay time and the decay speed are presented to quantify thetime-shifting characteristics (e effectiveness of the proposed method is verified from the perspective of the correlationbetween wind speed sequences (e time-shifting characteristics of wind speed sequences under the wind farms scale and thewind turbines scale are studied respectively (e results show that the proposed evaluation method can effectively achieve thequantitative analysis of time-shifting and could improve the results continuously according to the actual wind conditionsBesides it is suitable for any spatial scale(e calculation results can be directly applied to the wind power system to help obtainthe more accurate output of the wind farm

1 Introduction

A high proportion of renewable energy is one of the basiccharacteristics of the future electric power system and windpower is an important component of renewable energy [1]With the rapid expansion of wind power integration thevolatility of wind power has amore adverse impact on the griddispatching safe and stable operation of the electric powersystem and its power quality [2] In fact there is a certainspatiotemporal coupling relationship among the wind speedsequences at different spatial points [3] If we can grasp thiscoupling relationship at multiple temporal-spatial scales it isof great significance to analyze the clustering effect [4] dealwith the ramp events and extreme events obtain the realoutput of wind farm(s) [5] and so forth which can effectivelyreduce the negative effect of the wind power integration [6]

At present the research algorithms on the spatiotem-poral coupling relationship among wind speed time seriescan be divided into two categories the model-based physicalmethod and the data-based statistical method according tothe driving mode

(e physical method takes the wind speed data at acertain point as input and considers factors such as terrainroughness altitude and flow to establish the aerodynamicmodel to describe the coupling relationship among the windspeeds at different spatial locations Zheng [7] assumed thatthe wind spreads at the speed of the starting point andestablished the motion equation of wind speed to obtain theaverage delay time between the wind speed time series at twospatial positions She et al [8] used the rotor diameter andgeographic coordinate matrix of each wind turbine in thewind farm as the inherent characteristics and took the wind

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4057630 17 pageshttpsdoiorg10115520204057630

speed and direction of the wind tower rotor speed and pitchangle of each wind turbine as inputs to establish the wakeeffect model and the time-lag effect model then the dynamicspatial relationship among wind speed time series at eachwind turbine location was obtained Zeng et al [9] firstdeveloped the continuous partial differential equation ofeach wind turbine by calculating the wake effects accordingto the specific spatial locations and distributions of corre-lated wind turbines (en the finite volume method is usedto differentiate the differential equation at each grid pointand derive the spatial correlation matrix of wind speedFinally the wind speed at each wind turbine location isobtained by solving the above differential equation throughthe given boundary conditions Ye et al [10] proposed anovel spatial correlationmatrix by using computational fluiddynamics tomathematically describe the correlations amongwind speeds at different wind turbine locations

(e physical method studies the temporal and spatialdistribution and the coupling relationship of wind speeds byestablishing the aerodynamic model among the wind speedsat different spatial positions However this aerodynamicmodel cannot be optimized in real time according to theactual wind conditions Besides when the spatial scale in-creases the complexity of the constructed model increasessharply which leads to the increasing difficulty of modelingand solving the model simultaneously But it is difficult toensure the accuracy of the aerodynamic model if the factorsaffecting wind speed are simplified (erefore the appli-cation of this method is mainly concentrated in the interiorof the wind farm which is difficult to expand to a largerspatial scale

(e statistical method uses the actual wind data at dif-ferent spatial locations to analyze the spatiotemporal couplingrelationship among the wind speeds (e research methodscan be mainly divided into the following three types

(1) (e spatiotemporal coupling relationship is studiedby calculating the distribution of wind speeds atdifferent spatial locations from the perspectives ofaverage wind speed wind speed frequency distri-bution wind shear exponent and so forth [11] Liuet al [12] analyzed the spatiotemporal distributionsof the wind resource at the hub height in China fromthe aspects of average wind speed and average windpower density based on the large network of windspeed observations including 2430 meteorologicalstations from 2006 to 2015 (e results show thatnortheast China has the highest wind speed whileSouth China has the lowest wind speed Li and Yu[13] conducted the statistical analysis of the onshorenearshore and offshore wind energy conditionsfrom the aspects of Weibull shape and scale factorsaverage turbulence intensity and average windpower density (e results indicate that the averagewind speed at the offshore location is 1-2 timeshigher than that nearby onshore area at the height of50m above ground level Bosch et al [14] estimatedthe distributions of global offshore wind energypotential from the aspects of average wind speed

mean capacity factor which could be built across theviable offshore area and generation potential peryear (e results show that the wind speed increaseswith the distance to the coast

(2) (e spatiotemporal coupling relationship is studiedby establishing the mapping relationship amongwind speed time series at different spatial positionsthrough interpolation neural network principalcomponent analysis and so forth [15] Philip-popoulos and Deligiorgi [16] used two feedforwardneural network architectures to establish the rela-tionship among hourly wind speeds at six meteo-rological sites An insight into the underlying input-output function approximation of the neural net-works is obtained by examining their ability to in-corporate the mean wind variability characteristicsMiranda and Dunn [17] proposed a spatially cor-related wind speed model to characterize the windresources between various sites by using the mul-tivariate time-series approach with actual wind speeddata in different zones Park and Jang [18] presenteda modified Korean parameter-elevation regressionon independent slopes model for spatial interpola-tion of monthly wind speeds (e results show thatthe proposed model can performmore appropriatelyto represent the phenomena where similar windspeeds appear continuously along ridges andcoastlines compared with the cokriging model

(3) (e correlation coefficient is used to study the cor-relation characteristics of wind speed time series atdifferent spatial positions which is an important partof the spatiotemporal coupling relationship (ecalculation method can be divided into the followingcategories One is to directly calculate the correlationcoefficient based on the wind speed time series [19]Another is to use the Copula function to fit the jointprobability distribution of the wind speed sequencesand then calculate the correlation coefficient based onthe fitted Copula function [20] (e commonly usedcorrelation coefficients include the Pearson linearcorrelation coefficient [21] Kendall rank correlationcoefficient [22] and cross-correlation coefficient [23]

(e statistical method studies the spatiotemporal cou-pling relationship of wind speed sequences at differentspatial positions based on the actual wind data (e obtainedresults of this method can be continuously improvedaccording to actual conditions and the research method canbe applied to any spatial scale However only the correlationrelationship between wind speed sequences is quantified inthis method Compared with the physical method thetemporal relationship and the law of speed variation are bothimplicit in the spatial distribution and the mapping rela-tionship is not quantified yet

In conclusion when the physical method is used to studythe spatiotemporal coupling relationship among windspeeds at different spatial positions the established aero-dynamic model cannot be optimized in real time according

2 Mathematical Problems in Engineering

to the actual wind conditions and the application of thismethod mainly concentrated in the interior of the windfarm it is difficult to expand to a larger spatial scale Whenthe statistical method is applied only the correlation rela-tionship of wind speed sequences is quantified and thetemporal relationship and the law of speed variation amongwind speed sequences at different spatial positions have notbeen quantified yet In order to solve the above problems aquantification method of time-shifting characteristics basedon wind process is proposed in this paper (e results of theproposed method can be continuously improved accordingto the actual wind conditions and could be applied to anyspatial scale (e main contributions of this paper are asfollows

(1) (e concept of wind process is introduced to studythe time-shifting characteristics In order to copewith the complexity and variability of wind condi-tions the wind sequences at different spatial posi-tions are divided according to the certain timewindow at first and then the time-shifting charac-teristics are studied based on the wind processes

(2) Two evaluation indexes the delay time and the decayspeed are proposed to quantify the time-shiftingcharacteristics of the temporal relationship and thespeed variation in each wind process

(3) (e time-shifting characteristics are studied at twospatial scales the wind farms scale and the windturbines scale

