Data-driven Explicit Small-signal Stability Constraint Generation and Application Method

Abstract

The present invention discloses a data-driven explicit small-signal stability constraint generation and application method, and the steps are as follows: 1) Establish an optimal power flow model and calculate the initial economic dispatch calculation result; 2) Compress the sampling space based on sensitivity analysis and generate sampling samples by using the Latin hypercube sampling method; 3) Use the SVM method to train the sampling samples, distinguish stable samples and unstable samples, and establish an optimal power flow model including explicit small-signal stability constraints; 4) Input the samples into the optimal power flow model with small-signal stability constraints, and solve to obtain a rescheduling result that meets the small-signal stability requirements. The present invention proposes a data-driven explicit small-signal stability constraint generation method, takes voltage as the control variable, and studies its application in optimal power flow calculation, greatly improving the solution efficiency of optimal power flow considering small-signal stability constraints.

Description

Technical Field

[0001] The present invention relates to the field of power systems and their automation, and specifically to a method for generating and applying explicit small-signal stability constraints based on data-driven. Background Art

[0002] In power system operation calculations, power flow constraints and equipment operation constraints are usually considered first, and an economic dispatch result is obtained through optimal power flow calculation. Then, based on the economic dispatch result, the system is subjected to security checking, and the dispatch result is adjusted to ensure the safe operation of the system. In large interconnected power systems, the rapid development of ultra-high voltage AC transmission and the increasing popularity of renewable energy have posed new challenges to the small-signal stability of power systems. Small-signal instability of the system may lead to large-scale power outages. When performing security checking, if the economic dispatch result is difficult to ensure the small-signal stability of the system, the dispatch result needs to be adjusted.

[0003] However, the relationship between the small-signal stability index of the power system and the control variables is established by a set of highly nonlinear differential-algebraic equations. Therefore, the small-signal stability requirements of the system are difficult to be directly expressed as explicit constraints.

[0004] In existing optimization calculations considering small-signal stability, the eigenvalues of differential-algebraic equations need to be repeatedly calculated during the iteration process. And the eigenvalue calculation is very time-consuming, making the existing methods difficult to meet the application requirements. In practice, if the small-signal stability of the dispatch result is unstable, usually operators adjust the active power of the generator repeatedly according to experience to make the system reach small-signal stability. However, the active power output of the generator is the main factor affecting the system cost, and this means of adjusting the active power to meet the small-signal stability of the system needs to sacrifice economy. Summary of the Invention

[0005] The object of the present invention is to provide a method for generating and applying explicit small-signal stability constraints based on data-driven, including the following steps:

[0006] 1) Establish an optimal power flow model and calculate the initial economic dispatch calculation result; if the initial dispatch result cannot meet the small-signal stability requirements of the system, go to step 2);

[0007] The optimization objective of the optimal power flow model is as follows:

[0008]

[0009] In the formula, P g is the active power output of the g-th generator; Ξ is the set of generators; and are the coefficients of the generator cost function.

[0010] The constraint conditions of the optimal power flow model include node power balance constraints, branch power flow constraints, and system operation constraints;

[0011] The node power balance constraints are as follows:

[0012]

[0013] In the formula, Q g is the reactive power output of the gth generator; P i,d and Q i,d are the active power and reactive power of node i respectively; v i , v j are the voltage amplitudes of nodes i and j; the phase angle difference δ ij = δ i -δ j， δ i is the voltage phase angle of node i; G ij and B ij are the real part and imaginary part of the element Y ij of the node admittance equation respectively; i ∈ N; N is the set of nodes;

[0014] The branch power flow constraints are as follows:

[0015]

[0016]

[0017]

[0018]

[0019] In the formula, S ij,max is the maximum apparent power constraint of branch (i, j); (i, j) ∈ K; K is the set of branches; P ij , Q ij are the active power and reactive power of branch (i, j);

[0020] The system operation constraints are as follows:

[0021]

[0022]

[0023] v i,min ≤ v i ≤ v i,max i ∈ N (10)

[0024] In the formula, and They are the upper and lower limit constraints of the active power output of the g-th generator respectively; and They are the upper and lower limit constraints of the reactive power output of the g-th generator respectively; v i,min and v i,max are the upper and lower limit constraints of the voltage amplitude at node i.

[0025] 2) Compress the sampling space of the voltage value based on sensitivity analysis. Based on the compressed sampling space, generate samples by the Latin hypercube sampling method;

[0026] The sampling interval is set as the neighborhood of v ge,j [v ge,j -ε1, v ge,j +ε1]. The corresponding node voltage constraint is adjusted to:

[0027] v ge,j -ε1 ≤ v g,j ≤ v ge,j +ε1 j ∈ A v (11)

[0028] Taking the selected voltage amplitude as the input and the minimum damping ratio of the system as the output, generate samples by the Latin hypercube sampling method in the compressed sampling space

[0029] The steps to generate sampling samples include:

[0030] 2.1) Establish a small-signal stability analysis model of the power system, that is:

[0031]

[0032] 0 = g(x, u) (13)

[0033]

[0034] In the formula, x is the set of system state variables; u is the set of system voltage and phase angles; f is the system dynamic equation; g is the system power flow balance equation; the system state equation A s =(A - BD -1 C); parameter parameter parameter parameter The system characteristic roots of the small-signal stability analysis model of the power system are λ i =σ i ±jω i ; is the output; is 's correction amount; Δx is the correction amount of x; σ i 、ωi The real and imaginary parts of the characteristic root

[0035] 2.2) Calculate the damping ratio ζ of the system i , that is

[0036]

[0037] The small-signal stability requires ζ m ≥ζ c , ζ c is the damping ratio threshold, generally 3%-5%.

[0038] Among them, the minimum damping ratio ζ of the system m is as follows

[0039] ζ m = min(ζ) = h(v) (16)

[0040] In the formula, h() is the functional relationship between the minimum damping ratio of the system and the voltage

[0041] 2.3) Calculate the sensitivity s m of the damping ratio ζ g0,j to the voltage amplitude v 0,j , that is

[0042]

[0043] In the formula, λ m = σ m ±jω m is the eigenvalue corresponding to ζ m ; A v is the set of generator voltages to be adjusted

[0044] Among them, the sensitivity m of the eigenvalue λ g0,j to the voltage amplitude v is as follows

[0045]

[0046] In the formula, v m and u m are the left and right eigenvectors of the eigenvalue λ m respectively

[0047] The sensitivity g0,j of the generator voltage to be adjusted to the voltage amplitude v is as follows

[0048]

[0049] 2.4) Based on sensitivity analysis, propose a variable selection strategy, that is

[0050]

[0051] Wherein, s l is the sensitivity threshold of the selection variable;

[0052] The variable that satisfies Equation (20) is selected as the adjustment variable.

