An Optimal Fast Evaluation Method for Static Voltage Stability

Abstract

The present invention relates to the technology of static voltage stability assessment in power systems, and particularly to a fast assessment method for optimal static voltage stability. Based on the established static voltage stability boundary, this method proposes to use the norm from the current operating point to the nearest static voltage stability boundary point as the assessment index. Taking the minimum norm from the current operating point to the stability boundary point as the objective function and the power growth mode as the decision variable, a fast assessment model for static voltage stability is established. This model is solved by brute-force search or intelligent optimization to achieve the online fast assessment of static voltage stability. This method considers all possible power growth directions of voltage instability, and the online fast assessment time is independent of the system scale, which can be applied to the fast assessment of static voltage stability in large-scale power systems.

Description

Technical Field

[0001] The present invention belongs to the technical field of static voltage stability assessment of power systems, and particularly relates to a fast assessment method for optimal static voltage stability. Background Art

[0002] In various static voltage stability analysis methods, corresponding static voltage stability assessment indexes need to be established. [1-2] They are mainly divided into two types: state indexes and margin indexes. State indexes indirectly evaluate the static voltage stability of the system based on certain characteristics of the current system. For example, the sensitivity index uses the derivatives of active power and reactive power with respect to voltage as the evaluation index. This method is simple and convenient, but it is usually difficult to consider the complex constraints of the system. Margin indexes are based on the current operating state. Assuming that the power of the system changes in a certain way (usually the load increases), the system power flow is continuously calculated until voltage collapse occurs, and the distance from the current operating state to the voltage collapse state is used as the evaluation index. Compared with the evaluation of state indexes, the physical meaning of margin indexes is more direct, and various constraints can be conveniently considered, so they have been widely used.

[0003] In recent years, with the increasingly prominent problems such as shortage of fossil energy, climate change, and environmental pollution, the power system is developing towards the direction of green and low-carbon. A large number of new energy sources such as wind power and photovoltaic power are connected to the power generation side, and the randomness of system power changes is significantly enhanced, which has a huge impact on the safe and stable operation of the power system, among which the voltage stability problem is very prominent. The load margin calculated by the continuation power flow method often has little error in a system with weak randomness, but the error of this method will increase significantly in a system with strong randomness. At present, some literatures have proposed to consider the uncertainty of new energy output by using methods of random probability and interval analysis, and certain research has been done.

[0004] For the stochastic probability method, Reference [3] proposed a static voltage stability probability assessment method considering the stochastic output of distributed power in the distribution network. The continuous power flow method was used to calculate the determined voltage stability boundary value. Based on the calculation results of sample points, the Cornish-Fisher series was introduced to establish a probability distribution model of the static voltage stability limit, so as to obtain the statistical characteristics of the voltage stability limit and the probability distribution function of the load margin. Reference [4] calculated the semi-invariants of the L index according to the probability density function of wind power output, calculated the probability density function of the L index based on the Gram-Charlier series expansion, combined with the severity of static voltage instability to calculate the static voltage instability risk of each node, evaluated the static voltage stability of the system, and determined the weak nodes, which can be used for the static voltage stability probability assessment when wind power is integrated into the grid. Reference [5] obtained the samples of input random variables by introducing quasi-Monte Carlo simulation, improved the calculation efficiency of the simulation method, and accurately obtained the probability distribution function of the static voltage stability margin through the kernel density method based on the diffusion equation, and proposed a static voltage stability assessment method applicable to uncertain output. Probability assessment can solve the assessment problem caused by uncertain output to a certain extent, but the efficiency of obtaining the probability density function is relatively low, and it is difficult to ensure the accuracy of the assessment.

