Probabilistic Power Flow for Hybrid AC / DC Grids with Ninth-Order Polynomial Normal Transformation and Inherited Latin Hypercube Sampling

: This paper proposes a new probabilistic power ﬂow method for the hybrid AC / VSC-MTDC (Voltage Source Control-Multiple Terminal Direct Current) grids, which is based on the combination of ninth-order polynomial normal transformation (NPNT) and inherited Latin hypercube sampling (ILHS) techniques. NPNT is utilized to directly handle historical records of uncertain sources to build the accurate probability model of random inputs, and ILHS is adopted to propagate the randomness from inputs to target outputs. Regardless of whether the underlying probability distribution is known or unknown, the proposed method has the ability to adaptively evaluate the sample size according to a speciﬁc operational scenario of the power systems, thus achieving a good balance between computational accuracy and speed. Meanwhile, the frequency histograms, probability distributions, and some more statistics of the results can be accurately and e ﬃ ciently estimated as well. The modiﬁed IEEE 118-bus system, together with the realistic data of wind speeds and diverse consumer behaviors following irregular distributions, is used to demonstrate the e ﬀ ectiveness and superiority of the proposed method.


Introduction
The Voltage Source Converter based Multiple Terminal Direct Current (VSC-MTDC) technique has become the most feasible solution to the integration of remotely located large wind farms (WFs), as it can effectively support the AC grid, facilitate the integration of fluctuant wind power, and improve the transfer efficiency [1].With more and more WFs integrated into AC grids by using VSC-MTDC, the fluctuant wind power will significantly increase the stochastic nature of the power system, thus further exacerbating the operational condition of the hybrid AC/VSC-MTDC grids [2].Nevertheless, the deterministic power flow (DPF) cannot fully account for the random variables.Hence, the probabilistic power flow (PPF) methods for the hybrid AC/VSC-MTDC grid are rising with the aim of accurately evaluating the impact of probabilistic uncertainties.
In power system operation, the PPF analysis treats the uncertainties (such as wind speeds and loads) as input variables, and it focuses on accurately obtaining the statistical information of outputs including mean, standard deviation, frequency histogram, and even probability density function (PDF) based on the results within an acceptable computational time.To achieve this goal, the following problems are necessary to be considered and addressed: The MCS method could provide the "accurate" results (including statistical moments and PDFs), which are usually applied to validate the other methods.Unfortunately, the computational burden of such method is extremely heavy.In order to overcome this issue, MCS based on the conventional Latin hypercube sampling (CLHS) [19] is proposed, which generates the samples in a stratified way to cover the input PDFs more evenly and largely, thus getting a higher efficiency and accuracy in PPF analysis.Meanwhile, moments and PDFs of PPF results can be directly obtained by use of CLHS [11].The drawback of the CLHS lies in that the highly structured form of the sample set hiders it from directly adding an additional sample set to an already obtained sample set [20,21].That is to say, unlike conventional MCS, the size of CLHS cannot be increased simply by generating additional samples, since the new sample set (including the original and the additional sample set) cannot preserve the stratification properties that make CLHS so effective [22].This naturally results in a problem regarding how many samples are required for PPF of the complex hybrid AC/DC grids with diverse operational scenarios.In particular, for a practical power system operation, a small sample may not yield the accurate PPF results, while a large sample will decrease the efficiency.
In order to address the deficiency issues mentioned above, an inherited Latin hypercube sampling (ILHS) method [22] is proposed to extend sample size, making full use of the existing samples generated by CLHS.The ILHS does not only inherit the advantages of CLHS but also possesses higher computational efficiency and flexibility compared with CLHS.In this paper, a PPF method with combining the NPNT and ILHS is proposed for the first time for the power system, particularly such hybrid AC/VSC-MTDC grid, to overcome the limitations of PPF methods aforementioned.The main contributions of this paper are as follows: (1) Based on historical records, the proposed method has the ability to handle random variables following irregular distributions even with correlations.(2) Based on an acceptable computational accuracy for a specific operational scenario of the hybrid AC/VSC-MTDC grid, the proposed method could adaptively evaluate the sample size, achieving a good balance between computational accuracy and speed.(3) The statistical moments (such as means and standard deviations) and PDFs of the PPF results can be accurately and directly obtained by means of the proposed method.
The remainder of the paper is organized as follows.The DPF model of the hybrid AC/VSC-MTDC grid is given in Section 2. Section 3 introduces the formulation and procedure of the NPNT method.The CLHS and ILHS are introduced in Section 4. Section 5 describes the procedure of combining ILHS and NPNT for PPF analysis on a hybrid AC/VSC-MTDC grid.Section 6 presents the case studies and the discussions, followed by conclusions in Section 7.