[0053] 2.5) Damping ratio ζ m Perform a first-order Taylor expansion at the voltage amplitude v g0,j to obtain:

[0054]

[0055] Wherein, ζ m0 is the minimum damping ratio of the system corresponding to when the voltage takes v g0 ; R(v g ) is the remainder of the first-order Taylor expansion;

[0056] 2.6) Determine the voltage adjustment direction, that is:

[0057]

[0058]

[0059] Wherein, v g,jmax , v g,jmin are the upper and lower limits of voltage adjustment;

[0060] 2.7) Establish the available conditions for voltage adjustment, that is:

[0061]

[0062] 2.8) Establish an optimization model for solving the feasible point, that is:

[0063]

[0064] Wherein, the voltage difference Δv ge,j = v ge,j - v g0,j ; v ge is the voltage amplitude of the feasible point; j ∈ A v ;

[0065] The penalty function F(Δv ge , μ) for solving the optimization model of the feasible point is as follows:

[0066]

[0067] Wherein, μ is the penalty factor; j ∈ A v .

[0068] 2.9) According to the optimality condition F(Δv ge , μ) = 0, establish the voltage magnitude v of the feasible point ge solving framework, that is:

[0069]

[0070] where n a is the total number of voltages to be adjusted.

[0071] 2.10) Use the Newton iterative method to solve the voltage magnitude v of the feasible point ge , and obtain:

[0072]

[0073]

[0074]

[0075] where is the corresponding minimum damping ratio of the system;

[0076] 2.12) Use the Newton iterative method to solve and obtain the feasible voltage magnitude;

[0077] The steps of using the Newton iterative method to solve and obtain the feasible voltage magnitude include:

[0078] 2.12.1) Set the voltage magnitude damping ratio damping ratio threshold sensitivity s (k) = s0, iteration number k ← 0;

[0079] 2.12.2) Calculate the voltage magnitude using equations (28) and (29)

[0080] 2.12.3) Substitute the voltage magnitude into the modal analysis to calculate the damping ratio

[0081] 2.12.4) Judge whether the damping ratio holds. If it holds, the iteration stops, and set the feasible point as Otherwise, enter step 2.12.5);

[0082] 2.12.5) Judge whether the iteration number k is greater than the set maximum value. If it holds, the iteration stops. Otherwise, enter step 2.12.6);

[0083] 2.12.6) Recalculate the sensitivity of the voltage magnitude

[0084] 2.12.7) Update damping ratio threshold Δζ c >0, k ← k + 1, go back to step 2.1).

[0085] 3) Train the samples by SVM to generate explicit small-signal stability constraints, impose penalties on unstable samples, and reduce the misjudgment probability of unstable samples through a misclassification compensation strategy;

[0086] Stable samples refer to samples with damping ratio ζ ms ≥ζ c Unstable samples refer to samples with damping ratio ζ ms <ζ c of the samples.

[0087] The steps of training the sampling samples by the SVM method include:

[0088] 3.1) Establish the decision boundary of the SVM, that is:

[0089]

[0090] where the parameters w and b are the normal vector and intercept of the hyperplane respectively.

[0091] 3.2) Establish the equations for solving the parameters w and b, that is:

[0092]

[0093] where γ i ≥0 represents the slack variable; C is the misclassification penalty factor;

[0094] 3.3) Optimize the equations for solving the parameters w and b to obtain:

[0095]

[0096] where C + 、C - are the misclassification penalty factors for stable and unstable samples; I + 、I - are the stable and unstable sample sets; C + ≤C + ;

[0097] Calculate the penalty factor C + 、penalty factor C - The steps include:

[0098] a) Set the penalty factor Penalty factor Iteration times l ← 0;

[0099] b) Calculate the recall rate R of unstable samples using the SVM method, i.e.:

[0100]

[0101] where TP is the number of correctly classified unstable samples; FN is the number of unstable samples misjudged as stable.

[0102] c) Judge whether the recall rate R of unstable samples is greater than the set threshold R _thr holds. If it holds, the iteration terminates; otherwise, go to step d);

[0103] d) Judge whether the iteration number l is greater than the set maximum value. If it holds, the iteration terminates; otherwise, go to step e);

[0104] e) Update the penalty factor difference ΔC - > 0, l ← l + 1, and go back to step a).

[0105] 3.4) Establish a data - driven small - signal stability constraint, i.e.:

[0106] w T v ga + b ≥ 0 (35)

[0107] where v ga is a vector with respect to v g,j , j ∈ A v ;

[0108] 3.5) Substitute equations (11) and (35) into the optimal power flow model to establish an optimal power flow model considering explicit small - signal stability constraints.

[0109] 4) Establish an optimal power flow model containing explicit small - signal stability constraints and solve the rescheduling result. If the rescheduling result still does not meet the small - signal stability requirements, tighten the small - signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small - signal stability requirements.

[0110] Verify the rescheduling result. If the rescheduling result still does not meet the small - signal stability requirement ζ i ≥ ζ c , then tighten the small - signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small - signal stability requirements. The adjustment of the system's small - signal stability requirements is as follows:

[0111] w T v ga + b ≥ mτ (36)

[0112] The tools for solving the optimal power flow model with small-signal stability constraints include nonlinear solvers.

[0113] The technical effects of the present invention are beyond doubt. The present invention transforms the small-signal stability constraints expressed based on differential equations into an explicit constraint form that is easy to solve through data-driven methods, and applies it to the optimal power flow calculation considering small-signal stability constraints, providing a basis for the optimal dispatching to ensure the stability of the power system. Specifically, it involves small-signal stability analysis, data-driven methods, optimal power flow calculation, etc. The present invention proposes an explicit small-signal stability constraint generation method based on data-driven, and taking voltage as the control variable, studies its application in the optimal power flow calculation, greatly improving the solution efficiency of the optimal power flow considering small-signal stability constraints.

