Probabilistic Power Flow Calculation Method for Interconnected Power Grids Based on Static External Network Equivalence and Unscented Transformation Method

Abstract

The present invention relates to the technical field of probabilistic power flow calculation for interconnected power grids, and specifically discloses a probabilistic power flow calculation method for interconnected power grids based on static external network equivalence and unscented transformation method, including using the unscented transformation method to generate a sample point set X; obtaining the nodal admittance matrix Y of the internal network after equivalence eq ; performing AC power flow calculation on the sample point X0 representing the mean value to obtain the basic operating point, and then using the DC-like power flow method to obtain the operating points corresponding to other sample points; obtaining the sample point set Z of the equivalent parameters through equivalent calculation; based on Z and the subset X of X I , obtaining the sample point set R after equivalence. Then calculate the mean value #imgabs0# and covariance P of the input random variables of the internal network after equivalence RR ; using the unscented transformation method again to generate a new sample point set X eq ; taking Y eq and X eq as inputs, performing 2n2 + 1 deterministic AC power flow calculations to obtain the output sample point set Z eq , and calculating the mean value #imgabs1# covariance matrix #imgabs2# and standard deviation #imgabs3# of the output random variables of the internal network. This method realizes double dimensionality reduction in terms of the amount of calculation.

Description

Technical Field

[0001] The present invention relates to the technical field of probabilistic power flow calculation for interconnected power grids, and specifically discloses a method for calculating probabilistic power flow of interconnected power grids based on static external network equivalence and unscented transformation method. Technical Background

[0002] With the popularization of new energy power generation, the uncertain factors in the power system continue to increase, and deterministic power flow calculation can no longer meet the needs of power system analysis. At the same time, the distributed access of new energy to the power grid and the consumption of new energy have promoted the construction of the power grid and the connection between each sub-grid. Therefore, the modern power system is a large-scale interconnected power grid with uncertainty. At the same time, with the development of the power system, in order to take into account the influence of uncertain factors such as new energy power generation and load fluctuations, probabilistic power flow analysis of interconnected power grids is required, and the continuous expansion of the power grid scale will significantly increase the corresponding calculation burden. Therefore, it is necessary to study how to improve the calculation efficiency of probabilistic power flow of interconnected power grids to improve the efficiency of analyzing such scenarios.

[0003] Probabilistic power flow calculation methods include simulation method, analytical method and approximation method. The simulation method represented by the Monte Carlo method [2-4] needs to perform deterministic power flow calculations on a huge number of samples one by one to obtain the probability information of the output random variables. This type of method can handle relatively complex probabilistic power flow problems with high calculation accuracy, but low calculation efficiency. The analytical method calculates the probability information of the output random variables based on a linearized model, with high calculation efficiency. However, most of these methods are based on the assumptions of linearization and independent input random variables, and the calculation accuracy may be insufficient.

[0004] The approximation method better balances the accuracy and efficiency of probabilistic power flow calculation. This type of method performs deterministic power flow calculations on a series of points that approximately describe the probability distribution of input random variables to obtain the probability information of the output random variables. Among this type of methods, the unscented transformation method has the advantages of high accuracy and can directly handle correlated random variables, and is widely used. Therefore, this method can be used for probabilistic power flow analysis of interconnected power grids. The calculation amount of the unscented transformation method is closely related to the dimension of input random variables in the power system and the system scale. Directly using the unscented transformation method to perform full-network probabilistic power flow calculation on interconnected power grids with generally large system scales is a heavy burden.

[0005] Static external network equivalence methods include Ward equivalence method and REI equivalence method. For the traditional Ward equivalence method, when the internal network state changes, the calculated reactive power error is relatively large. Although the improved method of the Ward equivalence method has better reactive power response ability, it does not consider the sensitivity consistency before and after equivalence. Summary of the Invention

[0006] In view of the above problems, this paper proposes an equivalent method based on multiple operating points, and combines this method with the unscented transformation method to propose a probabilistic power flow calculation method applicable to interconnected power grids.

[0007] The probabilistic power flow calculation method for interconnected power grids based on static external network equivalence and unscented transformation method in the present invention is as follows:

[0008] Step S1: Using the unscented transformation method, based on the mean value of the input random variables of the whole network and covariance P XX , a sample point set X with 2n1 + 1 sample points is generated;

[0009] Step S2: Calculate the structural parameters of the equivalent model from the structural parameters of the whole network to obtain the node admittance matrix Y of the internal network after equivalence eq ;

[0010] Step S3: Conduct an AC power flow calculation on the sample point X0 representing the mean value to obtain the basic operating point, and then use the quasi - DC power flow method to obtain the operating points corresponding to other sample points; based on the obtained multiple operating points, perform equivalent calculation on each sample point in X to obtain a sample point set Z of equivalent parameters;

[0011] Step S4: Based on Z and the subset X of X I , obtain the sample point set R after equivalence. Then calculate the mean value of the input random variables of the internal network after equivalence and covariance P RR ;

[0012] Step S5: Use the unscented transformation method again, based on and P RR , generate a new sample point set X with 2n2 + 1 sample points eq ; Take Y eq and X eq as inputs, and perform 2n2 + 1 deterministic AC power flow calculations to obtain the output sample point set Z eq ;

[0013] Step S6: Based on Z eq , calculate the mean value of the output random variables of the internal network covariance matrix and standard deviation Thus, the probabilistic power flow calculation of the interconnected power grid is completed.

[0014] Furthermore, Step S1 includes:

[0015] Generate the input sample point set X = {X0, X1, …, X N-1} where N is the number of sample points. The 2n + 1 sampling points of the n - dimensional input random variable x are calculated as follows by the symmetric sampling strategy:

[0016] X0 = m

[0017]

[0018]

[0019] where: m is the mean of the random variable x; W0 is the weight corresponding to the center point X0; P XX is the covariance matrix of the input random variables; is the k-th column of the matrix, which can be obtained by performing a Cholesky decomposition on P XX ;

[0020] The weights of each sampling point are calculated as follows:

[0021] W0 = W0

[0022]

[0023]

[0024] where: W k and W k+n are the weights corresponding to X k and X k+n respectively.

