Random load flow calculation method based on improved adaptive sparse pseudo-spectral approximation
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-24
- Publication Date
- 2026-03-27
Smart Images
Figure CN121744682A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of power grid stochastic power flow calculation technology, and in particular to a stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation. Background Technology
[0002] The increasing number and proportion of new energy power plants connected to the grid are making the power grid development greener and more sustainable. However, the volatility and intermittency of their output are also highlighting the uncertainty of the grid's operating state. The uncertainty of the system state can be well described and measured using stochastic power flow (SPF). Since the complexity and computational cost of stochastic power flow calculation are much greater than those of deterministic power flow calculation, how to quickly and accurately perform stochastic power flow calculations for large-scale real-world systems has become a highly anticipated research topic.
[0003] Based on the different mechanisms employed, stochastic power flow methods to date can be broadly classified into: simulation methods, semi-invariant methods, point estimation methods, and generalized polynomial Chaos Expansion (PCE).
[0004] 1) Simulation methods. These include Monte Carlo Sampling (MCS), Latin Hypercube Sampling (LHS), and various improved methods. The advantage of these methods is their high computational accuracy, and their results are often used as a benchmark for other methods. However, the number of sampling points required is proportional to the inverse square of the accuracy; therefore, when the computational accuracy is high and the system size is large, the computation time is often very long.
[0005] 2) Semi-invariant method (CM). This type of method is relatively fast, but when the variance of the input variables is large, its linearization approximation of the original function will introduce large errors, resulting in poor accuracy of the calculation results.
[0006] 3) Point Estimation Method (PEM). This type of method requires only a small number of sampling points to complete the calculation, which has the advantage of fast calculation speed. However, since its calculation is based on low-value moment information, its overall calculation accuracy is relatively low.
[0007] 4) PCE class. This class of methods includes the stochastic Galerkin method, numerical integration method, and regression fitting method.
[0008] a) Stochastic Galerkin method. This method can obtain a highly accurate approximation function when the approximation function is chosen appropriately and the system size and the dimension of the random variables are both small. However, when the system size is large and the dimension of the input random variables is high, the process of deriving the nonlinear equation system with the coefficients to be determined becomes very complex, and there is a risk that the derivation will fail or the solution will fail.
[0009] b) Numerical integration method, also known as pseudospectral method. This type of method has strong theoretical support and high computational accuracy, but as the dimension of the input uncertainty variable increases, the number of basis function terms increases dramatically, and the computational cost also increases rapidly, resulting in the "curse of dimensionality" problem.
[0010] c) Regression fitting methods. This type of method has the advantage of high computational speed for large-scale practical systems and avoids the "curse of dimensionality" problem; however, there is no general strategy for constructing an accurate approximation function structure so far, and if the structure is not constructed properly, its computational accuracy cannot meet the requirements.
[0011] Existing stochastic power flow calculations suffer from the following problems: it is difficult to simultaneously meet the requirements of computational efficiency and accuracy for various stochastic power flow calculation methods. Among them, the semi-invariant method and the point estimation method have relatively fast computation speeds, but their computational accuracy is relatively low; although the pseudospectral method has high computational accuracy, its computation time is still very long for large-scale systems.
[0012] Therefore, a new calculation method is urgently needed to improve calculation accuracy. Summary of the Invention
[0013] The purpose of this invention is to address the problems existing in the background technology by proposing a stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation. A novel NA-SPAM method based on index set truncation is proposed and applied to stochastic power flow calculation. This new method first proposes a novel extended strategy for NA-SPAM based on index set truncation, sacrificing computational accuracy only slightly for a significant improvement in computational speed; then, it is applied to stochastic power flow calculation, achieving a substantial increase in computational speed for a given level of accuracy.
[0014] The technical solution of the present invention, in its first aspect, provides a stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation, comprising the following specific steps. S1. Improve the NA-SPAM method by setting index set extension constraints to limit the extension of high-dimensional and high-order coupling terms and reduce the computation of low-contribution basis functions; S2. Establish a stochastic power flow model that establishes the functional relationship between input random variables and output variables; S3. Calculate the probability distribution of each random variable model; S4. Verify the computational efficiency and accuracy using the results of the numerical examples.
[0015] Preferably, the constraints for expanding the index set in step S1 are as follows: In the formula, for dimensional index vector, The first indicator i One component; , The given truncation parameter.
[0016] Preferably, in step S2, the input random variables include load, wind power output, and photovoltaic power output; the output random variables include node voltage and line power.
