Alternating current optimal power flow calculation method and system of power system

Abstract

The invention discloses a power system AC optimal power flow calculation method and system, and relates to the technical field of power system analysis, and the method comprises the steps: deducing a simplified steady-state heat balance equation according to the electrothermal characteristics of an overhead transmission line in a power system; constructing an alternating current optimal power flow model considering the electric heating characteristics of the line by taking the lowest power generation cost as an objective function and combining constraint conditions, and deducing a KKT condition corresponding to the model; wherein the constraint condition comprises a line steady-state heat balance equation constraint and a line current-carrying upper limit constraint obtained by converting a line maximum allowable operating temperature constraint according to a steady-state heat balance equation; constructing a physical information neural network model, introducing a KKT condition into a loss function of the model, and training the model; and obtaining active and reactive power load requirements of the power system, inputting the trained physical information neural network model, and outputting AC optimal power flow distribution data of the power system, thereby effectively improving calculation precision and efficiency, and improving interpretability of model decision.

Description

Technical Field

[0001] This invention relates to the field of power system analysis technology, and in particular to a method and system for calculating optimal AC power flow in a power system. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Optimal power flow (OPF) refers to the optimal allocation of resources in a power system by optimizing controllable equipment while satisfying power balance constraints and various security constraints. The Alternating Current-Optimal Power Flow (AC-OPF) problem is a complex, non-convex, and nonlinear optimization problem, whose computation is difficult to converge or consumes a significant amount of time. Currently, common methods for solving AC-OPF involve simplifying or approximating the problem, such as constructing linear DC-OPF, convex relaxation, and other linear or convex approximations, to circumvent the non-convex and nonlinear characteristics of AC-OPF. However, this leads to insufficient computational accuracy, significant deviation from the actual system, and difficulty in balancing computational efficiency. Therefore, there is an urgent need for an effective computational method for AC-OPF that balances computational accuracy and speed.

[0004] In existing technologies, the construction of traditional AC optimal power flow models often ignores the dynamic impact of transmission line temperature on conductor electrical characteristics (such as resistance) during actual operation. The models struggle to accurately reflect physical reality, leading to deviations between optimization results and actual system conditions under significant load variations or extreme weather, thus compromising the accuracy of AC-OPF calculations. Furthermore, as model complexity increases, traditional calculation methods become costly and prone to iteration non-convergence, resulting in low computational efficiency. While machine learning algorithms, such as deep neural networks, are widely used in power systems and can improve computational efficiency to some extent, traditional neural networks suffer from high requirements for data quality and quantity and insufficient interpretability. This limits their application in solving AC optimal power flow problems in power systems. This approach relies solely on data fitting to construct the mapping relationship between input and output, lacking interpretability in the model's decision-making process. It cannot trace the physical rationality of the results and may even output infeasible solutions that violate physical rules, failing to meet the reliability and interpretability requirements of power systems as critical infrastructure. Summary of the Invention

[0005] To address the shortcomings of the existing technologies, this invention provides a method and system for calculating optimal AC power flow in power systems. Considering the electrothermal characteristics of transmission lines, a steady-state thermal balance equation for the lines is derived. This equation is incorporated into the solution framework to construct an optimal AC power flow model that takes into account the electrothermal characteristics of the lines. The model is then solved using a Physical Informed Neural Network (PINN) driven by both data and physics, achieving a balance between computational accuracy and efficiency, and effectively improving the interpretability of the model's decisions.

[0006] In a first aspect, the present invention provides a method for calculating the optimal AC power flow in a power system.

[0007] A method for calculating the optimal AC power flow in a power system includes: Based on the electrothermal characteristics of overhead transmission lines in power systems, a simplified steady-state thermal balance equation is derived. With the minimum power generation cost as the objective function, and in combination with constraints, an AC optimal power flow model considering the electrothermal characteristics of the line is constructed, and the corresponding KKT conditions are derived. Among them, the constraints include the line steady-state thermal balance equation constraint and the line current carrying limit constraint obtained by converting the line maximum allowable operating temperature constraint based on the steady-state thermal balance equation. A physical information neural network model is constructed, KKT conditions are introduced into the model's loss function, and the model is trained using the training set. The system obtains the active and reactive power load demands of the power system, inputs the trained physical information neural network model, and outputs the optimal AC power flow distribution data of the power system.

[0008] A further technical solution is that the derivation process of the simplified steady-state thermal balance equation is as follows: Based on the heat exchange principle of overhead transmission lines, an original heat balance equation is established. The formulas for calculating the solar radiation heat absorption, convective heat dissipation, and radiative heat dissipation per unit length of overhead transmission lines are derived respectively; among them, the convective heat dissipation is taken as the maximum value of forced convection heat dissipation and natural convection heat dissipation. Assuming the power system is in a steady state, ignoring the dynamic thermal balance process, taking the data of various meteorological conditions as typical values, and substituting the calculation formulas of solar radiation heat absorption, convective heat dissipation and radiative heat dissipation into the original thermal balance equation, we obtain a simplified form of the steady-state thermal balance equation, which is then converted into per-unit value form to clarify the relationship between line current and line temperature.

[0009] A further technical solution is that the AC optimal power flow model taking into account the line's electrothermal characteristics takes the minimum power generation cost as the objective function and includes constraints on the node power balance equation, the line steady-state thermal balance equation, the upper and lower limits of generator active and reactive power output, the node voltage amplitude constraint, and the upper limit of line current carrying capacity constraint.

[0010] A further technical solution is provided, wherein the physical information neural network model includes: The input layer is used to input the active power load demand and reactive power load demand of the power system. Multiple subnetworks, including a power prediction subnetwork, a voltage prediction subnetwork, a temperature prediction subnetwork, a current prediction subnetwork, and a Lagrange multiplier prediction subnetwork, are used to calculate the generator active and reactive power output setpoints, the real and imaginary parts of node voltages, line temperatures, the real and imaginary parts of line currents, and all Lagrange multipliers, respectively. The physical information layer is used to encapsulate the physical equations constructed based on KKT conditions, receive input samples and the prediction results output by the neural network, and output the physical information loss term. The output layer is used to output the predicted optimal AC power flow distribution data of the power system, including the active and reactive power output setpoints of generators in the power system, node voltages, transmission line currents and temperatures, and all Lagrange multipliers.

