Offshore wind farm static state output optimization method based on second-order cone relaxation

Abstract

The application provides a kind of offshore wind farm static operating mode output optimization method based on second-order cone convex relaxation.This method establishes the optimal power flow model of wind farm static operating mode, focuses on the content of constraint conditions and objective function involved in power flow optimization of wind farm under certain wind speed. Considering the stability requirements, safety margin, line capacity limit, node voltage modulus range, reactive power equipment capacity and wind turbine capacity involved in the optimal power flow process, the device operating state of the wind farm is adjusted to improve the power flow and reduce active power loss. The key optimization work of total power generation and line loss of wind farm is carried out, and the basic framework and content of the optimal power flow model under static operating condition are given to maximize the net power generation of offshore wind turbine.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of power systems, and particularly relates to a static working condition output optimization method for offshore wind farms. BACKGROUND

[0002] With the further increase of wind power penetration, wind power will have an undeniable impact on the safety, stability and economic operation of power systems. Therefore, it is an important task to establish a model capable of representing wind turbines in order to solve the problem of analysis and control of power systems containing wind power. At present, the modeling of wind turbines is relatively successful, and the mainstream models on the market, such as fixed-speed wind turbines, optimal slip wind turbines, doubly-fed induction wind turbines and permanent magnet synchronous wind turbines, have relatively detailed mathematical models, which have been widely used in theoretical research and actual analysis. However, the equivalent modeling of wind farms is relatively lagging behind, and the methods and ideas used are not unified, so more investment is needed for related research.

[0003] There are dozens or even hundreds of wind turbines in a wind farm, and although modeling each wind turbine can accurately represent the actual working state of the wind farm, it dramatically increases the calculation amount of the power system, prolongs the simulation time, and makes the power flow difficult to converge. In addition, the effectiveness of the detailed model and the correction of the data need further research. It is therefore necessary to study a modeling optimization method for the wind speed-power characteristics of wind farms that takes into account the differences in the working conditions of the wind turbines in the field and can be actually operated. SUMMARY

[0004] The purpose of the present application is to overcome the problem that the prior art method cannot efficiently solve the optimization problem of the output of offshore wind farms, and to provide an optimization solving method based on second-order cone convex relaxation, which is simple.

[0005] In order to achieve the above-mentioned purpose, the present method adopts the following technical solutions:

[0006] 1. A second-order cone convex relaxation-based static working condition output optimization method for offshore wind farms, characterized by comprising the following steps:

[0007] Step (1), establishing a mathematical optimization model of the offshore wind farm;

[0008] Step (2), based on the system mathematical model obtained in step (1), using the second-order cone convex relaxation optimization method to process the non-convex constraints to obtain a relaxed offshore wind farm optimization model;

[0009] Step (3), based on the optimization models in (1) and (2), the variable values in the original optimal power flow model before relaxation are solved by reducing the relaxation variables, and then the optimal variable solution of the optimization objective is obtained.

[0010] Further, step (1) comprises the following steps:

[0011] 1.1 Establish the optimization objective based on the maximum net power generation of offshore wind turbines;

[0012] 1.2 Establish power system constraints based on power system operating conditions.

[0013] The standard form of the optimal power flow problem is as follows:

[0014] min f(u,x)

[0015] s.t.g(u,x)=0,

[0016] h(u,x)≤0.

[0017] Where f(u,x) represents the optimization objective function, g(u,x) represents the equality constraints, and h(u,x) represents the inequality constraints. The variables in the optimal power flow problem are used to represent the operating state of the power system, generally including: node voltage amplitude and phase angle, node injected active power and reactive power.

