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Probabilistic load flow calculation based on sparse polynomial chaos expansion

2018; Institution of Engineering and Technology; Volume: 12; Issue: 11 Linguagem: Inglês

10.1049/iet-gtd.2017.0859

1751-8695

Xin Sun, Qingrui Tu, Jinfu Chen, Chengwen Zhang, Xianzhong Duan,

Structural Health Monitoring Techniques

IET Generation, Transmission & DistributionVolume 12, Issue 11 p. 2735-2744 Research ArticleFree Access Probabilistic load flow calculation based on sparse polynomial chaos expansion Xin Sun, Xin Sun School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorQingrui Tu, Qingrui Tu Electric Power Dispatching and Control Centre of Guangdong Power Grid, Guangzhou, People's Republic of ChinaSearch for more papers by this authorJinfu Chen, Corresponding Author Jinfu Chen chenjinfu@mail.hust.edu.cn School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorChengwen Zhang, Chengwen Zhang School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorXianzhong Duan, Xianzhong Duan School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this author Xin Sun, Xin Sun School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorQingrui Tu, Qingrui Tu Electric Power Dispatching and Control Centre of Guangdong Power Grid, Guangzhou, People's Republic of ChinaSearch for more papers by this authorJinfu Chen, Corresponding Author Jinfu Chen chenjinfu@mail.hust.edu.cn School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorChengwen Zhang, Chengwen Zhang School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this authorXianzhong Duan, Xianzhong Duan School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, People's Republic of ChinaSearch for more papers by this author First published: 18 April 2018 https://doi.org/10.1049/iet-gtd.2017.0859Citations: 14AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A probabilistic load flow (PLF) method based on the sparse polynomial chaos expansion (PCE) is presented here. Previous studies have shown that the generalised polynomial chaos expansion (gPCE) is promising for estimating the probability statistics and distributions of load flow outputs. However, it suffers the problem of curse-of-dimensionality in high-dimensional applications. Here, the compressive sensing technique is applied into the gPCE-based scheme, from which the sparse PCE is built as the surrogate model to perform the PLF in an accurate and efficient manner. The dependence among random input variables is also taken into consideration by making use of the Copula theory. Consequently, the proposed method is able to handle the correlated uncertainties of high-dimensionality and alleviate the computational effort as of popular methods. Finally, the feasibility and the effectiveness of the proposed method are validated by the case studies of two standard test systems. 1 Introduction Plenty of uncertainty sources exist in power system operations, such as variation of load demands, outages of network components. In power systems today and those in the future, randomness and intermittency of renewable energy sources (RES) like wind power and photovoltaic have brought additional uncertainties [1]. In order to assess the impacts of uncertainty on power system operation, probabilistic load flow (PLF) is proposed to enable dispatchers and operators to evaluate the weak points and potential crisis. The security level of system operation can be quantitatively assessed using the probabilistic metrics obtained through the PLF calculation. The security-economy control room decision-making is enhanced [2, 3]. Since the PLF was first proposed by Borkowska in 1974 [4], much technical literature can be found about this topic. As the mostly used one, Monte Carlo simulation using different sampling strategies, like simple random sampling (SRS), Latin supercube sampling [5], and Latin hypercube sampling (LHS) [6], is always adopted as the reference to validate other methods. The huge computational burden contributed by rounds of deterministic load flow (DLF) calculations makes it not suitable for the situation with the requirement of operational speed. Besides the MCS-like methods, analytical methods, i.e. convolution method [7] and cumulant method [8], are also introduced into the PLF calculation. Through the effective assumptions and approximations, high efficiency can be achieved due to the reduced number of repeated load flow calculations. However, the accuracy of the analytical methods is limited. As another category of PLF approaches, approximation methods execute fewer DLF calculations in the similar working mechanism of MCS, and then further mathematical treatments are taken to obtain the probabilistic results of load