Power system ac power flow solution method based on dynamic expansion of binary neighborhood

Abstract

The application discloses a power system alternating current flow solving method based on binary neighborhood dynamic expansion, and belongs to the field of power system optimization dispatching. The application is used for solving the difficulty caused by the product of double continuous variables, and can effectively improve the solvable performance of the problem. The application improves the mathematical model of alternating current flow, has fast convergence speed, and the average error of the calculation result is far lower than that of other traditional algorithms. The model provided by the application does not need to perform iterative calculation to generate the result, can be applied to larger scale power system dispatching calculation, and has excellent scalability.

Description

Technical Field

[0001] This invention relates to a method for solving AC power flow in power systems. In particular, it relates to a method for solving AC power flow in power systems based on dynamic expansion of binary neighborhoods. Background Technology

[0002] The safe and economical operation of the power system is one of the guarantees for my country's rapid development. The optimization objective is usually to minimize system operating costs while simultaneously satisfying power flow and security constraints. With the opening of the electricity market and the high proportion of renewable energy integration, power system operation is becoming increasingly complex, requiring the grid to calculate power flow more quickly and accurately.

[0003] Currently, the power grid uses DC power flow models, which achieve rapid power flow solutions by directly constraining the magnitude of node voltages and ignoring reactive power in branches. However, the resulting generation plans often deviate significantly from reality. In contrast, AC power flow models consider node voltage variations and the distribution of active and reactive power in branches, resulting in power flow calculations that are more realistic and reliable. However, from a mathematical perspective, AC power flow optimization is described as a non-convex, nonlinear mathematical programming problem, making it extremely difficult to solve. Therefore, researching more efficient methods for power system optimization and scheduling is of great significance and requires further research and improvement. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a power system AC power flow solution method based on binary neighborhood dynamic expansion that can effectively improve the solvability of the problem.

[0005] The technical solution adopted in this invention is: a method for solving AC power flow in a power system based on dynamic expansion of binary neighborhood, comprising the following steps:

[0006] 1) Based on binary extension theory, a dynamic binary neighborhood extension method is established for AC power flow; including:

[0007] (1.1) Establish a dynamic binary neighborhood expansion method for the product of linear continuous variables; including:

[0008] (1.1.a) The product of linear continuous variables can be expressed as:

[0009]

[0010] Where ζ is the product of any two linear continuous variables, ζ = x·y; x and y are both linear continuous variables with stable boundary conditions, and their ranges are x∈[x, y ... l ,x u ], y∈[yl ,y u ]; The superscripts u and l represent the upper and lower bounds of the constant value of a linear continuous variable, respectively; z n Let z be the nth variable in the sequence of 0-1 integer variables. n =0 or 1; The unit interval represents the discrete value of variable y. Let y be the distance between two discrete points; K is the scale parameter of the 0-1 integer variable, and y is expressed through 2... K Variable z n Complete discretization;

[0011] (1.1.b) Perform dynamic binary neighborhood expansion on newly added extended 0-1 integer variables;

[0012] Modify the linearization constraints and expand them as follows:

[0013]

[0014] Where ξ = χ·γ, χ and γ are both linear continuous variables, the boundary conditions of the variables are stable, and χ∈[χ... l ,χ u ], γ∈[γ l ,γ u ], ε m Let m be the m-th variable in a continuous sequence of variables taking values ​​in [0,1].

[0015] (1.2) Applying the binary neighborhood dynamic expansion method to the linear continuous variable product in the AC power flow model, including:

[0016] The product of linear continuous variables v in the formula for the product of linear continuous variables in the power flow model. i v j With ψ ij It is expressed as follows:

[0017]

[0018] Among them, P ij Q ij b represents the active and reactive power on branch i→j. ij g ij These represent the susceptance and conductance on branch i→j, respectively; i and j are nodes within the power grid, v i v j Let θ be the voltage magnitude at nodes i and j. i θ j Let ψ be the voltage phase angle at nodes i and j; N is the set of points, containing all generating and load nodes; E is the set of lines, containing all interconnected edges between nodes in the system, i.e., transmission lines; ψ ijThe voltage amplitude v at the nodes on both sides of branch i→j i v j The product of ψ ij =v i ·v j ;

[0019]

[0020] Among them, v i v j There are upper and lower bounds, v i v j ∈[v l ,v u ];w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; Represents variable v j The unit spacing is:

[0021]

[0022] 2) Establish an optimal scheduling model for AC power flow based on dynamic expansion of binary neighborhood, including:

[0023] (2.1) With economic efficiency as the objective, the objective function of the AC power flow optimization scheduling model based on dynamic expansion of binary neighborhood is established:

[0024] minfc = Cg + Clost

[0025] Where fc is the operating cost of the AC power flow system, Cg is the power generation cost of the thermal power unit, and Clost is the line loss cost of the entire network.