[0018] Further, the optimization objective in the mathematical optimization model of the offshore wind farm is to maximize the net power generation of offshore wind turbines; the equality constraints of the optimal power flow problem are usually non-convex power flow equations, which for a network of n nodes are described in polar coordinates:

[0019] The equality constraints of the optimal power flow problem are usually non-convex power flow equations, which for a network of n nodes are described in polar coordinates:

[0020]

[0021] I ij =(V i -V j )Y ij ,

[0022]

[0023] Where Y ij is the admittance between lines i,j, S ij is the complex power between lines i,j, I ij represents the line current. V i is the node voltage, and S iPower injection of nodes. The power flow equation equality constraint focuses on the power flow equation on each two nodes connected line, the equality constraint is constructed by the power flow relationship on the line and the node energy conservation law. The power flowing through the line is the product of voltage and current conjugate, the non-convexity comes from the product of voltage and current conjugate. The line current and the connected node voltage satisfy Kirchhoff's law. This optimal power flow model is branch power flow model, as shown in Figure 1 .

[0024] The above power flow equation constraint equation is written in the following form:

[0025]

[0026] This equality constraint does not contain line current variables and branch complex power variables, the main variables are node voltage variables and node injection power variables, this power flow constraint equation is expressed by node injection power, which hides the angle related information in the complex expression. Writing in the above form is helpful for related model transformation. The non-convexity comes from the quadratic term of voltage product. This optimal power flow model is node injection model.

[0027] The inequality constraints of the optimal power flow problem are as follows:

[0028] P min ≤P≤P max ,

[0029] Q min ≤Q≤Q max ,

[0030] V min ≤V≤V max ,

[0031] δ min ≤δ i -δ j ≤δ max ,

[0032] P min , P max are the upper and lower bounds of line active power constraints respectively, Q min , Q max are the upper and lower bounds of line reactive power constraints respectively, V min , V max are the upper and lower bounds of node voltage modulus constraints respectively, δ min , δ max are the upper and lower bounds of line phase angle difference constraints respectively.

[0033] Further, the second-order cone relaxation procedure of step (2) is as follows: first, corresponding transformation is performed on the variables and constraints, aiming to change the non-convex power flow equality equations into convex form and perform corresponding optimization solution. In the transformation process, the variables or corresponding constraints can be relaxed, so as to convert the optimization model into a convex optimization problem to solve the global optimal solution. Although this relaxation method expands the constraint feasible region, the optimal solution can be obtained on the boundary of the original feasible region due to the accuracy of the relaxation, as shown in Figure 2 .

[0034] The standard second-order cone programming form of a convex problem is as follows:

[0035] minimize f T x

[0036] subject to||A i x+b i ||2≤c i T x+d i ,i=1,K,m

[0037] Fx=g,

[0038] where x is the optimization variable, A i is the second-order cone constraint coefficient. The second-order cone programming is between linear programming and semi-definite programming, and belongs to convex optimization problems. When A i = 0, the second-order cone programming problem becomes a linear optimization; when c i = 0, the second-order cone programming problem becomes a quadratic constraint quadratic programming problem.

[0039] For the node injection model, define variables W ii , W ij respectively represent the square term of the voltage variable V i of node i and the voltage product of adjacent node voltage variable V i and V j , which is expressed in mathematical form as:

[0040]

[0041] The original equality power flow constraint is changed to:

[0042]

[0043] For the non-convex equality constraint , the second-order cone relaxation technique is used to relax it into the following inequality:

[0044] W ii W jj ≥|W ij2 |

[0045] Then the non-convex equality constraint can be relaxed to a second-order rotated cone.

[0046] For branch power flow model, introduce the following variables: node voltage magnitude square v i , line current square l ij and line power S ij , then the power flow equation can be expressed as

[0047] v i l ij = |S ij | 2

[0048]

[0049] where Z ij is the line impedance, the non-convex equality constraint v i l ij = |S ij | 2 is relaxed to the following inequality by using second-order cone relaxation technique:

[0050] v i l ij ≥ |S ij | 2

[0051] The node voltage magnitude square v i , line current square l ij and line complex power variable S ij can be constrained in second-order rotated cone. Thus, the second-order cone convex relaxation model of offshore wind farm is constructed. This model relaxes the power flow constraints by eliminating the phase angle information in voltage and current, and relaxes the power flow constraints to the relationship between node injection power and node voltage square, and the relationship between node injection power and line current square. In radial network topology, it can be proved by reductio ad absurdum that the second-order cone relaxation is exact when the active power and reactive power of load nodes have no upper bound. That is, the relaxation takes the equality at the optimal solution, so it is called exact convex relaxation.