[0114] The present invention proposes an efficient sample generation strategy for small-signal stability constraint modeling through sampling space compression. This helps to avoid the huge burden of data-driven methods in data generation and storage, and significantly improves the calculation efficiency. Sampling space compression is achieved through variable reduction and sampling interval compression. In this local sampling space, the non-linear relationship between the small-signal stability index and the generator voltage is greatly reduced, thereby reducing the training difficulty of data-driven methods.

[0115] The present invention proposes an optimal power flow method based on explicit small-signal stability constraints. This method uses SVM training to obtain explicit small-signal stability constraints, and ensures the small-signal stability of the rescheduling results through the unstable sample misclassification compensation strategy and the small-signal stability constraint verification strategy. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] Figure 1 is the algorithm flowchart;

[0117] Figure 2 is the eigenvalue distribution trajectory;

[0118] Figure 3 is the generator speed simulation diagram of Case 1;

[0119] Figure 4 is the generator speed simulation diagram of Case 3. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0120] The present invention will be further described below in conjunction with embodiments, but it should not be understood that the above-mentioned subject scope of the present invention is limited to the following embodiments. Without departing from the above technical ideas of the present invention, various substitutions and changes made according to ordinary technical knowledge and customary means in the art should be included within the protection scope of the present invention.

[0121] Embodiment 1:

[0122] See Figure 1 andFigure 2 , a data-driven explicit small-signal stability constraint generation and application method, comprising the following steps:

[0123] 1) Establish an optimal power flow model and calculate the initial economic dispatch calculation result; if the initial dispatch result does not meet the system small-signal stability requirements, go to step 2);

[0124] The optimization objective of the optimal power flow model is as follows:

[0125]

[0126] where P g is the active power output of the gth generator; Ξ is the generator set; and are the coefficients of the generator cost function.

[0127] The constraint conditions of the optimal power flow model include node power balance constraints, branch power flow constraints, and system operation constraints;

[0128] The node power balance constraint is as follows:

[0129]

[0130]

[0131] where Q g is the reactive power output of the gth generator; P i,d and Q i,d are the active load and reactive load of node i, respectively; v i , v j are the voltage amplitudes of nodes i and j; the phase angle difference δ ij =δ i -δ j， δ i is the voltage phase angle of node i; G ij and B ij are the real part and imaginary part of the element Y ij of the node admittance equation, respectively; i ∈ N; N is the node set;

[0132] The branch power flow constraint is as follows:

[0133]

[0134]

[0135]

[0136]

[0137] Wherein, S ij,max is the maximum apparent power constraint of branch (i, j); (i, j) ∈ K; K is the set of branches; P ij , Q ij are the active power and reactive power of branch (i, j);

[0138] The system operation constraints are as follows:

[0139]

[0140]

[0141] v i,min ≤ v i ≤ v i,max i ∈ N (10)

[0142] Wherein, and are the upper and lower limit constraints of the active power output of the g-th generator respectively; and are the upper and lower limit constraints of the reactive power output of the g-th generator respectively; v i,min and v i,max are the upper and lower limit constraints of the voltage amplitude of node i.

[0143] 2) Compress the sampling space of the voltage value based on sensitivity analysis. Based on the compressed sampling space, generate samples by the Latin hypercube sampling method;

[0144] The sampling interval is set as the neighborhood of v ge,j [v ge,j - ε1, v ge,j + ε1]. The corresponding node voltage constraint is adjusted to:

[0145] v ge,j - ε1 ≤ v g,j ≤ v ge,j + ε1 j ∈ A v (11)

[0146] Using the selected voltage amplitude as the input and the minimum damping ratio of the system as the output, generate samples by the Latin hypercube sampling method in the compressed sampling space

[0147] The steps to generate sampling samples include:

[0148] 2.1) Establish a small-signal stability analysis model of the power system, that is:

[0149]

[0150] 0 = g(x, u) (13)

[0151]

[0152] where x is the set of system state variables; u is the set of system voltage and phase angles; f is the system dynamic equation; g is the system power flow balance equation; the system state equation

[0153] A s = (A - BD -1 C); parameter parameter parameter parameter The system characteristic roots of the small-signal stability analysis model of the power system are λ i = σ i ± jω i ; is the output; is the correction of; Δx is the correction of x; σ i , ω i are the real and imaginary parts of the characteristic roots;

[0154] 2.2) Calculate the damping ratio ζ i , that is:

[0155]

[0156] The small-signal stability requires ζ m ≥ ζ c , ζ c is the damping ratio threshold, generally 3% - 5%.

[0157] Among them, the minimum damping ratio ζ m of the system is as follows:

[0158] ζ m = min(ζ) = h(v) (16)

[0159] where h() is the functional relationship between the minimum damping ratio of the system and the voltage;

[0160] 2.3) Calculate the damping ratio ζ m The sensitivity s g0,j to the voltage magnitude v 0,j , that is:

[0161]

[0162] where λ m = σ m ± jω m is the eigenvalue corresponding to ζ m ; Av is the set of generator voltages to be adjusted;

[0163] where the eigenvalue λ m sensitivity to the voltage magnitude v g0,j is as follows: as follows:

[0164]

[0165] In the formula, v m and u m are the left and right eigenvectors of the eigenvalue λ m respectively;

[0166] The sensitivity of the generator voltage to be adjusted to the voltage magnitude v g0,j is as follows: as follows:

[0167]

[0168] 2.4) A variable selection strategy is proposed based on sensitivity analysis, that is:

[0169]

[0170] In the formula, s l is the sensitivity threshold of the selected variable;

[0171] The variables that satisfy Equation (20) are selected as the adjustment variables.

[0172] 2.5) The damping ratio ζ m is expanded in the first-order Taylor series at the voltage magnitude v g0,j to obtain:

[0173]

[0174] In the formula, ζ m0 is the minimum damping ratio of the system corresponding to when the voltage takes v g0 ; R(v g ) is the remainder of the first-order Taylor series expansion;

[0175] 2.6) Determine the voltage adjustment direction, that is:

[0176]

[0177]

[0178] In the formula, v g,jmax and v g,jmin are the upper and lower limits of voltage adjustment;

[0179] 2.7) Establish the available conditions for voltage adjustment, that is:

[0180]

[0181] 2.8) Establish an optimization model for solving feasible points, i.e.:

[0182]

[0183] Where the voltage difference Δv ge,j = v ge,j - v g0,j ; v ge is the voltage magnitude of the feasible point; j ∈ A v ;

[0184] The penalty function F(Δv ge , μ) for solving the optimization model of the feasible point is as follows:

[0185]

[0186] Where μ is the penalty factor; j ∈ A v .