(4) (e improved clustering method of k-means x-means is employed to extract the typical windprocesses to explore the time-shifting characteristicsamong wind speeds under various typical windprocesses (e obtained results can be directly usedin the wind power system

(5) (e validity and practicability of the proposedmethod are verified by applying the calculation re-sults of time-shifting among wind speed sequences atdifferent spatial positions to the wind power system

(e remainder of this paper is organized as followsSection 2 describes the research process of wind speed time-shifting Section 3 studies the wind speed time-shiftingcharacteristics under two spatial scales the wind farms scaleand the wind turbines scale Section 4 verifies the effec-tiveness and practicability of the proposed time-shiftingevaluation method Section 5 concludes this paper

2 Research Process of Wind Speed Time-Shifting

(is section proposes an evaluation method of the time-shifting characteristics among wind speed sequences atdifferent spatial positions (e research process is as follows

21 Outline of the Research Process on Wind Speed Time-Shifting First the wind sequence at each point is pre-processed (e wind sequence is divided according to a

certain time window to obtain the wind processes at eachpoint (en some wind processes are screened out tocompose a new wind sequence that can be used to study thetime-shifting characteristics by judging whether the mainwind direction in the wind process is consistent with therelative position of points For example the wind sequenceat point p can be expressed as a composed sequence of windprocess αp wind process βp wind process mp and so forth

(en two evaluation indexes the delay time (Δt) and thedecay speed (Δv) are proposed to study the time-shiftingcharacteristics between wind processes at two points (efirefly optimization algorithm is used to calculate the pro-posed indexes in which the constraint conditions are thedistance between two points and the actual wind conditionsthe optimal objective is the minimum Euclidean distancebetween the corresponding wind processes Taking the windprocess αp and the wind process αq as an example the windprocess αp at point p and the wind process αq at point q can beexpressed as vαp(t minus x) vαp(t) vαp(t + y) andvαq(t minus x) vαq(t) vαq(t + y) respectively (e distancebetween two points and the wind conditions of the windprocess αp at point p are set as the constraint conditions andthe wind process αq is offset through the time axis and thespeed axis to achieve the objective of the minimum Euclideandistance between the wind process αp at point p and the newwind process αq

prime at point q(en the corresponding delay timeΔtα decay speed Δvα and the new wind process αq

prime (vαq(t minus

x + Δtα) + Δvα vαq(t + Δtα) + Δvα vαq(t + y + Δtα) +

Δvα) could be obtained at the same timeFinally the time-shifting characteristics of wind speed

sequences at two points are studied based on the proposedindexes (e delay time and the decay speed of each windprocess at point p and point q are calculated and then thestatistical analysis method is used to analyze the time-shifting Furthermore the improved clustering method of k-means x-means is employed to extract the typical windprocesses and the time-shifting characteristics of differenttypical wind processes are studied so that the research resultscan be applied to the actual planning and operation of theelectric power system

In order to verify the effectiveness of the proposed time-shifting analysis method the wind speed sequence at point q

is reconstructed after considering the time-shifting char-acteristics t-Copula function is used to fit the joint prob-ability distribution between wind speed sequences under twoconditions that is whether the time-shifting is considered(en the Kendall rank correlation coefficient τ the Spear-man rank correlation coefficient ρs and the tail correlationcoefficient λ are calculated based on the t-Copula function(e correlation coefficients increase with the increasingsimilarity between two wind speed sequences

(e outline of the research process on wind speed time-shifting is shown in Figure 1

It should be noted that due to the fact that the windspeed can gradually recover within a certain distance afterpassing through the wind turbine(s) or the wind farm(s) theevaluation indexes for time-shifting analysis should bedifferent according to the distance and the installed capacityof wind power that is whether the decay speed should be

Mathematical Problems in Engineering 3

contained when studying the time-shifting characteristics ofwind speed sequences at different spatial positions Accordingto the research results of Risoslash Laboratory at the TechnicalUniversity of Denmark even for a large-scale wind farm of30 kmtimes 30 km at the upwind direction the weakening effecton the wind speed at the downwind direction could vanishafter a distance of 30 km to 60 km [24] (erefore it can beconsidered that there is no influence of the decay speedamong wind speed sequences at different locations when thedistance between the wind farms exceeds 60 km

22 Calculation Method of the Proposed Indexes Based onFirefly Optimization Algorithm (e firefly optimizationalgorithm [25] is applied to calculate the proposed indexesthe delay time and decay speed Each firefly is regarded as apoint in the algorithm and the fireflies are distributedrandomly in the search space at first (ere are two pa-rameters in the firefly optimization algorithm the brightnessand the attractiveness(e brightness of the firefly is decidedby the objective function which indicates the position anddecides the next moving direction (e attractiveness be-tween two fireflies is inversely proportional to the distancewhich decides the next moving distance (e firefly isattracted and moves toward the relatively brighter oneaccording to the brightness and attractiveness then thefireflies gather through the movement In the optimizationprocess the brightness and attractiveness are updated

iteratively until the optimization objective is achieved (especific steps of the firefly algorithm are as follows

Step (a) to set the initial parameters of the algorithmincluding the maximum attraction β0 the light at-traction coefficient c the step factor α the maximumnumber of iterations and the number of firefliesStep (b) to initialize the position of firefliesStep (c) to calculate the objective function of eachfirefly and regard it as the maximum fluorescentbrightness I0Step (d) to define the relative brightness betweenfireflies as shown in the following equation

I I0eminus crij (1)

where rij is the relative distance between firefly i andfirefly j and the light attraction coefficient c reflects theweakening characteristics with the increase of the dis-tance and the absorption of the propagation mediumto define the attractiveness between fireflies as shownin the following equation

β β0eminus cr2

ij (2)

to calculate the relative brightness and attractivenessbetween fireflies according to equations (1) and (2) andsort them by the relative brightness

1 Wind sequence preprocessing

Wind sequence at each point is divided with a certain time window

Wind processes areobtained at each point

To calculate the main direction in each wind process

To screen out wind processes by judging whehter the main directionin the wind process is consistent with the relative position of points

New wind sequence at each pointPoint p wind processes αp βp mp

Point p wind processes αp βp mp

Point q wind processes αp βp mp

2 To propose evaluation indexesTaking wind process αp and wind process αq as an example

Wind process αp at point pvα

p(t ndash x) vαp (t) vα

p (t + y)

vαp(t ndash x + ∆tα) + ∆vα vα

p (t + ∆tα) + ∆vα vαp (t + y + ∆tα) + ∆vα

Wind process αq at point qvα

q(t ndash x) vαq (t) vα

q (t + y)

To offset αp through the time axis and the speed axis

Delay time ∆tα

Delay time ∆tα

Decay speed ∆vα Decay speed ∆vβ Decay speed ∆vm

Delay time ∆tβ Delay time ∆tm

Decay speed ∆vα

Firefly optimization algorithm Minimum Euclidean distance

To obtain the new wind process αpprime

To verify the proposed method

New wind sequence at point q aer considering time-shiing

Comparison of correlations on whether considering time-shiingWind sequence at point q

wind processes αp βp mp

Wind sequence at point pwind processes αp βp mp

Wind sequence at point qwind processes αpprime βpprime mpprime

3 To study time-shiing characteristics

To calculate delay time and decay speed of each wind process at points p amp q

Statistical analysisTo study time-shiing between wind sequences at two points

To analyze time-shiing of each typical wind process

x-meansTo extract typical wind processes

Typical set 1 Typical set 2 Typical set n

To study time-shiing of each typical wind process based ondelay time and decay speed

