C-Vine Pair Copula Based Wind Power Correlation Modelling in Probabilistic Small Signal Stability Analysis

Jin Xu,, Wei Wu,Keyou Wang,,Guojie Li,

Abstract—The increasing integration of wind power generation brings more uncertainty into the power system.Since the correlation may have a notable influence on the power system,the output powers of wind farms are generally considered as correlated random variables in uncertainty analysis.In this paper,the C-vine pair copula theory is introduced to describe the complicated dependence of multidimensional wind power injection,and samples obeying this dependence structure are generated.Monte Carlo simulation is performed to analyze the small signal stability of a test system.The probabilistic stability under different correlation models and different operating conditions scenarios is investigated.The results indicate that the probabilistic small signal stability analysis adopting pair copula model is more accurate and stable than other dependence models under different conditions.

I.INTRODUCTION

AS one of the most mature renewable energy techniques,wind power generation has been experiencing a dramatic growth of capacity in the past decade due to the global energy and environment problems.The impact of wind power integration on power system stability is always a research hotspot.Due to the intermittency and unpredictability of wind energy,the wind power (or wind speed)is generally regarded as random variable in uncertainty analysis of power system.In earlier research,the wind power models take no account of the correlation and are represented by independent probability distributions[1]?[4].Since this independent model may lead to a large error in uncertainty analysis,the dependence model is necessary.Linear correlation coefficient(LCC)isa common option [5]?[8],which reflects the linear relationship between variables.However,the wind power injections in different locations may have a more complicated dependence structure,and the information given by LCC is not enough to define this dependence structure[9].To describe this complex dependence,copula theory is introduced into the wind correlation modelling[10]?[17].Three different Archimedean copulas are used to fit the dependence structure of wind speeds at two sites in Netherlands in[9].However,in most cases[10]?[17],the dimension of random variables is higher than two and the choice of adequate copula models is limited to multivariate normal copula due to the mathematical complexity of multivariate Archimedean copulas.Compared with multivariate copula,pair copula theory provides a more fexible tool for correlation modelling in high dimension [18]?[23].In [18]and [19],the spatial dependences of wind power and photovoltaic generation are modelled with pair copula,respectively.Reference[20]models the spatiotemporal interdependencies of wind power with pair copula.The scatters of pair copula model show a good fitting effect to the empirical distribution.In this article,C-vine pair copula theory is utilized to model the wind power correlation and the generated samples are used in the probabilistic small signal stability analysis(PSSSA).The specific uncertainty analysis process is performed by Monte Carlo simulation (MCS)to make sure the analysis process itself is precise due to the input of enough samples.In order to testify whether pair copula model has a remarkable accuracy,this model is compared with independent model,LCC model and multivariate normal copula model in different scenarios.Furthermore,the probabilistic small signal stability under different load levels and wind power penetrations is studied.

II.WIND POWER MODELLING with PAIR COPULA

A. Basic Concepts

Copula is a function used to describe the dependence between random variables.Consider a vectorX=(X1,...,X n)of random variables and the corresponding marginal cumulative distribution functions(CDF)F1,...,Fn.Sklar’s theorem[24]states that the joint cumulative distribution functionFcan be expressed as

The functionC(·)is termed as copula function.The marginal distributions in (1)can be substituted by

Then,the copula can be seen as a multivariate joint distribution for which the marginal distribution of each variable is uniform.

Equation (3)also implies that,the dependence structure and the probability distributions are modelled separately in copula theory.In this paper,the probability distributions are obtained non-parametrically by using the empirical distribution functions,because the empirical distribution eliminates the risk of misspecification in the later pair copula construction [25].The copula function is obtained parametrically by selecting and estimating appropriate parametric copula model.Common parametric copula models can be classified into ellipse family and Archimedean family.Elliptical copulas,including normal copula and Student t copula,can easily be generalized to higher dimensions.On the other hand,Archimedean copulas,including Frank,Clayton and Gumbel copula,provide the capacity to represent more complicated dependence structures.The expressions of all these parametric copula models can be found in [24].

B.Pair Copula Constructions

Since Archimedean copulas cannot be easily generalized to higher dimensions,the available copula models are restricted to multivariate elliptical copulas(usually normal copula)in the correlation modelling of multiple wind power generations.However,the normal copula is inadequate to capture all the dependence structures of wind power generation caused by geographical and meteorological factors.Thus a decomposition method is proposed in [25],where then-dimensional dependence is modelled byn(n ?1)/2 bivariate copulas,socalled pair copulas,instead of onen-variable copula.Since all parametric copula models can be easily applied to bivariate copulas,the restriction is well avoided.These pair copulas constitute the building blocks of a specific structure.Fig.1 presents a common topology of this structure,named canonical vine (C-vine).