[0026] (2.2) Establish the constraints of the AC power flow optimization scheduling model based on binary neighborhood dynamic expansion, the constraints include AC power flow constraints, unit constraints and safety constraints embedded in binary neighborhood dynamic expansion;

[0027] 3) Solve the AC power flow power system optimization scheduling model based on binary neighborhood dynamic expansion to obtain the AC power flow power system optimization operation scheme.

[0028] The AC power flow solution method for power systems based on dynamic expansion of binary neighborhood, as proposed in this invention, has the following advantages:

[0029] 1. This invention proposes a binary neighborhood dynamic expansion algorithm to solve the difficulties caused by the product of two continuous variables, which can effectively improve the solvability of the problem;

[0030] 2. This invention improves the mathematical model of AC power flow, resulting in faster convergence. Compared with the Newton-Lager method for calculating nonlinear AC power flow models, the average error of the calculation results is much lower than that of other traditional algorithms.

[0031] 3. The model proposed in this invention does not require iterative calculations to generate results, making it applicable to larger-scale power system dispatch calculations. Furthermore, it exhibits excellent scalability, and its application in various special scenarios, such as multi-energy complementarity or cross-sectional quotas, can be further explored in the future, demonstrating significant theoretical and practical value. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of the binary neighborhood dynamic expansion method described in this invention;

[0033] Figure 2 This is a schematic diagram of the topology of the IEEE 39-node standard computational example of this invention;

[0034] Figure 3 This is a schematic diagram showing the deviation of the voltage amplitude of each node after power flow calculation in an example of the present invention. Detailed Implementation

[0035] The following detailed description of the power system AC power flow solution method based on binary neighborhood dynamic expansion of the present invention, with reference to embodiments and accompanying drawings, is provided in detail.

[0036] The present invention provides a method for solving AC power flow in a power system based on dynamic expansion of binary neighborhood, comprising the following steps:

[0037] 1) Based on binary extension theory, a dynamic binary neighborhood extension method is established for AC power flow, such as... Figure 1 As shown, including

[0038] (1.1) Establish a dynamic binary neighborhood expansion method for the product of linear continuous variables; including:

[0039] (a) The product of linear continuous variables is expressed as:

[0040]

[0041] Where ζ is the product of any two linear continuous variables, ζ = x·y; x and y are both linear continuous variables with stable boundary conditions, and their ranges are x∈[x, y ... l ,x u ], y∈[y l ,y u]; The superscripts u and l represent the upper and lower bounds of the constant value of a linear continuous variable, respectively; z n Let z be the nth variable in the sequence of 0-1 integer variables. n =0 or 1; The unit interval represents the discrete value of variable y. Let y be the distance between two discrete points; K is the scale parameter of the 0-1 integer variable, and y is expressed through 2... K Variable z n Complete discretization;

[0042] (b) Dynamically expand the binary neighborhood of the newly added extended 0-1 integer variables;

[0043] Modify the linearization constraints and expand them as follows:

[0044]

[0045] Where ξ = χ·γ, χ and γ are both linear continuous variables, the boundary conditions of the variables are stable, and χ∈[χ... l ,χu],γ∈[γ l ,γu],ε m Let m be the m-th variable in a continuous sequence of variables taking values ​​in [0,1].

[0046] (1.2) Applying the binary neighborhood dynamic expansion method to the linear continuous variable product in the AC power flow model, including:

[0047] The product of linear continuous variables v in the formula for the product of linear continuous variables in the power flow model. i v j With ψ ij It is expressed as follows:

[0048]

[0049] Among them, P ij Q ij b represents the active and reactive power on branch i→j. ij g ij These represent the susceptance and conductance on branch i→j, respectively; i and j are nodes within the power grid, v i v j Let θ be the voltage magnitude at nodes i and j. i θ j Let ψ be the voltage phase angle at nodes i and j; N is the set of points, containing all generating and load nodes; E is the set of lines, containing all interconnected edges between nodes in the system, i.e., transmission lines; ψ ij The voltage amplitude v at the nodes on both sides of branch i→j i v j The product of ψij =v i ·v j ;

[0050]

[0051] Among them, v i v j There are upper and lower bounds, v i v j ∈[v l ,v u ];w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable in [0, 1] after the node j is discretized; Represents variable v j The unit spacing has

[0052]

[0053] Need to adjust variable ψ ij Apply the following constraints:

[0054]

[0055]

[0056] Among them, w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; It is made by v j It comes from modifying the range of values, that is... To simplify the constraint equations, v j Transform it into a linear continuous variable starting from 0 to avoid adding unnecessary constants to the constraints.