Statisticalanalysis

t-Copula

t-CopulaComparison

τ1 ρs1 λ1

τ2 ρs2 λ2

Figure 1 Outline of the research process on wind speed time-shifting


Step (e) to update the position of fireflies as shown inthe following equation

xi xi + β xj minus xi1113872 1113873 + αεi (3)

where xi and xj are the positions of firefly i and firefly jrespectively (e step factor α is a disturbance termwhich is added to enlarge the search area and avoid theoptimization algorithm falling into partial convergenceprematurely In general α isin [0 1] and εi isin [minus 05 05]Step (f) to repeat Step (c) to Step (e) until the maxi-mum iterations number is reached and then the op-timal results can be obtained

23 Correlation Analysis Based on t-Copula Function (ecorrelation analysis between wind speed time series at dif-ferent spatial points is an importantmethod to test the validityof the proposed time-shifting indexes and evaluate the effectof the time-shifting characteristics At present most re-searchers use the linear correlation coefficient to measure thecorrelation relationship between wind speed sequencesHowever when the distribution of the variable follows thenonnormal distribution or nonelliptic distribution it may getwrong conclusions [26] (e distribution of wind speed ismore consistent with Weibull distribution so it is morereasonable to use the rank correlation coefficient to measurethe correlation relationship between wind speed sequencesFurthermore compared with the normal weather conditionsthe wind power output in the extreme wind conditions has agreater impact on the electric power system and the outputcorrelation among wind turbines or wind farms should bepaid more attention in these weather conditions

t-Copula [27] function is one of the commonly usedCopula functions It has a thicker tail and is more sensitive tothe changes of the tail correlation between random variableswhich can capture the symmetric tail correlation (ereforethe t-Copula function is used to fit the joint probabilitydensity of wind speed sequences under two conditions inthis paper including the original sequences and the se-quences considering time-shifting characteristics (en theKendall rank correlation coefficient Spearman rank corre-lation coefficient and tail correction coefficient can becalculated to verify the validity of the proposed time-shiftingindexes and evaluate the effect of the time-shifting char-acteristics(e specific steps of the correlation analysis basedon t-Copula function are as follows

(e binary t-Copula function is shown in equation (4)where the linear correlation coefficient between two vari-ables is ρ and the degree of freedom is k(e density functionand the distribution function of t-Copula are depicted inFigure 2

Ct(μ ] ρ k) 1113946

tminus 1k

(μ)

minus infin1113946

tminus 1k

(])

minus infin

1

2π

1 minus ρ21113969

middot 1 +s2 minus 2ρst + t2

k 1 minus ρ2( 11138571113890 1113891

minus (k+2)2

dsdt

(4)

where tminus 1k is the inverse function of the distribution function

of unary t distribution with freedom degree of k(e calculation equations of Kendall rank correlation

coefficient τ Spearman rank correlation coefficient ρs andtail correlation coefficient λ based on t-Copula function areshown in the following equations

τ 411139461

011139461

0C

t(μ ]) dC

t(μ ]) minus 1

2 arcsin ρ

π

(5)

ρs 1211139461

011139461

0C

t(μ ])dμd] minus 3

6arcsin(ρ2)

π

(6)

λ limμ⟶1minus

1 minus 2μ + Ct(μ μ)

1 minus μ

limμ⟶0+

Ct(μ μ)

μ

2 minus 2tk+1

k + 1

radic 1 minus ρ

1113968

1 + ρ

11139681113888 1113889

(7)

where tk+1((k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968) is the value of the

distribution function of unary twith freedom degree of k + 1at (

k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968

24ExtractionofTypicalWindProcessesBasedonthex-MeansClustering Method (e improved algorithm of k-meansalgorithm x-means [28] is adopted to extract the typicalwind processes when analyzing the time-shifting charac-teristics of each typical wind process Compared with k-means algorithm x-means algorithm is not required to givethe specific value of the clustering number n in advance butonly needs to give a range within which the true n reasonablylies (e clustering result is not only the set of centroids butalso the value for n which scores best in this range (especific steps of x-means are as follows

Step (a) to input m wind processes to be clustered andset the range of clustering number as [nmin nmax]where nmaxlemStep (b) to select nmin initial clustering centers fromoriginal wind processes randomlyStep (c) to calculate the Euclidean distance betweenother wind processes and each clustering center andassign the corresponding wind process to the nearestcategoryStep (d) to calculate the average value of wind pro-cesses in each clustering set and use the results as thenew clustering centersStep (e) to repeat Step (c) and Step (d) until theclustering centers no longer change and the optimalclustering results can be obtained with the clusteringnumber of nmin


Step (b) to Step (e) are the main steps of k-meansalgorithm when the clustering number is nminStep (f) to apply the 2-means algorithm in eachclustering set and the clustering results are evaluatedaccording to the Bayesian Information Criterion (BIC)(e calculation method of BIC can be seen in [28] Ifthe BIC score is larger when 2-means algorithm is usedin the clustering set the set will be divided into two setsotherwise the clustering set will be retainedStep (g) to repeat Step (f) in each new clustering setuntil the clustering number n is no longer increased ornge nmax

3 Case Study and Analysis

(is section analyzes the time-shifting characteristics ofwind speed time series in different points under the windfarms scale and the wind turbines scale respectively (edata includes wind speed and wind direction In the windfarms scale the data is the mesoscale data located in SouthGobi of Mongolian the time range is from January 1 2004to October 31 2018 and the temporal resolution is 1 h Allpoints are located in the flat terrain in this case In the windturbines scale the data is the real operation data of one windfarm in Northern China the time range is from January 12014 to December 30 2014 and the temporal resolution is1min All points are located in the hilly terrain in this case

31 Time-Shifting under Wind Farms Scale (e wind se-quence at each point is preprocessed at first (e wind se-quence is divided into different wind processes with a timewindow of 1 day in this case and then the wind process setsare obtained at each point (e relative distribution of eachpoint is shown in Figure 3 so the wind processes with themain wind direction between 0265deg and 275deg at the corre-sponding point should be screened out for the time-shiftingstudy For example when studying the time-shifting char-acteristics of wind speed sequences from point 1 to otherpoints (2ndash5) the wind processes with the main directionbetween 265deg and 275deg at point 1 and the contemporaneouswind processes at other points should be screened out

simultaneously (en the new wind speed sequence that canbe used to study the time-shifting characteristics at eachpoint is reconstructed (e average speeds of new windspeed sequences at points 1 to 4 are 832ms 844ms668ms and 606ms respectively

(en the time-shifting characteristics of new wind speedsequences between point 1 and points 2 to 5 between point 2and points 3 to 5 between point 3 and points 4 to 5 andbetween point 4 and point 5 are studied respectively Sincethe distances between two points are all more than 100 kmonly the delay time is used as the time-shifting evaluationindex in this case(e delay time distributions of wind speedtime series between different points are shown in Figure 4 Itcan be seen that with the increase of distance the mediandelay time between wind speed sequences at two pointsincreases and the frequency distribution becomes moredispersed (e delay time of the wind speed sequence frompoint 1 to other points (2ndash5) varies in 0 hndash26 h 0 hndash31 h0 hndash49 h and 0 hndash78 h respectively (e average delay timebetween wind speed sequences at two points has a corre-lation relationship with the average wind speed at thestarting point and the distance between two points but it isnot completely equal to the quotient between these twovariables Take the wind speed sequence at point 1 as anexample if the distances between point 1 and points 2 to 5are divided by the average wind speed at point 1 the cor-responding results are 3 h 7 h 10 h and 13 h respectivelyWhen the proposed time-shifting analysis method is appliedthe average delay time of wind speed sequence from point 1to other points (2ndash5) is 2 h 5 h 8 h and 14 h respectively(is is because even if the average speed of two windprocesses is the same the wind conditions can also be quite

15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)

Figure 2 (e density function and distribution function of binary t-Copula (a) Density function (b) Distribution function