flows. The point estimation method (PEM) [9] as well as the recently introduced stochastic interpolation method [10] and polynomial chaos expansion (PCE) method [11–13] are representatives of the approximation methods. In recent years, as the state-of-the-art approach in the field of uncertainty quantification, PCE and its extended form, generalised polynomial chaos expansion (gPCE) [14], have been investigated in-depth from the view of mathematics and widely introduced into solving many engineering problems [15–17]. With this method, the original model is analytically approximated in a suitable space spanned by the polynomial chaos basis which is a series of multivariate orthogonal polynomials with respect to the joint distribution of input variables [18]. Ren et al. [11] and Wu et al. [12] verified the accuracy and the efficiency of the gPCE method in PLF calculation problems for small-scale and medium-scale power systems. Nevertheless, this method encounters the remarkable degradation in efficiency in usages for large-scale power systems where the number of input variables significantly increases, because much more time-consuming evaluations on the original non-linear equations are required to determine the valid expansions. Regarding the high-dimensional PLF calculations, the sparse-adaptive scheme is one of suitable solutions to overcome this difficulty actually [13]. The motivation of this work is to find a novel and fast gPCE-based PLF approach that overcomes the limitations aforementioned. The compressive sensing (CS), also known as the compressed sensing or the compressive sampling, is a signal processing technique for recovering a sparse signal by finding the solutions to underdetermined problems [19–21]. It has been recently gained momentum in the context of polynomial expansions and proved useful in recovering a sparse response surface solution if the underlying physical system exhibits sparsity [22]. The contribution of this paper is applying the CS into the original gPCE-based scheme and providing a new PLF calculation method subsequently. The promising feature of the proposed method is that fewer rounds of DLFs are sufficient for building the effective surrogate model based on sparse polynomial expansions. Moreover, the correlation among random variables as another critical issue is also taken into consideration by employing the Copula theory. The sparsity of load flow outputs is demonstrated, and the performance of the proposed method is evaluated in the case studies. The remainder of this paper is organised as follows. Firstly, the theoretical background related to the gPCE approximation theory is introduced in Section 2. Section 3 presents the general procedure of PLF calculation based on the gPCE method, followed by the improved scheme employing the CS technique in Section 4. In Section 5, the performance of the proposed PLF method is studied in the test systems. Finally, conclusions are drawn in Section 6. 2 Generalised polynomial chaos expansion Suppose that the random input vector Z has the joint probability density function (PDF) , where is the marginal PDF of the univariate random variable Zi. The univariate generalised polynomial chaos (gPC) basis functions are the orthogonal polynomials satisfying (1) where δmn = 0 if m ≠n and δmn = 1 if m = n, is the Kronecker delta function, and S is the support of the density function. This orthogonality relationship keeps when the univariate PDF equals to the density function of the corresponding orthogonal polynomial. The well-known correspondences between probability distribution types and basis functions are listed in Table 1, which is also known as the Askey hypergeometric orthogonal polynomials scheme [14]. Table 1. Correspondence between gPC types and random variable distributions Distribution of Z gPC basis polynomials Support Gaussian Hermite (−∞, ∞) gamma Laguerre [0, ∞] beta Jacobi [a, b] uniform Legendre [a, b] Based on the univariate polynomials, the d -dimension p th-degree gPC basis function is defined as the products of univariate basis functions of total degree equal to p, that is (2) where i = (i1, …, id) is a multi-index satisfying |i | = i1 + ⋯ + id. The orthogonality of multivariate basis functions is expressed as (3) where and . Given the total degree multi-index collection , a function f (Z) in the random space could be approximated by the p order gPCE (4) where is the constant expansion coefficients, and the multivariate basis functions of degree up to p. The dimension, as well as the number of expansion terms, is (5) It has been proved that Pf (Z) is the best approximation in the weighted L2 norm with the mean-square convergence rate [18]. 