VSC Model
In general, the i th VSC in VSC-MTDC is viewed as a controllable voltage source U ci = U ci ∠δ ci with connecting an impedance Z ci = R ci + jX ci , as shown in Figure 1.Then, the converter is connected with a filter and a transformer.The AC bus i is connected with the filter B fi through the transformer represented by Z t f i = R t f i + jX t f i .The apparent power from the converter and flowing to AC bus U si = U si ∠δ si , respectively, are S ci = P ci + jQ ci and S si = P si + jQ si .The equations of the power flow injecting to AC bus can be expressed as [23]: where U f i = U f i ∠δ f i is the voltage at the filter.The corresponding equations of power flowing from the i th converter can be written as: The filter generates the reactive power which can be expressed as: The active power balance of the i th converter can be written as: Generally, the P DCi is determined as: where u di is the DC voltage at the i th converter, and Y dij is the conductance matrix of DC grids.
The can be obtained by: where K A , K B , and K C are constant parameters in [23], and I ci is the AC current magnitude.The filter generates the reactive power which can be expressed as: The active power balance of the i th converter can be written as: Generally, the P DCi is determined as: where is the DC voltage at the i th converter, and is the conductance matrix of DC grids.
The can be obtained by: where KA, KB, and KC are constant parameters in [23], and I ci is the AC current magnitude.

Control Modes of VSC
To keep the active power balance of VSC-MTDC system, at least one VSC should be selected as an active power regulator with using the constant u di control mode.The control modes of VSC are shown as follows: In a practical power system, the increasing WFs are integrated into the main AC grid through the advanced VSC-MTDC system.Unfortunately, the output power of WFs generally varies.The constant control modes shown above are not suitable for the VSCs on WF side (WFVSC).To handle this problem, a dual-mode control strategy (DMC) is introduced for WSVSC in [24].DMC control mode cannot only actively adjust the VSC to adapt the varying output of a WF but also has the ability to maintain the voltage of the WSVSC AC, thus improving the frequency and voltage stability of the WFs.Hence, the DMC control mode is adopted in case studies of this paper.The detailed control modes and methods can be accessed in [25,26].

Power Flow Calculation for Hybrid AC/VSC-MTDC Grids
The DPF problem of the hybrid AC/VSC-MTDC grid is usually solved by using the sequential method.In [24], the detailed procedures of DPF calculation for hybrid AC/VSC-MTDC were given.Actually, the DPF calculation for hybrid AC/DC grids can be viewed as an implicit function with multi-dimensional random input and output variables, which can be expressed as:

Control Modes of VSC
To keep the active power balance of VSC-MTDC system, at least one VSC should be selected as an active power regulator with using the constant u di control mode.The control modes of VSC are shown as follows: In a practical power system, the increasing WFs are integrated into the main AC grid through the advanced VSC-MTDC system.Unfortunately, the output power of WFs generally varies.The constant control modes shown above are not suitable for the VSCs on WF side (WFVSC).To handle this problem, a dual-mode control strategy (DMC) is introduced for WSVSC in [24].DMC control mode cannot only actively adjust the VSC to adapt the varying output of a WF but also has the ability to maintain the voltage of the WSVSC AC, thus improving the frequency and voltage stability of the WFs.Hence, the DMC control mode is adopted in case studies of this paper.The detailed control modes and methods can be accessed in [25,26].

Power Flow Calculation for Hybrid AC/VSC-MTDC Grids
The DPF problem of the hybrid AC/VSC-MTDC grid is usually solved by using the sequential method.In [24], the detailed procedures of DPF calculation for hybrid AC/VSC-MTDC were given.Actually, the DPF calculation for hybrid AC/DC grids can be viewed as an implicit function with multi-dimensional random input and output variables, which can be expressed as: where X represents input variables (such as loads, traditional generation and WFs' output power), while Y denotes output variables that mainly includes the voltages and branch flow of AC and DC grids.Therefore, if the randomness of wind power and load in hybrid AC/VSC-MTDC grid is considered, the model above will become a PPF problem.

Ninth-Order Polynomial Normal Transformation for Random Inputs Modeling
The uncertain sources like wind speeds are affected by diverse factors such as meteorological and geographical conditions, and hence they may not follow the common distributions but being subject to irregular distributions.Meanwhile, the correlations amongst these variables cannot be neglected.In this section, the NPNT is introduced to handle this problem based on historical records.

Polynomial Coefficients Evaluation for Modeling the Uncertainties
The key idea of polynomial normal transformation lies in that polynomial arithmetic of standard normal random variable is used to model random variable following arbitrary distribution.According to [13], the ninth-order polynomial normal transformation can be formulated as where x o is a continuous random variable in practical power grid like wind speeds, µ x and σ x are mean and standard deviation value of input uncertain variables, respectively, x denotes the normalized random variable.z is a standard normal random variable, and a k (k = 1, 2, . . ., 9) are the undetermined coefficients.If the suitable a k (k = 1, 2, . . ., 9) are obtained, the stochastic variables following irregular distributions can be stimulated.
In general, the statistical moments are used to denote the data characteristics of random variables.The probability weighted moment (PWM) in this paper is used to characterize the uncertainties.Reference [27] presents a method to obtain the PWMs of uncertain input variables by using historical records of uncertainties.Sorting the input random variables into an ascending order x n , the PWM β r can be solved by The first 10 PWMs (β 0 −β 9 ) of random variables can be calculated by Equation (9).By using Equations (10) and (11), the polynomial coefficients a k (k = 0, 1, . . ., 9) can be easily obtained.