[0052] Further, the relaxation variable reduction process of step (3) is that the second-order convex relaxation method eliminates the phase angle information in the voltage and current by introducing the square of the node voltage modulus and the square of the line current modulus, and relaxes the power flow constraint into the relationship between the node injection power and the square of the node voltage, and the relationship between the node injection power and the square of the line current. The relaxation variable is the square of the node voltage modulus variable and the square of the line current modulus variable, which needs to be restored into the optimization solution of the original optimal power flow problem. Specifically, the variables that need to be restored are the node voltage modulus and the phase angle information, and the line current modulus, wherein the node voltage modulus and the line current modulus can be obtained by taking the square root of the square of the node voltage modulus and the square of the line current modulus respectively. For the phase angle information of the node voltage, that is, the phase angle information of the voltage difference between the two ends of the line, the restoration process is as follows:

[0053]

[0054] Wherein β ij The phase angle difference of the voltage at both ends of the line can be restored by defining the voltage phase angle information of the root node as zero.

[0055] Due to the technical scheme of the present application, under the static working condition, the stability requirements and safety margin of the power grid involved in the optimal power flow process are considered, the operation state of the equipment of the wind farm is adjusted, the power flow is improved, and the active power loss is reduced. The method can quickly realize optimization solution, and has good convergence performance, and ensures the feasibility of the solution. BRIEF DESCRIPTION OF DRAWINGS

[0056] Figure 1 The figure is a schematic diagram of any branch in the offshore wind farm system, wherein for the branch connected to nodes i and j, all nodes connected to i are recorded as j, and there are n nodes in total.

[0057] Figure 2 The figure is a schematic diagram of the feasible region before and after the second-order convex relaxation. Although the second-order convex relaxation method expands the constraint feasible region, the accuracy of the relaxation can obtain the optimal solution on the boundary of the original feasible region. DETAILED DESCRIPTION

[0058] In order to more specifically describe the present application, the technical scheme of the present application will be described in detail in combination with the drawings and specific implementation cases.

[0059] The optimization process of the present application mainly includes the above-mentioned optimization model establishment, the second-order convex relaxation process, and the relaxation variable reduction.

[0060] I. Optimization model establishment: in the modeling of the power system model, according to the voltage variable V i= U i ∠θ i , active power variable P i g , reactive power variable Q active power variable P i c , reactive power variable Q establish equations of equality and inequality relationship; meanwhile, for the current variable I ij , line admittance constant Y ij , conductance constant G ij , susceptance constant B ij , active power variable P ij , reactive power variable Q ij , and complex power variable S ij establish equations thereof and variables between nodes. Wherein, the optimization model of the power system mainly involves power flow constraints, energy constraints, voltage constraints, and output constraints of each load and power generation; in addition, network topology, wiring mode, etc. of the power system also need to be considered. The equation is:

[0061]

[0062] The power flow equation constraint needs to satisfy the power flow relationship and the law of conservation of energy at the node. The power flowing through the line is the product of voltage and current conjugate, and the non-convexity comes from the product of voltage and current conjugate. The line current and the connected node voltage satisfy Kirchhoff's law:

[0063]

[0064] The optimization problem is to maximize the net power generation of the offshore wind farm, and the cost function is:

[0065] ∑P i gc (3)

[0066] In the above formula, P i gc active power of the offshore wind farm to the node of external power transmission.

[0067] Through modeling, the mathematical expression of the optimization problem is:

[0068]

[0069] This is the mathematical expression of the offshore wind farm output optimization problem before relaxation.

[0070] Second-order convex relaxation process: in power system, through variable definition, there are following equality and inequality constraints; among them, the relationship between voltage square term, current square term and line complex power is relaxed into second-order rotation cone constraint, and its specific expression is

[0071]

[0072] Among them, W ii , W ij respectively represent the square term of voltage variable V i of node i and the voltage product of adjacent node voltage variable V i and V j ;

[0073] At the same time, define the I ij square variable l ij of line current, and W ii , l ij and line complex power variable S ij constraint can be in second-order rotation cone.