[0011] A further technical solution is that the loss function of the physical information neural network model includes a data loss term and a physical information loss term corresponding to each sub-network; The physical information loss term is constructed based on KKT conditions and includes physical information loss terms corresponding to stationary conditions, original feasibility conditions, dual feasibility conditions, and complementary relaxation conditions.

[0012] A further technical solution is that the training process of the physical information neural network model is as follows: Acquire training data samples and construct a training set; the training data samples include the active power load demand and reactive power load demand of the power system, as well as the AC optimal power flow distribution data of the power system. The active power load demand and reactive power load demand of the power system are used as the input of the model, and the optimal AC power flow distribution data of the power system are used as the output of the model. The physical information neural network model is trained using the training set until the loss function is minimized, and the trained model is obtained.

[0013] Secondly, the present invention provides a power system AC optimal power flow calculation system.

[0014] A power system AC optimal power flow calculation system, comprising: The steady-state thermal balance derivation module is used to derive simplified steady-state thermal balance equations based on the electrothermal characteristics of overhead transmission lines in power systems. The optimal power flow model construction module is used to construct an AC optimal power flow model that takes into account the electrothermal characteristics of the line, with the objective function of minimizing power generation cost and in combination with constraints. The module also derives the corresponding KKT conditions for the model. The constraints include the line steady-state thermal balance equation constraint and the line current carrying limit constraint obtained by converting the line maximum allowable operating temperature constraint based on the steady-state thermal balance equation. The neural network model building module is used to build a physical information neural network model, introduce KKT conditions into the model's loss function, and train the model using the training set. The optimal power flow calculation module is used to obtain the active and reactive power load demand of the power system, input the trained physical information neural network model, and output the AC optimal power flow distribution data of the power system.

[0015] Thirdly, the present invention also provides an electronic device, comprising: a memory for storing executable instructions; and a processor for executing the executable instructions stored in the memory to implement the above-described method for calculating the optimal AC power flow of a power system.

[0016] Fourthly, the present invention also provides a computer-readable storage medium storing executable instructions for causing a processor to execute the executable instructions to implement the above-described method for calculating the optimal AC power flow of a power system.

[0017] Fifthly, the present invention also provides a computer program product comprising executable instructions stored in a computer-readable storage medium; wherein, when the processor of an electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the above-mentioned method for calculating the optimal AC power flow of a power system is implemented.

[0018] The above one or more technical solutions have the following beneficial effects: 1. This invention provides a method and system for calculating AC optimal power flow in a power system based on a physical information neural network (PINN) that considers the electrothermal characteristics of transmission lines. Building upon the traditional optimal power flow problem, this method considers the electrothermal characteristics of transmission lines, treating the resistance of the transmission lines as a function of temperature rather than a constant, and incorporating line temperature as a variable into the solution framework. This constructs an AC optimal power flow model that considers the electrothermal characteristics of the lines. The PINN is then used to solve the constructed optimal power flow problem. By encoding relevant KKT conditions and integrating them into the loss function term of the neural network as the physical guiding equation, a dual-driven neural network of data and physical equations is constructed. This enables the calculation of the optimal setpoint for generator power, node voltage, line current, and line temperature in the power system. This approach balances the accuracy and efficiency of AC optimal power flow calculation and effectively improves the interpretability of model decisions.

[0019] 2. In constructing the optimal power flow model, this invention breaks through the limitations of the original static parameter model, treats the line resistance as a function of temperature, incorporates the electrothermal balance equation into the model, and introduces the temperature variable, so that the resistance parameter values ​​used in the model are more in line with the values ​​in actual operation, thereby improving the calculation accuracy of the model.

[0020] 3. This invention constructs a corresponding physical information neural network to solve the AC optimal power flow model. It makes full use of the physical equations of the model and combines historical data to construct a physical + data dual-drive mode, reducing the dependence on the amount of data. Compared with traditional neural networks, this physical information neural network has higher solution accuracy and less training time. In addition, the neural network constructed in this invention adopts discrete sub-networks, which makes it easy to adjust the size of different sub-networks for different calculation problems, thereby improving the applicability and generalization of the network to different calculation problems.

[0021] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0022] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0023] Figure 1 This is a flowchart of the AC optimal power flow calculation method for a power system in Embodiment 1 of the present invention; Figure 2 This is a schematic diagram of heat exchange in an overhead power transmission line according to Embodiment 1 of the present invention; Figure 3 This is a schematic diagram of the physical information neural network model for solving the optimal power flow problem considering the line's electrothermal characteristics in Embodiment 1 of the present invention; Figure 4 This is the topology diagram of the IEEE 14-node standard computation example in Embodiment 1 of the present invention; Figure 5 The graph shows the loss curves of each variable during the network model training process in Embodiment 1 of the present invention; where (a) is the voltage prediction loss, (b) is the power prediction loss, (c) is the temperature prediction loss, and (d) is the current prediction loss. Detailed Implementation

[0024] It should be noted that the following detailed descriptions are exemplary and are intended only to describe specific embodiments and to provide further explanation of the invention, and are not intended to limit the scope of exemplary embodiments of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0025] The overall concept proposed in this invention is as follows: First, traditional optimal power flow models typically treat the resistance of overhead transmission lines as a constant. However, in actual operation, the temperature of the lines significantly affects their electrical characteristics. Ignoring the impact of temperature changes on line parameters leads to deviations between the optimized results and the actual system state under conditions of large load variations or extreme weather, ultimately causing inaccurate calculations. To improve the accuracy of power flow calculations, this invention considers the resistance of overhead transmission lines as a function of temperature rather than a constant, based on their electrothermal characteristics. A heat balance equation is used to link the current and temperature of the transmission lines, constructing an optimal power flow model that takes into account the electrothermal characteristics of the lines. Based on this design, by considering the temperature-resistance variation characteristics, the resistance is updated in real-time according to the operating temperature of the conductors during the calculation process. This improves calculation accuracy, reduces the deviation between the model and reality, and introduces temperature variables to construct temperature constraints on the lines. This fully explores the transmission potential of the transmission lines and avoids safety hazards caused by excessively high equipment temperatures, ensuring the feasibility of the solution to the AC optimal power flow problem and the safety of power system operation.