[0025] Furthermore, step 2 also includes:

[0026] The interconnected power system can be divided into an internal network including a boundary system and an internal system, and an external network including an external system. The internal nodes, boundary nodes, and external nodes are represented by I, B, and E respectively, and the generator nodes and non-generator nodes are represented by G and L respectively;

[0027] The nodes of the entire network are divided into: G = {G E , G I} and L = {L E , L B , L I}, where G E represents the external generator nodes, and G I , L E , L B and L I and so on;

[0028] After equivalence, the non-generator nodes and generator nodes in the internal network are respectively divided into L = {L I , L eq} and G = {G I , G eq}, where L eq and Geq respectively represent the boundary nodes and equivalent generator nodes after equal value.

[0029] Furthermore, step S3 includes:

[0030] S31: Calculate the operating point corresponding to the sample point

[0031] Perform AC power flow calculation on the sample X0 representing the mean value to obtain the basic operating point;

[0032] For the operating points corresponding to other sample points, the following DC-like power flow model is used for calculation:

[0033] P *(s) = B * θ (s)

[0034] In the formula: the superscript (s) all represents the power flow calculation result of X s ; P *(s) and B * respectively represent the modified nodal injected power matrix and the modified admittance matrix, and θ (s) represents the voltage phase angle of all network nodes;

[0035] Calculate the voltage phase angles of all network nodes corresponding to each sample point, and combine with the voltage amplitude of the basic operating point to obtain the operating point U corresponding to each sample point (s) ;

[0036] S32: Perform equivalent calculation on the sample point

[0037] Take the injected active power P L of the generator node as the input random variable, and use the unscented transformation method to generate a sample point set X according to the probability characteristics of the input random variables of the whole network;

[0038] According to the internal and external network division situation, the sample point set X = [X ET X BT X IT T For the (s + 1)-th sample point, where and respectively correspond to the injected active powers of the external, boundary and internal non-generator nodes and X s The corresponding operating point is the voltage phasor U of all network nodes (s) , and U (s) is obtained by deterministic power flow calculation;

[0039] For X s , the equivalent load at the boundary is calculated as follows: ​

[0040]

[0041]

[0042] In the formula: The superscript (s) all represents X s corresponding parameters; and are the active and reactive powers of the equivalent load at the boundary respectively; and are calculated from U (s) ; and are the reactive power injections at the boundary and external non - generator nodes respectively;

[0043] The output of the equivalent generator at the boundary and its voltage phasor are calculated as follows:

[0044]

[0045]

[0046] In the formula: F1 (s) and F2 (s) are calculated from U (s) ; P GE is the output of the external network generator; is the voltage phasor of the external network generator node; Y trans is obtained by calculating from the structural parameters.

[0047] Furthermore, step S4 includes:

[0048] S41: Calculate the sample point set R after equivalence

[0049] By performing equivalence calculation on each sample point in X through step S32, a sample point set Z of equivalent parameters is obtained. Combining Z with the subset X of the internal input random variables in X I and establishing the probability model of the internal network input random variables after equivalence by the unscented transformation method as follows:

[0050] R = [Z T X IT T ,

[0051] In the formula: R is the sample point set obtained by combining Z and X I ;

[0052] S42: Calculate the mean and covariance of the internal network input random variables after equivalence

[0053] Calculate the mean and covariance of the internal network input random variables by the following formula:​

[0054]

[0055]

[0056] In the formula: is the mean value corresponding to the random variable input in the internal network after equivalence; n1 is the dimension of the random variable input in the whole network; R s is the (s + 1)-th sample point of R; P RR is the covariance matrix of the random variable input in the internal network after equivalence.

[0057] Furthermore, step S5 includes:

[0058] Again, according to the principle of unscented transformation, generate the sample point set X after equivalence in the same way as in step 1 eq , where the number of sample points in this sample point set is 2n2 + 1, n2 is the dimension of the random variable input in the internal network after equivalence, and n2 < n1;

[0059] Take Y eq and X eq as inputs, and perform 2n2 + 1 times of deterministic AC power flow calculations through the following formula to obtain the output sample point set Z eq :

[0060]

[0061] In the formula: is the output sample point of the corresponding AC power flow model; is the (p + 1)-th sample point of X eq ; Y eq is the nodal admittance matrix of the internal network after equivalence; h(·) is the nonlinear relationship between the output random variable and the input random variable in the AC power flow model.

[0062] Furthermore, step S6 includes:

[0063] Calculate the mean value covariance matrix and standard deviation of the random variable output in the internal network as follows:

[0064]

[0065]

[0066]

[0067] In the formula: is the corresponding weight; and are the mean, covariance matrix, and standard deviation of the output random variable, respectively; denotes taking The diagonal elements of form a column vector.

[0068] The present invention proposes an equivalent method based on multiple operating points, and combines this method with the unscented transformation method to propose a probabilistic power flow calculation method applicable to interconnected power grids. This method has made the following progress:

[0069] (1) By using the unscented transformation method, according to the probability characteristics of the input random variables of the entire network, a sample point set for probabilistic modeling of equivalent parameters at the boundary is obtained;

[0070] (2) Aiming at the probability scenario, an equivalent method based on multiple operating points is proposed to calculate the equivalent parameters at the boundary of each sample point, and a probability model of the input random variables of the internal network after equivalence is established by weighted processing with the unscented transformation method, thereby reducing the dimension of the input variables to reduce the time of power flow calculation;

[0071] (3) The class DC power flow method is used to obtain multiple operating points for equivalence, reducing the calculation time of deterministic power flow calculation. Thus, the complexity of the entire power flow calculation is reduced.