Correlation Coefficients Estimation in Standard Normal Space
Generally, the correlations among uncertain sources cannot be ignored.Suppose that x 1 , x 2 are two correlated and normalized random variables following arbitrarily distributions.Using the procedure of polynomial coefficients calculation introduced above, x 1 and x 2 can be expressed as If the correlation coefficient ρ z between z 1 , z 2 (following standard normal distributions) can be determined, the correlated x 1 , x 2 can be simulated by using correlated standard normal random variables.To obtain the ρ z the functional relationship between ρ x and ρ z is established, which is where µ r and σ r are the means and standard deviations of x r (r = 1, 2), respectively.
The product moments E(z i 1 z j 2 ) can be expressed as a polynomial of ρ z which is introduced in detail in [13].Note that the correlation coefficient ρ x between x 1 , x 2 can be obtained by historical records.Hence, the correlation coefficient ρ z can be determined by solving Equation (13).

NPNT for Modeling Multiple Random Variables in Hybrid AC/VSC-MTDC Grid
Actually, the normalized m-dimension random variable x i in X can be formulated as can be easily obtained.Note that R Z is the equivalent correlation matrix of R X in the standard normal space.Then, random matrix X can be simulated by: (1) obtain the standard normal random variables Z with the correlation matrix R Z ; (2) calculate the correlated multivariate random vector X by using correlated standard normal vector Z.
The detailed procedures of modeling multiple random variables by NPNT can be accessed in [13].The key advantage of using NPNT in PPF analysis for AC/VSC-MTDC grids lies in that NPNT merely uses historical records of uncertain sources without assuming the PDFs of uncertainties in advance.That is to say, it is able to accurately handle correlated random variables, regardless of whether the underlying distribution is known or unknown.

Inherited Latin Hypercube Sampling Technique
The powerful NPNT can be adopted to build an accurate probability model of random inputs based on practical recordings.Then, the next crucial step is to propagate the randomness from inputs to target outputs by using the sampling technique with the DPF model of hybrid AC/VSC-MTDC grid.In this section, the CLHS is briefly introduced, followed by the detailed introduction of ILHS.

Conventional Latin Hypercube Sampling Technique
In general, the sampling and permutation are the two major steps of CLHS [19].The purpose of sampling is to obtain representative samples that could reflect the distribution regions of input random variables, and then the permutation technique is used to control the correlation of generated samples.The q th CDF F q value of input variables X p,q ranges from 0 to 1, which is divided into P num non-overlapping intervals.Note that P num (1, 2, . . ., p) donates sample size, and Q num (1, 2, . . ., q) represents the number of input variables.The procedure of CLHS is given below.(2) Obtain the sample point matrix on uniform distribution R = [r p,1 , r p,2 , . . ., r p,q ] by using r p,q = (p − u p,q ).( 3) The sample values on the q th original distribution can be obtained by X p,q = F −1 q (r p,q ).

Permutation
(1) Cholesky decomposition is adopted herein to obtain the permutation matrix of sample points, making the correlations trend closer to theoretical values [28].

Inherited Latin Hypercube Sampling Design
It is well-known that CLHS exhibits higher computational speed than the MCS method.However, the drawback of CLHS lies in that its highly structured form makes it difficult to directly add extra samples to an already generated sample matrix [21].This naturally leads to a question regarding how many samples are sufficient in PPF analysis for the complex hybrid AC/VSC-MTDC grid based on CLHS.Actually, a small sample may not guarantee accurate PPF results, while a large sample will decrease the computational efficiency.Therefore, the extension work makes sense to further improve the computational flexibility of CLHS.
Figure 2 shows the schematic diagram of the ILHS method.The basic idea of ILHS lies in that it is desirable to start with a small sample size for conducting PPF analysis of a hybrid AC/VSC-MTDC grid as an initial step and then gradually extend the sample size if deemed necessary.As well known, refining the existing sample matrix instead of recreating a totally new one could reuse obtained CLHS's results, leading to a significant improvement in the computational efficiency for PPF analysis.This is extremely meaningful for PPF analysis of the hybrid AC/VSC-MTDC grid since DPF simulation of the hybrid AC/VSC-MTDC grid is more expensive than the pure AC grid.Meanwhile, based on a given convergence criterion for a specific operational scenario of the hybrid AC/VSC-MTDC grid, the ILHS could adaptively change the sample size, thus obtaining higher computational accuracy and efficiency.Furthermore, the moments and PDFs can also be accurately and directly obtained by use of ILHS.