[0026] Secondly, considering the limitations of traditional neural networks in model solving applications, such as difficulty in balancing computational accuracy and efficiency, and a lack of interpretability, this invention introduces a Physically Informed Neural Network (PINN). By incorporating physical equations into the training process, a dual-driven training method of data and physical equations is constructed on top of the original pure data training method. This allows the neural network to use physical laws as guiding equations for optimization parameters while referencing given data. This leverages the powerful function fitting ability of neural networks while also giving the model a certain degree of interpretability. This invention optimizes the PINN model structure and incorporates the KKT conditional encoding related to the optimal power flow model into the loss function term of the neural network as the physical guiding equation. This dual-driven neural network of data and physical equations is used to solve the constructed optimal power flow problem, realizing the calculation of the optimal setpoint of generator power, node voltage, line current, and line temperature in the power system, effectively improving computational accuracy and efficiency.

[0027] Example 1 To address the problems of insufficient calculation accuracy, high data dependence, and slow calculation speed in existing power system optimal power flow calculation methods, this embodiment proposes a new AC optimal power flow calculation method for power systems, such as... Figure 1 As shown, the specific steps include: Step S1: Based on the electrothermal characteristics of overhead transmission lines in the power system, derive a simplified steady-state thermal balance equation.

[0028] Specifically, such as Figure 2 As shown, based on the heat exchange principle of overhead transmission lines, the original heat balance equation for overhead transmission conductors / lines can be established, which can be expressed as: (1) in, The mass per unit length of an overhead transmission line; The temperature of overhead power transmission lines; Specific heat capacity of overhead transmission lines; For time; This represents the current value passing through the transmission line; This is the series resistance value of the overhead transmission line; The length of the transmission line; It is the amount of solar radiation absorbed per unit length of transmission line; It is the convective heat dissipation per unit length of transmission line; It is the amount of heat dissipated by radiation per unit length of transmission line.

[0029] The solar radiation heat absorption, convective heat dissipation, and radiative heat dissipation per unit length of the above-mentioned overhead transmission lines These are all parameters closely related to environmental parameters. The derivation and description of their specific functional relationships are as follows: (1) Derive the solar radiation heat absorption per unit length of overhead transmission line The calculation formula is: (2) in, Where is the diameter of the conductor's cross-section. The light absorption rate of the conductor. The intensity of solar radiation heat due to weather conditions; Effective solar incidence angle; The solar altitude correction parameter is expressed as follows: (3) In the above formula, For altitude, according to relevant statistics, the altitude range of locations where overhead cables are erected is approximately 0-5000 meters. Within this range, the solar altitude correction parameter can be determined. With altitude It increases monotonically with the increase of .

[0030] Effective angle of solar incidence The expression is: (4) In the above formula, The solar altitude angle, ranging from 0 to 90°, is the angle between the direct sunlight and the Earth's horizontal plane. The solar azimuth angle is calculated using the following formula: (5) (6) (7) in, The azimuth angle of the overhead transmission line is taken as due south in the standard. The solar azimuth constant is determined by the time angle. and conversion factor The relevant calculation formula is given by equation (7); The latitude of the location of the overhead power transmission line. Let be the declination angle of the sun, which varies between -23.43° and 23.43°. The corresponding calculation formula is: (8) in, Indicates the day of the year, for example, January 21st is the 21st day of the year; weather and solar radiation intensity. Given by the following expression: (9) in, The solar altitude angle, Solar altitude angle of Power of 1 The range is 1-6. As its coefficient, it represents the solar radiation heat intensity factor. The specific value depends on the external environment of the location of the overhead transmission line. It is mainly divided into clean environment and industrial environment. The reference values ​​are given in Table 1 below.

[0031] Table 1. Polynomial factors of solar radiation heat intensity under different external environments

[0032] (2) Deriving the convective heat dissipation The calculation formula is as follows: Typically, convective heat loss is taken as the maximum of forced convection and natural convection, and the calculation formula is: (10) in, To facilitate forced convection cooling, For natural convection heat dissipation, The specific expression is: (11) in, For the temperature of the conductor; The ambient temperature surrounding overhead transmission lines; The atmospheric density at the location of the overhead power line is calculated using the following formula: (12) in, The average temperature of the overhead transmission line and its surrounding environment is calculated using the following formula: .

[0033] Forced convection cooling is a function of the wind speed around the conductor, and its calculation formula differs depending on the wind speed. At low wind speeds... and high wind speed The calculation formulas are as follows: (13) In the above formula (13), The Reynolds number is used to measure the ratio of inertia to viscous forces in an airflow, and its magnitude depends on the wind speed. The thermal conductivity of air; This is the correction factor. The specific expressions for the above parameters are as follows: (14) (15) (16) (17) in, Let be the dynamic viscosity coefficient of air, and its calculation formula is shown in equation (17); This is the wind direction angle.

[0034] (3) Derive the formula for calculating the radiative heat dissipation of overhead transmission lines. Radiative heat dissipation refers to the dissipation of energy in the form of radiation when there is a temperature difference between the conductor and the surrounding environment. Its essence is the Stefan-Boltzmann law, and the relevant expression is as follows: (18) in, The emissivity of a conductor is related to the surface material properties of the conductor. Both radiative heat dissipation and radiative heat absorption involve energy exchange through radiation; however, radiative heat absorption leads to an increase in conductor temperature, while radiative heat dissipation leads to a decrease in conductor temperature. The two types of radiation have opposite effects on conductor temperature.