[0017] Preferably, the stochastic power flow model established in step S2 is expressed as: In the formula, The input is a vector of random variables; To input uncertain variables, ,in , These represent the active power injection vectors of the wind farm and the photovoltaic farm in the system, respectively. This represents the uncertain vector of active power load demand. ,in , representing the reactive power injection and reactive power load demand vectors of the wind farm, respectively; These represent the active power output vector, node voltage vector, and active and reactive power demand vectors of the deterministic load at the PV nodes of the traditional units in the system, respectively. To output a vector of random variables, ,in , Separate the phase angle and magnitude vector of the system node voltages; It is an implicit function based on the relationship between input and output variables in the power flow equation.
[0018] Preferably, the probability distribution of the load in step S2 is calculated using the following formula: In the formula, the active component of the load at each node is... The reactive component follows a normal distribution with a mean of 5% of the ground state value and a standard deviation of 5% of the ground state value. Based on its power factor in the ground state Calculate the active component.
[0019] Preferably, the probability distribution of wind power output in step S2 is calculated using the following formula: In the formula, wind speed Represented by the Weibull distribution. and These are shape and scale parameters, extracted from historical wind speed data using the maximum likelihood method. Wind power active power output Generally, it has a piecewise functional relationship with wind speed: in, Is it the cutting-in wind speed, It is to cut off the wind speed, That is the rated wind speed. This refers to the rated power; the reactive power output of wind power. Maintaining a constant power factor .
[0020] Preferably, the probability distribution of photovoltaic power output in step S2 is calculated using the following formula: Solar radiation intensity Represented by a general Beta distribution. and It is the shape parameter of the distribution, obtained from historical data using the maximum likelihood method; The maximum solar radiation intensity; the photovoltaic active power output and solar radiation intensity have a piecewise functional relationship: in, This is the solar radiation intensity value under standard conditions, typically 1000 ppm. , This is the rated power of photovoltaic power generation; reactive power output from photovoltaics is generally not considered.
[0021] Preferably, in step S3, the input random variable is converted into an independent random variable using the Nataf transform and sampled to generate sample point-function value pairs for iterative calculation of adaptive sparse pseudospectral approximation; In step S3, in each iteration, the index with the largest local error estimate is selected from the external index set and added to the internal index set. The external index set is then dynamically expanded according to the constraints until the convergence condition is met. In step S3, after the calculation is completed, the approximation function is expressed as a linear combination of basis functions, and the expected value, variance, probability density function and probability distribution function of the output random variable are calculated based on this combination.
[0022] Preferably, in step S4, the IEEE standard test system is used for simulation verification, and the proposed method is compared with Latin hypercube sampling, point estimation method and standard adaptive sparse pseudospectral approximation method, with computation time and computation accuracy as evaluation indicators; the improved NA-SPAM method is used for stochastic power flow calculation.
[0023] A second aspect of the present invention provides a stochastic power flow calculation system, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the stochastic power flow calculation method described above.
[0024] Compared with the prior art, the present invention has the following beneficial technical effects: The proposed stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation improves the NA-SPAM method by introducing index set extension constraints, effectively limiting the expansion of high-dimensional and high-order coupling terms, thereby reducing the computational cost of low-contribution basis functions. This improvement not only increases computational speed but also significantly enhances the efficiency of stochastic power flow calculation while maintaining computational accuracy. Compared with traditional Latin hypercube sampling, point estimation methods, and standard adaptive sparse pseudospectral approximation methods, the proposed method exhibits superior performance in both computation time and accuracy. Especially when dealing with large-scale practical systems, the proposed method can provide accurate calculation results more quickly, providing more efficient and reliable technical support for stochastic power flow calculation in power grids. Attached Figure Description
[0025] Figure 1 This is a flowchart of the stochastic power flow calculation method in an embodiment of the present invention; Figure 2 Different methods are used to obtain the results in the embodiments of the present invention. The probability density curve; Figure 3 Different methods obtained in the embodiments of the present invention Probability density curve. Detailed Implementation
[0026] Example 1 This invention proposes a stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation, comprising the following steps: (1) An improved NA-SPAM method is proposed, which extends the boundary by using a constraint-based index set to limit the extension of high-dimensional and high-order coupling terms, reduce the computation of low-contribution basis functions, and thus improve efficiency; (2) Establish a stochastic power flow model that establishes the functional relationship between input random variables (such as load, wind power / photovoltaic output) and output variables (such as node voltage, line power).
[0027] (3) Calculate the probability distribution of each random variable model; (4) Verify the computational efficiency and accuracy of the algorithm by analyzing the results of the numerical examples.