Fig.1.C-vine structure of pair copulas.

In C-vine structure,the nodesF1,...,Fnin levelL0are the marginal distributions of historical observationsx1,...,xnfrom wind farm datasets.The edgesu1,...,unconnectingL0andL1are observations transformed from historical ones by(2).The nodesC1,2,...,C1,nin level L1 are bivariate copulas evaluated at the observationsu1andu1+i(i=1,...,n ?1).According to the conditional probability theory,the expression of observationsu2|1,...,un|1in the next level can be deduced:

Then,the copulas in levelL2can be evaluated at the observationsu2|1andu2+i|1(i=1,...,n ?2).Analogously,the observationsuj+i|1,...,j(i=1,...,n ?j)are available from the evaluated copulas and transformed observations in the last level recursively using(5).

The partial derivative in (5)can be derived from the expressions of parametric copula models in [24].Given (5)and the original observationsx1,...,xn,we are able to select appropriate copula models,estimate parameters and compute the transformed observations for the next level until the whole pair copula construction (PCC)is completed.If every pair copula is evaluated specifically,the calculation is in proportion withn2,and grows outrageous in high dimension.In light of the fact that dependence in higher levels is much weaker,the pair copula construction above a certain level,e.g.LK,can be approximately represented by a multivariate normal copula[26],which can be estimated much faster and the calculation time is neglected here.Once the levelLKis determined,the evaluation time of the simplified PCC(sPCC)is proportional ton.It is far less than the calculation of the fully evaluated PCC,especially in high dimension.

C.Wind Power Modelling with Pair Copula

The procedure of the simplified PCC of wind power is outlined as follows.

It should be noted that,in Fig.2,the parameters for each copula model are estimated by maximum likelihood estimate(MLE),which is aimed to find the best-fitting parameters to the empirical joint distribution.MLE method has been discussed a lot[27],so it is not illustrated here.In addition,the goodnessof-fit for each copula model in a GOF test is measured by their Euclidean distances to the empirical joint distribution.More details can be found in [27]. With a constructed pair copula model,samples obeying this dependence structure can be generated by the following procedure.

2)Based on the samples generated in step 1)and the topology in Fig.1,the other samples on the edges of C-vine structure can be calculated according to(5).

3)The original samples of wind powerX1,...,X ncan be transformed inversely from the samplesu1,...,unaccording to(2).

Fig.2.Procedure of the simplified PCC.

III.PROBABILISTIC SMALL SIGNAL STABILITY ANALYSIS CONSIDERING WIND POWER CORRELATION

Modal analysis utilizes the eigenvalues of system state matrix to describe the dynamic properties[28].Since power system is nonlinear,its eigenvalues are related to the current operating state.The mapping relationship from wind farm output poweruto eigenvaluesλcan be described by

where the mapping functionhdenotes a series of operations including power flow calculation,system linearization and eigen-decomposition.In order to explore the probabilistic small signal stability of power system considering wind power correlation,Monte Carlo simulation (MCS)is performed by repeating modal analysis numerous times.The entire procedure is displayed in the flowchart of Fig.3.

Fig.3. Flow chart of PSSSA.

This procedure begins with the pair copula construction of the wind power data.As the MCS inputs,samples of wind power injections are generated according to the constructed pair copula model.Then,the power flow under the wind power injections of a set of generated samples is calculated and the modal analysis is performed.Repeat the simulation with the next set of generated samples until all the samples are simulated.Finally,the statistical results of the small signal stability are carried out.It should be noted that,at various operating points,the order of modes in the eigenvalue vector may vary accordingly.Different modes may have similar eigenvalues,while their eigenvectors usually differ a lot.Therefore,eigenvectors are utilized to track the modes in a MCS.Modes computed at different random operating points are matched if the 2-norm distance of their normalized right eigenvectors is small enough.By this way,the inconsistency of modal analysis in MCS is avoided.

IV.CASE STUDY

A.Wind Power Modelling

The wind power data used in case study come from the Wind Integration Datasets of NREL [29].These datasets provide 10-minute time-series wind data for 2004,2005,and 2006.More than 157 824 wind plant data points are available across the eastern United States.In this section,we start with the wind power modelling of wind farms at 6 different sites.following the procedure in Section II,a simplified pair copula model is constructed here.Reference[19]has showed that the dependence of fourteen random variables can be well captured in 3 levels.SoK=3 is accurate enough for the simplified pair copula model of six random variables in this case.The optimal copula at each node selected by GOF test is displayed in Fig.4.