[0035] Finally, a simplified steady-state heat balance equation is obtained. Specifically, for a system with slowly changing power load, assuming the system is in a steady state, the dynamic heat balance process of the transmission line can be ignored. Based on this, the heat balance equation of the overhead transmission line can be simplified. According to the above equation (1), the steady-state heat balance equation is constructed as follows: (19) Furthermore, to simplify the calculation equations and highlight the relationship between temperature and resistance, typical values ​​were taken from various meteorological conditions. It was assumed that all three phases of the overhead transmission line were under the same meteorological conditions, all conductors were made of the same material (i.e., they had the same thermal properties), the internal and surface temperatures of each transmission line were the same, the altitude was 10m, and the latitude was 37°N. The atmospheric environment was set as clean air, with solar radiation absorptivity and emissivity both taken as 0.8. The line azimuth angle was taken as 1.57° (unitless), the solar azimuth angle was taken as a typical value of 90°, the wind speed was taken as 0.61m / s, and the wind direction angle was 0.78° (unitless). The transmission line type was steel-cored aluminum stranded wire with a cross-sectional diameter of 28.1mm. 2 The ambient temperature is 25℃. Based on the above settings, the expressions for convective and radiative heat dissipation, which are easy to calculate and analyze, are as follows: (20) (twenty one) in, The convective heat dissipation coefficient, Let be the radiation heat dissipation coefficient; considering the electrothermal characteristics of the circuit, the relationship between conductor resistance and temperature is: (twenty two) in, For reference temperature, 20℃ is generally taken as the reference temperature. The reference resistance value at the reference temperature is provided by the overhead transmission line manufacturer. This is the temperature correction factor for the resistor. This represents the current actual temperature of the line. This is the corrected resistance value.

[0036] Based on the simplified formula and parameter values ​​mentioned above, the coefficients and parameter values ​​of the simplified steady-state thermal equilibrium equation can be determined as shown in Table 2 below.

[0037] Table 2. Coefficients of the heat balance equation

[0038] To facilitate calculations, the steady-state thermal balance equation is transformed into per-unit form, which clarifies the relationship between line current and line temperature, and can be expressed as: (twenty three) (twenty four) (25) in, These are the per-unit values ​​for line current and resistance, respectively, with subscripts... Representing the One line; This is the system's power reference value. These are the reference values ​​for voltage and current, respectively. This is the impedance reference value; It is the first The length of the line.

[0039] Step S2: Using the minimum power generation cost as the objective function and combining the constraints, construct an AC optimal power flow model that takes into account the line's electrothermal characteristics, and derive the corresponding KKT conditions for the model.

[0040] Specifically, the objective function of the AC optimal power flow problem, which aims to minimize power generation costs, can be expressed as: (26) in, These represent the vectors of the generator's active power output and reactive power output, respectively. These represent the linear cost coefficients for the generator's active power output and reactive power output, respectively. Common models assume that adjusting reactive power is costless, so the cost coefficient for reactive power is usually 0.

[0041] For a given power system topology, the optimal setpoint for generator output depends on the active and reactive load demands at various points in the system, and is determined by... The injected power at each node can be expressed as follows: For a given load demand, the injected power at each node can be expressed as: (27) In formula (27), Represents a node in the system. This represents the total number of nodes in the system. These represent the nodes respectively. The active and reactive power of the generator at that location, Representing nodes respectively The load power required at the location.

[0042] According to the nodal power balance equation, the injected power at the system nodes can be expressed as: (28) (29) in, Representing nodes respectively The real and imaginary parts of the voltage at the point; Representing nodes respectively and nodes As can be seen from equation (22), in a system considering the electrothermal characteristics of the line, the resistance of the line is a function of the line temperature. When constructing the admittance matrix, the line resistance will affect the admittance matrix parameters of the power grid. Therefore, the admittance matrix of the overhead transmission line is a function of the line temperature.

[0043] For each overhead transmission line, there is a steady-state thermal balance constraint, which is used to represent the constraint relationship between the line current and the line temperature. This steady-state thermal balance constraint can be represented by equation (23).

[0044] In addition, the above model also includes constraints: 1) Upper and lower limits of generator active and reactive power output: (30) Among them, subscript Representing the One generator, Representing the first The lower and upper limits of the active power output of the generator. Representing the first The lower and upper limits of reactive power output of the generator; This indicates the number of nodes connected to the generator in the system.

[0045] 2) Node voltage amplitude constraints: (31) in, Represents a node Voltage amplitude at that point Representing nodes respectively The lower and upper limits of the voltage amplitude. This represents the total number of nodes in the system.

[0046] 3) Maximum allowable operating temperature constraint for the line: (32) in, Indicates the first The temperature of the line, Indicates the first The maximum allowable operating temperature of the line, This represents the total number of lines in the system.

[0047] Furthermore, in traditional optimal power flow models, line overload is determined by comparing the line current with the maximum allowable current; therefore, it is necessary to specify the maximum allowable current carrying capacity of the line in steady state. From the steady-state thermal balance equation, it is known that there is an equality constraint between line current and line temperature. Therefore, the line current can be indirectly constrained by specifying the maximum allowable steady-state operating temperature of the line. This is achieved by selecting the maximum allowable operating temperature for each overhead transmission conductor. The maximum current carrying capacity of each line can be calculated using the following formula: (33) (34) in, To ensure the line temperature reaches the upper limit Current at that time To ensure the line temperature reaches the upper limit The resistance (per unit value) at that time can be used to convert the upper limit of line temperature into the upper limit of line current carrying capacity. The upper limit of line current carrying capacity is constrained as follows: (35) By introducing a temperature variable to recalculate the maximum current carrying capacity constraint of the line, the transmission potential of the line can be maximized while ensuring the safe and stable operation of the power system. In this embodiment, the temperature constraint is transformed into the maximum current carrying capacity constraint, and the same physical quantities are used as in the traditional model, which helps to make intuitive comparisons and better reflects the additional current carrying capacity of the line after considering the dynamic characteristics of the transmission line. This avoids wasting the power transmission capacity of the line due to overly conservative model parameter settings.