[0057] The variable ψ ij The derivation process of the corresponding constraint proof is as follows:

[0058] Let ω = βα p ω is an intermediate variable with no practical meaning, β is any continuous variable, and α p Let be the p-th variable in a continuous sequence of variables with values ​​in [0,1]. The superscripts u and l represent the upper and lower bounds of the constant value of the variable, respectively. To prove that the derivation holds, we need to discuss different cases.

[0059] Case 1: If 0 ≤ α p <1, it is easy to see that ω-M·β≤0, M is a very large constant. In order to make the envelope as tight as possible, we can take M=ω u =β u Then ω≤β u ·α p And ω≤β;

[0060] Case 2: If α p =1, then ω = β, split the equality sign; for case 2a: ω ≥ β, that is, β - ω - M(1 - α) p If ) < 0, still take M = β u We can obtain β-ω-β u (1-α p ) < 0; For case 2b: ω ≤ β, β - ω - M(1 - α) p )≥0; still take M=β u We can obtain β-ω-β u (1-α p If α ≥ 0; then if α p =1, then ω≥β and ω≤β, which ensures the constraint, but since β-ω-M(1-α) p The ω≤β constraint derived from )≥0 is the same as the conclusion of case 1, and is no longer represented in the constraint group.

[0061] 2) Establish an optimal scheduling model for AC power flow based on dynamic expansion of binary neighborhood, including:

[0062] (2.1) With economic efficiency as the objective, the objective function of the AC power flow optimization scheduling model based on dynamic expansion of binary neighborhood is established:

[0063] minfc = Cg + Clost

[0064] Where fc is the operating cost of the AC power flow system, Cg is the power generation cost of the thermal power unit, and Clost is the line loss cost of the entire network.

[0065] (a) The power generation cost of the aforementioned thermal power unit is:

[0066]

[0067] Among them, C g The power generation cost of thermal power units within the grid; P g Let G be the active power output of the g-th device; G is the set of points, which includes all generator sets in the system; a, b, and c are the economic parameters of coal consumption for power generation of thermal power units. In engineering, piecewise linearization is used to transform this part from mixed integer programming to mixed integer linear programming in order to improve the solution efficiency.

[0068] (b) The total network line loss cost is:

[0069]

[0070] Among them, C lost Cost of heat loss across the entire network; ij r is the square value of the current on branch i→j; ij Let be the resistance value of branch i→j, and E be the set of lines, which includes all the edges that connect the nodes in the system, i.e., the transmission lines.

[0071] (2.2) Establish the constraints for the AC power flow optimization scheduling model based on binary neighborhood dynamic expansion. These constraints include AC power flow constraints embedded with binary neighborhood dynamic expansion, unit constraints, and security constraints; wherein,

[0072] (a) The AC power flow constraint for the dynamic expansion of the embedded binary neighborhood is represented as follows:

[0073]

[0074]

[0075] Among them, P ij Q ij b represents the active and reactive power on branch i→j. ij g ij These represent the susceptance and conductance on branch i→j, respectively; i and j are nodes within the power grid; v i v j The voltage magnitudes at nodes i and j; v i v j There are upper and lower bounds, v i v j ∈[v l ,v u ];ψ ij The voltage amplitude v at the nodes on both sides of branch i→j i v j The product of ψ ij =v i ·v j ;θ i θ j Let be the voltage phase angles of nodes i and j; N is the set of points, containing all generating and load nodes; E is the set of lines, containing all edges connecting the nodes in the system, i.e., transmission lines; the superscripts u and l represent the upper and lower bounds of the constant values ​​of the linear continuous variables, respectively; w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =vi ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; Represents variable v j The unit spacing has It is by

[0076] v j This comes from modifying the range of values.