1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km

Figure 3 (e relative distribution of points (wind farms scale)


different and the time-shifting characteristics are variousunder different wind processes (erefore it is more rea-sonable to calculate the delay time between wind speedsequences at two points based on the similarity of windprocesses

In order to preliminarily analyze the mapping rela-tionship among the time-shifting characteristics of windspeed sequences at different points the distances betweenpoints and the actual wind conditions the wind processes atpoint 1 are divided according to the average wind speed ofeach process with the interval of 1ms (e average delaytime of point 1 to points 2ndash5 is calculated in each wind speedbin respectively and the results are depicted in Figure 5 Ascan be seen the delay time increases with the distance whenthe wind speed is almost the same and decreases with thewind speed when the distance is basically the same How-ever since the wind speed is inevitably affected by variousfactors such as terrain and obstacles when it propagates fromone point to another point in space the propagation time isnot equal to the quotient of distance and its value Fur-thermore the wind process is the time series composed ofdifferent wind speeds so the propagation law of the windprocess is more complicated (is is why some dots inFigure 5 look like the ldquoabnormal dotsrdquo such as the averagedelay time from point 1 to point 2 when the average speed ofthe wind process is between 8ms and 9ms and the averagedelay time from point 1 to point 3 when the average speed ofthe wind process is between 11ms and 12ms (erefore itis necessary to further analyze the time-shifting character-istics of different wind processes at different spatial scales

x-Means algorithm is applied to cluster the wind pro-cesses at point 1 to extract typical wind processes and thenthe typical wind process sets are constructed (e range ofclustering number is set as in [3 15] when using x-meansalgorithm (e optimal clustering number is 24 in this case(e average delay time of each wind process set from point 1to other points (2ndash5) is calculated respectively as shown inFigure 6 Six typical wind process sets are selected randomlyas depicted in Figure 7 (e red solid line is the clusteringcenter and the black-dashed lines are the actual wind

processes of each set (e average delay time from point 1 toother points and the average wind speed of these six typicalwind process sets are listed in Table 1

(e new wind speed sequence at each point consideringthe time-shifting characteristics is reconstructed respec-tively according to the calculation results of the delay time(e wind speed time series of whether considering the time-shifting is shown in Figure 8 As can be seen the wind speedsequences at different points could be more similar if thetime-shifting characteristics are considered

In order to further verify the validity of the proposedmethod the correlation analysis is used to test the effec-tiveness of the time-shifting characteristics t-Copulafunction is used to fit the original wind speed sequences oftwo points and the new wind speed sequences consideringthe time-shifting characteristics of two points respectively(en the Kendall correlation coefficient τ Spearman cor-relation coefficient ρs and tail correlation coefficient λ are

0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5

Figure 5 Average delay time between different points in each windspeed bin (wind farms scale)

1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)

Figure 4 Delay time distributions of wind speed sequences be-tween different points (wind farms scale)

0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)

4 8 12 16 20 240Wind process set

1ndash21ndash3

1ndash41ndash5

Figure 6 Average delay time in each wind process set betweenpoint 1 and other points (wind farms scale)


calculated based on the fitted function and the results arelisted in Table 2 It can be seen that the correlation of windspeed sequences at different points is enhanced significantlywhen the time-shifting characteristics are considered Sinceall points are located in the flat terrain in this case the effect

of time-shifting becomes more obvious as the distance in-creases For example taking point 1 as the starting point ifthe time-shifting characteristics are considered the corre-lations of wind speed sequences between point 1 and points2 to 5 are increased by 0048 0097 0123 and 0227

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)

Wind processesClustering center

(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )

Figure 7 Six typical wind process sets at point 1 (wind farms scale) (a) Typical wind process set 6 (b) Typical wind process set 10 (c)Typical wind process set 15 (d) Typical wind process set 18 (e) Typical wind process set 22 (f ) Typical wind process set 23


respectively when the Kendall correlation coefficient τ isused as the evaluation index the correlations are increasedby 0046 0107 0149 and 0288 respectively when theSpearman correlation coefficient ρs is used as the evaluationindex Furthermore the tail correlation of wind speed se-quences is also improved when the time-shifting

characteristics are taken into account which means theappearance times under high wind conditions and smallwind conditions are more consistent (e tail correlationcoefficients of wind speed sequences between point 1 andpoints 2 to 5 are increased by 0090 0050 0025 and 0045respectively

Table 1 Average delay time in six wind process sets between point 1 and other points (wind farms scale)

Typical wind process setAverage delay time (h)

Average wind speed (ms)1-2 1ndash3 1ndash4 1ndash5

6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)

Figure 8Wind speed sequences of whether considering time-shifting characteristics (wind farms scale) (a) Original (b) Considering time-shifting

Table 2 Correlation of wind speed sequences between points under two conditions (wind farms scale)

τ 1-2 1ndash3 1ndash4 1ndash5 2-3 2ndash4 2ndash5 3-4 3ndash5 4-5Original 0569 0479 0399 0283 0641 0485 0464 0681 0555 0615Considering time-shifting 0616 0577 0522 0510 0655 0543 0567 0712 0611 0648Difference 0048 0097 0123 0227 0014 0058 0103 0030 0056 0033ρs 1-2 1ndash3 1ndash4 1ndash5 2-3 2ndash4 2ndash5 3-4 3ndash5 4-5Original 0756 0660 0563 0411 0831 0670 0640 0862 0743 0805Considering time-shifting 0802 0767 0712 0699 0845 0735 0758 0886 0801 0834Difference 0046 0107 0149 0288 0014 0065 0118 0024 0058 0029λ 1-2 1ndash3 1ndash4 1ndash5 2-3 2ndash4 2ndash5 3-4 3ndash5 4-5Original 0348 0226 0093 0040 0095 0106 0241 0463 0318 0301Considering time-shifting 0438 0276 0118 0085 0138 0119 0244 0507 0333 0390Difference 0090 0051 0025 0045 0043 0013 0003 0044 0015 0089


32 Time-Shifting under Wind Turbines Scale ContainingWake Effect (e wind sequence at each point is pre-processed at first(e wind sequence is divided into differentwind processes with a time window of 1 hour in this caseand then the wind process sets are obtained at each point(e relative distribution of each point is shown in Figure 9so the wind processes with the main wind direction between130deg and 140deg at the corresponding point should be screenedout for the time-shifting study (e average speeds of newwind speed sequences at points 1 to 3 are 1145ms 1017ms and 1049ms respectively

(en the time-shifting characteristics of new wind speedsequences between point 1 and points 2 to 4 between point 2and points 3 to 4 and between point 3 and point 4 arestudied respectively Since the data used in this case is theactual operation data of the wind farm the factors that affectwind speed sequences at different locations should includeboth the delay time and the decay speed(e distributions ofdelay time and decay speed of wind speed time series be-tween different points are shown in Figure 10 It can be seenthat with the increase of distance the median delay timebetween wind speed sequences at two points increases andthe frequency distribution becomes more dispersed (ewind speed at the downstream location is not always lowerthan that at the upstream location (e average decay speedsof the wind speed sequence from point 1 to other points(2ndash4) are 062ms minus 016ms and 087ms respectively(is is because the points are all located in the hilly terrain inthis case and the propagation of wind speed between dif-ferent points is greatly affected by the terrain Furthermorethe wake effect of wind turbines is weakened with the in-crease of the distance and the relief of the terrain

In order to preliminarily analyze the mapping relation-ship among the time-shifting characteristics of wind speedsequences at different points the distances between pointsand the actual wind conditions the wind processes at point 1are divided according to the average wind speed of eachprocess with the interval of 1ms(e average delay time andaverage decay speed of wind processes from point 1 to points2ndash4 are calculated in each wind speed bin respectively andthe results are depicted in Figure 11 As can be seen the delaytime increases with the distance when the wind speed is al-most the same and decreases with the wind speed when thedistance is basically the same However due to the influenceof the small distance between two points and the low timeresolution of the data the delay time of wind speed sequencebetween two points does not decrease with the increase ofwind speed in some situations (is situation happens morefrequently with the shortening of the distance between twopoints (e decay speed of wind speed sequences betweendifferent points varies with the value of wind speed