[0048] Combining all the objective functions and constraints mentioned above, the AC optimal power flow model considering the line electrothermal characteristics can be reformulated into the following intensive form: (36) (37) (38) (39) (40) In the above formula, formula (36) represents the linear cost objective function of the optimal AC power flow; formulas (37) and (38) represent the power balance equation constraint and the steady-state heat balance equation constraint, respectively, both of which are equality constraints; formula (39) represents the various inequality constraints, where ,when hour, ; and when hour, ,in Indicates the relevant upper limit constraint, Formula (40) represents all lower limit constraints; Formula (40) represents line current constraints.

[0049] In this way, this embodiment incorporates the relationship between resistance and temperature and the heat balance equation to solve the AC-OPF problem on the basis of conventional AC-OPF, making the constructed model more in line with physical reality, correcting the calculation deviation caused by the traditional model treating resistance as a constant value, and further exploring the power transmission capacity wasted due to the overly conservative setting of model parameters.

[0050] As a further implementation, the KKT conditions (Karush-Kuhn-Tucker conditions, necessary conditions for the optimal solution of nonlinear programming) corresponding to the above-mentioned AC optimal power flow model considering the line electrothermal characteristics are derived. Specifically, based on the above-mentioned AC optimal power flow model considering the line electrothermal characteristics, its corresponding Lagrange equation can be obtained as follows: (41) in, and These are the Lagrange multipliers for equality constraints and inequality constraints, respectively. Let represent the Lagrange function. Based on this, KKT conditions of the following form can be constructed: (42) (43) (44) (45) (46) (47) (48) (49) Among them, formula (42) constitutes the stationary point condition in the KKT conditions, formulas (43)-(46) constitute the original feasibility conditions, formula (47) is the dual feasibility condition, and formulas (48)-(49) constitute the complementary relaxation conditions. All are Lagrange multipliers corresponding to the inequality constraints, with subscripts... This indicates that the constraint originates from node-related physical quantity constraints, subscript This indicates that the constraint originates from the physical quantity constraints related to the line. The above KKT conditions are necessary conditions for satisfying optimality and will be used as guiding equations to construct the PINN framework.

[0051] Step S3: Construct a physical information neural network model, introduce KKT conditions into the model's loss function, and train the model using the training set.

[0052] Specifically, considering the increased model complexity due to the introduction of additional constraints and variables, this embodiment adopts PINN to construct an end-to-end solution method in order to reduce computational costs and improve computational efficiency. It constructs and trains a physical information neural network to solve the optimal power flow problem considering the line's electrothermal characteristics. That is, physical constraints are incorporated into the network to guide network training, thereby improving the optimization solution speed for various similar scenarios, further enhancing the interpretability of the model, and ensuring the reliability of the calculation results.

[0053] First, the PINN network model structure is constructed. Building upon previous data-driven deep neural networks, PINN incorporates existing physical rules and equations, integrating physical knowledge into the neural network framework. Physical equations can be integrated into various modules of the neural network. A widely used and straightforward method is to construct corresponding loss function terms using physical equations. These loss function terms constrain PINN's behavior, ensuring the results satisfy both data requirements and physical realities, exhibiting strong interpretability and excellent generalization ability. Furthermore, using physical equations as guidance transforms the neural network from a single data-driven model to a dual-driven model of physical rules and data, reducing dependence on the amount of data. The neural network can be trained with a smaller dataset to achieve performance comparable to traditional neural networks. Considering PINN's data + physical dual-driven approach, for the training set, it's unnecessary to determine the optimal power generation scheduling arrangement for every sample point; instead, a certain proportion of matching points are set. For these matching points, the neural network's prediction is considered the optimal value, and the corresponding data loss term is all 0. Therefore, the loss function incorporating the physical equation loss term can be expressed as: (50) in, This represents the number of physical equations incorporated. This represents the loss term in the constructed physical equations. Represents the number of samples. Represents the actual label value. This represents the calculated value (also known as the predicted value) of PINN; These are the weighting coefficients of the data loss term and the physical equation loss term, respectively, which determine the reference weight values ​​of the optimization parameters during backpropagation.

[0054] In this embodiment, for the optimal power flow problem considering the line's electrothermal characteristics, the corresponding physical information neural network is designed as follows: Figure 3 As shown, it includes: (1) Input layer, used to input the active power load demand and reactive power load demand of the power system, wherein, These represent the active power load demand and reactive power load demand in the power network, respectively.

[0055] (2) Multiple subnetworks, including These are subnetworks used to calculate the generator's active and reactive power output setpoints, the real and imaginary parts of node voltages, line temperature, the real and imaginary parts of line currents, and all Lagrange multipliers. Specifically, these are the power prediction subnetwork, voltage prediction subnetwork, temperature prediction subnetwork, current prediction subnetwork, and Lagrange multiplier prediction subnetwork. In this embodiment, subnetworks are used instead of a unified network for design. The advantage of this approach is that the PINN architecture, composed of various subnetworks, can be viewed as multi-objective task training. This effectively shortens the training time while maintaining an effective network size, improving the learning accuracy of the physical information neural network. Furthermore, the optimal number of neurons in each subnetwork can be set according to different example sizes.

[0056] (3) Physical information layer, which is used to encapsulate the physical equations that guide the training of the neural network, receive input samples and the prediction results of the neural network and output the loss term of the physical equation. It is the core module of the physical information neural network. In this embodiment, the physical equation is derived from the KKT conditions of the model built above.