[0187] 2.9) According to the optimality condition F(Δv ge , μ) = 0, establish a voltage magnitude v ge solution framework for the feasible point, i.e.:

[0188]

[0189] Where n a is the total number of voltages to be adjusted.

[0190] 2.10) Use the Newton's method iterative method to solve the voltage magnitude v ge , and obtain:

[0191]

[0192]

[0193]

[0194] Where is the corresponding minimum damping ratio of the system;

[0195] 2.12) Use the Newton's method iterative method to solve and obtain the feasible voltage magnitude;

[0196] The steps of using the Newton's method iterative method to solve the feasible voltage magnitude include:

[0197] 2.12.1) Set the voltage magnitude damping ratio damping ratio threshold Sensitivity s (k) = s0, iteration number k ← 0;

[0198] 2.12.2) Calculate the voltage amplitude using equations (28) and (29)

[0199] 2.12.3) Substitute the voltage amplitude into the modal analysis to calculate the damping ratio

[0200] 2.12.4) Determine whether the damping ratio holds. If it holds, the iteration stops and the feasible point is set to Otherwise, go to step 2.12.5);

[0201] 2.12.5) Determine whether the iteration number k is greater than the set maximum value. If it holds, the iteration stops. Otherwise, go to step 2.12.6);

[0202] 2.12.6) Recalculate the sensitivity of the voltage amplitude;

[0203] 2.12.7) Update the damping ratio threshold Δζ c > 0, k ← k + 1, return to step 2.1).

[0204] 3) Train the samples through SVM to generate explicit small-signal stability constraints, impose penalties on unstable samples, and reduce the misjudgment probability of unstable samples through the misclassification compensation strategy;

[0205] Stable samples refer to samples with damping ratio ζ ms ≥ ζ c Unstable samples refer to samples with damping ratio ζ ms <ζ c of the samples.

[0206] The steps of training the sampling samples using the SVM method include:[[]]

[0207] 3.1) Establish the decision boundary of SVM, that is:

[0208] {x: f(x) = w T x + b = 0} In equation (31), the parameters w and b are the normal vector and intercept of the hyperplane respectively.

[0209] 3.2) Establish the equations for solving the parameters w and b, that is:

[0210]

[0211] In the formula, γ i≥0 represents the slack variable; C is the misclassification penalty factor;

[0212] 3.3) Solve and optimize the equations for parameters w and b to obtain:

[0213]

[0214] In the formula, C + , C - are the misclassification penalty factors for stable samples and unstable samples; I + , I - are the stable and unstable sample sets; C + ≤C + ;

[0215] Calculate the penalty factor C + , the penalty factor C - The steps include:

[0216] a) Set the penalty factor The penalty factor The iteration number l ← 0;

[0217] b) Use the SVM method to calculate the recall rate R of unstable samples, that is:

[0218]

[0219] In the formula, TP is the number of correctly classified unstable samples; FN is the number of unstable samples misjudged as stable.

[0220] c) Judge whether the recall rate R of unstable samples is greater than the set threshold R _thr If it holds, the iteration terminates; otherwise, go to step d);

[0221] d) Judge whether the iteration number l is greater than the set maximum value. If it holds, the iteration terminates; otherwise, go to step e);

[0222] e) Update the penalty factor The difference ΔC - > 0, l ← l + 1, go back to step a).

[0223] 3.4) Establish a small-signal stability constraint based on data-driven, that is:

[0224] w T v ga + b ≥ 0 (35)

[0225] In the formula, v ga is a vector about v g,j , j ∈ A v ;

[0226] 3.5) Substitute equations (11) and (35) into the optimal power flow model to establish an optimal power flow model considering explicit small-signal stability constraints.

[0227] 4) Establish an optimal power flow model with explicit small-signal stability constraints and solve the rescheduling result. If the rescheduling result still does not meet the small-signal stability requirements, tighten the small-signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small-signal stability requirements.

[0228] Verify the rescheduling result. If the rescheduling result still does not meet the small-signal stability requirement ζ i ≥ζ c , then tighten the small-signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small-signal stability requirements. The adjustment of the system's small-signal stability requirements is as follows:

[0229] w T v ga +b≥mτ (36)

[0230] Tools for solving the optimal power flow model with small-signal stability constraints include nonlinear solvers.

[0231] Example 2:

[0232] See Figure 1 and Figure 2 , a data-driven method for generating and applying explicit small-signal stability constraints, includes the following steps:

[0233] 1) Calculate the optimal power flow without considering small-signal stability constraints to obtain the initial economic dispatch calculation result. If the initial dispatch result does not meet the system's small-signal stability requirements, execute the method proposed in the present invention.

[0234] 2) Compress the sampling space based on sensitivity analysis. Based on the compressed sampling space, generate samples by the Latin hypercube sampling method.

[0235] 3) Train the samples by SVM to generate explicit small-signal stability constraints. Reduce the misjudgment probability of unstable samples through the misclassification compensation strategy.

[0236] 4) Establish an optimal power flow model with explicit small-signal stability constraints and solve the rescheduling result. If the rescheduling result still does not meet the small-signal stability requirements, tighten the small-signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small-signal stability requirements.