(e improved algorithm of k-means x-means is employedto cluster the wind processes at point 1 to extract typical windprocesses and then the typical wind process sets are con-structed(e range of clustering number is set as in [4 20] andthe optimal clustering number is 32 in this case (e averagedelay time and the average decay speed of eachwind process setfrom point 1 to other points (2ndash4) are calculated respectivelyand the results are shown in Figure 12 Six typical wind process

sets are selected randomly as depicted in Figure 13 (e av-erage delay time and the average decay speed from point 1 toother points and the average wind speeds of these six typicalwind process sets are listed in Table 3

(e new wind speed sequence at each point consideringthe time-shifting characteristics is reconstructed respec-tively according to the calculation results of the delay timeand decay speed (e wind speed time series of whetherconsidering the time-shifting is shown in Figure 14 As canbe seen the wind speed sequences at different points couldbe more similar if the time-shifting characteristics areconsidered

In order to further verify the validity of the proposedmethod the correlation analysis is used to test the effec-tiveness of the time-shifting characteristics t-Copula functionis used to fit the original wind speed sequences of two pointsand the new wind speed sequences considering the time-shifting characteristics of two points respectively (en theKendall correlation coefficient τ Spearman correlation co-efficient ρs and tail correlation coefficient λ are calculatedbased on the fitted function and the results are listed inTable 4 It can be seen that the correlation of wind speedsequences at different points is enhanced significantly whenthe time-shifting characteristics are considered For exampletaking point 1 as the starting point if the time-shiftingcharacteristics are taken into account the correlations of windspeed time series between point 1 and points 2 to 4 are in-creased by 0142 0103 and 0202 respectively when theKendall correlation coefficient τ is used as the evaluationindex the correlations are increased by 0065 0046 and0124 respectively when the Spearman correlation coefficientρs is used as the evaluation index Furthermore the tailcorrelations of wind speed sequences between point 1 andpoints 2 to 4 are increased by 0374 0147 and 0802 re-spectively It should be noted that different from the case inwind farms scale the points are all located in the hilly terrainin this case the mapping relationship among the wind speedsequences at different points is more complicated and thetime-shifting effect does not show a more obvious trend withthe increase of distance

4 Verification of the Proposed Time-ShiftingEvaluation Method

(is section takes the data on December 31 2014 under thewind turbines scale as an example to verify the calculationresults of time-shifting and illustrate the effectiveness of the

140deg

130deg

492km215km

087km

123

4

Figure 9 (e relative distribution of points (wind turbines scale)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)

Figure 11 (e time-shifting characteristics between different points in each wind speed bin (wind turbines scale) (a) Average delay time(b) Average decay speed

1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4

ndash4ndash3ndash2ndash1

012345

Dec

ay sp

eed

(ms

)

(b)

Figure 10(e time-shifting distributions of wind speed sequences between different points (wind turbines scale) (a) Delay time (b) Decay speed

0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)

Figure 12 (e time-shifting characteristics in each wind process set between point 1 and other points (wind turbines scale) (a) Averagedelay time (b) Average decay speed


proposed evaluation method in this paper (e wind speedsequences at points 2 to 4 can be considered the same as thatat point 1 if the time-shifting characteristics are not takeninto account When the time-shifting characteristics areconsidered the wind speed sequences at points 2 to 4 shouldbe calculated respectively based on the research results of

time-shifting (e specific steps are as follows first to dividethe wind speed sequence and the wind direction sequence atpoint 1 with a time window of 1 hour and second to judgewhether the main wind direction at point 1 is consistent withthe relative position of wind turbines in each wind processthat is whether the main direction of the wind process at

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )

Figure 13 Six typical wind process sets at point 1 (wind turbines scale) (a) Typical wind process set 2 (b) Typical wind process set 4 (c)Typical wind process set 7 (d) Typical wind process set 11 (e) Typical wind process set 23 (f ) Typical wind process set 31


point 1 is between 130deg and 140deg If so it should be judged atfirst which wind process set the wind process belongs to andthen the wind process at points 2 to 4 should be constructedaccording to the above calculation results of time-shifting Ifnot the corresponding wind speed at points 2 to 4 can beconsidered the same as that at point 1(e actual wind speedsequence and wind speed sequences of whether consideringtime-shifting at points 2 to 4 are shown in Figure 15 It canbe seen that the wind speed sequence is more similar to theactual sequence at each point if the time-shifting charac-teristics are considered

To further verify the calculation results of time-shifting the average absolute deviations between thecalculated wind speed sequences of whether consideringthe time-shifting and the actual wind speed sequence areused as the evaluation index to quantify the effectivenessof time-shifting (e average absolute deviations betweenthe calculated wind speed sequence and the actual windspeed sequence at points 2 to 4 are 127 ms 118ms and126 ms respectively if the time-shifting is not takeninto account If the time-shifting is considered the av-erage absolute deviations are 106 ms 102 ms and104 ms respectively (e average absolute deviations arereduced by 165 136 and 175 respectively when

the time-shifting is considered (e average absolutedeviations between wind speed sequences of whetherconsidering the time-shifting and the actual wind speedsequence in each time window at points 2 to 4 aredepicted in Figure 16 It can be seen that the wind speed

Table 3 Average delay time and average decay speed in six wind process sets between point 1 and other points (wind turbines scale)

Typical wind process setAverage delay time (min) Average decay speed (ms)

Average wind speed (ms)1-2 1ndash3 1ndash4 1-2 1ndash3 1ndash4

2 0 2 4 minus 167 minus 081 183 16894 0 2 5 minus 161 minus 033 209 15757 0 1 3 minus 134 minus 004 minus 021 201211 2 5 9 minus 051 minus 050 minus 005 68323 2 4 8 minus 044 minus 034 018 88131 1 2 5 minus 091 009 159 1267

20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)

Figure 14 Wind speed sequences of whether considering time-shifting characteristics (wind turbines scale) (a) Original (b) Consideringtime-shifting

Table 4 Correlation of wind speed sequences between pointsunder two conditions (wind turbines scale)τ 1-2 1ndash3 1ndash4 2ndash3 2ndash4 3ndash4Original 0758 0794 0651 0827 0587 0649Considering time-shifting 0900 0897 0853 0912 0853 0872

Difference 0142 0103 0202 0086 0266 0223ρs 1-2 1ndash3 1ndash4 2-3 2ndash4 3-4Original 0920 0938 0842 0959 0783 0839Considering time-shifting 0985 0983 0966 0988 0965 0974

Difference 0065 0046 0124 0029 0182 0135λ 1-2 1ndash3 1ndash4 2-3 2ndash4 3ndash4Original 0467 0703 0004 0517 0000 0058Considering time-shifting 0841 0850 0806 0872 0828 0813

Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time

Not considering time-shiftingConsidering time-shiftingMeasured

(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)

Figure 15 Wind speed sequences of whether considering the time-shifting characteristics at each point

2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

Not considering time-shiftingConsidering time-shifting

(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued


sequences are more similar to the actual wind speedsequences in all time windows if the time-shifting isconsidered