[0074]

[0075] In the formula, Z ij is line impedance, S ij and S ji are respectively the complex power of node i flowing to node j and the complex power of node j flowing to node i.

[0076] After relaxation, the optimization problem is

[0077]

[0078] Third, the relaxation variable restoration process is that the second-order convex relaxation method eliminates the phase angle information in voltage and current by introducing the square of node voltage modulus and the square of line current modulus, and relaxes the power flow constraint into the relationship between node injection power and node voltage square, and the relationship between node injection power and line current square. The relaxation variables are node voltage modulus square variable and line current modulus square variable, which need to be restored into the optimization solution of the original optimal power flow problem. The specific variables to be restored are node voltage modulus and phase angle information, and line current modulus information, among them, the node voltage modulus and line current modulus can be obtained by taking the square root of the square of node voltage modulus and the square of line current modulus, that is, the node voltage modulus and line current modulus. For the phase angle information of node voltage, that is, the phase angle information of the voltage difference between the two ends of the line, the restoration process is as follows:

[0079]

[0080] Among them, β ijThe phase angle difference between the voltages at both ends of the line can be recovered by defining the voltage phase angle information of the root node as zero.

[0081] Taking the Putuo Wind Farm in Zhoushan as an example, the wind farm was modeled as a radial network with 132 nodes and 131 lines as a simulation test system, containing 63 wind turbines, each with a rated capacity of 4000kW. Using the second-order conical relaxation optimization method of this invention, the active and reactive power of the generators were optimized under different operating conditions to reduce line losses, thereby maximizing the active power delivered to the grid. The optimization results were compared with the case where the generator power factor was 1, i.e., no reactive power was generated, showing a significant improvement while ensuring the feasibility of the optimized solution. When the wind turbine output was 100%, each turbine generated a maximum of 4MW of active power, and the entire wind farm generated a maximum of 252MW of active power. Through optimization, the active power delivered to the grid was 250831.3878kW, while the control group result was 244880.2268kW, an improvement of 5951.161kW. When the wind turbines operate at 75% capacity, each turbine generates a maximum of 3MW of active power, resulting in a maximum active power output of 189MW for the entire wind farm. Through optimization, the active power transmitted to the grid is 188,343.6299kW, compared to 183,893.5419kW in the control group, representing an increase of 4,450.088kW. When the wind turbines operate at 50% capacity, each turbine generates a maximum of 2MW of active power, resulting in a maximum active power output of 126MW for the entire wind farm. Through optimization, the active power transmitted to the grid is 125,708.3765kW, compared to 122,751.1559kW in the control group, representing an increase of 2,957.2206kW. When the wind turbines operate at 25% capacity, each turbine generates a maximum of 1MW of active power, resulting in a maximum active power output of 63MW for the entire wind farm. Through optimization, the active power delivered to the grid side was 62927.1001kW, while the control group result was 61452.9038kW, an increase of 1474.1963kW.

[0082] The above description of the embodiments is provided to enable those skilled in the art to understand and apply the present invention. It will be apparent to those skilled in the art that various modifications can be made to the above embodiments, and the general principles described herein can be applied to other embodiments without inventive effort. Therefore, the present invention is not limited to the above embodiments, and any improvements and modifications made to the present invention by those skilled in the art based on the disclosure thereof should be within the scope of protection of the present invention.