Inherited Latin Hypercube Sampling Design
It is well-known that CLHS exhibits higher computational speed than the MCS method.However, the drawback of CLHS lies in that its highly structured form makes it difficult to directly add extra samples to an already generated sample matrix [21].This naturally leads to a question regarding how many samples are sufficient in PPF analysis for the complex hybrid AC/VSC-MTDC grid based on CLHS.Actually, a small sample may not guarantee accurate PPF results, while a large sample will decrease the computational efficiency.Therefore, the extension work makes sense to further improve the computational flexibility of CLHS. Figure 2 shows the schematic diagram of the ILHS method.The basic idea of ILHS lies in that it is desirable to start with a small sample size for conducting PPF analysis of a hybrid AC/VSC-MTDC grid as an initial step and then gradually extend the sample size if deemed necessary.As well known, refining the existing sample matrix instead of recreating a totally new one could reuse obtained  To illustrate the key idea of ILHS, the two variables u ma , u mb following uniform distributions are defined.It is assumed that three sample points are generated in the first step by using CLHS, as shown in Figure 3a.The next step for ILHS is to generate new sample points while inheriting obtained sample points.It is assumed that six sample points are needed.That is to say, three additional sample points should be generated by ILHS.The variables u ma , u mb can be divided into six non-overlapping intervals, as shown in Figure 3b.The spaces represented by the inherited sample points are shaded with green.If shaded spaces are removed from Figure 3b, the unrepresented variable space will emerge as a blank space, as shown in Figure 3c. Figure 2 shows the schematic diagram of the ILHS method.The basic idea of ILHS lies in that it is desirable to start with a small sample size for conducting PPF analysis of a hybrid AC/VSC-MTDC grid as an initial step and then gradually extend the sample size if deemed necessary.As well known, refining the existing sample matrix instead of recreating a totally new one could reuse obtained CLHS's results, leading to a significant improvement in the computational efficiency for PPF analysis.This is extremely meaningful for PPF analysis of the hybrid AC/VSC-MTDC grid since DPF simulation of the hybrid AC/VSC-MTDC grid is more expensive than the pure AC grid.Meanwhile, based on a given convergence criterion for a specific operational scenario of the hybrid AC/VSC-MTDC grid, the ILHS could adaptively change the sample size, thus obtaining higher computational accuracy and efficiency.Furthermore, the moments and PDFs can also be accurately and directly obtained by use of ILHS.To illustrate the key idea of ILHS, the two variables uma, umb following uniform distributions are defined.It is assumed that three sample points are generated in the first step by using CLHS, as shown in Figure 3a.The next step for ILHS is to generate new sample points while inheriting obtained sample points.It is assumed that six sample points are needed.That is to say, three additional sample points should be generated by ILHS.The variables uma, umb can be divided into six non-overlapping intervals, as shown in Figure 3b with green.If shaded spaces are removed from Figure 3b, the unrepresented variable space will emerge as a blank space, as shown in Figure 3c.Actually, the three additional sample points can be filled into the blank space in Figure 4a.By using CLHS, the positions of the three additional sample points can be easily determined.It can be observed that inherited sample points (denoted by red dots) and the additional sample points (represented by black square dots) form the new sample set covering the input probability space more evenly and largely, as shown in Figure 4b.Then, these new sample sets can be permutated and transformed into the original probability space.It is valuable to note that, in PPF analysis based on ILHS, the obtained PPF results corresponding to inherited sample points can be reused, thus resulting in the improvement of accuracy while retaining the computational efficiency.The main implement steps of ILHS in handling multiple variables are elaborated as follows [22].
Algorithm: ILHS NOTE: it is assumed that the PPF analysis based on CLHS has been executed as the first step with using the sample size P num .Hence, the inherited sample set U = [u , , u , , … , u , ] is viewed as the input variables in ILHS.
(  Actually, the three additional sample points can be filled into the blank space in Figure 4a.By using CLHS, the positions of the three additional sample points can be easily determined.It can be observed that inherited sample points (denoted by red dots) and the additional sample points (represented by black square dots) form the new sample set covering the input probability space more evenly and largely, as shown in Figure 4b.Then, these new sample sets can be permutated and transformed into the original probability space.It is valuable to note that, in PPF analysis based on ILHS, the obtained PPF results corresponding to inherited sample points can be reused, thus resulting in the improvement of accuracy while retaining the computational efficiency.The main implement steps of ILHS in handling multiple variables are elaborated as Algorithm 1 [22].
Energies 2019, 12, 3088 9 of 21 Algorithm 1: ILHS NOTE: it is assumed that the PPF analysis based on CLHS has been executed as the first step with using the sample size P num .Hence, the inherited sample set U = u p,1 , u p,2 , . . ., u p,q is viewed as the input variables in ILHS.

Combination of NPNT and ILHS for PPF Analysis of Hybrid AC/VSC-MTDC Grid
The procedures of combining NPNT and ILHS for conducting PPF analysis on a hybrid AC/VSC-MTDC grid is presented in Figure 5.As the initial step for PPF analysis, modeling of random inputs is critical.However, the direct information about probabilistic characteristics of inputs is rather limited in realistic engineering applications, while the raw statistical data is more or less available in practice; moreover, even if raw data is sufficient for fitting the PDF but not always in a common PDF type, thus leading to an accuracy loss.Therefore, NPNT is used to handle the raw statistical data and build the probability model of random inputs.
Once the accurate input probability model is built, the next step is to propagate the randomness from the input probability model to target outputs by means of ILHS.The steps of combining TPNT and CLHS methods are introduced in [11].The similar idea can be applied to the combination of NPNT and ILHS herein as well.In addition, the basic idea of combining NPNT and CLHS or ILHS methods is summarized as (1) obtain the samples on uniform distributions; (2) transform these samples into standard normal space; (3) transform the samples of standard normal space into original space by using ninth-order polynomial.The core steps are highlighted in the green part of Figure 5.The advantages of the proposed method are shown below.
(1) The proposed method can deal with the random variables following irregular distributions even with correlations merely based on historical records.(2) The results obtained in the previous simulations can be reused by the proposed method, thus improving the computational speed and efficiency.(3) The proposed method has a better adaptability and flexibility for diverse operational scenarios of the complex AC/VSC-MTDC grids as it is able to adaptively evaluate sample size adequacy, based on an acceptable accuracy.(4) The statistical moments (such as means and standard deviations) and PDFs of PPF results could be accurately and directly obtained by use of the proposed method.
thus improving the computational speed and efficiency.
3) The proposed method has a better adaptability and flexibility for diverse operational scenarios of the complex AC/VSC-MTDC grids as it is able to adaptively evaluate sample size adequacy, based on an acceptable accuracy.4) The statistical moments (such as means and standard deviations) and PDFs of PPF results could be accurately and directly obtained by use of the proposed method.