Fig.4.Copula selection at each node.

In Fig.4,among the copulas in the first level,some are Frank and the others are Gumbel.In higher level,the dependence structures become fickler,covering every Archimedean copula.After the pair copula construction,wind power samples with a size of 15 000*6 are generated.For comparison,wind power samples with the same size are generated by the linear correlation coefficient (LCC)model and multivariate normal copula model respectively.The scatters of wind power samples at site 1 and site 2 generated by three different models are compared with the scatter of actual wind power data as shown in Fig.5.

From the four scatters above,it is obvious that pair copula model has a better fitting effect to the actual dependence than the others.The scatters of the other sites lead to the same conclusion,which are not listed here.As the Fig.4 shows,none of the best-fitting copulas are normal in the first three levels.Hence,it is understandable that the 6-dimensional normal copula may not fit the actual dependence structure well compared with pair copula model.As for the LCC model,it is too simple to reflect the complex dependence structure between wind power data,which makes it less accurate than the other two.

B.Probabilistic Stability Under Different Dependence Models

In order to testify whether pair copula model has a remarkable improvement in the accuracy of PSSSA,the probability distributions of electromechanical modes under different dependence models are compared.The PSSSA is performed on the New England system with six wind plants integrated at Bus 32,33,...,37.The power injection deviations caused by random wind power are balanced by conventional generations nearby.The generators are represented by 4-order transient models,and loads are regarded as constant impedances.Other controllers,such as power system stabilizer,are neglected.The power flow calculation and modal analysis are done by power system toolbox(PST)[30]in Matlab.The four different wind power correlation models are listed as follows.

Model 1:the dependence of wind power is neglected,and the wind power samples are generated independently according to their empirical distributions of the original wind power data.

Model 2:the dependence is described by linear correlation coefficient matrix,and the samples are generated by orthogonal transformation method [5],[8].

Model 3:the dependence is modeled by 6-dimensional normal copula,and the samples are generated by the expression of normal copula.

Model 4:the dependence is modeled by simplified pair copula with C-vine structure,and the samples are generated according to procedure in Section II.By performing MCS with a sample size of 3000*6,the CDFs of the electromechanical modal damping ratios under different dependence models are obtained.They are compared with the MCS result of actual data with a sample size of 15 000*6.Fig.6 describes the empirical probability density functions(PDF)of damping ratios of four typical modes.

The results indicate that the dependence models have different influences on the damping ratios for different electromechanical modes.For most modes likeλ7andλ13,independent model is the least accurate,and always has a smaller or larger confidence interval than other models.That means the independent model underestimates or overestimates the extreme damping ratio of some modes.For some modes likeλ15andλ17,LCC model is almost as inaccurate as independent model.For some other modes,the influence of dependence models is limited,possibly because the wind power has little effect on the dominated generators of these modes.In conclusion,independent model and LCC model may result in obvious error in the PDFs of the damping ratios.To further compare the accuracy of different models quantitatively,especially the normal copula model and pair copula model,the numerical average root mean square error is given by

Fig.6.PDFs of damping ratios under different dependence models.

whereis thekth value of a certain CDF of damping ratio under a particular model,is thekth value of the empirical CDF of actual data,andNis the sample number in a CDF curve.The CDF errors of all electromechanical modal damping ratios under different dependence models are calculated by(7)and the mean CDF errors are shown in Table I.

TABLE I MEAN CDF ERRORS UNDER DIFFERENT DEPENDENCE MODELS

On average,the CDFs of pair copula model have the smallest distance to the empirical CDFs of actual data.

C.Comparison of Dependence Models in Different Scenarios

Three sets of wind farm data are selected and used in Scenarios 1-3 respectively.Each of these datasets comes from 6 wind farms which have a completely different distribution density with those of the other datasets.

As shown in Fig.7,the Dataset 1 comes from the most adjacent wind farms,therefore has the strongest correlation.On the contrary,the Dataset 3 comes from the most decentralized wind farms,therefore has the weakest correlation.In each scenario,6 wind farms are assumed to be integrated into the same 10-machine-39-bus system.Every dataset is modeled by Independent model,LCC model,Normal Copula model and sPCC model in turn.PDFs of the damping(real part)of the critical eigenvalues in Scenario 1?3 are shown in Figs.8?10,respectively.