[0072] This method reduces the system scale on the one hand, thereby reducing the time required for each deterministic power flow calculation; on the other hand, it reduces the dimension of the input random variables, thereby reducing the number of times of deterministic power flow calculation, and finally realizing the double dimension reduction in terms of the amount of calculation for this application. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 is a schematic flow chart of the probabilistic power flow calculation method for interconnected power grids based on static external network equivalence and unscented transformation method in an embodiment of the present invention.

[0074] Figure 2 is a schematic diagram of the static external network equivalent model of sample points in an embodiment of the present invention.

[0075] Figure 3 is a schematic diagram of the correlation coefficient matrix between equivalent parameters and input random variables of the internal network in an embodiment of the present invention.

[0076] Figure 4 is a graph of the average relative error of equivalent parameters under different load conditions in an embodiment of the present invention.

[0077] Figure 5 is a schematic diagram of the relative error of the mean value of the internal network output variables of the 118-node system by different methods in an embodiment of the present invention.

[0078] Figure 6Schematic diagram of standard deviation of output variables of intranet of 118-node system in different methods in embodiments of the present invention.

[0079] Figure 7 Schematic diagram of the absolute error of output variables calculated on a 1354-node system in an embodiment of the present invention.

[0080] Figure 8 Schematic diagram of simulation time of different methods in a 118-node system according to an embodiment of the present invention. DETAILED DESCRIPTION

[0081] The method for calculating the probability flow of interconnected power grids based on the static external network equivalent and traceless transformation method in the present invention is basically as follows: Figure 1 The specific steps are as follows:

[0082] Step S1: Use the unscented transformation method based on the mean of the random variables input from the entire network With covariance P XX , generate a sample point set X with 2n1+1 sample points;

[0083] Furthermore, the characteristics of step S1 are as follows:

[0084] Generate input sample point set X = {X0, X1, ..., X N-1}, N is the number of sample points. According to the symmetric sampling strategy, the 2n+1 sampling points of the n-dimensional input random variable x are calculated as follows:

[0085] X0=m (1)

[0086]

[0087]

[0088] Where: m is the mean of the random variable x; W0 is the weight corresponding to the center point X0; P XX is the covariance matrix of the input random variables;

[0089] for The kth column of the matrix, Through P XX Perform Cholesky decomposition to obtain .

[0090] The weight of each sampling point is calculated as follows:

[0091] W0=W0 (4)

[0092]

[0093]

[0094] Where: W k and W k+n are the weights corresponding to X k and X k+n respectively.

[0095] Step S2: Calculate the structure parameters of the equivalent model from the structure parameters of the whole network, and obtain the node admittance matrix Y of the internal network after equivalence eq ;

[0096] In the theory of static external network equivalence, an interconnected power system can be divided into an internal network (including the boundary system and the internal system) and an external network (external system), and the internal nodes, boundary nodes, and external nodes are represented by I, B, and E respectively, and the generator nodes and non-generator nodes are represented by G and L respectively. Since the boundary nodes in the actual network are generally not generator nodes, the nodes of the whole network can be divided into: G = {G E , G I} and L = {L E , L B , L I}, where G E represents the external generator node, and G I , L E , L B and L I and so on;

[0097] The improved static external network equivalence method proposed in this example is based on the equivalent model proposed in the literature YU Juan, LIN Wei, KAMEL S, et al. Equivalent model considering frequency characteristics and renewable uncertainties for probabilistic power flow. The model is as Figure 2 shown, and the meanings of the parameters in the figure are all referenced from this literature.

[0098] After equivalence, the non-generator nodes and generator nodes of the internal network are divided into L = {L I , L eq} and G = {G I , G eq} respectively, where L eq and G eq represent the boundary node and the equivalent generator node after equivalence respectively. Figure 2The equivalent model at the medium boundary contains two types of parameters: (1) Structural parameters: the admittance of each equivalent branch; (2) Operating parameters: the equivalent injection power at the boundary nodes, the power generated by the equivalent generator nodes, and the node voltages. It can be known from the existing literature that the calculation of the structural parameters is only related to the structural parameters of the entire network and will not change with the operating state of the internal network; the calculation of the operating parameters is related to the selection of the basic operating point. When the basic operating point deviates greatly from the actual operating point, the accuracy of the calculated operating parameters is low. Therefore, the equivalent method applicable to probability scenarios proposed in this example performs equivalent for each sample point based on its corresponding operating point to improve the equivalent accuracy.

[0099] Step S3: Perform AC power flow calculation on the sample point X0 representing the mean value to obtain the basic operating point, and then use the quasi-DC power flow method to obtain the operating points corresponding to other sample points; based on the obtained multiple operating points, perform equivalent calculation on each sample point in X to obtain the sample point set Z of the equivalent parameters.

[0100] In some embodiments, this step specifically includes:

[0101] S31: Calculate the operating points corresponding to the sample points

[0102] Perform AC power flow calculation on the sample X0 representing the mean value to obtain the basic operating point.

[0103] For the operating points corresponding to other sample points, use the quasi-DC power flow model proposed in the literature "Li Hucheng, Yu Yijun, Gao Zonghe, etc. A quasi-DC power flow algorithm for improving calculation accuracy [J]. Automation of Electric Power Systems, 2013, 37(12): 128-133" for calculation, and its model is as follows:

[0104] P *(s) =B * θ (s) (7)

[0105] In the formula: the superscript (s) all represents the power flow calculation result of X s ; P *(s) and B * are defined in reference

[18] ; θ (s) represents the voltage phase angle of all nodes in the network.

[0106] After calculating the voltage phase angles of all nodes in the network corresponding to each sample point and combining with the voltage amplitude of the basic operating point, the operating point U (s) .

[0107] S32: Perform equivalent calculation on the sample points

[0108] The injected active power P LAs the input random variable, the probability characteristics of the random variable input from the whole network are used to generate the sample point set X using the untraceable transformation method. According to the division of the internal and external networks, the sample point set X = [X ET X BT X IT ] T For the s+1th sample point, in and Active power is injected into external, boundary and internal non-generator nodes respectively and X s The corresponding operating point is the voltage phasor U of the whole network node (s) , and U (s) Need to be obtained by deterministic power flow calculation.