[0237] Example 2:

[0238] See Figure 1 and Figure 2, A data-driven explicit small-signal stability constraint generation and application method, including the following steps:

[0239] First, based on the economic dispatch result of small-signal instability, through sensitivity analysis, only select the voltage magnitudes that have a significant impact on the small-signal stability of the system as the control variables for establishing constraints, thereby reducing the sampling space dimension. Secondly, based on the first-order Taylor series expansion, approximately calculate the feasible point closest to the boundary and the initial point, and set the sampling interval around this feasible point, thereby further compressing the sampling space. In this local sampling space, the non-linear relationship between the small-signal stability index and the generator voltage is greatly reduced, thereby reducing the learning difficulty of the data-driven method. Based on the compressed sampling space, generate samples by the Latin hypercube sampling method. Then, use the support vector machine method (SVM) to train the generated samples to form an explicit surrogate constraint for small-signal stability driven by data, and reduce the misclassification probability of small-signal unstable samples by adjusting the penalty factor of the unstable samples in the support vector machine method. Substitute the proposed explicit constraint of small-signal stability driven by data into the optimal power flow analysis to obtain a rescheduling result that satisfies small-signal stability. Finally, conduct simulations on the IEEE 39-bus system in the Digsilent / Power Factory and Matlab software environments, which proves that the invention can efficiently solve the optimal power flow considering small-signal stability constraints and can effectively ensure the economic and stable operation of the system. The specific method steps are as follows:

[0240] 1 Optimal power flow calculation method

[0241] The general optimal power flow model is as follows:

[0242] Optimization objective:

[0243]

[0244] Among them, P g is the active power output of the gth generator; Ξ is the set of generators; and are the coefficients of the generator cost function.

[0245] Constraints:

[0246] 1) Node power balance constraint, i ∈ N:

[0247]

[0248]

[0249] Among them, Q g is the reactive power output of the gth generator; P i,dand Q i,d are the active and reactive power loads of node i, respectively; v i is the voltage magnitude of node i; δ ij = δ i - δ j， δ i is the voltage phase angle of node i; G ij and B ij are the real and imaginary parts of the element Y ij of the nodal admittance equation, respectively; N is the set of nodes.

[0250] 2) Branch power flow constraint, (i, j) ∈ K:

[0251]

[0252]

[0253]

[0254]

[0255] where S ij,max is the maximum apparent power constraint of branch (i, j); K is the set of branches.

[0256] 3) System operation constraint

[0257]

[0258]

[0259] v i,min ≤ v i ≤ v i,max i ∈ N (10)

[0260] where and are the upper and lower limits of the active / reactive power output of the gth generator, respectively; v i,min and v i,max are the upper and lower limits of the voltage magnitude of node i.

[0261] 2 Sampling space compression technology

[0262] 1) Small-signal stability analysis

[0263] The small-signal stability analysis model of a power system can generally be described by the following equation:

[0264]

[0265] 0 = g(x, u) (12)

[0266]

[0267] Among them, x is the set of system state variables; u is the set of system voltage and phase angles; f is the system dynamic equation; g is the system power flow balance equation; A s is the system state equation,

[0268] A s =(A - BD -1 C),

[0269] Based on the system state matrix A s the system characteristic roots λ i =σ i ±jω i can be obtained. From the characteristic roots, the damping ratio of the system can be obtained:

[0270]

[0271] According to the small-signal analysis principle, the system is stable when the real part of the characteristic root is less than 0, and the system is more stable when the real part of the characteristic root is far from the imaginary axis and the imaginary part is close to the real axis. Therefore, when the damping ratio ζ i >0, the larger the damping ratio, the more stable the system.

[0272] The small-signal stability requires ζ m ≥ζ c , ζ c is the damping ratio threshold, generally 3%-5%.

[0273] 2) Variable selection method based on sensitivity analysis

[0274] The present invention selects the voltage amplitude as the adjustment variable to avoid conflicts between stability and economy. A large power system contains a large number of generators, but only a few generator voltage amplitudes have a significant impact on small-signal stability. In this section, based on sensitivity analysis, the generator voltage amplitude quantities that have a significant impact on small-signal stability are screened out as the variables for establishing constraints and sampling objects.

[0275] The variable selection strategy is shown in Equation (15):

[0276]

[0277] Among them, s l is the sensitivity threshold of the selected variable, and the voltage amplitudes with sensitivities greater than s l will be selected as adjustment variables. The value of s l should ensure that the variable with the maximum sensitivity and variables of the same order of magnitude as it are selected. ζ mis the minimum damping ratio of the system, which can be used as an index to evaluate the minimum stability of the system. Its value can be expressed as a function related to the voltage amplitude:

[0278] ζ m =min(ζ)=h(v) (16)

[0279] Generally, it is considered that when the system damping ratio is greater than a certain threshold ζ c (3% - 5%), it can be considered that the system meets the small disturbance stability. v g0,j is the voltage amplitude of generator j in the initial economic dispatch calculation result. s 0,j is ζ m 's sensitivity to v g0,j , and its value can be calculated by Equation (17):

[0280]

[0281] where λ m =σ m ±jω m is the eigenvalue corresponding to ζ m ; A v is the set of voltages of the generators to be adjusted. The sensitivity can be obtained from

[0282]

[0283] where v m and u m are the left and right eigenvectors of the eigenvalue λ m respectively; the sensitivity can be expressed as:

[0284]

[0285] ζ m When performing a first-order Taylor expansion at v g0,j , we can get:

[0286]

[0287] where ζ m0 is the minimum damping ratio of the system corresponding to when the voltage is v g0 ; R(v g ) is the remainder of the first-order Taylor expansion formula.

[0288] To ensure the small disturbance stability of the system, after voltage regulation, ζ m should be greater than ζ m0 . Therefore, each voltage adjustment should satisfy:

[0289]

[0290] The generator voltage should satisfy its limit [v g,jmin , v g,jmax . The maximum voltage adjustment can be expressed as:

[0291]

[0292] Ignoring R(v g ), according to (17), (20) and (22), the selected generator voltage should at least satisfy:

[0293]

[0294] 3) Sampling interval compression method

[0295] In the sampled space after dimension reduction, the sampling interval is further compressed. The sampled space after sampling interval compression should meet the following requirements: 1) Contain the small-signal stability boundary (ζ m = ζ c ); 2) The number of stable samples and unstable samples is balanced; 3) Be close to the initial scheduling point without considering the small-signal stability constraint. For this purpose, the present invention needs to first determine a feasible point that ensures small-signal stability and is close to the initial scheduling point. According to (20), this feasible point can be determined by the optimization model formula (24):

[0296]

[0297] where Δv ge,j = v ge,j - v g0,j ; v ge is the voltage amplitude of the feasible point; j ∈ A v .

[0298] Based on the interior point method, the penalty function of formula (24) is:

[0299]

[0300] where μ is the penalty factor; j ∈ A v .