[0005] For the interval estimation assessment method, Reference [6] proposed a new interval-form static voltage stability assessment model for the situation of uncertain grid load injection level and growth. Reference [7] used interval numbers to describe the stochastic fluctuation characteristics of wind farm output, segmented the fluctuation interval of the wind farm, and calculated the stability margin of each segment respectively. Compared with probability assessment, the calculation efficiency was improved. Reference [8] proposed a polynomial time algorithm for calculating the fluctuation interval of the static voltage stability limit for data uncertainty, which was very fast in small-scale systems. Reference [9] established a two-layer optimization model for the upper and lower bounds of interval assessment for the AC-DC hybrid power grid, where the uncertain fluctuations of the output of new energy power stations were described by interval numbers. The calculated upper and lower bounds were more accurate and efficient, and had certain engineering value. The essence of interval analysis is an optimization problem with the maximum and minimum stability margins as the objective functions. This optimization problem needs to consider various complex power system constraint conditions, and the online fast calculation is limited to a certain extent.

[0006] The method of static voltage stability region can give the power limit information under all growth modes and become an effective means to solve the randomness of power change.

[10] , based on the already constructed static voltage stability boundary, the present invention proposes a fast static voltage stability assessment method considering the randomness of system power change. Summary of the Invention

[0007] Aiming at the problems in the background technology, the present invention provides an optimal fast static voltage stability assessment method.

[0008] To solve the above technical problems, the present invention adopts the following technical solution: An optimal static voltage stability rapid evaluation method, comprising the following steps:

[0009] Step 1, construct a static voltage stability boundary;

[0010] Step 2, determine a static voltage stability margin index;

[0011] Step 3, establish a static voltage stability rapid evaluation model;

[0012] Step 4, solve the static voltage stability rapid evaluation model.

[0013] In the above optimal static voltage stability rapid evaluation method, the implementation of Step 1 includes:

[0014] Step 1.1, construct a static voltage stability boundary mathematical model based on the continuation power flow method:

[0015] u i = u b + Δu i = u b + b i λ imax (9)

[0016] In the formula, u i is an arbitrary point on the static voltage stability boundary, u b is the initial operating base state; b i is the power change mode from the initial operating base state u b to the static voltage stability boundary point u i , λ imax is the maximum power increment under this power change mode; u i and u b are both high-dimensional column vectors, and the components of the column vectors are active and reactive powers in the injection power space;

[0017] Step 1.2, considering that when the initial state u b is fixed, there is a one-to-one correspondence between the power growth mode and the maximum power growth amount, construct a general mathematical model of the static voltage stability boundary point:

[0018] u i = u b + b i F(b i ) (10)

[0019] In the formula, u i is an arbitrary point on the static voltage stability boundary, u b is the initial operating base state; b i is the initial operating base state u bTo the static voltage stability boundary point u i The power change mode, u i and u b are both high-dimensional column vectors, and the components of the column vectors are the active and reactive powers in the injection power space; F represents the mapping relationship between b i and λ imax :

[0020] λ max = F(b) (3).

[0021] In the above optimal static voltage stability rapid evaluation method, the implementation of step 2 includes:

[0022] Step 2.1, Define the static voltage stability margin index L as:

[0023] L = ||u h - u0||2, u h ∈ SVSRB (11)

[0024] where u h represents the static voltage stability boundary point closest to the current operating state u0, and the current operating state u0 and the boundary point u h both represent column vectors in the injection power space;

[0025] Step 2.2, Define the power growth mode b h that is most likely to cause static voltage instability as:

[0026]

[0027] where u h represents the static voltage stability boundary point closest to the current operating state u0, and the current operating state u0 and the boundary point u h both represent column vectors in the injection power space.

[0028] In the above optimal static voltage stability rapid evaluation method, the implementation of step 3 includes:

[0029] Step 3.1, Establish a static voltage stability rapid evaluation optimization model:

[0030]

[0031] In the formula, L is the static voltage stability margin index, and the objective function f is the norm from the current operating point u0 to the stable boundary point u i :

[0032]

[0033] Step 3.2. Substitute into the static voltage stability boundary expression, and the static voltage fast evaluation optimization model can be obtained as follows:

[0034]

[0035] where u b is the initial operating base state; b is the power change mode, F represents the mapping relationship between b and λ max , and L is the static voltage stability margin index.