[0109] For X s , the equivalent load at the boundary is calculated as follows:

[0110]

[0111]

[0112] In the formula: superscript (s) means X s Corresponding parameters; and are the active and reactive powers of the equivalent load at the boundary respectively; and According to the literature YUJuan, LIN Wei, KAMEL S, et al. Equivalent model considering frequency characteristics and renewable uncertainties for probabilistic power flow by U (s) calculate; and Reactive power is injected into boundary and external non-generator nodes respectively.

[0113] Output P of generators in interconnected system G In this paper, it is not considered as a random variable, and the output of the equivalent generator at the boundary and its voltage phasor The calculation is as follows:

[0114]

[0115]

[0116] Where: F1 (s) and F2(s) Calculated according to the literature YU Juan, LIN Wei, KAMEL S, et al. Equivalent model considering frequency characteristics and renewable uncertainties for probabilistic power flow by U (s) P GE is the output of the external network generator; is the voltage phasor of the external network generator node; Y trans It can refer to formula (41) in this literature and be calculated from the structural parameters.

[0117] In the equivalent method based on multiple operating points proposed in this paper, the operating parameters of each sample point need to be continuously updated based on its corresponding operating point, specifically reflected in: for each sample point, in formulas (8) - (11) F1 (s) F2 (s) and take different values, so the operating parameters of the equivalent model calculated are all random variables. While the equivalent method adopted in the literature YU Juan, LIN Wei, KAMEL S, et al. Equivalent model considering frequency characteristics and renewable uncertainties for probabilistic power flow is only based on the basic operating point (the operating point corresponding to the mean value of the random variable), and A1, A2, F1, F2 and U GE are all constants, making P Geq in the operating parameters of the equivalent model and U Geq always be constants.

[0118] Step S4: Based on the subset X I of Z and X, obtain the equivalent sample point set R. Then calculate the mean value of the input random variables of the internal network after equivalence and the covariance P RR .

[0119] In some embodiments, this step specifically includes:

[0120] S41: Calculate the equivalent sample point set R

[0121] Perform equivalent calculation on each sample point in X by formulas (8) - (11) to obtain the sample point set Z of the equivalent parameters. Combine Z with X I(Combined with the subset corresponding to the internal input random variable in X), the probability model of the internal network input random variable after equivalent value is established by the unscented transformation method as follows:

[0122] R = [Z T X IT T (12)

[0123] In the formula: R is the sample point set obtained by combining Z and X I ;

[0124] S42: Calculate the mean value and covariance of the internal network input random variable after equivalent value

[0125] The mean value and covariance of the internal network input random variable are calculated by the following formula:

[0126]

[0127]

[0128] In the formula: is the mean value corresponding to the internal network input random variable after equivalent value; n1 is the dimension of the whole network input random variable; R s is the (s + 1)-th sample point of R; P RR is the covariance matrix of the internal network input random variable after equivalent value.

[0129] Step S5: Again adopt the unscented transformation method, based on and P RR , generate a new sample point set X eq with the number of sample points being 2n2 + 1; Take Y eq and X eq as inputs, and perform 2n2 + 1 times of deterministic AC power flow calculations to obtain the output sample point set Z eq .

[0130] In some embodiments, this step specifically includes:

[0131] Again according to the principle of unscented transformation, from formulas (1) to (6), generate the sample point set X eq after equivalent value, and the number of sample points in this sample point set is 2n2 + 1 (n2 is the dimension of the internal network input random variable after equivalent value, and n2 < n1).

[0132] Take Y eq and X eq as inputs, and perform 2n2 + 1 times of deterministic AC power flow calculations through the following formula to obtain the output sample point set Z eq .

[0133]

[0134] ​Wherein: is the output sample point of the corresponding AC power flow model; is the (p + 1)-th sample point of X eq ; Y eq is the nodal admittance matrix of the internal network after equivalence; h(·) is the nonlinear relationship between the output random variable and the input random variable in the AC power flow model.

[0135] Step S6: Based on Z eq , calculate the mean value covariance matrix and standard deviation

[0136] In some embodiments, this step specifically includes:

[0137] Calculate the mean value covariance matrix and standard deviation

[0138]

[0139]

[0140]

[0141] Wherein: is the corresponding weight; and are respectively the mean value, covariance matrix and standard deviation of the output random variable; means to take the diagonal elements of to form a column vector; thus completing the probabilistic power flow calculation of the interconnected power grid.

[0142] Calculation example

[0143] 1. Calculation of equivalent parameters at the boundary in the random scenario

[0144] (1) Example introduction

[0145] In this example, the IEEE 118-bus standard system is selected and modified as the test system. The original internal and external network division methods and the wind farm connection nodes of the test system are shown in Table 1. The modifications to the standard system include removing the branches 77-80 and 79-80 in the system. The wind speed of the wind farm follows the Weibull distribution with scale parameter and shape parameter of 7.218 and 1.837 respectively. The wind power conversion relationship refers to NASSAR ME, SALAMA MMA. Probabilistic power flow using novel wind and solar probabilistic models[C] / / IEEE General Meeting Power&Energy Society, July 17-21, 2016, Boston, America: 1-5., and the cut-in wind speed, rated wind speed, cut-out wind speed and rated power are 4m / s, 15m / s, 25m / s and 20MW respectively. The loads in the system all follow the normal distribution, and their mean values are the same as the original load values in the standard example, and the standard deviation is 5% of the mean value. The correlation coefficient between the load and the wind speed in the system is 0.0667, and the correlation between the wind speeds of different wind farms is 0.15; the correlation coefficient between the loads belonging to the same region is 0.4, and the correlation coefficient between the loads belonging to different regions is 0.2.