System, Data, and Scenarios
The modified IEEE 118-bus system was applied in this paper to demonstrate the effectiveness and superiority of the proposed method.As shown in Figure 6, seven additional wind farms (including WF 1 , WF 2 , WF 3 , WF 4 , WF 5 , WF 6 , and WF 7 ) were added to the modified IEEE 118-bus system respectively at buses 35, 36, 43, 44, 45, 46, and 90.Meanwhile, the wind farms WF 8 , WF 9 , and WF 10 were connected to the AC grid at bus 30, 38, 84, and 115 through a five-terminal DC grid incorporating VSCs.The parameters of the IEEE 118-bus system can be found in Matpower 6.0 [29].The parameters of the VSCs and DC lines are given in [3].In this paper, the control modes of the VSCs are presented in Table 1.Note that the base power of the AC and DC grids were set to be 100 MVA.
In a practical power system, different areas may have different types of loads with diverse characteristics and following various distributions.In this paper, it was assumed that the loads in the test system can be categorized into three groups: commercial load, industrial load, and residential load.Assumed that the same type of load has a similar probability characteristic.In order to simulate the probability characteristic of such practical loads, the realistic historical records of commercial load, industrial load, and residential load in a southern provincial power grid of China were used.The procedures for modeling different types of loads are given in Appendix A. The types of loads in the test system are presented in Table 2.The modified IEEE 118-bus system was applied in this paper to demonstrate the effectiveness and superiority of the proposed method.As shown in Figure 6, seven additional wind farms (including WF1, WF2, WF3, WF4, WF5, WF6, and WF7) were added to the modified IEEE 118-bus system respectively at buses 35, 36, 43, 44, 45, 46, and 90.Meanwhile, the wind farms WF8, WF9, and WF10 were connected to the AC grid at bus 30, 38, 84, and 115 through a five-terminal DC grid incorporating VSCs.The parameters of the IEEE 118-bus system can be found in Matpower 6.0 [29].The parameters of the VSCs and DC lines are given in In this paper, the control modes of the VSCs are presented in Table 1.Note that the base power of the AC and DC grids were set to be 100 MVA.
In a practical power system, different areas may have different types of loads with diverse characteristics and following various distributions.In this paper, it was assumed that the loads in the test system can be categorized into three groups: commercial load, industrial load, and residential load.Assumed that the same type of load has a similar probability characteristic.In order to simulate the probability characteristic of such practical loads, the realistic historical records of commercial load, industrial load, and residential load in a southern provincial power grid of China were used.The procedures for modeling different types of loads are given in Appendix A. The types of loads in the test system are presented in Table 2.
The realistic wind speeds may not follow common distributions like Weibull distributions.The data of wind speeds are based on historical records in a southern provincial power grid of China, which is applied to the wind farms in this case study.An emphasis is put on that the historical records of wind speed and load are collected from the same area.It was assumed that the correlation coefficients amongst wind speeds at wind farms WF1, WF2, WF3, WF4, WF5, WF6, and WF7 were 0.3, and correlation coefficients amongst wind speeds at wind farms WF8, WF9, and WF10 were 0.4.Moreover, assuming that the correlation coefficients among commercial loads were set to be 0.2, the correlation coefficients among residential loads were 0.25, and the correlation coefficients between any two industrial loads were set to be 0.1.The rated capacities of these wind farms were set to 100  The realistic wind speeds may not follow common distributions like Weibull distributions.The data of wind speeds are based on historical records in a southern provincial power grid of China, which is applied to the wind farms in this case study.An emphasis is put on that the historical records of wind speed and load are collected from the same area.It was assumed that the correlation coefficients amongst wind speeds at wind farms WF 1 , WF 2 , WF 3 , WF 4 , WF 5 , WF 6 , and WF 7 were 0.3, and correlation coefficients amongst wind speeds at wind farms WF 8 , WF 9 , and WF 10 were 0.4.Moreover, assuming that the correlation coefficients among commercial loads were set to be 0.2, the correlation coefficients among residential loads were 0.25, and the correlation coefficients between any two industrial loads were set to be 0.1.The rated capacities of these wind farms were set to 100 MW, and the power factors of these wind farms were set to be 0.95.The wind farm output model in 7 was applied in this case study.