[0077] (b) The unit constraints mentioned above include:

[0078] The upper and lower limits of the unit's output are constrained as follows:

[0079] P g.min ≤P g ≤P g.max ,

[0080] Q g.min ≤Q g ≤Q g.max ,g∈G

[0081] Among them, P g Q g These represent the active and reactive power outputs of the g-th unit, respectively; P g.max P g.min These represent the upper and lower bounds of the active power output of the g-th unit, respectively; Q g.max Q g.min Let G be the upper and lower bounds of the reactive power output of the g-th unit, and G be the set of points containing all generator units in the system.

[0082] The unit's standby constraints are as follows:

[0083] ∑ g∈G (P g.max -P g )≥ρ∑ i∈N P d

[0084] Where ρ is the system hot standby parameter; P d Let be the load of the i-th node;

[0085] (c) The security constraints mentioned above include:

[0086] The node voltage constraints are as follows:

[0087] v l ≤v i ≤v u

[0088] The upper and lower bounds of the phase angle are constrained as follows:

[0089]

[0090] The thermal constraints on branch power flow are as follows:

[0091]

[0092] The upper and lower bound constraints of the branch current are as follows:

[0093] l ij.min ≤l ij ≤l ij.max

[0094] in, t represents the maximum permissible phase difference between the voltages at nodes i and j on both sides of branch i→j; ij The thermal limit of branch i→j; l ij Let l be the square of the current on branch i→j; ij.min l ij.max This represents the upper and lower bounds of the square value of the current on branch i→j.

[0095] 3) Solve the AC power flow power system optimization scheduling model based on binary neighborhood dynamic expansion to obtain the AC power flow power system optimization operation scheme.

[0096] Specific examples are given below.

[0097] Example:

[0098] Using the IEEE 39-node standard example as the simulation scenario, its topology diagram is as follows: Figure 2 As shown. A comparative analysis was conducted using simulations with other methods. Method details are as follows:

[0099] Method 1: DC Power Flow Calculation Method (DCPF) for Power Systems;

[0100] Method 2: A power system AC power flow calculation method based on quadratic convex relaxation (QC-ACPF);

[0101] Method 3: A power system AC power flow solution method based on dynamic expansion of binary neighborhood (BE-ACPF).

[0102] To demonstrate the accuracy of the proposed method, the power flow calculation results obtained by PYPOWER are taken as the assumed accurate reference value, denoted as PY-ACPF. After obtaining corresponding calculation results using other algorithms, these values ​​are compared with this value. The accuracy of the relevant power flow quantities is described as the relative error with PY-ACPF; that is, the smaller the deviation from PY-ACPF, the more accurate the method. The correctness of the proposed power flow model is demonstrated by comparing it with several existing power flow algorithms. The average deviation of the power flow calculation results is expressed as... This indicates that x is the corresponding variable.

[0103] Table 1 shows the average deviation of each variable after power flow calculation.

[0104] Table 1

[0105]

[0106]

[0107] in, This represents the average deviation of the voltage amplitude and phase angle at the nodes within the network. The table shows the average deviations of active and reactive power in the branch power flow within the network. Analysis of the results in Table 1 reveals that the DCPF algorithm forcibly constrains voltage values ​​and approximates reactive power flow, leading to inaccurate calculations. The average deviation of the branch active power from the precise value calculated by PY-ACPF reaches as high as 10.62%. The QC-ACPF algorithm, due to its inherent limitation of continuously using the M-envelope, also exhibits some deviations in its calculations. The table shows that the average deviations of the node parameters in the QC-ACPF algorithm are 0.23% and 0.1379 degrees, respectively, while the average deviations of the branch power flow are smaller, at 0.81% and 3.12%, respectively.

[0108] The BE-ACPF algorithm is an improvement and optimization of the QC-ACPF algorithm. Internally, it avoids continuous approximation using the M-envelope and instead employs a more accurate binary neighborhood dynamic expansion to linearize node parameters. Final simulation results show that the calculation results of the power system AC power flow solution method based on binary neighborhood dynamic expansion are closest to those calculated by the nonlinear algorithm. The per-unit voltage amplitude of each node is constrained between 0.94 and 1.06, meeting the 6% voltage fluctuation limit of IEEE 39 examples. The average node voltage deviation is approximately 1 / 10 of that of the QC-ACPF algorithm, and the average branch active power deviation and reactive power deviation are 0.59% and 0.56%, respectively. It can be seen that the power flow results calculated by the BE-ACPF algorithm are more accurate than those of the QC-ACPF algorithm.