[0146] Table 1 Original internal and external network division methods and wind power connection conditions of the test system

[0147]

[0148] (2) Equivalent parameter calculation

[0149] The calculation of the equivalent parameters at the boundary of the test system is realized by using different equivalent methods to compare their calculation accuracies. The equivalent methods used include the equivalent methods based on multiple operating points and single operating point (i.e., the basic operating point); the methods for calculating the operating points include the AC power flow method, the traditional DC power flow method and the quasi-DC power flow method. Therefore, the following four equivalent parameter calculation methods are used:

[0150] 1) E0: Multiple operating points are calculated based on the AC power flow method;

[0151] 2) E1: Multiple operating points are calculated based on the quasi-DC power flow method;

[0152] 3) E2: Multiple operating points are calculated based on the traditional DC power flow method;

[0153] 4) E3: The basic operating point is calculated based on the AC power flow method.

[0154] Among the four equivalent methods, equivalent method E0 can provide accurate equivalent results. Therefore, E0 is used as the reference method. In this subsection, the unscented transformation method is used to generate the sample point set for calculating equivalent parameters, and the calculation results of equivalent parameters are shown in Table 2 below.

[0155] Table 2 Maximum relative error of equivalent parameters

[0156]

[0157] In Table 2, P Leq , Q Leq , P Geq and U Geq respectively represent the injected active power and injected reactive power of non-generator nodes at the boundary after equivalence, the active power generated by the equivalent generator nodes, and the voltage amplitude of the equivalent generator nodes. e μ represents the relative error of the mean value, and e σ represents the relative error of the standard deviation. It can be seen from Table 2 that the relative errors of the mean values of the equivalent parameters calculated by equivalent methods E1 and E3 are both less than 1%, while the maximum relative error of the mean value of P Geq calculated by E2 is 18.5%. In terms of the calculation of the standard deviation of equivalent parameters, the maximum relative errors of E1, E2, and E3 are 17.64%, 67.83%, and 100% respectively. To sum up: First, obtaining multiple operating points using the quasi-direct current power flow method is more accurate than using the traditional direct current method; Second, compared with the equivalent methods around the basic operating point, the equivalent method based on multiple operating points proposed in this paper has higher calculation accuracy of equivalent parameters when using the quasi-direct current power flow method to obtain multiple operating points; Finally, when calculating equivalent parameters using the equivalent method E1 proposed in this paper, certain errors will be introduced, but the impact of these errors on the calculation results of the internal network probabilistic power flow after equivalence is not significant.

[0158] The correlation coefficients between the equivalent parameters at the boundary and the input random variables of the internal system are shown in Figure 3 (a)–(d) of Leq and are given by the matrix color block diagram; in the figure, P Leq represents the equivalent injected active power at the boundary, Q LI represents the equivalent injected reactive power at the boundary, P Geq represents the injected active power of the internal system nodes, PWF represents the injected active power of the nodes connected to the wind farms, U Geq represents the voltage amplitude of the equivalent generator nodes, and P Figure 3 It can be seen from Figure 3The part circled by the red curve (as shown in [Figure]) differs significantly from the reference method; the correlation coefficient between the equivalent generator parameters calculated by E3 and other random variables is far from that of the reference method (as shown in the part circled by the blue curve in [Figure]). Therefore, the equivalent method based on multiple operating points can more accurately describe the correlation coefficient between the equivalent parameters and the internal system random variables, and calculating multiple operating points using the quasi-direct current power flow method is more accurate than using the traditional direct current power flow method. Figure 3 (as shown in the part circled by the blue curve in [Figure]). Therefore, the equivalent method based on multiple operating points can more accurately describe the correlation coefficient between the equivalent parameters and the internal system random variables, and calculating multiple operating points using the quasi-direct current power flow method is more accurate than using the traditional direct current power flow method.

[0159] To discuss the influence of different numbers of internal and external network nodes on the calculation accuracy of the equivalent method E1 proposed in this paper, five different internal and external network divisions and wind power access conditions are given in this example, as shown in Table 3.

[0160] Table 3 Internal and external network divisions and wind power access conditions of the test system

[0161]

[0162] Table 4 shows the maximum relative errors of the mean value and standard deviation of the equivalent parameters obtained by E1 for the test system under five internal and external division methods.

[0163] Table 4 Numbers of internal and external network nodes of the test system under five division methods and the maximum relative errors of the equivalent parameters calculated by E1

[0164]

[0165] It can be seen from Table 4 that the number of internal and external network nodes has little influence on the accuracy of the mean value of the equivalent parameters and has a greater influence on the accuracy of the standard deviation. However, it is not necessarily the case that the fewer the number of internal network nodes, the higher or lower the accuracy of the standard deviation of the equivalent parameters; and under different division methods, the errors introduced by E1 are all moderate, and have a low impact on the accuracy of the internal network probabilistic power flow calculation.

[0166] Since the selection of the power flow calculation method in the equivalent method will affect the calculation accuracy of multiple operating points, and thus affect the equivalent result. Therefore, to analyze the sensitivity of the influence of different power flow calculation methods on the equivalent result, the basic operating point load (i.e., the load mean value) in the test system is changed to 0.5 - 2 times the original load, and the standard deviation of the load is constantly 5% of the mean value (i.e., the load fluctuation will increase with the increase of the basic operating point load); and the equivalent parameters are calculated by the equivalent method E1 (using the quasi-direct current power flow method) and E2 (using the traditional direct current power flow method). The Figure 4 average relative errors of the mean value and standard deviation of the equivalent parameters are given. From Figure 4It can be seen that the relative error obtained from E1 is always smaller than that from E2; however, the standard deviation relative error obtained from E1 increases with the increase of the basic operating point load. This is because the accuracy of the DC-like power flow algorithm will decrease with the increase of the load fluctuation. Therefore, it can be concluded that: First, when the load fluctuation is small or large, the traditional DC power flow method is not considered for obtaining the operating point; Second, since the moderate error of the external network equivalent has a low impact on the accuracy of the internal network probabilistic power flow calculation, the probabilistic power flow calculation method proposed in this paper is still applicable to the case of heavy load and large fluctuation; However, if the user has a high requirement for the accuracy of the calculation results, a more advanced improved DC power flow method needs to be adopted to improve the calculation accuracy of the equivalent parameters.