3 PLF based on gPCE The physical load flows of power systems are determined by solving the load flow equations: (6) where Pis and Qis are the power injections at bus i ; Vi and θi the voltage magnitude and angle at bus i ; θij is the angle difference between two involved buses; Gij and Bij are the network parameters related to the branch connecting bus i and bus j. The above equations are denoted as the original model and abstracted as Y = f (Z), in which Z = [Z1, …, Zd]T is the vector of input variables, and Y is the state (e.g. bus voltage magnitude or angle) or load flow (e.g. branch active power or reactive power) of interest, also referred to as the model output response. As the input variables, like load demands PL and renewable energy outputs PRES, are assumed to be affected by uncertainty [23], the conventional DLF is combined with the stochastic analysis scheme to bring about the PLF. In this section, the general procedure of the gPCE-based PLF method is outlined below step by step. In this method, the surrogate model, i.e. gPCE of the output response, is utilised instead of the original load flow equations for executing PLF calculations. 3.1 Dealing with the stochastic input variables The input variables of gPCE are required to be independent. However, the random inputs are sometimes correlated in PLF calculations, e.g. demands of the load in the same area, power outputs of neighbouring wind farms (WFs). The Copula theory is adopted in this paper to decorrelate the random input vector, so that the surrogate model could be applied in the case of dependent inputs. Taking two random variables as an example, for the joint cumulative distribution function (CDF) F12 of variables Z1 and Z2, there is a Copula function Cf satisfying [24] (7) where Fi is the marginal CDF of Zi. Typical Copula functions include the elliptical Copula, e.g. Gaussian Copula and t Copula, and Archimedean Copula, e.g. Clayton Copula and Gumbel Copula. The joint PDF can be obtained by differentiating the joint CDF: (8) where cf denotes the Copula density function and ρi is the marginal PDF of Zi. According to (8), the joint PDF of the multivariate random variable vector is the product of each variable's marginal PDF and Copula density function. Therefore, the joint PDF of correlated random variables can be obtained by (i) determining the marginal PDFs of random variables, (ii) selecting a suitable Copula function to represent the dependence structure of multiple variables. Based on the marginal PDFs of input variables, the proper orthogonal polynomials can be selected for the gPCE. If the probability distribution of a univariate variable is out of the Askey scheme, the existing iso-probabilistic transform techniques are useful for transforming it to the desired distribution type. 3.2 Functional approximation of load flow outputs Using the full gPCE as the surrogate model, the output response is functionally approximated by: (9) where is the multi-dimensional orthogonal polynomial of degree p. The above surrogate model can also be expressed as the matrix form: (10) where is the vector of expansion coefficients, and the vector of basis functions. A higher degree of orthogonal polynomial reflects the greater accuracy of the output but also results in more coefficients to be estimated and as such an increased number of system evaluations. In the case of PLF calculations, polynomials of degree give negligible increased accuracy. Through considering synthetically with the computational burden and calculation accuracy, a second-order surrogate model is adopted in this paper. 3.3 Estimating the expansion coefficients in surrogate model After building the surrogate model of the output response, the expansion coefficients must be determined immediately thereafter. Both the intrusive methods, e.g. Galerkin projection, and non-intrusive methods, e.g. least-square regression, can be used for coefficient estimation through several times of deterministic evaluations on the original model. Considering the feasibility and the simplicity for PLF problems with the complex and non-linear form in high-dimension, the regression method is the better choice for estimating the expansion coefficients. In the regression method, let be a set of N samples of input vector Z, and be the corresponding results of DLF calculations . The collection is called the experimental design (ED). The unknown coefficients in (10) are estimated by directly solving the least-square regression problem to minimise the residual between the output responses of two models: (11) which is equivalent to (12) where the ED matrix D is defined by: (13) Although a variety of sampling strategies can be adopted in generating D, the linear oversampling rate, saying with , is always required to ensure the numerical stability of the regression problem [25]. 