In order to study the influence of time-shifting on thewind power generation system the measured wind speedsequence and the calculated wind speed sequences undertwo conditions (considering time-shifting or not) at eachpoint are converted to the power sequences based on thetheoretical power curve of the wind turbine respectivelythen the total output of wind turbines at points 1 to 4 isobtained by adding the output of wind turbine at each point(e total power sequences of four wind turbines calculatedby the measured wind speed sequence and the wind speedsequences of whether considering time-shifting are shown inFigure 17 It can be seen that the calculated total power ismore similar to the actual power when the time-shifting isconsidered To further illustrate the necessity of time-shifting the influence of time-shifting is quantified from twoperspectives one is the volatility of the total power sequenceand the other is the generation of four wind turbinesFluctuation ratio (FR) [29] is used to evaluate the volatility ofthe power sequence in this paper (e calculation method ofFR is shown in the following equation

FR 1113936

Nminus 1t1 (P(t + 1) minus P(t))Pcap

11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)

where P (t) is the power of wind turbines at time t and Pcap isthe rated power of wind turbines 802MW in this study

(e total power generation and the volatility of thepower sequence of four wind turbines are calculatedthrough the wind speed sequences under two conditionsand the results are compared with the results when usingthe measured wind speed sequences as shown in Table 5As can be seen from the table if the time-shifting is taken

into account the calculated daily power generation iscloser to the actual generation and the volatility of thecalculated power sequence is more similar to that of theactual power sequence (e aggregation characteristics ofwind turbines output are reflected Compared with theactual daily power generation the deviation of the cal-culated generation is 86 if the time-shifting is con-sidered and is 106 if the time-shifting is notconsidered Compared with the volatility of the actualpower sequence the deviation of the volatility of thecalculated power sequence is 702 if the time-shifting isconsidered and is 772 if the time-shifting is notconsidered

4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)

Figure 16 Average absolute deviations of wind speed sequences under two conditions at each point

0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)

Wind speed not considering time-shiingWind speed considering time-shiingMeasured wind speed

Figure 17 (e power sequences of four wind turbines of whetherconsidering time-shifting


5 Conclusions

A quantification method of wind speed time-shifting atdifferent spatial positions based on wind process is proposedin this paper As a result of this study the following con-clusions can be drawn

(1) (e proposed evaluation method can effectivelyachieve the quantitative analysis of the time-shiftingcharacteristics of wind speed sequences at differentspatial positions Furthermore the results of theproposed method can be continuously improvedaccording to the actual wind conditions and could beapplied to any spatial scale

(2) (e time-shifting effect between wind speed se-quences increases with the distance when points arelocated in the flat terrain But when points are lo-cated in the hilly terrain the time-shifting effect doesnot show a more obvious trend with the increase ofdistance

(3) (e median delay time of wind speed sequences in-creases and the frequency distribution of the delay timebecomes more dispersed with the increase of distanceno matter what terrain the points are located in

(4) (e wind speed at the downstream location could belarger than that at the upstream location in the windfarm when points are located in the hilly terrain

(5) x-Means clustering method can achieve the effectivedivision of wind processes and the time-shiftingcharacteristics of different wind processes are quitevarious

(6) (e calculated output of wind turbines is more similarto the actual output in the wind power system if thetime-shifting characteristics are taken into account

(e time-shifting characteristics of wind speeds arestudied when the wind direction is consistent with therelative position of points in this paper In future researchthe proposed evaluation method will be applied to analyzethe time-shifting characteristics of wind speeds in full winddirections to better grasp the spatiotemporal coupling re-lationship of wind speeds at multiple scales

Data Availability

(e data used to support the findings of this paper areavailable upon request from the corresponding author

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

(is work was supported by the National Key Research andDevelopment Program of China (2017YFE0109000) and theNational Natural Science Foundation of China (U1765201and 51707063)

References

[1] Y Liu Y Ye X Chen H Li and Y Huang ldquoRobust day-ahead dispatch for integrated power-heat-gas microgridconsidering wind power uncertaintyrdquoMathematical Problemsin Engineering vol 2020 no 9 Article ID 4215906 12 pages2020

[2] Y Zhu X Liu Y Zhai and R Deng ldquoMonthly unit com-mitment model and algorithm with renewable energy gen-eration considering system reliabilityrdquo MathematicalProblems in Engineering vol 2019 Article ID 38352969 pages 2019

[3] C Lucheroni J Boland and C Ragno ldquoScenario generationand probabilistic forecasting analysis of spatio-temporal windspeed series with multivariate autoregressive volatilitymodelsrdquo Applied Energy vol 239 pp 1226ndash1241 2019

[4] M Cellura G Cirrincione A Marvuglia and A MiraouildquoWind speed spatial estimation for energy planning in sicily aneural kriging applicationrdquo Renewable Energy vol 33 no 6pp 1251ndash1266 2008

[5] S Z Moghaddam ldquoGeneration and transmission expansionplanning with high penetration of wind farms consideringspatial distribution of wind speedrdquo Electrical Power andEnergy Systems vol 106 pp 232ndash241 2019

[6] C M Grams R Beerli S Pfenninger I Staffell andH Wernli ldquoBalancing Europersquos wind-power output throughspatial deployment informed by weather regimesrdquo NatureClimate Change vol 7 no 8 pp 557ndash562 2017

[7] W Zheng Research on Wind Speed Forecasting of RegionalWind Farm Group Based on Spatio-Temporal CorrelationNorth China Electric Power University Beijing China 2014

[8] S She Z Li and X Cai ldquoResearch on wind speed distributionmodel of wind farm based on its dynamic space-time rela-tionrdquo Power System Technology vol 38 no 6 pp 1432ndash14382014

[9] C Zeng L Ye and Y Zhao ldquoSpatial model for short termwind power prediction considering wake effectsrdquo PowerSystem Protection and Control vol 40 no 24 pp 59ndash1422012

[10] L Ye Y Zhao C Zeng and C Zhang ldquoShort-term windpower prediction based on spatial modelrdquo Renewable Energyvol 101 pp 1067ndash1074 2017

[11] Y L Pichugina R M Banta T Bonin et al ldquoSpatial vari-ability of winds and HRRR-NCEP model error statistics atthree Doppler-lidar sites in the wind-energy generation re-gion of the Columbia river basinrdquo Journal of Applied Mete-orology and Climatology vol 58 no 8 pp 1633ndash1656 2019

Table 5 Comparison of wind power output under two conditions

Power generation (MW middot h) Power volatilityNot considering time-shifting

Considering time-shifting Measured Not considering time-

shiftingConsidering time-

shifting Measured

9651 9477 8728 00101 00097 00057


[12] F Liu F Sun W Liu et al ldquoOn wind speed pattern andenergy potential in Chinardquo Applied Energy vol 236pp 867ndash876 2019

[13] J Li and X Yu ldquoOnshore and offshore wind energy potentialassessment near Lake Erie shoreline a spatial and temporalanalysisrdquo Energy vol 147 pp 1092ndash1107 2018

[14] J Bosch I Staffell and A D Hawkes ldquoTemporally explicitand spatially resolved global offshore wind energy potentialsrdquoEnergy vol 163 pp 766ndash781 2018

[15] Y Chen S ZhangW Zhang J Peng and Y Cai ldquoMultifactorspatio-temporal correlation model based on a combination ofconvolutional neural network and long short-term memoryneural network for wind speed forecastingrdquo Energy Conver-sion and Management vol 185 pp 783ndash799 2019

[16] K Philippopoulos and D Deligiorgi ldquoApplication of artificialneural networks for the spatial estimation of wind speed in acoastal region with complex topographyrdquo Renewable Energyvol 38 no 1 pp 75ndash82 2012

[17] M S Miranda and R W Dunn ldquoSpatially correlated windspeed modelling for generation adequacy studies in the UKrdquoin Proceedings of the 2007 IEEE Power Engineering SocietyGeneral Meeting Tampa FL USA June 2007