[0036] In the above optimal static voltage stability fast evaluation method, the implementation of step 4 includes:

[0037] Step 4.1. Search for the power growth direction b p that makes the optimization model reach an extreme value, and calculate the stable boundary point u h nearest to the operating state u0;

[0038] Step 4.2. Calculate the static voltage stability margin index L under the current operating state u0; if the power growth direction b p has been successfully searched and the corresponding nearest stable boundary point u h has been calculated, then directly calculate the stability margin index L;

[0039] Step 4.3. Solve the power growth mode b h that is most likely to cause static voltage instability; after obtaining the nearest boundary point u h , calculate the power growth mode that is most likely to cause static voltage instability:

[0040]

[0041] where u h represents the static voltage stability boundary point nearest to the current operating state u0, b p represents the power growth direction that makes the optimization model reach an extreme value, and F represents the mapping relationship between b and λ max .

[0042] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The present invention considers all possible power growth directions that may cause voltage instability; 2. The present invention can effectively realize the online fast evaluation of static voltage stability; 3. The online fast evaluation time of the present invention is independent of the system scale and can be applied to the fast evaluation of static voltage stability of large-scale power systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 FIG. is a schematic diagram of detecting the static voltage stability boundary by the continuous power flow method according to an embodiment of the present invention;

[0044] Figure 2Schematic diagram of the static voltage stability margin index for an embodiment of the present invention;

[0045] Figure 3 Schematic diagram of the rapid assessment of static voltage stability for an embodiment of the present invention;

[0046] Figure 4 IEEE 9 - bus test system for an embodiment of the present invention;

[0047] Figure 5 IEEE 39 - bus test system for an embodiment of the present invention;

[0048] Figure 6 Search result of the baffle method for an embodiment of the present invention. Detailed implementation manners

[0049] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0050] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments may be combined with each other.

[0051] The present invention will be further described below in conjunction with specific embodiments, but it is not limited to the present invention.

[0052] In this embodiment, an optimal static voltage stability rapid assessment method is proposed. Based on the constructed static voltage stability boundary, a static voltage stability margin index considering the randomness of power change is proposed. A rapid assessment model of static voltage stability is established based on the index, and the problem is solved by brute - force search or intelligent optimization.

[0053] This embodiment is implemented through the following technical solutions. An optimal static voltage stability rapid assessment method specifically includes the following steps:

[0054] S1. Construct a static voltage stability boundary;

[0055] S1.1. Based on the idea of detecting the boundary by the continuous power flow method, as Figure 1 shown, the mathematical model of the static voltage stability boundary point can be expressed as:

[0056] u i = u b + Δu i = u b + b i λ imax (1)

[0057] where \(u\) i is an arbitrary point on the static voltage stability boundary, and \(u\) b is the initial operating base state; \(b\) i is the power change mode from the initial operating base state \(u\) b to the static voltage stability boundary point \(u\) i , and \(\lambda\) imax is the maximum power increment under this power change mode. Both \(u\) i and \(u\) b are high-dimensional column vectors, and the components of the column vectors are the active and reactive powers in the injection power space.

[0058] S1.2. Considering that when the initial state \(u\) b is fixed, there is a one-to-one correspondence between the power growth mode and the maximum power growth amount. Therefore, a general mathematical model for the static voltage stability boundary point is constructed as:

[0059] \(u\) i = \(u\) b + \(b\) i F(b i ) (17)

[0060] where \(u\) i is an arbitrary point on the static voltage stability boundary, \(u\) b is the initial operating base state; \(b\) i is the power change mode from the initial operating base state \(u\) b to the static voltage stability boundary point \(u\) i , both \(u\) i and \(u\) b are high-dimensional column vectors, and the components of the column vectors are the active and reactive powers in the injection power space. F represents the mapping relationship between \(b\) i and \(\lambda\) imax :

[0061] \(\lambda\) max = F(b) (18).

[0062] S2. Propose a static voltage stability margin index;

[0063] S2.1. Define the static voltage stability margin index as:

[0064] L = ||u h - u0||2, \(u\) h ∈ SVSRB (19)

[0065] where \(u\) h represents the static voltage stability boundary point closest to the current operating state u0. Both the current operating state u0 and the boundary point \(u\) h represent column vectors in the injection power space. The physical meaning of the margin index is as Figure 2as shown

[0066] S2.2. Define the power growth mode that is most prone to static voltage instability as:

[0067]

[0068] where u h represents the static voltage stability boundary point closest to the current operating state u0. Both the current operating state u0 and the boundary point u h are column vectors in the injection power space.