3.4 Calculation of probabilistic indices of output response The samples of the output response can be computed directly by the conventional Monte Carlo method through the surrogate model. In this way, only the algebraic operations are involved. Therefore, much computational effort is saved compared with directly calculating outputs through solving load flow equations that requires several times of iterative operations. Finally, the probability distribution and statistical moments such as mean μ and standard deviation σ can be evaluated by the output samples. Moreover, the statistical moments of output can also be directly calculated by the expansion coefficients (14) 4 Sparse polynomial chaos expansion As described in the previous section, the main idea of the gPCE-based PLF is using the surrogate model instead of the original load flow equations to generate samples of load flow outputs. According to (5), the required number of DLF calculations increases along with the number of expansion terms in the surrogate model which itself dramatically increases with the dimension of input variables. Such sensitivity to the dimension circumscribes the application of this method to high-dimensional PLF problems. However, it argues that the number of important expansion terms is relatively small, because of two points as follows [17]. Firstly, high-order interaction effects are usually negligible compared to main effect and low-order interaction effects. The interaction order is defined as the number of non-zero univariate elements in the multi-index i. The interaction of the expansion terms describes the simultaneous influence of several input variables on the output response, and usually, low-order ones are physically meaningful in common practice. Secondly, the output response is not affected by all input variables equally. For example, the load power change has more significant effects on branches or buses near the load rather than those remote. The influence of expansion terms on the output response is quantified by their coefficients. The aforementioned two reasons lead to small coefficients of expansion terms that are of high-order interactions or contain trivial input variables, and could be approximately zero consequently. From this perspective, the classical gPCE and its corresponding expansion coefficient vector a is of sparsity. In this section, the expansion terms with non-zero coefficients are regarded as active and retained for representing the output response. The expansion coefficients are estimated by the CS technique, an advanced regression method. As the consequence, a new surrogate model based on the sparse PCE is formed. 4.1 CS for expansion coefficients estimation In the traditional scheme of the gPCE-based method, the size of input variable samples should be at least greater than the dimension of the full gPCE. However, it is much more desirable to take fewer samples without compromising the effective surrogate model, wherein the ordinary regression problem becomes underdetermined because N < M and consequently cannot be used for the coefficient estimation. However, the sparsity of gPCE makes it compressible, which means the output response can be represented by those expansion terms having the most obvious impacts on itself. Therefore, the CS technique, which is originally applied to recover signals or images from fewer samples or measurements by exploiting their sparse representations in data processing problems, is applicable here for estimating the expansion coefficients in the surrogate model. The purpose of applying the CS technique is to build the effective surrogate model with limited samples of input variables whose size is smaller than the full gPCE's dimension. With the sparsity prerequisite, the optimisation problem is formulated to select the most relevant expansion terms with respect to the given evaluations y of the original model. Mathematically, this objective is expressed by minimising the number of non-zero elements in the expansion coefficient vector a, i.e. l0 -norm ||a ||0 : (15) This problem can be solved by greedy pursuit approaches, e.g. least angle regression [17], least absolute shrinkage and selection operator [26], and orthogonal matching pursuit (OMP) [27], or convex relaxation algorithms which replace the problem with a convex optimisation problem relying on the close approximation of the l0 -norm to the l1 -norm (the absolute value norm of a) when the vector is sparse. In this paper, the OMP algorithm is used to solve the CS-based optimisation problem by choosing those polynomial bases which have the greatest impact on the output response among all expansion terms. In the k th iteration