[0109] Figure 3 This is a schematic diagram illustrating the deviation of the voltage amplitude at each node after power flow calculation in an example of the present invention.

[0110] from Figure 3As can be seen, the voltage results at each node are consistent with the AC power flow results calculated by the nonlinear method. A slight deviation occurs at node 34, where the voltage amplitude deviates from the accurate result by 0.047%. This deviation mainly stems from the algorithm itself. When the power flow through the branch is large, the difference between the lower bound value obtained by the algorithm using the M-envelope and the actual value will widen. Although this deviation is unavoidable, it is still within an acceptable range since only one M-envelope approximation is used.

Claims

1. A method for solving AC power flow in a power system based on dynamic expansion of binary neighborhood, characterized in that, Includes the following steps: 1) Based on binary extension theory, a dynamic binary neighborhood extension method is established for AC power flow. include; (1.1) Establish a dynamic binary neighborhood expansion method for the product of linear continuous variables; include; (1.1.a) The product of linear continuous variables can be expressed as: Where ζ is the product of any two linear continuous variables, ζ = x·y; x and y are both linear continuous variables with stable boundary conditions, and their ranges are x∈[x, y ... l ,x u ], y∈[y l ,y u ]; The superscripts u and l represent the upper and lower bounds of the constant value of a linear continuous variable, respectively; z n Let z be the nth variable in the sequence of 0-1 integer variables. n =0 or 1; The unit interval represents the discrete value of variable y. Let y be the distance between two discrete points; K is the scale parameter of the 0-1 integer variable, and y is expressed through 2... K Variable z n Complete discretization; (1.1.b) Perform dynamic binary neighborhood expansion on newly added extended 0-1 integer variables; Modify the linearization constraints and expand them as follows: Where ξ = χ·γ, χ and γ are both linear continuous variables, the boundary conditions of the variables are stable, and χ∈[χ... l ,χ u ], γ∈[γ l ,γ u ], ε m Let m be the m-th variable in a continuous sequence of variables taking values ​​in [0,1]. (1.2) Applying the binary neighborhood dynamic expansion method to the linear continuous variable product in the AC power flow model, including: The product of linear continuous variables v in the formula for the product of linear continuous variables in the power flow model. i v j With ψ ij It is expressed as follows: Among them, P ij Q ij b represents the active and reactive power on branch i→j. ij g ij These represent the susceptance and conductance on branch i→j, respectively; i and j are nodes within the power grid, v i v j Let θ be the voltage magnitude at nodes i and j. i θ j Let ψ be the voltage phase angle at nodes i and j; N is the set of points, containing all generating and load nodes; E is the set of lines, containing all interconnected edges between nodes in the system, i.e., transmission lines; ψ ij The voltage amplitude v at the nodes on both sides of branch i→j i v j The product of ψ ij =v i ·v j ; Among them, v i v j There are upper and lower bounds, v i v j ∈[v l ,v u ];w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; Represents variable v j The unit spacing is: 2) Establish an optimal scheduling model for AC power flow based on dynamic expansion of binary neighborhood, including: (2.1) With economic efficiency as the objective, the objective function of the AC power flow optimization scheduling model based on dynamic expansion of binary neighborhood is established: minfc = Cg + Clost Where fc is the operating cost of the AC power flow system, Cg is the power generation cost of the thermal power unit, and Clost is the line loss cost of the entire network. (2.2) Establish the constraints of the AC power flow optimization scheduling model based on binary neighborhood dynamic expansion, the constraints include AC power flow constraints, unit constraints and safety constraints embedded in binary neighborhood dynamic expansion; 3) Solve the AC power flow power system optimization scheduling model based on binary neighborhood dynamic expansion to obtain the AC power flow power system optimization operation scheme.

2. The power system AC power flow solution method based on dynamic expansion of binary neighborhood as described in claim 1, characterized in that, Step 1) In step (1.2), it is necessary to adjust the variable ψ. ij Apply the following constraints: w jk ≤(v u -v l )·ε jk Among them, w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; It is made by v j It comes from modifying the range of values, that is... To simplify the constraint equations, v j Transform it into a linear continuous variable starting from 0 to avoid adding unnecessary constants to the constraints.