[0301] According to the optimality condition F(Δv ge , μ) = 0, Δv ge can be obtained from formula (26):

[0302]

[0303] where n a is the total number of voltages to be adjusted.

[0304] Thus, v ge can be calculated from formula (27):

[0305] vge,j = v g0,j + Δv ge,j j ∈ A v (27)

[0306] Considering Δv ge may be large, and the relationship between the damping ratio and the voltage amplitude is non - linear. However, Equation (24) based on the first - order Taylor series expansion is linear. Therefore, it is difficult to directly obtain a feasible point through (24) - (27). Thus, the present invention iteratively solves for this feasible point based on the Newton method.

[0307] Equations (26) and (27) are rewritten as:

[0308]

[0309]

[0310]

[0311] where is the corresponding minimum damping ratio of the system.

[0312] The steps to solve for the feasible point are as follows:

[0313] 0: Set s (k) = s0, k ← 0;

[0314] 1: Calculate

[0315] through Equations (28) and (29) Substitute

[0316] 2: into the modal analysis calculation 3: If

[0317] the iteration stops, set the feasible point as

[0318] 4: If the number of iterations is greater than the set maximum value, the algorithm stops;

[0319] 5: Recalculate the sensitivity of the voltage amplitude; Δζ c > 0, k ← k + 1, go back to step 1.

[0320] After obtaining the feasible point v ge,j , the sampling interval can be set as the neighborhood of v ge,j

[0321] [v ge,j - ε1, vge,j + ε1]. The corresponding node voltage constraint is adjusted to:

[0322] v ge,j -ε1 ≤ v g,j ≤ v ge,j +ε1 j ∈ A v (31)

[0323] Equation (23) and the above step 4 together constitute the available conditions of the present invention. If the above two conditions cannot be satisfied, the present invention is not available.

[0324] 4) Generate samples

[0325] Taking the selected voltage amplitude as the input and the minimum damping ratio of the system as the output, samples are generated by the Latin hypercube sampling method in the compressed sampling space.

[0326] 3 Explicit small-signal stability constraint generation method based on data-driven

[0327] 1) Explicit small-signal stability constraint generation method based on SVM

[0328] The present invention uses SVM to fit the small-signal stability boundary. For a binary classification problem, the data set can, according to the classification target, label the samples as y i = +1 or y i = -1. The present invention is a typical binary classification problem, and the generated samples are classified according to whether they meet the small-signal stability requirements. Among them, if the sample ζ ms ≥ ζ c , then the sample is considered a stable sample and is labeled as (vgs, +1); if the sample ζ ms <ζ c , then the sample is considered an unstable sample and is labeled as (v gs , -1).

[0329] In the compressed local sampling space, the nonlinearity of the stability boundary is greatly reduced. To avoid increasing the difficulty of optimization solution, the present invention uses a linear kernel function to fit the stability boundary. The decision boundary of SVM can be represented by the hyperplane described by Equation (32).

[0330] {x: f(x) = w T x + b = 0} (32)

[0331] Among them, the parameters w and b are the normal vector and intercept of the hyperplane, respectively.

[0332] Using the above samples as input data, the hyperplane parameters can be optimized and solved by Equation (33).

[0333]

[0334] Among them, γ i ≥0 represents the slack variable; C is the misclassification penalty factor.

[0335] It can be obtained from Equation (32) that the small-signal stability constraint based on data-driven can be expressed as:

[0336] w T v ga +b≥0 (34)

[0337] Among them, v ga is a vector with respect to v g,j ,j∈A v vectors.

[0338] 2) Misclassification compensation strategy

[0339] The error of the data-driven method is inevitable. For small-signal stability, in order to ensure system safety, it is intolerable that unstable samples are misjudged as stable. On the contrary, misjudging stable samples as unstable may lead to economic losses in the final optimization result, but it does not threaten system safety, and this kind of misclassification is considered acceptable.

[0340] Therefore, the present invention introduces the recall rate R of unstable samples to analyze the situation of misjudging unstable samples, and its definition is as follows:

[0341]

[0342] Among them, TP is the number of correctly classified unstable samples; FN is the number of unstable samples misjudged as stable.

[0343] To ensure the recall rate of unstable samples, the penalty for misclassifying unstable samples should be more strict, so Equation (33) is rewritten as:

[0344]

[0345] Among them, C + / C - is the misclassification penalty factor for stable / unstable samples; I + / I - is the stable / unstable sample set; C + ≤C + .

[0346] C + / C + The value of can be calculated by the following steps.

[0347] 0: Set l←0;

[0348] 1: Execute SVM and calculate R;

[0349] 2: If R is greater than the set threshold R _thr , the algorithm terminates;

[0350] 3: If the number of iterations is greater than the set maximum value, the algorithm stops;

[0351] 4: ΔC - > 0, l ← l + 1, go back to step 1.

[0352] 4 Optimal power flow model considering explicit small-signal stability constraints

[0353] Substitute equations (31) and (34) into the optimal power flow model (1)-(10) to establish an optimal power flow model considering explicit small-signal stability constraints. This model can be directly solved by traditional non-linear solvers such as the interior point method and ipopt. Although the misclassification compensation strategy can significantly reduce the probability of misclassifying unstable samples, it still cannot completely avoid the errors brought by data-driven methods. To ensure that the rescheduling results can guarantee the stable operation of the power system, the present invention further proposes a small-signal stability constraint verification strategy. If the rescheduling results after adding the small-signal stability constraint (34) still cannot meet the small-signal stability requirements, the small-signal stability constraint needs to be further tightened, that is:

[0354] w T v ga + b ≥ mτ (37)

[0355] where m is the number of times to recalculate the optimal power flow considering explicit small-signal stability constraints; τ is the iteration step size, τ > 0.

[0356] Example 4:

[0357] Verification experiment on the method for generating and applying explicit small-signal stability constraints based on data-driven, including:

[0358] 1) Test system

[0359] In this example, the IEEE 39-bus system is used as the test system, which has a total of 10 generators. The dynamic models of the generators, excitation systems, and governors are described by the built-in dynamic models of DIgSILENT / PowerFactory. The IEEE type I excitation model is used for the excitation system, and the IEEE type I governor model is used for the governor. The parameters involved in this paper are shown in the following table:

[0360] Table 1 Parameter settings

[0361] <![CDATA[k max > <![CDATA[Δζ c > <![CDATA[ε1]]> n ΔC_ R <![CDATA[l max > τ 10 0.001 0.01 100 0.5 100％ 50 1

[0362] Set 70% of the samples as training samples and 30% of the samples as test samples.