[0028] Furthermore, step (1) proposes an improved NA-SPAM algorithm based on index set truncation constraints. This algorithm implements the expansion restriction of higher-order cross terms by adding index set expansion constraints, and correspondingly increases the expansion of lower-order terms with large contributions. Under the premise of ensuring that the approximation function has a given (or close to a given) computational accuracy, the computational efficiency is greatly improved.
[0029] (1.1) Constraints are proposed for the expansion of the indicator set: (1) In the formula, for dimensional index vector, The first indicator i One component; , The truncation parameter is given. This constraint improves computational efficiency by suppressing the expansion of higher-order cross terms and correspondingly increasing the contribution of lower-order terms.
[0030] (1.2) Steps of the improved NA-SPAM algorithm based on the index truncation constraint of equation (1): 1) Set of indicators Divide into mutually exclusive sets of external indicators and internal indicator set , , , External backup indicator set Given truncation parameters and .
[0031] 2) From Selecting the one with the largest local error estimation Indicators That is, select the index that maximizes the reduction of the global error of the approximation function according to equation (2): (2) In the formula: The difference operator is expressed as follows: (3) In the formula: ; This represents the degree vector of d-dimensional basis functions. express The degree of the basis functions; For d-dimensional basis functions, To and The corresponding number of times is The basis functions, whose form is based on Selection of distribution type; Indicates the number of times Inner product of basis functions: (4) In the formula: express The joint probability density function of all components in. express The probability density function; express and The inner product of , which is expressed based on the Gaussian integral, is: (5) In the formula: for The samples, whose values are basis functions The One zero point; For the corresponding sample points The weight.
[0032] 3) Indicators from move in In, that is, to execute and ; 4) External indicator set To expand: 4a) Let , ; 4b) Calculation indicators Previous neighbor indicators ,in For the first dimension 1 1-dimensional unit vector; 4c) Judgment Indicators Does the given constraint (Equation (1)) satisfy? If it does, proceed to step 4d; otherwise, Proceed to step 4b). 4d) Calculation index The next neighbor index ,in For the first dimension 1 A 1D unit vector, if ,make ;like for If all are true, then... Add to indicator set In, that is , ; 4e) If , End the expansion and proceed to step 5); otherwise Proceed to step 4b). 5) If ,Will Add to external backup indicator set ,Right now ; 6) Calculate the global error according to equation (6). ,like or The calculation then ends, where (Given a threshold value; otherwise, proceed to step 2).
[0033] (6) In the formula: This represents the correction coefficient, based on experience. .
[0034] Furthermore, step (2) established a stochastic power flow model based on various random variables such as load and wind power output; (2.1) The stochastic power flow model can be expressed in the following form: (7) In the formula: The input is a vector of random variables; To input uncertain variables, ,in , These represent the active power injection vectors of the wind farm and the photovoltaic farm in the system, respectively. This represents the uncertain vector of active power load demand. ,in , representing the reactive power injection and reactive power load demand vectors of the wind farm, respectively; These represent the active power output vector, node voltage vector, and active and reactive power demand vectors of the deterministic load at the PV nodes of the traditional units in the system, respectively. To output a vector of random variables, ,in , Separate the phase angle and magnitude vector of the system node voltages; It is an implicit function based on the relationship between input and output variables in the power flow equation.
[0035] (2.2) The probability distribution of each input random variable is generally calculated as follows: 1) Probability distribution of load: Active components of load at each node It typically follows a normal distribution with a mean of the ground state value and a standard deviation of 5% of the ground state value. Assuming its reactive component... Based on its power factor in the ground state Calculate with active component: (8) 2) Probability distribution of wind power output: wind speed This can be represented by the Weibull distribution, whose probability density function is: (9) in, and These are shape and scale parameters, which can be extracted from historical wind speed data using methods such as maximum likelihood estimation.
[0036] Wind power has active power output. Generally, it has a piecewise functional relationship with wind speed: (10) in, Is it the cutting-in wind speed, It is to cut off the wind speed, That is the rated wind speed. This is the rated power. Assume the reactive power output of the wind power. Maintaining a constant power factor .
[0037] 3) Probability distribution of photovoltaic power output: Solar radiation intensity This is usually represented by the Beta distribution: (11) in, and It is the shape parameter of the distribution, which can be obtained from historical data using the maximum likelihood method; This represents the maximum solar radiation intensity. Generally, the photovoltaic active power output has a piecewise functional relationship with the solar radiation intensity: (12) in, This is the solar radiation intensity value under standard conditions, typically 1000 ppm. , This is the rated power of photovoltaic power generation; reactive power output from photovoltaics is generally not considered.