Claims

1. A method for optimizing the output of offshore wind farms under static conditions based on second-order conical relaxation, characterized in that... Includes the following steps: Step (1): Establish a mathematical optimization model for offshore wind farms; Step (2): Based on the system mathematical model obtained in step (1), the non-convex constraints are processed by the second-order cone convex relaxation optimization method to obtain the relaxed offshore wind farm optimization model. Step (3): Based on the optimization models in (1) and (2), the variable values ​​in the original optimal power flow model before relaxation are solved by restoring the slack variables, and then the optimal variable solution of the optimization objective is obtained. In step (2), the variables and constraints are first transformed accordingly. The goal is to transform the non-convex power flow equation into a convex form and perform corresponding optimization solutions, thereby converting the optimization model into a convex optimization problem and solving the global optimal solution. For the node injection model, define variable W. ii W ij V represents the voltage variable at node i. i The squared term and the voltage variable V of adjacent nodes i With V j The voltage product, expressed mathematically, is: The original equal power flow constraints change to: For nonconvex equality constraints Using the second-order cone relaxation technique, the relaxation yields the following inequality: IN ii IN jj ≥|In ij 2 | By performing second-order cone relaxation on the non-convex equation, a second-order rotating cone is obtained. For the branch power flow model, the following variable is introduced: the square of the node voltage magnitude v i Line current square l ij and line power S ij Then express the current equation as v i l ij =|S ij | 2 In the formula, Z ij For the line impedance, constrain v using the non-convex equality. i l ij =|S ij | 2 Using the second-order cone relaxation technique, the relaxation yields the following inequality: v i l ij ≥|S ij | 2 The square of the node voltage magnitude v i Line current square l ij With line complex power variable S ij The constraints are placed in a second-order rotating cone, thus constructing a second-order cone-convex relaxation model for offshore wind farms. This model relaxes the power flow constraints into the relationship between node injected power and the square of node voltage, and the relationship between node injected power and the square of line current, by eliminating the phase angle information in voltage and current.

2. The method for optimizing the static operating condition output of offshore wind farms based on second-order conical relaxation as described in claim 1, characterized in that... Step (1) Includes the following steps: 1.1 Establish an optimization objective based on maximizing the net power generation of offshore wind turbines; 1.2 Based on the operating conditions of the power system, establish power system constraints.

3. The method for optimizing the static operating condition output of offshore wind farms based on second-order conical relaxation as described in claim 1, characterized in that... The standard form of the optimal power flow problem is as follows: min f(u,x) stg(u,x)=0, h(u,x)≤0. Where f(u,x) represents the optimization objective function, g(u,x) represents equality constraints, and h(u,x) represents inequality constraints; The optimization objective in the mathematical optimization model of offshore wind farms is to maximize the net power generation of offshore wind turbines; the equality constraints of the optimal power flow problem are non-convex power flow equations, which are described in polar coordinates for an n-node network: I ij =(V i -V j )Y ij , Where Y ij S is the admittance between lines i and j. ij Let I be the complex power between lines i and j. ij V represents the line current; i For node voltage, S i Inject power into the nodes; this optimal power flow model is a branch power flow model. The power flow constraint equation (1) can be simplified to the following form: This equation represents the node injection model; The inequality constraints for the optimal power flow problem are as follows: P min ≤P≤P max , Q min ≤Q≤Q max , In min ≤V≤V max , d min ≤δ i -d j ≤δ max , In the formula P min ,P max These are the upper and lower bounds of the active power constraint for the line, Q. min Q max These are the upper and lower bounds of the reactive power constraint for the line, V. min V max These are the upper and lower bounds of the nodal voltage magnitude constraint, δ. min ,δ max These are the upper and lower bounds of the line phase angle difference constraint, respectively.

4. The method for optimizing the static operating condition output of offshore wind farms based on second-order conical relaxation as described in claim 1, characterized in that... In step (3), the relaxation variable restoration process is as follows: The second-order cone-convex relaxation method eliminates the phase angle information in the voltage and current by introducing the square of the node voltage magnitude and the square of the line current magnitude, thus relaxing the power flow constraint into the relationship between the node injected power and the square of the node voltage, and the relationship between the node injected power and the square of the line current. The node voltage magnitude and the line current magnitude are obtained by taking the square root of the square of the node voltage magnitude and the square of the line current magnitude, respectively. For the node voltage phase angle information, i.e., restoring the phase angle information of the voltage difference between the two ends of the line, the restoration process is as follows: Where β ij The phase angle difference between the voltages at both ends of the line can be recovered by defining the voltage phase angle information of the root node as zero.

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