[0363] This method was simulated by Digsilent / Power Factory. The initial scheduling results without considering the small-signal stability constraints are shown in the following table:

[0364] Table 2 Initial scheduling results without considering small-signal stability constraints

[0365] Operating cost <![CDATA[ζ c (%)]]> <![CDATA[ζ m0 (%)]]> Adjusted voltage Sensitivity <![CDATA[Δv gmax (p.u.)]]> <![CDATA[v eg (p.u.)]]> 213016 3 -1.12 <![CDATA[v 38 (Gen9)]]> 6.34 0.105 0.962

[0366] Including the initial scheduling results, three cases were tested in the present invention:

[0367] Case 1: Optimal power flow calculation without considering small-signal stability constraints

[0368] Case 2: Traditional optimal power flow calculation with small-signal stability constraints based on eigenvalue sensitivity iteration

[0369] Case 3: Optimal power flow calculation using the method of the present invention

[0370] 2) Comparison of optimal power flow calculation results

[0371] Table 3 Comparison of optimal power flow calculation results

[0372]

[0373] As can be seen from Table 3, the method proposed in the present invention can make the rescheduling results meet the system small-signal stability requirements and can sacrifice relatively less system economy.

[0374] 3) Validation of effectiveness

[0375] A three-phase short-circuit current fault was set at node 21. The small-signal stability analysis results of Case 1 and Case 3 are as Figures 2 - 4 shown. In Case 1, there is a mode with a positive real part of the eigenvalue, so the system oscillation diverges and the small-signal is unstable. In Case 3, the real parts of all eigenvalues are negative and the system oscillation converges. This proves that the method proposed in the present invention can effectively ensure the small-signal stability of the rescheduling results.

[0376] 4) Analysis of calculation efficiency

[0377] Since the main computational burden of the optimal power flow with small-signal stability constraints lies in eigenvalue calculations, this paper mainly analyzes the steps that require eigenvalue calculations. The comparative analysis of the computational efficiency of Case 2 and Case 3 is as follows. Since the generation of each sample is relatively independent during the sample generation process, parallel computing can be used in Case 3 to obtain samples. Since eigenvalue calculations do not need to be repeated during the optimization iteration process, the computational efficiency of Case 3 is significantly improved compared to Case 2. The computational times of other steps of the present invention are as follows: the SVM training time is 0.0057 s; the optimal power flow calculation time is 0.1304 s.

[0378] Table 4 Comparison of Computational Efficiency

[0379]

[0380] It can be seen from the experimental results that when the data-driven explicit small-signal stability constraint generation method proposed in the present invention is applied to the optimal power flow calculation, the rescheduling result can sacrifice relatively less system economy while meeting the system small-signal stability requirements, and significantly improve the computational efficiency.

[0381] In summary, the present invention proposes a data-driven explicit small-signal stability constraint generation method and analyzes its effectiveness in the optimal power flow calculation. First, a sampling space compression technique is proposed by reducing the number of variables and compressing the sampling interval to achieve efficient sample generation. In the compressed sampling space, small-signal stability surrogate constraints are generated through SVM training and substituted into the optimal power flow for rescheduling calculation. The simulation results on the IEEE 39-bus system show that this method has little impact on economy while ensuring small-signal stability and has high computational efficiency.

Claims

1. A data-driven explicit small-signal stability constraint generation and application method, characterized in that It includes the following steps: 1) Calculate the optimal power flow without considering small-signal stability constraints to obtain the initial economic dispatch calculation result; If the initial dispatch result cannot meet the small-signal stability requirements of the system, go to step 2); 2) Compress the sampling space of voltage values based on sensitivity analysis. Based on the compressed sampling space, generate samples by the Latin hypercube sampling method; 3) Train the samples by SVM to generate explicit small-signal stability constraints, impose penalties on unstable samples, and reduce the misjudgment probability of unstable samples through the misclassification compensation strategy; 4) Establish an optimal power flow model including explicit small-signal stability constraints and solve the rescheduling result; if the rescheduling result still does not meet the small-signal stability requirements, tighten the small-signal stability constraints and repeat the optimal power flow calculation process until the rescheduling result meets the small-signal stability requirements; The sampling interval is set to v ge,j in the domain of [v ge,j -ε1, v ge,j +ε1]; the corresponding node voltage constraint is adjusted to: v ge,j -ε1 ≤ v g,j ≤ v ge,j +ε1 for j ∈ A v (30) where, v ge,j is the voltage magnitude of the feasible point; Using the selected voltage amplitude as the input and the minimum damping ratio of the system as the output, samples are generated by the Latin hypercube sampling method in the compressed sampling space. The steps of training the samples by SVM include: 3.1) Establish the decision boundary of SVM, that is: {x: f(x) = w T x + b = 0}, where in equation (31), the parameter w and the parameter b are the normal vector and the intercept of the hyperplane, respectively; 3.2) Establish the equations for solving parameters w and b, that is: In the formula, γ i ≥0 represents the slack variable; C is the misclassification penalty factor; 3.3) Optimize the equations for solving parameters w and b to obtain: where C + , C - are the misclassification penalty factors for stable samples and unstable samples; I + , I - are the stable and unstable sample sets; C + ≤C + ; 3.4) Establish data-driven small-signal stability constraints, that is: w T v ga +b≥0 (34) where v ga is a vector with respect to v g,j , j ∈ A v ; 3.5) Substitute Eqs. (30) and (34) into the optimal power flow model to establish an optimal power flow model considering explicit small-signal stability constraints.

2. The method for generating and applying explicit small-signal stability constraints based on data driving according to claim 1, wherein: The optimization objective of the optimal power flow model is as follows: where P g is the active power output of the g-th generator; Ξ is the set of generators; and are the coefficients of the generator cost function.