Performance Evaluation on Probability Model of Random Inputs
The frequency histogram was applied as a major metric to evaluate the accuracy of NPNT for modeling of uncertain inputs.A frequency histogram similarity index (FHSI) [30] was used in this paper to denote the degree of overlap between any two frequency histograms obtained by means of reference method and the methods in comparison, which is denote percentages of the number of samples fall in each Bin interval.N s represents the number of Bin intervals.Actually, the number of Bins in frequency histogram must be identical, and 100 was applied for all tests in this paper.The range of FHSI is between 0 and 1.A larger FHSI represents that the obtained frequency histogram is closer to the reference's, indicating the higher accuracy is achieved in terms of modeling uncertain variables.FHSI >90% is regarded as "accurate" in this paper.
In order to evaluate the performance of NPNT, the frequency histograms of loads and wind speeds obtained by historical records were regarded as the references for validation.To further demonstrate the superiority of NPNT, the results fit by NPNT were compared with those obtained from TPNT, FPNT, Gaussian, and Weibull distributions.The Gaussian and Weibull distributions were frequently used in the probabilistic analysis of the power system, which were directly applied to fit the historical records in this paper.Note that the number of samples of historical records is 9000; accordingly, in Figure 7, NPNT, FPNT, TPNT, Gaussian, and Weibull distributions generate 9000 samples as well for a fair comparison.
Energies 2019, 12, x FOR PEER REVIEW 12 of 21 distributions.Obviously, the frequency histograms obtained by historical records cannot be fitted well with Gaussian and Weibull distributions.The reason for this lies in that the random variables in the practical power system may not follow common distributions but irregular distributions.Furthermore, frequency histograms obtained using FPNT and TPNT are biased compared with the reference results obtained by using historical records.Meanwhile, it is obvious that the frequency histograms of NPNT can fit very well with the frequency histograms of reference results.Compared with FPNT and TPNT, NPNT has the ability to control the first to tenth moments of historical records, leading to the improvement in the accuracy of modeling random inputs.and Weibull distributions.The reason for this lies in that the random variables in the practical power system may not follow common distributions but irregular distributions.Furthermore, frequency histograms obtained using FPNT and TPNT are biased compared with the reference results obtained by using historical records.Meanwhile, it is obvious that the frequency histograms of NPNT can fit very well with the frequency histograms of reference results.Compared with FPNT and TPNT, NPNT has the ability to control the first to tenth moments of historical records, leading to the improvement in the accuracy of modeling random inputs.
The accuracy index FHSIs obtained by NPNT, FPNT, TPNT, Gaussian, and Weibull distributions also support the conclusions given above, as shown in Table 3.Using common distributions (such as Gaussian and Weibull distributions) may not fit well with frequency histograms obtained by historical records in practice.For example, the maximum and average values of FHSIs obtained Gaussian and Weibull distributions respectively are 85.74%, 78.45% (for Gaussian distribution) and 86.42%, 82.85% (for Weibull distribution).The average values of FHSIs obtained by NPNT, FPNT, and TPNT, respectively, are 93.92%,89.40%, and 86.66%.Actually, most FHSIs of NPNT are greater than 93.5%, qualifying the NPNT as "accurate."NPNT allows the accuracy control of higher-order moments, which yields a more accurate simulation of input random variables.Hence, in the next section, NPNT is applied to simulate correlated random variables in PPF analysis.

Performance Evaluation of Proposed PPF Method
In this section, the performance of the proposed PPF method will be comprehensively evaluated.The reference results are obtained by using the MCS based NPNT method, in which the MCS generates 50,000 sample points.It is sufficient to yield reliable PPF results.The proposed method in this paper is combined NPNT with ILHS, which is denoted as NPNT-ILHS.

Comparison with the CLHS Method
In this subsection, the CLHS is combined with NPNT to form the NPNT-CLHS method for comparison.The proposed method NPNT-ILHS adds 100 samples in each iteration until convergence is reached.The average value of the first to ninth order raw moments of VSC3 s output active power is set to be the convergence criterion; meanwhile, the threshold value is set to 5%.
The relative errors of the first to ninth raw moments of VSC3 s output active power obtained by using NPNT-ILHS and NPNT-CLHS methods are shown in Figure 8.Meanwhile, Figure 9 presents the average values of FHSIs of DC bus voltages corresponding to different sample sizes.It can be investigated that the computational accuracy of the NPNT-CLSH method and proposed method