[0167] 2. Probabilistic Power Flow Calculation of Interconnected Power Grids

[0168] (1) Evaluation of Calculation Accuracy

[0169] 1) Accuracy Comparison of Different Calculation Methods

[0170] The methods used for the internal network probabilistic power flow calculation of the test system in Table 1 are shown in Table 5, including the benchmark method as a reference, the method proposed in this example, and 8 comparison methods (Comparison Method 1 - Comparison Method 8, hereinafter referred to as Method 1 - Method 8 for short); The calculation results of the internal network output variables are as Figure 5 and Figure 6 shown, Figure 5 where e μU , e μθ , e μP and e μQ respectively represent the relative errors of the internal network node voltage amplitude, phase angle, and the mean values of branch active and reactive powers. When using the Monte Carlo method for probabilistic modeling or probabilistic power flow calculation of equivalent parameters in Table 5, the number of samples is 10,000, and the number of samples when using the Latin hypercube sampling method for calculation is 500. The Nataf transformation is used to handle the correlation between input random variables. Among them, MCS represents the Monte Carlo method, UT represents the unscented transformation method, LHS represents the Latin hypercube sampling method, and \ means that no probabilistic equivalent method is adopted due to not performing equivalence. The equivalent methods in Table 5 use E1 and E3 to compare the influence of the equivalent method based on multiple operating points and the equivalent method based on a single operating point on the accuracy of the internal network probabilistic power flow calculation. The deterministic power flow calculation method in the probabilistic power flow in Table 5 means: when performing probabilistic power flow calculation, the method used for multiple deterministic power flow calculations; when not using equivalence and using equivalence, the calculation objects are the entire network and the internal network after equivalence respectively.

[0171] Table 5 Calculation Methods for Probabilistic Power Flow of Interconnected Power Grids

[0172]

[0173] According to the accuracies of Method 6 and Method 7, evaluate the accuracy of the deterministic power flow calculation in the probabilistic power flow calculation that does not perform equivalence and directly adopts the improved DC power flow method. As Figure 5 shown, the maximum errors of the mean values of the output variables calculated by Method 6 and Method 7 are both about 5%; while Figure 6 in subgraphs (a) and (d) of

[0174] the standard deviation curves of the output variables calculated by the two methods are quite different from the reference method. Therefore, directly adopting the improved DC power flow method for the deterministic power flow calculation in the probabilistic power flow calculation has low calculation accuracy and cannot accurately describe the probability characteristics of the output variables. Figure 5 It can be seen that the maximum relative error of the mean value of the output variables calculated by the proposed method is less than 1%, while the maximum relative error corresponding to Method 4 is 18.24%. At the same time, as Figure 6 shown, the standard deviation curve of the output variables obtained by the proposed method is close to the reference method; while the standard deviation curves of the internal network node voltage phase angle and branch active power of Method 4 are quite different from the reference method. It can be seen from this that: the proposed method in this example has good calculation accuracy; this method adopts Equivalence Method E1, and certain errors will be introduced in the equivalence link, but the accuracy of the probabilistic power flow calculation of the internal network after equivalence is good; while Method 4 adopting Equivalence Method E3 introduces larger errors in the equivalence link, resulting in larger errors in the mean value and standard deviation of the output variables of the internal network after equivalence.

[0175] Since the application research of the existing equivalence methods in the probabilistic power flow calculation does not consider the correlation between the internal and external network input random variables, and this correlation exists in the actual power grid, therefore, compare the calculation results of the proposed method and Method 5 to evaluate the necessity of considering the correlation between the internal and external networks. As Figure 5 shown, the relative errors of the mean values of the output variables calculated by the proposed method and Method 5 are both less than 1%. And it can be seen from Figure 6 that the standard deviation curves of the internal network voltage phase angle and branch reactive power calculated by Method 5 are quite different from the reference method; the standard deviation curves of each output variable calculated by the proposed method are close to the reference curve. It can be obtained from this that: ignoring the correlation between the internal and external networks has little impact on the accuracy of the mean value of the output variables and has a greater impact on the standard deviation accuracy. Therefore, it is necessary to consider the correlation between the internal and external networks.

[0176] To further analyze the high-order statistic errors of the proposed method, Table 6 gives the average values of the absolute errors of the third and fourth order statistics of the output random variables calculated by the proposed method with equivalence and Method 3 without equivalence, and both the proposed method and Method 3 adopt the unscented transformation method. In Table 6, ε U 、ε θ 、εP and ε Q respectively represent the average values of the absolute errors of the corresponding items in the internal network node voltage amplitude, phase angle, and branch active and reactive power meters.

[0177] As can be seen from Table 6, the errors of the third- and fourth-order central moments calculated by the proposed method and Method 3 are similar. Therefore, it can be considered that the equivalent method will not significantly increase the errors of the third- and fourth-order central moments.

[0178] Average values of the absolute errors of the third- and fourth-order central moments of the output variables calculated by the proposed method and Method 3 in Table 6

[0179]

[0180] 2) Precision analysis of the proposed method in different test systems

[0181] To discuss the influence of different internal and external network division methods on the calculation precision, under three internal and external network division methods, the proposed method in this paper is used to calculate the internal network probabilistic power flow of the 118-node test system. The average values of the absolute errors of the internal network output variables are shown in Table 7, where ε μU , ε μθ , ε μP and ε μQ respectively represent the average values of the absolute errors of the internal network node voltage amplitude, phase angle, and the mean values of branch active and reactive power; ε σU , ε σθ , ε σP and ε σQ respectively represent the average values of the absolute errors of the internal network node voltage amplitude, phase angle, and the standard deviations of branch active and reactive power. It can be seen from Table 7 that the errors of the internal network output random variables are similar under different internal and external network division methods and are all very small, that is, the influence of different division methods on the calculation precision of the proposed method in this example is relatively weak.