[0057] (4) Output layer, used to output the predicted AC optimal power flow distribution data of the power system, including the active and reactive power output setpoints of generators in the power system, node voltages, current and temperature of transmission lines, and all Lagrange multipliers.

[0058] in, These are the predicted power values, predicted voltage values, predicted line temperature values, predicted current values, and predicted Lagrange multipliers values; correspondingly, , , , , These are the true label values ​​of each variable, used to calculate the data prediction error; , , , , These are the weights of the data loss error terms corresponding to each variable; These are the data loss terms for each sub-neural network. The physical loss term in the physical information layer, together with the loss term in the physical information layer, constitutes the total loss (LOSS) of the entire physical information neural network. During training, the neural network determines whether the current LOSS has reached a set condition. If it has, the training will terminate; otherwise, the weights and biases in the hidden layer will be optimized using the backpropagation algorithm based on the LOSS value.

[0059] Secondly, the hidden layer sub-networks are optimized. In this embodiment, the ReLU function is selected as the activation function for neurons in the hidden layer. Due to its unique nonlinear structure, the ReLU function maps all output values ​​to 0 on the negative half-axis of the independent variable, while exhibiting the properties of a linear activation function on the positive half-axis. This accelerates neural network training. Its mathematical expression is: (51) (52) in, For this intermediate variable, These represent the weights and biases of the neuron, respectively. This is the output value of the previous neuron. This is the output value of this neuron.

[0060] For the output layer neurons of the hidden layers, since the number of neurons in the output layer must be equal to the number of output variables, different output layer sizes need to be set for different computational examples. Furthermore, to avoid interference from the activation function on the output values, a linear activation function is chosen for each output layer. The expression for the linear activation function is: (53) In addition, to prevent overfitting during model training, L2 regularization is added to the neurons in the hidden layers. Overfitting is prevented by constraining the magnitude of all neuron weights. The core principle is to add an extra penalty term to the loss function, which is the sum of the squares of all neuron weights. The formula is: (54) in, For the total loss function, For the original loss function, is the control coefficient for L2 regularization, and is a hyperparameter of the neural network used to control the degree of regularization; This represents the sum of squares of all weights in a physical information neural network.

[0061] Next, the physical information layer and the total loss function are optimized. The KKT conditions for the optimal power flow optimization problem, considering the line's electrothermal characteristics, are necessary conditions for the optimal solution. Therefore, the KKT conditions are used as the physical information loss term for this PINN, and a unified loss function is constructed by combining them with the data loss term. The relevant formula can be expressed as: (55) (56) (57) (58) Among them, all those with superscript ( The variable ) represents the prediction result of the neural network; This represents the physical information loss term corresponding to the stationary conditions in the KKT conditions; This refers to the physical information loss item corresponding to the original feasibility conditions; This refers to the physical information loss term corresponding to duality feasibility. This refers to the physical information loss term corresponding to the complementary relaxation condition; Let ReLU be the function used in this scenario. It utilizes the one-sided truncation property to logically express inequality constraints. Specifically, when the left-hand side of the inequality constraint is negative (the constraint is satisfied), the ReLU function maps it to zero, eliminating the penalty contribution of that constraint term. When the left-hand side is positive (i.e., the constraint is violated), the ReLU function retains the original value as the penalty measure. This design not only conforms to the physical principle that the degree of constraint violation is positively correlated with the penalty value, but also automatically filters out invalid negative values ​​through the nonlinear characteristics of the activation function, ensuring the non-negativity of the loss function. Furthermore, the linear response characteristic of ReLU (gradient is 1 in the positive region) provides a stable gradient propagation path for the optimization process, making it more conducive to solving constrained optimization problems compared to other nonlinear functions.

[0062] Based on the above physical information loss term, and combined with the data loss term of each sub-network, a unified total loss equation for the physical information neural network is constructed as follows: (59) in, This indicates the number of data points in the training set. This indicates the number of collocations in the training set; , , , , These represent the average absolute losses for power, voltage, temperature, and current, respectively. This represents the mean absolute loss under the KKT conditions; , , , , , These represent the weight values ​​of the relevant loss terms. It should be noted that the pairwise data does not participate in the calculation of the data loss term; during training, its loss term is considered to be zero, and only its physical information loss term is calculated.

[0063] After the model and the physical information neural network architecture for solving are built, training data samples are obtained to construct a training set. The training data samples include the active power load demand and reactive power load demand of the power system, as well as the AC optimal power flow distribution data of the power system. The active power load demand and reactive power load demand of the power system are used as the input of the model, and the AC optimal power flow distribution data of the power system is used as the output of the model. The physical information neural network model is trained using the training set until the loss function is minimized, and the trained model is obtained.

[0064] Step S4: Obtain the active and reactive power load demand of the power system, input the trained physical information neural network model, and output the optimal AC power flow distribution data of the power system.

[0065] This embodiment focuses on constructing an optimal power flow model that considers the electrothermal characteristics of power lines and builds the PINN framework to solve the optimal power flow problem, establishing a mapping relationship between load characteristics and output variables. Specifically, based on the traditional optimal power flow model, a steady-state thermal balance equation is incorporated, adding line temperature as a variable to the solution framework. The line resistance is relaxed to be a function of temperature rather than a constant value, thus constructing an optimal power flow model that considers the electrothermal characteristics of power lines and providing relevant KKT conditions. Subsequently, a corresponding physical information neural network is constructed to solve the proposed problem. A data + physics dual-driven training mode is adopted, and relevant sub-networks are designed. The KKT conditions are added as physical guiding equations to the neural network framework, and a corresponding loss function is designed for the optimal power flow model that considers the electrothermal characteristics of power lines to fully utilize the physical equations of the optimal power flow model.