[0069] S3. Establish a fast evaluation model for static voltage stability;

[0070] S3.1. Establish a fast evaluation optimization model for static voltage stability:

[0071]

[0072] In the formula, L is the static voltage stability margin index, and the objective function f is the norm of the distance from the current operating point u0 to the stability boundary point u i :

[0073]

[0074] S3.2. Substitute the static voltage stability boundary expression to obtain the fast evaluation optimization model for static voltage as:

[0075]

[0076] where u b is the initial operating base state; b is the power change mode, F represents the mapping relationship between b and λ max and L is the static voltage stability margin index.

[0077] Equation (8) represents an optimization problem of finding the maximum or minimum value of the decision variable b. The decision variable b is an R-dimensional vector. Its physical meaning is to find a certain power growth mode b so that the system operates in this power growth mode at the current operating state and reaches the minimum norm of the static voltage stability boundary.

[0078] S4. Solve the fast evaluation model for static voltage stability.

[0079] S4 is essentially to solve an optimization problem, which can be solved by common brute-force search or heuristic intelligent optimization methods. The solution steps are shown in Figure 3 as follows, specifically including:

[0080] 1) Search for the power growth direction b p that can make the optimization model reach the extreme value, and calculate the stability boundary point u h that is closest to the operating state u0

[0081] 2) Calculate the static voltage stability margin index L under the current operating state u0. If the power increase direction b has been successfully searched p , and the nearest stable boundary point u h is calculated accordingly, then the stability margin index L can be directly calculated from the definition.

[0082] 3) Solve the power increase mode b that is most prone to voltage collapse h . After obtaining the nearest boundary point u h , the power increase mode that is most prone to voltage collapse can be directly calculated as follows:

[0083]

[0084] where u h represents the static voltage stability boundary point closest to the current operating state u0, b p represents the power increase direction that makes the optimization model reach the extreme value, and F represents the mapping relationship between b and λ max .

[0085] In specific implementation, this example takes the IEEE9 and IEEE39 bus systems as examples. The IEEE9 bus system is as Figure 3 shown, and the IEEE39 bus system is as Figure 4 shown.

[0086] In the IEEE9 bus system, select the active power of buses 5, 6, and 8 to form the injection power space of the three-dimensional static voltage stability boundary, that is:

[0087] S = [P5, P6, P8] (25)

[0088] The current system operating state is S = [-1.25, -0.9, -1.0] p.u., and the negative sign indicates power absorption. Use the brute-force search method to solve the fast evaluation model of static voltage stability, and use the parameter D to control the number of search points M. The relationship between the two is:

[0089]

[0090] Respectively take D = 10, 20, 40, 60, 80, 100. The number of search points, search results, and search time are shown in Table 1. Draw a line chart of the stability margin index and search time varying with the baffle parameter D, as Figure 5 shown;

[0091] Table 1 Brute-force search results

[0092]

[0093] When the baffle parameter D increases from 10 to 100, the margin index changes by 0.02304, 0.0028, 0.00274, 0.00044, and 0.00012, respectively. As the baffle parameter D increases, the margin index changes less and less, indicating that the approximate value is gradually approaching the optimal value. At the same time, as parameter D gradually increases, the computational time cost increases significantly, rising roughly exponentially with parameter D. In the four-dimensional static voltage stability boundary assessment constructed in this system, if the margin index L is accurate to three decimal places, the baffle parameter D can be set between 80 and 100.