[0182] Average values of the absolute errors of the output variables calculated by the proposed method in different systems in Table 7

[0183]

[0184] To further verify the applicable range of the proposed method, in the modified IEEE 39-node system and 1354-node system, the benchmark method and the proposed method in this paper are used to calculate the internal network probabilistic power flow. The internal and external network division methods of the IEEE 39-node system are shown in Table 3, and the internal and external network division of the IEEE 1354-node system is as follows:

[0185] 1) Disconnect the branch circuits: branch circuits 7328 - 6921, 7119 - 7873, 7831 - 3346, 2732 - 3346, 4550 - 3346, 1709 - 3346, 2132 - 2303, 1883 - 2132;

[0186] 2) Boundary nodes: nodes 3499 and 4541;

[0187] 3) Connection lines between the external network and the boundary nodes: branch circuits 3499 - 2197, 3499 - 7640, 5007 - 4541;

[0188] 4) Connection lines between the internal network and the boundary nodes: branch circuits 4541 - 3346, 4541 - 7464, 1249 - 4541, 2101 - 4541, 3919 - 4541, 3499 - 7988, 3499 - 4231, 3499 - 5144, 3499 - 3513, 3499 - 7289.

[0189] The average value of the absolute errors of the probabilistic power flow calculation results of the internal network of the 39 - node and 1354 - node systems is shown in Table 7; the absolute errors of the voltages of each node and the branch powers in the internal network of the 1354 - node system are shown in Figure 7 . It can be seen from Table 7 that: the calculation accuracy of the proposed method in the 118 - node system and the 39 - node system is very high. Combining Table 7 and Figure 7 it can be known that: the overall situation of the calculation accuracy of the probabilistic power flow of the internal network of the proposed method in the 1354 - node system is good, and only the accuracies of the voltages of very few nodes and the powers of very few branch circuits are poor; this is because when the number of input random variables is large, a certain sample point set generated by the unscented transformation method is difficult to accurately represent the probability information of all input random variables. Therefore, the proposed method is also applicable to the case of a large - scale power system and the overall accuracy is good, but compared with medium - and small - scale systems, the calculation accuracy in large - scale systems is more significantly affected by the probability modeling accuracy of the unscented transformation method.

[0190] (2) Evaluation of calculation efficiency

[0191] It can be known from the foregoing analysis that: the accuracy of the deterministic power flow calculation in the probabilistic power flow calculation directly using the improved DC power flow method is poor, and it is necessary to consider the correlation between the input random variables of the internal and external networks. Therefore, when evaluating the calculation efficiency in this part, only the calculation times of the benchmark method, the proposed method, and Methods 1 - 4 and Method 8 in the 118 - node system are compared, and the calculation times of each method are obtained by Figure 8As shown in the figure. The annotation "equivalent value" in the figure represents the time required for equivalence (including the time for improving the DC power flow calculation to obtain the operating point during equivalence), "deterministic power flow" represents the time required for multiple deterministic power flow calculations involved in probabilistic power flow analysis using the unscented transformation method, and "others" represents the time required for generating samples, dividing the internal and external networks, and statistically calculating probability results, etc.

[0192] First of all, the benchmark method, Method 3, and Method 8 all directly perform probabilistic power flow calculations for the entire network without equivalence, as Figure 8 shown: they respectively require 783.66 s, 13.46 s, and 44.3 s. It can be seen from this that in the test system, compared with the simulation method, using the unscented transformation method for probabilistic power flow calculation can effectively improve the efficiency of probabilistic power flow calculation. Secondly, the time required for deterministic probabilistic power flow calculations for Method 1 - Method 4 and the proposed method in probabilistic power flow is 1.09 s, 1.13 s, 11.48 s, 1.03 s, and 1.24 s respectively. From the comparison and analysis of Method 3 and the proposed method in Table 8, it can be known that when equivalence is adopted in the test case, reducing the system scale reduces the time for each deterministic power flow calculation from 0.098 s to 0.026 s (i.e., a reduction of 73.4%), and reducing the dimension of input random variables reduces the number of deterministic power flow calculations from 117 times to 47 times (i.e., a reduction of 59.8%). Then, the time required for equivalence for Method 1, Method 2, Method 4, and the proposed method is 31.61 s, 11.34 s, 0.7 s, and 1.41 s respectively. Among them, Method 1 and 2 use the Monte Carlo method to perform probability modeling of internal network input variables after equivalence, and the proposed method and Method 4 use the unscented transformation method, that is, using the unscented transformation method for probability modeling of internal network input variables after equivalence is more efficient.

[0193] Table 8 Deterministic power flow calculation time and number of times of two probabilistic power flow methods in the 118 - node system

[0194]

[0195] From the above analysis: The method proposed in the present invention uses the unscented transformation method twice. The first time, the unscented transformation method is used for probability modeling of internal network input random variables after equivalence. Compared with the simulation method, it reduces the number of operating points to be obtained and the number of equivalence calculations. The second time, the unscented transformation method is used for probabilistic power flow calculation. Compared with the simulation method, it reduces the number of deterministic power flow calculations and significantly improves the calculation efficiency. At the same time, using this method to obtain the probability information of the output random variables of the regional power grid, compared with directly using the unscented transformation method for probabilistic power flow calculation of the entire network, the probability equivalence technology reduces the power grid scale and the dimension of input random variables, thereby reducing both the time for each deterministic power flow calculation and the number of times of performing deterministic power flow calculations in probabilistic power flow, and finally achieving a double reduction in the amount of calculation for this application.