[0038] Furthermore, step (3) is based on the stochastic power flow calculation of the improved NA-SPAM, using... truncation or The truncation method accelerates computational efficiency. The complete steps for stochastic power flow calculation based on this method are given below: 1) For equation (7) (hereinafter referred to as the original model), firstly, the relevant input random variable vectors are... The random variables are transformed into independent vectors using Nataf transform, denoted as [vectors of random variables]. Let the transformed and The functional relationship is The stochastic power flow model becomes: (13) For a vector of random variables Sampling was performed to obtain M indivual d dimensional sampling points ( The corresponding actual input sampling points should be obtained through... calculate ( ), and then Only by substituting these values into deterministic power flow calculations can the corresponding output random variables be obtained. ( That is, to obtain the sample point-function value pairs required for NA-SPAM. ( ).
[0039] 2) Define two disjoint sets of indices. and and external backup indicator set ,initialization (d elements are all 1) , For the empty set, and the index set Given truncation parameters , Set the convergence threshold TOL.
[0040] 3) Press The approximation function corresponding to the initial index is calculated sequentially. : (14) 4) Calculate the global error estimate : (15) in, (16) 5) If and If yes, proceed to step 6; otherwise, skip to step 10).
[0041] 6) In Select the one with the largest local error estimate Indicators Local error The calculation is as follows: (17) 7) Indicators from Centralized In the middle, update the approximation function .
[0042] 8) Expand the indicator set : 8a) Order , ; 8b) Find the indicators exist i Previous neighbor index in direction ,in, For the first i Dimension 1, all others 0 d 1D unit vector.
[0043] 8c) Judgment Indicators Does it satisfy the constraint as in equation (1)? If it does, proceed to step 8d). If it does not, (Skip to step 8b) 8d) Definition exist r The next neighbor index in the direction is , and when season ,like for If all are true, then... Add to indicator set middle, ; 8e) If End the expansion; otherwise (Skip to step 8b). 9) If Then the indicator Also added to the external indicator backup indicator set In, that is ; Jump back to 4).
[0044] 10) Combine the basis function terms to transform it into a linear combination of basis function terms: (18) Then calculate the expected value and variance of the output variable according to the following formula: (19) (20) 11) Use replace Monte Carlo sampling is performed, and then the probability density and probability distribution function of each output are obtained through statistics.
[0045] Furthermore, in step (4), in order to verify the effectiveness of the improved NA-SPAM method based on the index set truncation strategy proposed in this paper, the effectiveness of the proposed power system stochastic power flow calculation method is tested using the IEEE-39 node system.
[0046] (4.1) Stochastic power flow simulation system The effectiveness of the proposed stochastic power flow method was tested using an IEEE-118 node system. Generator, load, and network parameters can be found at https: / / git-hub.com / tjykm / Data-for-stochastic-PFC / Case118.xlsx. Wind farms with a rated capacity of 30MW were connected to nodes 1, 2, 3, and 4, and photovoltaic power plants with a rated capacity of 30MW were connected to nodes 5, 6, 7, 8, and 9. All loads followed a normal distribution; wind speeds followed a Weibull distribution. The cut-in wind speed, rated wind speed, and cut-out wind speed are 2, 12, and 25 m / s, respectively; the solar radiation intensity follows a Beta distribution, and the parameters are... The peak value of solar radiation intensity is , , The system's stochastic power flow problem has 108 input random variables and 186 output random variables.
[0047] To verify the effectiveness and superiority of the proposed method, the system was computed using LHS, PEM, NA-SPAM, and a modified NA-SPAM. The computational parameters for each method were set as follows: 1) NA-SPAM convergence threshold was 0.001, and the maximum number of samples was 10,000; 2) PEM used 3n-point estimation, with a total of 53 sampling points; 3) LHS used 20,000 sampling points; 4) the modified NA-SPAM cutoff parameter was set to... , .
[0048] (4.2) Comparison and verification of calculation results The computational accuracy and computation time are shown in Table 1. It can be seen that the computational accuracy of NA-SPAM and the improved NA-SPAM is much higher than that of the PEM method. Although the improved NA-SPAM loses some computational accuracy compared to NA-SPAM, from 0.0053 to 0.0063, this accuracy is actually fully satisfactory. However, its computation time is reduced by nearly 1000 seconds compared to NA-SPAM, and the computational efficiency is greatly improved.
[0049] Figure 2 The node voltages obtained by different methods are given. probability density curve, Figure 3 The branch power obtained by different methods is given. Probability density curves. It can be seen that the probability density curves obtained by NA-SPAM and the improved NA-SPAM are significantly closer to the probability density curve obtained by LHS than those obtained by PEM.