[0094] Particle swarm optimization search is used, and the particle swarm size is set to 56. The search accuracy, number of iterations, and search time are shown in Table 2;

[0095] Table 2 Particle swarm search results

[0096]

[0097] Compared to the speed of the baffle search method in Table 1, the computation time does not increase exponentially with increasing search accuracy. The baffle search increased the result from 1.93822 to 1.90908 by 16.77 seconds, while the particle swarm search increased the result from 1.91313 to 1.90866 by only 2.46 seconds. This is because the baffle method searches for the optimal solution with the same precision within the range of non-optimal solutions, significantly increasing the computational resources consumed. In contrast, the particle swarm algorithm gradually approaches the optimal range and searches within it with full force. Combining these two characteristics, in practical applications, the baffle method can be used to first search for a good initial value, followed by a precise search around the initial value using the particle swarm algorithm.

[0098] In the IEEE 39-node system, the active power of nodes 3, 23, 30, and 32 is selected to form the injection power space of the three-dimensional static voltage stability boundary, which is:

[0099] S=[P3,P 23 P 30 P 32 ] (27)

[0100] Ten groups of typical operating conditions are selected, as shown in Table 3. Group 1 is the data of the IEEE39 standard example, and the remaining nine groups are obtained by randomly floating 10% based on Group 1.

[0101] Table 3 Typical operating status of IEEE39 system

[0102]

[0103] Perform a rapid assessment of the static voltage stability for the above 10 groups of operating states respectively, and optimize using an algorithm that combines the baffle method and the particle swarm optimization. Set the particle swarm size to 56 and the search accuracy to 0.0001. Finally, obtain the static voltage stability indicators and the time consumption for the above 10 groups of operating states as shown in Table 4.

[0104] Table 4 IEEE39 Rapid Assessment Results

[0105]

[0106] It can be seen from the above results that under the condition of ensuring an accuracy of 0.00001, the evaluation time of the IEEE9 bus system is 2.94 s, and for the IEEE39 bus system, the time for the ten groups of experiments is between 1 s and 3 s, with a maximum of 2.54 s. The rapid assessment tasks are achieved for both examples, and by comparison, it can be found that the online rapid assessment time has nothing to do with the system scale.

[0107] The above is only a preferred embodiment of the present invention, and it does not limit the implementation manner and protection scope of the present invention accordingly. For those skilled in the art, it should be able to realize that all the equivalent replacements and obvious changes made by using the content of the specification of the present invention should be included within the protection scope of the present invention.

[0108] References:

[0109] [1] Jiang Tao, Li Guoqing, Jia Hongjie, Chen Houhe, Miao Weiwei. Simplified L-index and its sensitivity analysis method for online monitoring of voltage stability [J]. Automation of Electric Power Systems, 2012, 36(21): 13 - 18.

[0110] [2] Qi Weifu, Yang Honglei, Li Junming, Xu Jianyuan, Teng Yun, Li Jiajue. Simulation evaluation index for adapting to multi-bus voltage stability [J]. Power System Technology, 2013, 37(06): 1639 - 1644.

[0111] [3] Hu Lijuan, Liu Keyan, Sheng Wanxing, Meng Xiaoli. Fast probabilistic assessment method for static voltage stability of distribution networks with distributed generation of random output [J]. Power System Technology, 2014, 38(10): 2766 - 2771.

[0112] [4] Chen Lei, Min Yong, Hou Kaiyuan. Probabilistic assessment of static voltage stability considering wind power randomness [J]. Proceedings of the CSEE, 2016, 36(03): 674 - 680.

[0113] [5] Zhao Jianwei, Li Yupeng, Yang Zenghui, Yan Zheng, Xu Xiaoyuan, Feng Nan, Cui Yong. Probabilistic static voltage stability calculation method based on quasi-Monte Carlo simulation and kernel density estimation [J]. Power System Technology, 2016, 40(12): 3833 - 3839.

[0114] [6] Xiong Ning, Cheng Haozhong, Li Manli, Xue Yingcheng. Static Voltage Stability Evaluation Based on Confidence Interval [J]. Automation of Electric Power Systems, 2009, 33(09): 16 - 19.

[0115] [7] Lu Yuan, Lin Shunjiang, Liu Mingbo, Yang Zhibin. Calculation of Static Voltage Stability Margin of Power System Considering the Fluctuation Interval of Wind Farm Output [J]. Automation of Electric Power Systems, 2018, 42(08): 92 - 100.