Claims

1. A probabilistic power flow calculation method for interconnected power grids based on static external network equivalent and unscented transformation method, characterized in that, The steps are as follows: Step S1: Using the unscented transformation method, based on the mean of the input random variables of the entire network and covariance P XX , generate a sample point set X with 2n1 + 1 sample points; Step S2: Calculate the structural parameters of the equivalent model from the structural parameters of the entire network to obtain the node admittance matrix Y of the intranet after equivalent eq ; Step S3: Perform an AC power flow calculation on the sample point X0 representing the mean to obtain the basic operating point, and then use the quasi-DC power flow method to obtain the operating points corresponding to other sample points; based on the obtained multiple operating points, perform equivalent calculation on each sample point in X to obtain the sample point set Z of the equivalent parameters; Step S4: Based on Z and the subset X of X I , obtain the sample point set R after equal value, and then calculate the mean value of the random variables input to the intranet after equal value and the covariance P RR ; Step S5: Again, using the unscented transformation method, based on and P RR , generate a new sample point set X eq with 2n2 + 1 sample points; Using Y eq and X eq as inputs, perform 2n2 + 1 deterministic AC power flow calculations to obtain the output sample point set Z eq ; Step S6: Based on Z eq , calculate the mean value of the random variables of the intranet output covariance matrix and standard deviation Thus, the probabilistic power flow calculation of the interconnected power grid is completed.

2. The method according to claim 1, wherein Step S1 includes: Generate the input sample point set X = {X0, X1, …, X N-1}, where N is the number of sample points. According to the symmetric sampling strategy, the 2n + 1 sampling points of the n-dimensional input random variable x are calculated as follows: X0 = m Where: m is the mean of the random variable x; W0 is the weight corresponding to the center point X0; P XX is the covariance matrix of the input random variables; is the k-th column of the matrix, obtained by performing a Cholesky decomposition on P XX ; The weights of each sampling point are calculated as follows: W0 = W0 Where: W k and W k+n are the weights corresponding to X k and X k+n respectively.

3. The method according to claim 2, wherein Step 2 also includes: The interconnected power system can be divided into an internal network including a boundary system and an internal system, and an external network including an external system. The internal nodes, boundary nodes, and external nodes are represented by I, B, and E respectively, and the generator nodes and non-generator nodes are represented by G and L respectively; Divide all the nodes in the network into: G = {G E , G I} and L = {L E , L B , L I}, where G E represents the external generator nodes, G I , L E , L B and L I and so on; After equivalence, the non-generator nodes and generator nodes in the internal network are respectively divided into L = {L I , L eq} and G = {G I , G eq}, where L eq and G eq respectively represent the boundary nodes and equivalent generator nodes after equivalence.

4. The method according to claim 3, characterized in that, Step S3 includes: S31: Calculate the operating point corresponding to the sample point Perform an AC power flow calculation on the sample X0 representing the mean to obtain the basic operating point; For the operating points corresponding to other sample points, the following quasi-DC power flow model is used for calculation: P *(s) = B * θ (s) In the formula: The superscript (s) all represents X s The power flow calculation result of *(s) and B * respectively represent the corrected nodal injection power matrix and the corrected admittance matrix, θ (s) represents the voltage phase angle of all nodes in the network; By calculating the voltage phase angles of all network nodes corresponding to each sample point and combining with the voltage magnitudes at the basic operating points, the operating point U corresponding to each sample point can be obtained. (s) ; S32: Perform equivalent calculation on the sample points Take the injected active power P of the generator node L as the input random variable. According to the probability characteristics of the input random variables of the entire network, use the unscented transformation method to generate the sample point set X; According to the division of the internal and external networks, the sample point set X = [X ET X BT X IT T , for the (s + 1)-th sample point, where and correspond to the active power injections of the external, boundary, and internal non-generator nodes respectively and X s The corresponding operating point is the voltage phasor U of all network nodes (s) , and U (s) is obtained by deterministic power flow calculation;​ For X s , the equivalent load at the boundary is calculated as follows: In the formula: The superscript (s) all represents X s corresponding parameter; and are respectively the active and reactive powers of the equivalent load at the boundary; and are calculated from U (s) ; and are respectively the reactive powers injected at the boundary and the non - generator nodes outside; Output of the equivalent generator at the boundary and its voltage phasor are calculated as follows: Where: F1 (s) and are calculated from U (s) ; P GE is the output of the external network generator; is the voltage phasor of the external network generator node; Y trans is calculated from the structural parameters.

5. The method according to claim 4, characterized in that, Step S4 includes: S41: Calculate the sample point set R after equivalence By performing equivalent value calculations on each sample point in X through step S32, a sample point set Z of equivalent value parameters is obtained. Combining Z with the subset X of the internal input random variables in X I and establishing the probability model of the internal network input random variables after equivalent value by the unscented transformation method as follows: R = [Z T X IT T ,​ Where: R is the sample point set obtained by combining Z and X I ; S42: Calculate the mean and covariance of the input random variables of the internal network after equivalence The mean and covariance of the input random variables of the internal network are calculated by the following formula: In the formula: is the mean value corresponding to the internal network input random variable after equalization; n1 is the dimension of the whole network input random variable; R s is the (s + 1)-th sample point of R; P RR is the covariance matrix of the internal network input random variable after equalization.

6. The method according to claim 5, characterized in that, Step S5 includes: Again, according to the principle of unscented transformation, generate the sample point set \(X\) after equivalent transformation in the same way as in Step 1 eq , where the number of sample points in the sample point set is \(2n_2 + 1\), \(n_2\) is the dimension of the internal network input random variable after equivalent transformation, and \(n_2 < n_1\); Take Y eq and X eq as inputs, and perform 2n2 + 1 deterministic AC power flow calculations through the following formula to obtain the output sample point set Z eq : In the formula: is the output sample point of the corresponding AC power flow model; is the (p + 1)-th sample point of X eq ; Y eq is the nodal admittance matrix of the internal network after equivalence; h(·) is the nonlinear relationship between the output random variable and the input random variable in the AC power flow model.

7. The method according to claim 6, wherein Step S6 includes: Calculate the mean of the output random variable of the internal network Covariance matrix and standard deviation as follows: Wherein: is the corresponding weight; and are respectively the mean, covariance matrix and standard deviation of the output random variable; denotes taking the diagonal elements of which form a column vector.

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