of OMP, column nk of ED matrix D, which is most correlated with the approximation error denoted by the residue at current step, is identified and added to the active set . Corresponding to the columns selected via correlation, the indices in also indicate the locations of the non-zero entries in the coefficient vector a. The values of these non-zero coefficients are estimated by least square method only with the columns of ED matrix D present in . Since the sparsity information of the coefficient vector is not known a prior, the iteration would be terminated when a sufficiently small residue magnitude ||rk ||2 has been reached while the number of iterations is not greater than N. Finally, the expansion terms in the are retained to formulate the surrogate model. The pseudocode of the OMP algorithm is provided below: OMP algorithm.Input: , Output: A sparse coefficient vector 1: Initialise. Set residual , coefficient vector , index set , and counter 2: while and do 3: Identity. Find column of D that is most strongly correlated with the residual : , and set 4: Estimate. Find the best values with the columns chosen so far: 5: Update. 6: Set. 7: end while 4.2 Procedure of proposed gPCE-CS-based PLF method With the joint strength of gPCE and CS, a novel PLF method based on the sparse PCE is proposed. The performance of the method in high-dimensional PLF problems is supposed to be improved for the reason that the effective surrogate model could be obtained even only with a limited number of input variable samples. The computational procedure of the proposed gPCE-CS-based PLF calculation is shown in Fig. 1 and summarised as follows: Read the basic data, such as the necessary data for the PLF calculation, data of the multiple random inputs, and so on. Select a suitable Copula function to express the dependent structure of the inputs if they are correlated. Generate the ED set by SRS. Build the two-order surrogate model based on the multivariate gPC basis functions corresponding to the distribution types of input variables. Then produce the ED matrix D. Execute a batch of DLF calculations with each ED sample in the experimental set to get the output response . Determine the sparse expansion coefficients by solving the CS optimisation problem with the OMP algorithm described in (15). Generate enough inputs samples, and calculate the output response with each sample through the surrogate model. Evaluate the statistical moments and probability distributions of the output. Fig. 1Open in figure viewerPowerPoint Flowchart of the gPCE-CS-based PLF calculation 5 Case study In this section, the benchmark IEEE 39-bus system is used to test the performance of the proposed PLF method. The basic data, including network topology and parameters, load and generation powers etc., keeps same with the data file in MATPOWER [28]. Four 450 MW WFs are located at bus 30 (WF1 and WF2) and bus 32 (WF3 and WF4) to replace the conventional generators in the original case. The penetration level of the wind power is ∼30%. The forecast outputs of WF1 and WF2 are assumed as 0.7 p.u. to make the power injection at bus 33 similar to the original case, while the values of WF3 and WF4 are set to 0.3 p.u. for the same purpose. The WF output is assumed following the Beta distribution B (α, β), which is a proper choice for simulating the short-term (1–48 h) forecast uncertainty of WF output [29] owing to the bounded nature of power produced by the WF. The specific distribution parameters can be decided based on historical data and forecast tools. As it is not within the purpose of this paper to model the uncertainty factors in detail, the results in [29] are used directly. The associated parameters of four WFs used in this case study are listed in Table 2. Besides, the uncertainty of load behaviour is simulated in two parts: their normalised active powers are assumed to follow the standard Gaussian distribution G (1, 0.032) with power factors described by the Uniform distribution U (0.95, 1). In the cases considering the dependent variables, the non-linear correlation described by the Spearman rank correlation coefficient (rho) and handled by the Gaussian Copula function is adopted. The rho of load and load, WF and WF, and WF and load are assumed as 0.8, 0.69, and 0.12, respectively [10]. Two test scenarios are considered: (i) independent scenario S1, in which the stochastic inputs are assumed to be independent of each other. (ii) Dependent scenario S2, in which the stochastic inputs are correlated with the assumed correlation relationship. Table 2. Parameters of WFs WF Rated power, MW , p.u. Parameters 1, 2 450 0.7 α = 3.78, β = 1.62 3, 4 450 0.3 α = 6.70, β = 15.62 The DLF is solved using the functions of MATPOWER, and