Fig.7. Locations of wind farms providing the three datasets.

Fig.8.PDFs of the damping of the critical eigenvalue in Scenario 1.

In Scenario 1,the independent model results in the largest error in the PDF of the damping of the critical eigenvalues.The LCC model also leads to a big error.The sPCC model is the most precise,and the Normal copula model comes second.In Scenario 2,the independent Model remains the most inaccurate but less inaccurate than that in Scenario 1.The LCC model almost has the same accuracy with Normal copula model and sPCC model.In Scenario 3,the differences between each model become rather small.To analyze the errors of these correlation models quantitatively,the percentage errors of the mean,standard deviation,skewness and kurtosis of the PDF curves in Figs.8–10 are listed in Table II.

Fig.9.PDFs of the damping of the critical eigenvalue in Scenario 2.

Fig.10.PDFs of the damping of the critical eigenvalue in Scenario 3.

TABLE II PERCENTAGE ERRORS OF THE MEAN,STANDARD DEVIATION,SKEWNESS AND KURTOSIS OF THE PDF CURVES

From Figs.8–10 and the Table II,the following conclusions can be drawn.1)A ll these four dependence models have a small error in the mean value of the critical eigenvalue.2)Taking the overall errors into consideration,the order of accuracy from high to low is:sPCC>Normal Copula>LCC>Independent.3)The superiority of sPCC and Normal Copula model becomes unobvious as the correlation between wind farms becomes weak.

D.Probabilistic Stability Under Different Operating Conditions

Furthermore,the impact of load level and wind power penetration on the probabilistic small signal stability is investigated and the stability of PSSSA adopting pair copula model is verified under different conditions.Different load levels:The original load level is selected as the basic level.In high level case,all the active and reactive powers of loads are magnified by a coefficient of 1.3,and output powers of all the synchronous generators are increased with the same ratio to meet the power demand.The generated wind power samples remain unchanged.If the coefficient is assigned with 0.7,we get the low level case.By MCS,we obtain the distributions of their eigenvalues in the complex plane as shown in Fig.11.

Fig.11.Eigenvalue distributions of electromechanical modes in the complex plane (green:low load level,blue:basic load level,red:high load level).

In Fig.11,the mean damping ratios generally decrease with the improving load level and the distribution areas of their eigenvalues move towards right in the complex plane as the load level improves.The model is an exception,which is the inter-area oscillation mode between the local grid and the equivalent generator of an adjoining grid.Different wind power penetrations:The original wind power penetration is selected as the basic level.In high penetration case,all the wind power samples are doubled to simulate the enlarging of the wind power installed capacity,and the synchronous generators nearby reduce their output power correspondingly.The load powers remain unchanged.Similarly,halving the wind power samples,we can get the low penetration case.By MCS,we obtain the distributions of their eigenvalues in the complex plane as shown in Fig.12.

In Fig.12,the mean damping ratios generally increase with the improving penetration and the distribution areas of their eigenvalues expand towards left in the complex plane as the penetration improves.The mode 1 is still an exception to the others.The influence of load level and wind power penetration on small signal stability can be explained by the state changes of synchronous generators in this case study.The high load level decreases the damping ratios by increasing all the generator outputs,while the high wind power penetration increases the damping ratios by reducing the generator outputs nearby.After the MCSs under different load levels and wind power penetrations,the mean CDF errors defined by(7)under different operating conditions are calculated and listed in Table.3.

Fig.12.Eigenvalue distribution of electromechanical modes in the complex plane (green:low penetration,blue:basic penetration,red:high penetration).

TABLE III MEAN CDF ERRORS UNDER DIFFERENT OPERATING CONDITIONS

From the table above,the result of PSSSA with pair copula model is the most accurate under any operating conditions and the error is relatively insensitive to the change of operating state.

V.CONCLUSION

In order to describe the wind power correlation in the probabilistic small signal stability,simplified pair copula construction (sPCC)with C-vine structure is introduced.In the case study,sPPC model proves to have a better reflection of the actual dependence than the linear correlation coefficient(LCC)model and multivariate normal copula model.By applying probabilistic small signal analysis on the modified New England test system in different scenarios,the results proved that the sPCC model has the highest accuracy among these dependence models,but the superiority becomes unobvious when the correlation become weak.Hence the sPCC is very fit for the probabilistic modelling of the wind power with strong correlation.