[0066] To further verify the effectiveness of the method proposed in this embodiment, the performance of the constructed algorithm was tested in the IEEE 14-node standard computational test. The topology diagram of the IEEE 14-node standard computational test is shown below. Figure 4 As shown, the IEEE 14-node system has 14 nodes, of which node 1 is the balancing node, serving as the phase angle reference for the entire system, and its voltage imaginary part is 0; there are 20 lines, of which 3 are transformer branches; there are five generators, with nodes 1, 2, 3, 6, and 8 being generator nodes connected to the generators; and nodes 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, and 14 being load nodes connected to the loads, and each load is independent of the others.

[0067] For optimal power flow problems considering line electrothermal characteristics, it is necessary to specify data such as the upper and lower limits of generator output, the upper and lower limits of voltage amplitude, and the upper limit of current amplitude in the calculation example. A system power reference value is then selected. Table 3 shows the basic parameters of the generator.

[0068] Table 3 Generator Parameters for the IEEE 14-Node Example

[0069] Where Node represents the generator number, which is also the node number of the node where the generator is located; the upper and lower limits of active power and reactive power output are Pmax, Pmin, Qmax, and Qmin, respectively, all in per-unit form, with the unit being pu; , These represent the active and reactive power output costs of the generator, respectively. To simplify the model, only the active power cost is considered, while the reactive power output cost of the generator is ignored.

[0070] Table 4. Branch data for the IEEE 14-node example.

[0071] Table 4 above shows the basic data used in the example, where the Name column is the line number, and Busi and Busj are the start and end node numbers of the line, respectively; the line uses... An equivalent circuit model is used, where R and X represent the series resistance and series reactance of the line, respectively, both per-unit values ​​in pu; R represents the initial series resistance, used as an initial value in the calculation of the line's electrothermal characteristics; Bhalf is half the parallel susceptance of the line, also per-unit value in pu; BaseKV is the voltage reference value of the nodes connected to each line, in kV; LenKm is the length of each line, in km; Imax is the per-unit value of the maximum amplitude of the line current calculated based on the steady-state thermal balance equation, in pu; and, considering line safety, the maximum operating temperature of all lines is set. The upper limit of the line current Imax is calculated according to equation (33). The upper limit of the voltage of all nodes is set to... The lower voltage limit is set to All voltage values ​​are per-unit values, and the unit is pu.

[0072] Next, the hyperparameters of the physical information neural network are set. Hyperparameters are parameters set before the neural network runs and are used to control the training behavior and performance of the neural network. In all samples, assuming that the active and reactive loads of each node are independent and their values ​​are specified as 60%-100% of their respective maximum power, 10,000 sets of load data are randomly collected in the input domain using Latin hypercube sampling. 3,000 sets are allocated to the test set, 2,000 sets are used to generate data points, and 5,000 sets are used as pairing data points. In the training set, the data labels for data points are calculated using the interior point method, while the data labels for pairing points do not need to be calculated. This data partitioning method effectively reduces the amount of data required.

[0073] Table 5 below shows the size of each sub-network in this example, Table 6 below shows the hyperparameter values ​​required during network training, and Table 7 below shows the calculated weights of each loss term.

[0074] Table 5. Size of different sub-networks

[0075] Table 6 Hyperparameters of PINN

[0076] Wherein, Learning Rate is the learning rate, Regularization Rate is the regularization rate, Epochs is the number of iterations (here, it means 1000 iterations), and Batch Size is the batch size. In each iteration, the neural network is trained by splitting the dataset according to the Batch Size.

[0077] Table 7 Weights of each loss item

[0078] The loss curve during the final training process is as follows: Figure 5 As shown (the horizontal axis Epochs represents the period), for a test set consisting of 3000 samples, the mean absolute error (MAE) and mean absolute percentage error (MAPE) of active power, reactive power, real part of voltage, imaginary part of voltage, temperature, real part of current, and imaginary part of current are calculated respectively, and compared with the test set. The calculation formulas for MAE and MAPE are shown in equation (60) and equation (61) respectively: (60) (61) in, For the sample size, and These are the label values ​​and predicted values ​​of the relevant variables, respectively. The calculated results are shown in Tables 8 and 9 below.

[0079] Table 8 MAE and MAPE for Voltage and Power

[0080] Table 9 MAE and MAPE for Temperature and Current

[0081] Among them, the mean absolute error of voltage, power and current are all in per-unit form, with the unit being pu; while the MAE of temperature is the denormalized value, with the unit being ℃.

[0082] Based on the results of the above examples, the PINN constructed in this embodiment shows good comprehensive performance in the optimal power flow problem that takes into account the line's electrothermal characteristics. From the perspective of error indicators, the model has high prediction accuracy in core indicators such as the real part of voltage and active power. Moreover, the difference between MAE and MAPE of the test set and the training set is not significant, indicating that the overfitting phenomenon of the model is within an acceptable range and the model has good generalization performance.

[0083] Example 2 This embodiment provides a power system AC optimal power flow calculation system, including: The steady-state thermal balance derivation module is used to derive simplified steady-state thermal balance equations based on the electrothermal characteristics of overhead transmission lines in power systems. The optimal power flow model construction module is used to construct an AC optimal power flow model that takes into account the electrothermal characteristics of the line, with the objective function of minimizing power generation cost and in combination with constraints. The module also derives the corresponding KKT conditions for the model. The constraints include the line steady-state thermal balance equation constraint and the line current carrying limit constraint obtained by converting the line maximum allowable operating temperature constraint based on the steady-state thermal balance equation. The neural network model building module is used to build a physical information neural network model, introduce KKT conditions into the model's loss function, and train the model using the training set. The optimal power flow calculation module is used to obtain the active and reactive power load demand of the power system, input the trained physical information neural network model, and output the AC optimal power flow distribution data of the power system.

[0084] Example 3 This embodiment provides an electronic device, including: a memory for storing executable instructions; and a processor for executing the executable instructions stored in the memory to implement the method provided in this embodiment.

[0085] Example 4 This embodiment also provides a computer-readable storage medium storing executable instructions, which, when executed by a processor, will cause the processor to execute the method described above in this embodiment.