other analysis programs are developed with MATLAB R2014a. The case studies are carried out on the computer with Intel Core I5 CPU at 2.60 GHz and 4 GB of RAM. In the whole case study, the calculation results of PLF based on the Monte Carlo simulation using SRS (MCS-SRS) with 50,000 random samples are assumed to be accurate and used as the reference herein. In order to assess the accuracy of the proposed method, the relative error index regarding the probabilistic statistics evaluation is adopted: (16) where refers to the type of output variables, s represents the type of statistical properties including mean and standard deviation . and are the results calculated by the compared method and the referential method. 5.1 Performance evaluation of PLF calculations based on gPCE-CS In this part, the performance of the proposed method is examined. The error indices of branch active power P are chosen as representatives and used as the output responses here to illustrate the effectiveness of the proposed method. 5.1.1 Evaluation of the surrogate model The total number of random inputs is 42, including the active power and power factor of 19 loads, together with the output of four WFs. Under this condition, the two-order gPCE has 946 expansion terms. The ordinary least square method with 2000 inputs samples is used to determine the expansion coefficients. A measure for the sparsity of full gPCE is defined by the sparsity index IS: (17) where the coefficient ai is regarded as the non-zero coefficient if |ai |/||a ||∞ > 0.005. The IS indexes of all output responses are visualised as histograms in Fig. 2. Fig. 2Open in figure viewerPowerPoint Sparsity index of full gPCE in two test scenarios From the results in the figure, the sparsity of full gPCE is illustrated. In the independent scenario S1, the average IS index of all outputs is 0.0122, and the maximum IS index is 0.0307. It means even for the least sparse one, there are only 35 expansion terms marked by the large coefficients having the dominant influence on the output response. Such sparsity character is the foundation of the CS optimisation. While in the dependent scenario S2, the average and maximum IS increase to 0.0388 and 0.2801, respectively, which indicates that the correlation among the random inputs might decrease the sparsity of full gPCE. Based on the sparsity of full gPCE, the OMP algorithm () is used to recover the sparse PCE with limited ED sizes by solving the CS optimisation problem. To demonstrate the effectiveness of the OMP, three different ED sizes, 100, 500, and 1000, are conducted individually, and the Euclidean distance between the coefficients and estimated by the least square method and the CS method, respectively, are calculated to show their similarity. The results of branch 3–18, 4–14, and 16–19 in scenario S1, branch 4–5, 1–2, and 17–18 in scenario S2 are selected to present in Table 3, because they are typical of full gPCE in the system corresponding to three different levels of sparsity, with 3–18/4–5 least sparse and 16–19/17–18 most. Table 3. Comparison of expansion coefficients for several lines in the test system Scenario Branch N = 100 N = 500 N = 1000 S1 3–18 0.2058 0.0961 0.0500 4–14 0.1180 0.0613 0.0375 16–19 0.0351 0.0128 0.0070 S2 4–5 0.6891 0.1415 0.0804 1–2 0.3509 0.1246 0.0626 17–18 0.1702 0.1034 0.0553 As indicated by the results on similarity, the accuracy of coefficient estimation by OMP is improved with more ED samples, regardless of the sparsity of full gPCE or the correlation structure of inputs. On the other hand, with a particular ED size, the increasing sparsity of full gPCE would increase the similarity of the coefficients estimated by two methods, while the correlation among input variables would deteriorate the performance of OMP on the expansion coefficients recovery. Other kinds of output responses, like voltage magnitude V, voltage angle θ, and branch reactive Q, follow the same pattern of estimation similarity. 5.1.2 Evaluation of PLF calculation accuracy The calculation accuracy of the proposed PLF method is studied from the view that whether it can produce the same probabilistic statistics as the reference method. Since the number of output responses is more than one, the mean value of is used to indicate the accuracy degree of the entire system. Owing to the random sampling strategy used in forming the ED matrix, the proposed gPCE-CS with a particular ED size will run 100 times for the purpose of evaluating the performance of this approach accurately. The mean value of the 100 error indices is adopted as the final error index, and the maximum value is also recorded to show the robustness. The error curves of and are shown in Fig. 3. Fig. 3Open in figure viewerPower