[18] J Park and D-H Jang ldquoApplication of MK-PRISM for in-terpolation of wind speed and comparison with co-kriging inSouth Koreardquo GIScience amp Remote Sensing vol 53 no 4pp 421ndash443 2016

[19] G Ren J Wan J Liu and D Yu ldquoAssessing temporalvariability of wind resources in China and the spatial cor-relation of wind power in the selected regionsrdquo Journal ofRenewable and Sustainable Energy vol 12 Article ID 133022020

[20] Z Q Xie T Y Ji M S Li and Q HWu ldquoQuasi-Monte Carlobased probabilistic optimal power flow considering the cor-relation of wind speeds using copula functionrdquo IEEETransactions on Power Systems vol 33 no 2 pp 2239ndash22472018

[21] X Hui Calculation of Available Transfer Capability Consid-ering Temporal and Spatial Correlation in Wind Power In-tegrated System Northeast Electric Power University JilinChina 2018

[22] G Ren J Wan J Liu and D Yu ldquoSpatial and temporalassessments of complementarity for renewable energy re-sources in Chinardquo Energy vol 177 pp 262ndash275 2019

[23] M Zeng J-h Li Q-h Meng and X-n Zhang ldquoTemporal-spatial cross-correlation analysis of non-stationary near-surface wind speed time seriesrdquo Journal of Central SouthUniversity vol 24 no 3 pp 692ndash698 2017

[24] S T Frandsen H E Joslashrgensen R Barthelmie et al ldquo(emaking of a second-generation wind farm efficiency modelcomplexrdquo Wind Energy vol 12 no 5 pp 445ndash458 2009

[25] M B Imamoglu M Ulutas and G Ulutas ldquoA new reversibledatabase watermarking approach with firefly optimizationalgorithmrdquo Mathematical Problems in Engineering vol 2017Article ID 1387375 14 pages 2017

[26] Y Kang Z Zhang and X Guo ldquoPortfolio risk analysis inelectricity market based on copula approachrdquo Power SystemProtection and Control vol 40 no 6 pp 50ndash56 2012

[27] Z Jiang and J He ldquoMethod of fusion diagnosis for damservice status based on joint distribution function of multiplepointsrdquo Mathematical Problems in Engineering vol 2016Article ID 9049260 10 pages 2016

[28] D Pelleg and A Moore ldquoX-means extending K-means withefficient estimation of the number of clustersrdquo in Proceedings

of the Seventeenth International Conference on MachineLearning Stanford CA USA July 2000

[29] S Han L-n Zhang Y-q Liu et al ldquoQuantitative evaluationmethod for the complementarity of wind-solar-hydro powerand optimization of wind-solar ratiordquo Applied Energyvol 236 pp 973ndash984 2019


speed and direction of the wind tower rotor speed and pitchangle of each wind turbine as inputs to establish the wakeeffect model and the time-lag effect model then the dynamicspatial relationship among wind speed time series at eachwind turbine location was obtained Zeng et al [9] firstdeveloped the continuous partial differential equation ofeach wind turbine by calculating the wake effects accordingto the specific spatial locations and distributions of corre-lated wind turbines (en the finite volume method is usedto differentiate the differential equation at each grid pointand derive the spatial correlation matrix of wind speedFinally the wind speed at each wind turbine location isobtained by solving the above differential equation throughthe given boundary conditions Ye et al [10] proposed anovel spatial correlationmatrix by using computational fluiddynamics tomathematically describe the correlations amongwind speeds at different wind turbine locations

(e physical method studies the temporal and spatialdistribution and the coupling relationship of wind speeds byestablishing the aerodynamic model among the wind speedsat different spatial positions However this aerodynamicmodel cannot be optimized in real time according to theactual wind conditions Besides when the spatial scale in-creases the complexity of the constructed model increasessharply which leads to the increasing difficulty of modelingand solving the model simultaneously But it is difficult toensure the accuracy of the aerodynamic model if the factorsaffecting wind speed are simplified (erefore the appli-cation of this method is mainly concentrated in the interiorof the wind farm which is difficult to expand to a largerspatial scale

(e statistical method uses the actual wind data at dif-ferent spatial locations to analyze the spatiotemporal couplingrelationship among the wind speeds (e research methodscan be mainly divided into the following three types

(1) (e spatiotemporal coupling relationship is studiedby calculating the distribution of wind speeds atdifferent spatial locations from the perspectives ofaverage wind speed wind speed frequency distri-bution wind shear exponent and so forth [11] Liuet al [12] analyzed the spatiotemporal distributionsof the wind resource at the hub height in China fromthe aspects of average wind speed and average windpower density based on the large network of windspeed observations including 2430 meteorologicalstations from 2006 to 2015 (e results show thatnortheast China has the highest wind speed whileSouth China has the lowest wind speed Li and Yu[13] conducted the statistical analysis of the onshorenearshore and offshore wind energy conditionsfrom the aspects of Weibull shape and scale factorsaverage turbulence intensity and average windpower density (e results indicate that the averagewind speed at the offshore location is 1-2 timeshigher than that nearby onshore area at the height of50m above ground level Bosch et al [14] estimatedthe distributions of global offshore wind energypotential from the aspects of average wind speed

mean capacity factor which could be built across theviable offshore area and generation potential peryear (e results show that the wind speed increaseswith the distance to the coast

(2) (e spatiotemporal coupling relationship is studiedby establishing the mapping relationship amongwind speed time series at different spatial positionsthrough interpolation neural network principalcomponent analysis and so forth [15] Philip-popoulos and Deligiorgi [16] used two feedforwardneural network architectures to establish the rela-tionship among hourly wind speeds at six meteo-rological sites An insight into the underlying input-output function approximation of the neural net-works is obtained by examining their ability to in-corporate the mean wind variability characteristicsMiranda and Dunn [17] proposed a spatially cor-related wind speed model to characterize the windresources between various sites by using the mul-tivariate time-series approach with actual wind speeddata in different zones Park and Jang [18] presenteda modified Korean parameter-elevation regressionon independent slopes model for spatial interpola-tion of monthly wind speeds (e results show thatthe proposed model can performmore appropriatelyto represent the phenomena where similar windspeeds appear continuously along ridges andcoastlines compared with the cokriging model

(3) (e correlation coefficient is used to study the cor-relation characteristics of wind speed time series atdifferent spatial positions which is an important partof the spatiotemporal coupling relationship (ecalculation method can be divided into the followingcategories One is to directly calculate the correlationcoefficient based on the wind speed time series [19]Another is to use the Copula function to fit the jointprobability distribution of the wind speed sequencesand then calculate the correlation coefficient based onthe fitted Copula function [20] (e commonly usedcorrelation coefficients include the Pearson linearcorrelation coefficient [21] Kendall rank correlationcoefficient [22] and cross-correlation coefficient [23]

(e statistical method studies the spatiotemporal cou-pling relationship of wind speed sequences at differentspatial positions based on the actual wind data (e obtainedresults of this method can be continuously improvedaccording to actual conditions and the research method canbe applied to any spatial scale However only the correlationrelationship between wind speed sequences is quantified inthis method Compared with the physical method thetemporal relationship and the law of speed variation are bothimplicit in the spatial distribution and the mapping rela-tionship is not quantified yet

In conclusion when the physical method is used to studythe spatiotemporal coupling relationship among windspeeds at different spatial positions the established aero-dynamic model cannot be optimized in real time according















prime (vαq(t minus













I I0eminus crij (1)


β β0eminus cr2

ij (2)













p (t + y)





q (t + y)


Delay time ∆tα

Delay time ∆tα



Decay speed ∆vα
















Statisticalanalysis

t-Copula

t-CopulaComparison

τ1 ρs1 λ1

τ2 ρs2 λ2









Ct(μ ] ρ k) 1113946

tminus 1k

(μ)

minus infin1113946

tminus 1k

(])

minus infin

1

2π

1 minus ρ21113969


k 1 minus ρ2( 11138571113890 1113891

minus (k+2)2

dsdt

(4)