3. The power system AC power flow solution method based on dynamic expansion of binary neighborhood as described in claim 2, characterized in that, The variable ψ ij The derivation process of the corresponding constraint proof is as follows: Let ω = βα p ω is an intermediate variable with no practical meaning, β is any continuous variable, and α p Let be the p-th variable in a continuous sequence of variables with values ​​in [0,1]. The superscripts u and l represent the upper and lower bounds of the constant value of the variable, respectively. To prove that the derivation holds, we need to discuss different cases. Case 1: If 0 ≤ α p <1, it is easy to see that ω-M·β≤0, M is a very large constant. In order to make the envelope as tight as possible, we can take M=ω u =β u Then ω≤β u ·α p And ω≤β; Case 2: If α p =1, then ω = β, split the equality sign; for case 2a: ω ≥ β, that is, β - ω - M(1 - α) p If ) < 0, still take M = β u We can obtain β-ω-β u (1-α p ) < 0; For case 2b: ω ≤ β, β - ω - M(1 - α) p )≥0; still take M=β u We can obtain β-ω-β u (1-α p If α ≥ 0; then if α p =1, then ω≥β and ω≤β, which ensures the constraint, but since β-ω-M(1-α) p The ω≤β constraint derived from )≥0 is the same as the conclusion of case 1, and is no longer represented in the constraint group.

4. The power system AC power flow solution method based on dynamic expansion of binary neighborhood as described in claim 1, characterized in that, Step 2) In step (2.1): The power generation cost of the aforementioned thermal power unit is: Among them, C g The power generation cost of thermal power units within the grid; P g Let G be the active power output of the g-th device; G is the set of points, which includes all generator sets in the system; a, b, and c are the economic parameters of coal consumption for power generation of thermal power units. In engineering, piecewise linearization is used to transform this part from mixed integer programming to mixed integer linear programming in order to improve the solution efficiency. The total network line loss cost is as follows: Among them, C lost Cost of heat loss across the entire network; ij r is the square value of the current on branch i→j; ij Let be the resistance value of branch i→j, and E be the set of lines, which includes all the edges that connect the nodes in the system, i.e., the transmission lines.

5. The method for solving AC power flow in a power system based on dynamic expansion of binary neighborhood as described in claim 1, characterized in that, Step 2) In step (2.2): (a) The AC power flow constraint for the dynamic expansion of the embedded binary neighborhood is represented as follows: Among them, P ij Q ij b represents the active and reactive power on branch i→j. ij g ij These represent the susceptance and conductance on branch i→j, respectively; i and j are nodes within the power grid; v i v j The voltage magnitudes at nodes i and j; v i v j There are upper and lower bounds, v i v j ∈[v l ,v u ];ψ ij The voltage amplitude v at the nodes on both sides of branch i→j i v j The product of ψ ij =v i ·v j ;θ i θ j Let be the voltage phase angles of nodes i and j; N is the set of points, containing all generating and load nodes; E is the set of lines, containing all edges connecting the nodes in the system, i.e., transmission lines; the superscripts u and l represent the upper and lower bounds of the constant values ​​of the linear continuous variables, respectively; w jk To be the k-th intermediate variable after discretizing node j after expanding the product, we have w jk =v i ·ε jk , ε jk Let j be the k-th continuous variable between [0, 1] after discretization of node j; Represents variable v j The unit spacing has It is made by v j This comes from modifying the range of values. (b) The unit constraints mentioned above include: The upper and lower limits of the unit's output are constrained as follows: P g.min ≤P g ≤P g.max , Q g.min ≤Q g ≤Q g.max ,g∈G Among them, P g Q g These represent the active and reactive power outputs of the g-th unit, respectively; P g.max P g.min These represent the upper and lower bounds of the active power output of the g-th unit, respectively; Q g.max Q g.min Let G be the upper and lower bounds of the reactive power output of the g-th unit, and G be the set of points containing all generator units in the system. The unit's standby constraints are as follows: ∑ g∈G (P g.max -P g )≥ρ∑ i∈N P d Where ρ is the system hot standby parameter; P d Let be the load of the i-th node; (c) The security constraints mentioned above include: The node voltage constraints are as follows: in l ≤in i ≤in u The upper and lower bounds of the phase angle are constrained as follows: The thermal constraints on branch power flow are as follows: The upper and lower bound constraints of the branch current are as follows: l ij.min ≤l ij ≤l ij.max in, t represents the maximum permissible phase difference between the voltages at nodes i and j on both sides of branch i→j; ij The thermal limit of branch i→j; l ij Let l be the square of the current on branch i→j; ij.min l ij.max This represents the upper and lower bounds of the square value of the current on branch i→j.

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