Comparison of PPF Methods with Using Different Input Probability Models
The test in this subsection is mainly used for comparison in terms of computation accuracy.The comparison methods include CDNT-MCS, TPNT-MCS, and FPNT-MCS, as shown in Table 4.These methods use the same sampling method MCS, and the sample size is 50,000.The methods of building input probability model are different.Common distributions and NATAF transformation are used to build the input probability model in CDNT-MCS, while input random variables are handled by using TPNT and FPNT in TPNT-MCS and FPNT-MCS.Note that the convergence criterion and the threshold value are same as the ones of the NPNT-ILHS method presented in Section 6.3.1.Figure 11 presents the contours of frequency histograms of the VSC3 s active power output and AC bus 38 s voltage magnitude obtained by NPNT-ILHS, CDNT-MCS, TPNT-MCS, and FPNT-MCS methods.Evidently, compared with CDNT-MCS, TPNT-MCS, and FPNT-MCS methods, the proposed method NPNT-ILHS can fit very well with reference's frequency histogram.As shown in Table 5, the FHSIs obtained by using NPNT-ILHS, CDNT-MCS, FPNT-MCS, and FPNT-MCS are presented.The average FHSI values of NPNT-ILHS, CDNT-MCS, FPNT-MCS, and FPNT-MCS method respectively are 93.35%,83.33%, 88.43%, and 85.45%.The maximum FHSI value of the proposed method NPNT-ILHS achieves 94.16%.Actually, the good performance of the proposed method NPNT-ILHS in PPF analysis is guaranteed, which mainly lies in two reasons.Firstly, the accurate probability model of input variables following irregular distributions can be built based on the NPNT method.This provides a good base for ILHS to propagate the randomness from inputs to target outputs.Secondly, based on a given convergence precision for a specific operational scenario of the hybrid AC/VSC-MTDC grid, the ILHS could adaptively assess the sample size, thus getting higher computational accuracy while holding a good efficiency.However, frequency histograms obtained by CDNT-MCS, TPNT-MCS, and FPNT-MCS methods are biased compared with those obtained by the reference method in Figure 11.The computation errors of these methods are mainly caused by inaccurate probability model of random input variables.It is demonstrated in Section 6.2 of this paper that the historical records may not follow the common distributions.Hence, the CDNT-MCS method, based on the assumption of input variables following common distributions, will bring error into PPF analysis.Although TPNT-MCS and FPNT-MCS methods build the probability model according to historical records, the frequency histogram of the reference's result cannot be fitted very well by using these two methods.The FHSIs of voltage magnitude of AC bus 45 of TPNT-MCS and FPNT-MCS methods respectively are 85.67% and 87.93%, which are lower than the FHSI value (93.13%) obtained by proposed method NPNT-ILHS.Moreover, the NPNT has the ability to control the first to the tenth moments of historical records and it is able to build more accurate probability model of inputs, thus leading to an increase in computational accuracy.

Comparison with other PPF Methods
The point estimation method (PEM) is a popular method for solving the PPF problem.In [31], the PEM is combined with TPNT to deal with the PPF problem in pure AC grid.The similar idea can be applied in this paper as well.In this subsection, two PEMs [32] (including 2n+1 PEM and 4n+1 PEM) are combined with NPNT to form the NPNT-2PEM and NPNT-4PEM methods for comparing with proposed method NPNT-ILHS.Meanwhile, two convergence criterions are defined for NPNT-ILHS method to obtain different PPF results with different purposes.
(2) NPNT-ILHS(b): The average value of the first to third order raw moments of VSC3 s output active power is set to be convergence criterion; meanwhile, the threshold value equals to 5%.
The relative errors of the first to ninth raw moments of DC bus 2 s voltage magnitude is given in Table 6.Table 7 presents the average values of relative errors of DC bus and AC bus voltage magnitudes obtained by means of NPNT-2PEM, NPNT-4PEM, NPNT-ILHS(a), and NPNT-ILHS(b) methods.Meanwhile, the computational time and the number of samples of these methods are given in Table 8.The relative errors of the ninth order raw moment of DC bus 2 s voltage magnitude obtained using NPNT-2PEM, NPNT-4PEM, NPNT-ILHS(a), and NPNT-ILHS(b) methods, respectively, are 70.14%,39.39%, 10.46%, and 68.45%.The average values of relative errors of the third order raw moment of AC bus voltage magnitudes obtained by using NPNT-2PEM, NPNT-4PEM, NPNT-ILHS(a), and NPNT-ILHS(b) methods, respectively, are 15.69%, 9.56%, 1.84%, and 12.56%.The computational time of NPNT-2PEM, NPNT-4PEM, NPNT-ILHS(a), and NPNT-ILHS(b) methods respectively are, 49.64, 99.06, 1024.13, and 88.26 s.The NPNT-ILHS(a) shows the highest accuracy with the lowest computational efficiency.The computational accuracy of NPNT-ILHS(b) is higher than NPNT-2PEM's but lower than NPNT-4PEM's.Accordingly, the computational time of NPNT-ILHS(b) stands between NPNT-2PEM and NPNT-4PEM methods.The proposed method NPNT-ILHS also shows more flexibility compared with NPNT-2PEM and NPNT-4PEM method as it can adaptively assess the sample size according to the given convergence level in a specific operational scenario of the hybrid AC/VSC-MTDC grids.For example, if the accurate PDFs of PPF results are required in an operational scenario, a more strict convergence target can be set for the proposed method NPNT-ILHS (like NPNT-ILHS(a)) to achieve the satisfactory accuracy while keeping the relatively low computational cost.However, the PEM methods have no ability to accurately estimate the PDFs in the complex AC/DC grids, and the MCS method is extremely time-consuming (the computational time of the reference method is 11,132.75s).On the other hand, if the accurate high order moments of PPF results are not required for an operational scenario, the proposed method could flexibly set a relaxed convergence (like NPNT-ILHS(b)) to save computational time.Unfortunately, the PEM method seems rigid compared with the proposed method NPNT-ILHS, as the PEM method cannot smoothly adjust to select the number of sample points.Hence, the proposed method in this paper has a better adaptability and flexibility for diverse operational scenarios of the complex AC/VSC-MTDC grids.