[0116] [8] Chen Chang, Wan Kaiyao, Jiang Tong. Second - Order Cone Optimization Algorithm for Static Voltage Stability Limit of Radial Network Considering Uncertainty [J]. Proceedings of the CSEE, 2022, 37(15): 1 - 8.

[0117] [9] Chen Gang, Liu Wanbin, Yang Yuerong, Zheng Waisheng, Tu Sijia, Lin Shunjiang, Zhao Ligang, Zhou Baorong, Yao Wenfeng. Calculation of Static Voltage Stability Margin Interval of AC - DC Hybrid Power Grid Considering Uncertain Fluctuations of New Energy [J]. Power System Technology, 2022, 02(15): 1 - 11.

[0118]

[10] Ff W U, Tsai Y K. Probabilistic Dynamic Security Assessment of Power Systems - I: Basic Model [J]. IEEE Transactions on Circuits & Systems, 1983, 30(03): 148 - 159.

Claims

1. An optimal static voltage stability rapid assessment method, characterized in that: It includes the following steps: Step 1. Construct the static voltage stability boundary, which includes the following steps: Step 1.

1. Construct the mathematical model of the static voltage stability boundary based on the continuation power flow method: (1) In the formula, u i is an arbitrary point on the static voltage stability boundary, u b is the initial operating base state; b i is the initial operating base state u b to the power change mode from the initial operating base state to the static voltage stability boundary point u i , and λ imax is the maximum power increment under this power change mode; u i and u b are both high-dimensional column vectors ， The components of the column vector are the active and reactive powers in the injection power space. Step 1.

2. Considering the initial state u b When it is fixed, there is a one-to-one correspondence between the power growth mode and the maximum power increment, and a general mathematical model of the static voltage stability boundary point is constructed: (2) wherein, u i is an arbitrary point on the static voltage stability boundary, u b is the initial operating base state; b i is the initial operating base state u b to the power change mode from the initial operating base state to the static voltage stability boundary point u i ; u i and u b are both high - dimensional column vectors ， The components of the column vector are the active and reactive powers in the injection power space; F represents b i and λ imax mapping relationship: （3）； Step 2. Determine the static voltage stability margin index, which includes the following steps: Step 2.

1. Define the static voltage stability margin index L as: (4) Among them, u h represents the static voltage stability boundary point closest to the current operating state u 0, the current operating state u 0, the boundary point u h both represent column vectors in the injection power space; Step 2.2: Define the power growth mode that is most prone to static voltage instability as follows: (5) Among them, u h represents the static voltage stability boundary point closest to the current operating state u 0, the current operating state u 0, the boundary point u h both represent the column vectors of the injection power space; Step 3. Establish a fast evaluation model for static voltage stability, which includes the following steps: Step 3.

1. Establish the fast evaluation optimization model for static voltage stability: (6) where L is the static voltage stability margin index, and the objective function f is the current operating point u from 0 to the stability boundary point u i is the norm of: (7)； Step 3.

2. Substitute the static voltage stability boundary expression, and the fast evaluation optimization model for static voltage can be obtained as: (8) Among them, u b is the initial operating ground state; b is the power change mode, F represents b and λ max the mapping relationship, and L is the static voltage stability margin index; Step 4. Solve the fast evaluation model for static voltage stability, which includes the following steps: Step 4.1: Search for the power growth direction b that makes the optimization model reach an extreme value, and calculate the stable boundary point closest to the operating state p , and calculate the distance to the operating state u 0 u h ; Step 4.2, calculate the current operating state u The static voltage stability margin index L under 0; if the power increase direction has been successfully searched b p , and calculate the nearest stable boundary point accordingly u h , then directly calculate the stability margin index L; Step 4.3, Solve the power growth mode most prone to static voltage instability b h ; Obtain the nearest boundary point u h After that, calculate the power growth mode most prone to static voltage instability: (9) Among them, u h represents the static voltage stability boundary point closest to the current operating state u 0, b p represents the power growth direction that enables the optimization model to achieve an extreme value, F represents b and λ max mapping relationship of.

Images