[0086] Example 5 This embodiment provides a computer program product including executable instructions, which are computer instructions; the executable instructions are stored in a computer-readable storage medium. When the processor of an electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, the electronic device performs the method described in this embodiment.

[0087] The steps and methods involved in Embodiments 2 to 5 above correspond to those in Embodiment 1. For specific implementation details, please refer to the relevant description section of Embodiment 1. The term "computer-readable storage medium" should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.

[0088] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0089] The above description is only a preferred embodiment of the present invention. Although the specific implementation of the present invention has been described in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that, based on the technical solution of the present invention, various modifications or variations that can be made by those skilled in the art without creative effort are still within the scope of protection of the present invention.

Claims

1. A method for calculating optimal AC power flow in a power system, characterized in that, include: Based on the electrothermal characteristics of overhead transmission lines in power systems, a simplified steady-state thermal balance equation is derived. With the minimum power generation cost as the objective function, and in combination with constraints, an AC optimal power flow model considering the electrothermal characteristics of the line is constructed, and the corresponding KKT conditions are derived. Among them, the constraints include the line steady-state thermal balance equation constraint and the line current carrying limit constraint obtained by converting the line maximum allowable operating temperature constraint based on the steady-state thermal balance equation. A physical information neural network model is constructed, KKT conditions are introduced into the model's loss function, and the model is trained using the training set. The system obtains the active and reactive power load demands of the power system, inputs the trained physical information neural network model, and outputs the optimal AC power flow distribution data of the power system.

2. The method for calculating optimal AC power flow in a power system as described in claim 1, characterized in that, The derivation process of the simplified steady-state thermal balance equation is as follows: Based on the heat exchange principle of overhead transmission lines, an original heat balance equation is established. The formulas for calculating the solar radiation heat absorption, convective heat dissipation, and radiative heat dissipation per unit length of overhead transmission lines are derived respectively; among them, the convective heat dissipation is taken as the maximum value of forced convection heat dissipation and natural convection heat dissipation. Assuming the power system is in a steady state, ignoring the dynamic thermal balance process, taking the data of various meteorological conditions as typical values, and substituting the calculation formulas of solar radiation heat absorption, convective heat dissipation and radiative heat dissipation into the original thermal balance equation, we obtain a simplified form of the steady-state thermal balance equation, which is then converted into per-unit value form to clarify the relationship between line current and line temperature.

3. The method for calculating optimal AC power flow in a power system as described in claim 1, characterized in that, The AC optimal power flow model, which takes into account the electrothermal characteristics of the line, takes the minimum power generation cost as the objective function and includes constraints on the node power balance equation, the line steady-state thermal balance equation, the upper and lower limits of generator active and reactive power output, the node voltage amplitude constraint, and the upper limit of line current carrying capacity constraint.

4. The method for calculating optimal AC power flow in a power system as described in claim 1, characterized in that, The physical information neural network model includes: The input layer is used to input the active power load demand and reactive power load demand of the power system. Multiple subnetworks, including a power prediction subnetwork, a voltage prediction subnetwork, a temperature prediction subnetwork, a current prediction subnetwork, and a Lagrange multiplier prediction subnetwork, are used to calculate the generator active and reactive power output setpoints, the real and imaginary parts of node voltages, line temperatures, the real and imaginary parts of line currents, and all Lagrange multipliers, respectively. The physical information layer is used to encapsulate the physical equations constructed based on KKT conditions, receive input samples and the prediction results output by the neural network, and output the physical information loss term. The output layer is used to output the predicted optimal AC power flow distribution data of the power system, including the active and reactive power output setpoints of generators in the power system, node voltages, transmission line currents and temperatures, and all Lagrange multipliers.

5. The method for calculating the optimal AC power flow in a power system as described in claim 4, characterized in that, The loss function of the physical information neural network model includes a data loss term and a physical information loss term for each sub-network. The physical information loss term is constructed based on KKT conditions and includes physical information loss terms corresponding to stationary conditions, original feasibility conditions, dual feasibility conditions, and complementary relaxation conditions.

6. The method for calculating optimal AC power flow in a power system as described in claim 1, characterized in that, The training process of the physical information neural network model is as follows: Acquire training data samples and construct a training set; the training data samples include the active power load demand and reactive power load demand of the power system, as well as the AC optimal power flow distribution data of the power system. The active power load demand and reactive power load demand of the power system are used as the input of the model, and the optimal AC power flow distribution data of the power system are used as the output of the model. The physical information neural network model is trained using the training set until the loss function is minimized, and the trained model is obtained.

7. A power system AC optimal power flow calculation system, characterized in that, include: The steady-state thermal balance derivation module is used to derive simplified steady-state thermal balance equations based on the electrothermal characteristics of overhead transmission lines in power systems. The optimal power flow model construction module is used to construct an AC optimal power flow model that takes into account the electrothermal characteristics of the line, with the objective function of minimizing power generation cost and in combination with constraints. The module also derives the corresponding KKT conditions for the model. The constraints include the line steady-state thermal balance equation constraint and the line current carrying limit constraint obtained by converting the line maximum allowable operating temperature constraint based on the steady-state thermal balance equation. The neural network model building module is used to build a physical information neural network model, introduce KKT conditions into the model's loss function, and train the model using the training set. The optimal power flow calculation module is used to obtain the active and reactive power load demand of the power system, input the trained physical information neural network model, and output the AC optimal power flow distribution data of the power system.

8. An electronic device, characterized in that, include: Memory, used to store executable instructions; The processor, when executing executable instructions stored in the memory, implements the AC optimal power flow calculation method for the power system as described in any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, The device stores executable instructions that, when executed by a processor, implement the AC optimal power flow calculation method for a power system as described in any one of claims 1-6.

10. A computer program product, characterized in that, The computer program product includes executable instructions stored in a computer-readable storage medium; When the processor of the electronic device reads the executable instructions from the computer-readable storage medium and executes the executable instructions, it implements the AC optimal power flow calculation method for the power system according to any one of claims 1-6.

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