Performance Evaluation with Different Correlation Levels
The correlations between wind speeds are non-negligible in modern power systems.It is assumed that the correlation coefficients between wind speeds at WF8, WF9, and WF10 increase from 0.0 to 1.0 at intervals of 0.1.Note that the defined simulation conditions given in Section 6.1 of case studies will be still used in subsequent calculations, and the convergence criterion for NPNT-ILHS was introduced in Section 6.3.1.
Tables 9 and 10 present the maximum and average errors of the first to third raw moments of DC bus voltages obtained using the proposed method against the reference results when the correlation coefficients between wind speeds at wind farms WF8, WF9, and WF10 increase from 0.0 to 1.0 at intervals of 0.1.These results above reveal that the average errors (regarding the first to third raw moments) of DC bus voltages are lower than 4%, while the maximum errors of the first to third raw moments of DC bus voltages are lower than 5%.Obviously, the test results illustrate the good accuracy and robustness of the proposed method in handling various correlation levels.

Conclusions
A new PPF method based on NPNT and ILHS, particularly for the hybrid AC/DC grids, was proposed in this paper.The correlated random variables following arbitrary distributions can be accurately considered and handled by means of the proposed method.Meanwhile, based on an acceptable computational accuracy for a specific operational scenario of the AC/VSC-MTDC hybrid grid, the proposed method could adaptively evaluate the sample size, achieving a good balance

Figure 1 .
Figure 1.A steady-state model of a Voltage Source Converter station.

Figure 1 .
Figure 1.A steady-state model of a Voltage Source Converter station.

Figure 3 .
Figure 3. Inherited design points and underrepresented variable space.(a): Three samples generated in uniform distributions; (b): The uniform distributions with six non-overlapping intervals ; (c): The unrepresented variable space .

Figure 3 .
Figure 3. Inherited design points and underrepresented variable space.(a): Three samples generated in uniform distributions; (b): The uniform distributions with six non-overlapping intervals ; (c): The unrepresented variable space .Energies 2019, 12, x FOR PEER REVIEW 8 of 21

Figure 4 .
Figure 4. Mapping the new sample points in unrepresented space.(a): The samples generated in the unrepresented variable space; (b): Six samples in uniform distributions.

1 Figure 4 .
Figure 4. Mapping the new sample points in unrepresented space.(a): The samples generated in the unrepresented variable space; (b): Six samples in uniform distributions.

( 2 )
k = 2, where k represents the iterations.(3) While C_criterion = 1.(4) Divide the uniform distributions into kP intervals with equal probability.(5) Generate sample points in unrepresented variable space.(6) Obtain the new sample set U new = u kp,1 , u kp,2 , . . ., u kp,q on uniform distributions.(7) The new sample set is permutated by using Cholesky decomposition.(8) Select the samples in the unrepresented variable space and transform these samples into original probability space.(9) Feed the transformed samples into DPF module of hybrid AC/VSC-MTDC grids.(10) Analyze the results (the results include the newly obtained and inherited PPF outputs).(11) If (I k Y − I k−1 Y )/I k Y ≤ β, where I k Y denotes the statistical moments of PPF outputs after the k th extension, β represents the threshold value.(12) C_criterion = 0 (13) Else (14) k = k + 1; (15) End (16) End

Figure 5 .
Figure 5.The flow chart of combining ninth-order polynomial normal transformation (NPNT) and ILHS for probabilistic power flow (PPF) analysis on the AC/DC grid.

Figure 5 .
Figure 5.The flow chart of combining ninth-order polynomial normal transformation (NPNT) and ILHS for probabilistic power flow (PPF) analysis on the AC/DC grid.

Figure 6 .
Figure 6.The modified IEEE-118 bus system with a VSC-MTDC system.

Figure 6 .
Figure 6.The modified IEEE-118 bus system with a VSC-MTDC system.
values of locations of each Bin interval, while

Figure 7
Figure7presents the frequency histograms of the active power of load at bus 1, bus 20, bus 75, and wind speeds obtained by historical records, NPNT, FPNT, TPNT, Gaussian, and Weibull distributions.Obviously, the frequency histograms obtained by historical records cannot be fitted well with Gaussian

Figure 9 .
Figure 9.The average values of FHSIs of DC bus voltage obtained by using ILHS and CLHS methods.

Figure 10 .
Figure 10.The computational time of NPNT-ILHS and NPNT-CLHS with gradually increasing sample points (the sample size is increased from 100 to 3800 at an interval 100).

Figure 10 .
Figure 10.The computational time of NPNT-ILHS and NPNT-CLHS with gradually increasing sample points (the sample size is increased from 100 to 3800 at an interval 100).

Figure 11 .
Figure 11.Frequency histograms of PPF results: (a) the frequency histograms of the VSC3′s active power output; (b) the frequency histograms of voltage magnitude of AC bus 38.

Figure 11 .
Figure 11.Frequency histograms of PPF results: (a) the frequency histograms of the VSC3 s active power output; (b) the frequency histograms of voltage magnitude of AC bus 38.

Table 2 .
Types of loads at different buses in the modified IEEE 118-bus system.

Table 4 .
The comparison methods.

Table 4 .
The comparison methods.

Table 6 .
The relative errors (%) of the first to the ninth raw moments of DC bus 2 s voltage magnitude.

Table 7 .
The average values of relative errors (%) of DC bus and AC bus voltage magnitudes.

Table 8 .
The computational time and the number of samples.

Table 9 .
The average errors of the first to third raw moments of DC bus voltages with different correlation coefficients.

Table 10 .
The maximum errors of the first to third raw moments of DC bus voltages with different correlation coefficients.