3. The data-driven explicit small-signal stability constraint generation and application method according to claim 1, characterized in that: The constraint conditions of the optimal power flow model without considering small-signal stability constraints include node power balance constraints, branch power flow constraints, and system operation constraints; The node power balance constraints are as follows: where Q g is the reactive power output of the g-th generator; P i,d and Q i,d are the active power load and reactive power load of node i, respectively; v i , v j are the voltage amplitudes of nodes i and j; the phase angle difference δ ij = δ i -δ j， δ i is the voltage phase angle of node i; G ij and B ij are the real part and imaginary part of the element Y ij of the nodal admittance equation, respectively; i ∈ N; N is the set of nodes; The branch power flow constraints are as follows: where S ij,max is the maximum apparent power constraint of branch (i, j); (i,j) ∈ K; K is the set of branches; P ij , Q ij are the active power and reactive power of branch (i,j); The system operation constraints are as follows: v i,min ≤ v i ≤ v i,max i ∈ N(10) where and are the upper and lower bound constraints of the active power output of the g-th generator respectively; and are the upper and lower bound constraints of the reactive power output of the g-th generator respectively; v i,min and v i,max are the upper and lower bound constraints of the voltage magnitude at node i.

4. The method for generating and applying explicit small-signal stability constraints based on data-driven according to claim 1, wherein: The steps of generating sampling samples include: 1) Establish a small-signal stability analysis model for the power system, that is: 0 = g(x,u) (12) Where x is the set of system state variables; u is the set of system voltage and phase angles; f is the system dynamic equation; g is the system power flow balance equation; the system state equation A s =(A - BD -1 C); Parameter Parameter Parameter Parameter The system characteristic roots of the small-signal stability analysis model of the power system are λ i =σ i ±jω i ; σ i , ω i are the real and imaginary parts of the characteristic roots; is the output; is 's correction amount; Δx is the correction amount of x; 2) Calculate the damping ratio ζ of the computing system i , namely: Among them, the system minimum damping ratio ζ m is as follows: ζ m = min(ζ i ) = h(v) (15) In the formula, h() is the functional relationship between the minimum damping ratio of the system and voltage; 3) Calculate the damping ratio ζ m For the voltage amplitude v g0,j The sensitivity s 0,j That is: where λ m = σ m ± jω m are the eigenvalues corresponding to ζ m ; A v is the set of generator voltages to be adjusted; Among them, the eigenvalue λ m The sensitivity of g0,j to the voltage amplitude v is as follows: where, v m and u m are the left and right eigenvectors of the eigenvalue λ m respectively; Sensitivity of the adjusted generator voltage to the voltage amplitude v g0,j is as follows: as shown below: 4) Propose a variable selection strategy based on sensitivity analysis, that is: where s l is the sensitivity threshold of the selection variable; the variable that satisfies Equation (19) is selected as the adjustment variable; 5) Damping ratio ζ m Perform a first-order Taylor expansion at the voltage amplitude v g0,j to obtain: where ζ m0 is the minimum damping ratio of the system corresponding to when the voltage takes v g0 ; R(v g ) is the remainder of the first-order Taylor expansion; 6) Determine the voltage adjustment direction, that is: where v g,jmax and v g,jmin are the upper and lower limits of voltage adjustment; 7) Establish the available conditions for voltage adjustment, that is: where ζ c is the damping ratio threshold value; 8) Establish an optimization model for solving the feasible point, that is: where the voltage difference Δv ge,j = v ge,j - v g0,j ; v ge is the voltage magnitude of the feasible point; j ∈ A v ; The penalty function F(Δv ge , μ) for solving the optimization model of the feasible point is as follows: where μ is the penalty factor; j ∈ A v ; 9) According to the optimality condition F(Δv ge , μ) = 0, establish the voltage magnitude v of the feasible point ge solution framework, that is: Where n a is the total number of voltages to be adjusted; 10) Use Newton's iterative method to solve for the voltage magnitude v of the feasible point ge , and obtain: In the formula, is the corresponding minimum damping ratio of the system.

5. The method for generating and applying explicit small-signal stability constraints based on data-driven according to claim 4, characterized in that: The steps of using Newton's iterative method to solve for the feasible voltage amplitude include: 1) Set the voltage amplitude Damping ratio Damping ratio threshold Sensitivity s (k) = s0, iteration number k ← 0; 2) Calculate the voltage amplitude using equations (27) and (28) 3) Using the voltage amplitude to calculate the damping ratio 4) Determine the damping ratio If it holds, the iteration stops and the feasible point is set to Otherwise, go to step 5); 5) Judge whether the iteration number k is greater than the set maximum value. If it holds, the iteration stops; otherwise, go to step 6); 6) Recalculate the sensitivity of the voltage amplitude; 7) Update the damping ratio threshold Δζ c > 0, k ← k + 1, return to step 1).

6. The method for generating and applying explicit small-signal stability constraints based on data-driven according to claim 1, wherein: A stable sample refers to the damping ratio ζ ms ≥ζ c of the sample, and an unstable sample refers to the damping ratio ζ ms <ζ c of the sample.

7. The method for generating and applying explicit small-signal stability constraints based on data-driven according to claim 1, characterized in that: The steps of imposing penalties on unstable samples include: 1) Set the penalty factor Penalty factor The number of iterations l ← 0; 2) Use the SVM method to calculate the recall rate R of unstable samples, that is: In the formula, TP is the number of correctly classified unstable samples; FN is the number of unstable samples misjudged as stable; 3) Determine whether the recall rate R of unstable samples is greater than the set threshold R _thr If it holds, the iteration terminates; otherwise, go to step 4); 4) Judge whether the iteration number l is greater than the set maximum value. If it holds, the iteration terminates; otherwise, go to step 5); 5) Update penalty factor Difference ΔC - > 0, l ← l + 1, go back to step 1).

8. The method for generating and applying explicit small-signal stability constraints based on data-driven according to claim 1, wherein: Verify the rescheduling result. If the rescheduling result still does not meet the small-signal stability requirement ζ i ≥ζ c , then tighten the small-signal stability constraint and repeat the optimal power flow calculation process until the rescheduling result meets the small-signal stability requirement; the adjustment of the system small-signal stability requirement is as follows: w T v ga +b≥mτ (36) In the formula, m is the number of times of repeatedly calculating the optimal power flow considering explicit small-signal stability constraints; τ is the iteration step size, τ > 0.

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