τ 411139461

011139461

0C

t(μ ]) dC

t(μ ]) minus 1

2 arcsin ρ

π

(5)

ρs 1211139461

011139461

0C


6arcsin(ρ2)

π

(6)

λ limμ⟶1minus


1 minus μ

limμ⟶0+

Ct(μ μ)

μ

2 minus 2tk+1

k + 1

radic 1 minus ρ

1113968

1 + ρ

11139681113888 1113889

(7)

where tk+1((k + 1

radic 1 minus ρ

1113968)

1 + ρ



k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968










15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)


1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km









0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057



































prime (vαq(t minus













I I0eminus crij (1)


β β0eminus cr2

ij (2)













p (t + y)





q (t + y)


Delay time ∆tα

Delay time ∆tα



Decay speed ∆vα
















Statisticalanalysis

t-Copula

t-CopulaComparison

τ1 ρs1 λ1

τ2 ρs2 λ2









Ct(μ ] ρ k) 1113946

tminus 1k

(μ)

minus infin1113946

tminus 1k

(])

minus infin

1

2π

1 minus ρ21113969


k 1 minus ρ2( 11138571113890 1113891

minus (k+2)2

dsdt

(4)




τ 411139461

011139461

0C

t(μ ]) dC

t(μ ]) minus 1

2 arcsin ρ

π

(5)

ρs 1211139461

011139461

0C


6arcsin(ρ2)

π

(6)

λ limμ⟶1minus


1 minus μ

limμ⟶0+

Ct(μ μ)

μ

2 minus 2tk+1

k + 1

radic 1 minus ρ

1113968

1 + ρ

11139681113888 1113889

(7)

where tk+1((k + 1

radic 1 minus ρ

1113968)

1 + ρ



k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968










15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)


1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km









0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057


























I I0eminus crij (1)


β β0eminus cr2

ij (2)













p (t + y)





q (t + y)


Delay time ∆tα

Delay time ∆tα



Decay speed ∆vα
















Statisticalanalysis

t-Copula

t-CopulaComparison

τ1 ρs1 λ1

τ2 ρs2 λ2









Ct(μ ] ρ k) 1113946

tminus 1k

(μ)

minus infin1113946

tminus 1k

(])

minus infin

1

2π

1 minus ρ21113969


k 1 minus ρ2( 11138571113890 1113891

minus (k+2)2

dsdt

(4)




τ 411139461

011139461

0C

t(μ ]) dC

t(μ ]) minus 1

2 arcsin ρ

π

(5)

ρs 1211139461

011139461

0C


6arcsin(ρ2)

π

(6)

λ limμ⟶1minus


1 minus μ

limμ⟶0+

Ct(μ μ)

μ

2 minus 2tk+1

k + 1

radic 1 minus ρ

1113968

1 + ρ

11139681113888 1113889

(7)

where tk+1((k + 1

radic 1 minus ρ

1113968)

1 + ρ



k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968










15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)


1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km









0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057




























Ct(μ ] ρ k) 1113946

tminus 1k

(μ)

minus infin1113946

tminus 1k

(])

minus infin

1

2π

1 minus ρ21113969


k 1 minus ρ2( 11138571113890 1113891

minus (k+2)2

dsdt

(4)




τ 411139461

011139461

0C

t(μ ]) dC

t(μ ]) minus 1

2 arcsin ρ

π

(5)

ρs 1211139461

011139461

0C


6arcsin(ρ2)

π

(6)

λ limμ⟶1minus


1 minus μ

limμ⟶0+

Ct(μ μ)

μ

2 minus 2tk+1

k + 1

radic 1 minus ρ

1113968

1 + ρ

11139681113888 1113889

(7)

where tk+1((k + 1

radic 1 minus ρ

1113968)

1 + ρ



k + 1

radic 1 minus ρ

1113968)

1 + ρ

1113968










15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)


1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km









0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057




























15

10

5

0

c (u

v)

108

0604

020 0 02 04 06 08 1

V U

(a)

c (u

v)

108

0402

0 0 02 04 0606 08 1

V U

1

08

06

04

02

0

(b)


1 2 3 4 5

265deg

275deg1038km2031km

3026km4035km









0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057




























0

5

10

15

20

25

30

Aver

age d

elay

tim

e (h)

4-5 6-7 8-9 10-11 12-13 14-152-3Wind speed (ms)

1ndash21ndash3

1ndash41ndash5


1-2

1-3

1-4

1-5

2-3

2-4

2-5

3-4

3-5

4-5

0

20

40

60

80

Del

ay ti

me (

h)


0

5

10

15

20

25

30Av

erag

e del

ay ti

me (

h)


1ndash21ndash3

1ndash41ndash5





21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057
























21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(a)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(b)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(c)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(d)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(e)

21

18

15

12

9

6

3

0

Win

d sp

eed

(ms

)

0 4 8 12 16 20 24Time series (h)


(f )








6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057



























6 2 4 6 9 118410 4 10 15 23 55415 3 7 11 16 72718 2 5 8 12 94922 2 5 11 14 83723 3 7 13 20 626

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(a)

0

2

4

6

8

10

12

14

16

18

Win

d sp

eed

(ms

)

12 24 36 48 60 720Time series (h)

123

45

(b)














140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057































140deg

130deg

492km215km

087km

123

4



0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057






















0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

Aver

age d

ecay

spee

d (m

s)

5-6 7-8 9-10 11-12 13-14 15-16 17-18 19-203-4Wind speed (ms)

1-21-31-4

(b)


1-2 1-3 1-4 2-3 2-4 3-40

5

10

15

20D

elay

time (

min

)

(a)

1-2 1-3 1-4 2-3 2-4 3-4


012345

Dec

ay sp

eed

(ms

)

(b)


0

2

4

6

8

10

Aver

age d

elay

tim

e (m

in)


1-21-31-4

(a)

ndash2

ndash1

0

1

2

3

4

Aver

age d

ecay

spee

d (m

s)


1-21-31-4

(b)





20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057
























20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(a)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(b)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(c)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(d)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(e)

20

16

12

8

4

0 0 10 20 30 40 50 60Time series (min)

Win

d sp

eed

(ms

)


(f )










20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057





























20 40 60 80 100 1200Time series (min)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

12

34

(a)

5

6

7

8

9

10

Win

d sp

eed

(ms

)

20 40 60 80 100 1200Time series (min)

12

34

(b)





Difference 0374 0147 0802 0354 0828 0754


2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057






















2

3

6

9

12W

ind

spee

d (m

s)

0400 0800 1200 1600 2000 24000000

Time


(a)3

0400 0800 1200 1600 2000 24000000

Time


3

6

9

12

Win

d sp

eed

(ms

)

(b)4


369

1215

Win

d sp

eed

(ms

)

0400 0800 1200 1600 2000 24000000

Time

(c)


2

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time


(a)

3


0400 0800 1200 1600 2000 24000000Time

0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

(b)

Figure 16 Continued




FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057
























FR 1113936


11138681113868111386811138681113868

11138681113868111386811138681113868

N minus 1 (8)




4


0

1

2

3

4

Aver

age a

bsol

ute d

evia

tion

(ms

)

0400 0800 1200 1600 2000 24000000Time

(c)


0400 0800 1200 1600 2000 24000000Time

0

2000

4000

6000

8000

Win

d po

wer

(kW

)




5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057






















5 Conclusions









Data Availability




Acknowledgments


References
















shifting Measured

9651 9477 8728 00101 00097 00057






















Evaluation Method of Wind Speed Time-Shifting ...

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Transcript of Evaluation Method of Wind Speed Time-Shifting ...