[0050] The above calculation results demonstrate that the proposed improved NA-SPAM significantly reduces the number of basis function terms required for computation by expanding the boundary through the constraint index set, thereby greatly reducing computation time while maintaining high computational accuracy. The improved NA-SPAM algorithm achieves a significant improvement in computational speed with only a small sacrifice in computational accuracy. Furthermore, its application to stochastic power flow calculation demonstrates a substantial increase in computational speed for a given level of accuracy.
[0051] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited thereto. Various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention.
Claims
1. A stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation, characterized in that, The specific steps include the following: S1. Improve the NA-SPAM method by setting index set extension constraints to limit the extension of high-dimensional and high-order coupling terms and reduce the computation of low-contribution basis functions; S2. Establish a stochastic power flow model that establishes the functional relationship between input random variables and output variables; S3. Calculate the probability distribution of each random variable model; S4. Verify the computational efficiency and accuracy using the results of the numerical examples.
2. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 1, characterized in that, The constraints for expanding the index set in step S1 are as follows: In the formula, for dimensional index vector, The first indicator i One component; , The given truncation parameter.
3. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 1, characterized in that, In step S2, the input random variables include load, wind power output, and photovoltaic power output; the output random variables include node voltage and line power.
4. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 1, characterized in that, The stochastic power flow model established in step S2 is expressed as follows: In the formula, The input is a vector of random variables; To input uncertain variables, ,in , These represent the active power injection vectors of the wind farm and the photovoltaic farm in the system, respectively. This represents the uncertain vector of active power load demand. ,in , representing the reactive power injection and reactive power load demand vectors of the wind farm, respectively; These represent the active power output vector, node voltage vector, and active and reactive power demand vectors of the deterministic load at the PV nodes of the traditional units in the system, respectively. To output a vector of random variables, ,in , Separate the phase angle and magnitude vector of the system node voltages; It is an implicit function based on the relationship between input and output variables in the power flow equation.
5. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 4, characterized in that, The probability distribution of the load in step S2 is calculated using the following formula: In the formula, the active component of the load at each node is... The reactive power component follows a normal distribution with a mean of 5% of the ground state value and a standard deviation of 5% of the ground state value. Based on its power factor in the ground state Calculate the active component.
6. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 4, characterized in that, The probability distribution of wind power output in step S2 is calculated using the following formula: In the formula, wind speed Represented by the Weibull distribution. and These are shape and scale parameters, extracted from historical wind speed data using the maximum likelihood method. Wind power active power output Generally, it has a piecewise functional relationship with wind speed: in, Is it the cutting-in wind speed, It is to cut off the wind speed, That is the rated wind speed. This refers to the rated power; the reactive power output of wind power. Maintaining a constant power factor .
7. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 4, characterized in that, The probability distribution of photovoltaic power output in step S2 is calculated using the following formula: Solar radiation intensity Represented by a general Beta distribution. and It is the shape parameter of the distribution, obtained from historical data using the maximum likelihood method; This represents the maximum solar radiation intensity. The relationship between photovoltaic active power output and solar radiation intensity is a piecewise function: in, This is the solar radiation intensity value under standard conditions, typically 1000 ppm. , This is the rated power of photovoltaic power generation; reactive power output from photovoltaics is generally not considered.
8. A stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to any one of claims 4-7, characterized in that, In step S3, the Nataf transform is used to convert the input random variable into an independent random variable, and sampling is performed to generate sampling point-function value pairs for iterative calculation of adaptive sparse pseudospectral approximation. In step S3, in each iteration, the index with the largest local error estimate is selected from the external index set and added to the internal index set. The external index set is then dynamically expanded according to the constraints until the convergence condition is met. In step S3, after the calculation is completed, the approximation function is expressed as a linear combination of basis functions, and the expected value, variance, probability density function and probability distribution function of the output random variable are calculated based on this combination.
9. The stochastic power flow calculation method based on improved adaptive sparse pseudospectral approximation according to claim 1, characterized in that, In step S4, the IEEE standard test system is used for simulation verification. The proposed method is compared with Latin hypercube sampling, point estimation method and standard adaptive sparse pseudospectral approximation method, with computation time and computational accuracy as evaluation indicators. The improved NA-SPAM method is used for stochastic power flow calculation.
10. A stochastic power flow calculation system, characterized in that, It includes a memory and a processor, the memory storing a computer program, and the processor executing the computer program to implement the steps of the stochastic power flow calculation method as described in any one of claims 1 to 9.