Name: power flow analysis in electric networks
Uploaded: 2024-11-21T14:54:26.000Z
Duration: 1 min 26 s
Channel: JoVE

潮流计算问题用于求解平衡三相稳态电力系统中的电压幅值、相角以及有功和无功功率潮流。考虑其单线图提供的三种输入数据。每个节点包含四个变量：其中两个为输入数据，另外两个为计算所得。
输送到母线的功率被分为发电机功率和负荷功率两部分。母线分为三种类型。对于每种类型，某些变量作为输入给定，而其他变量则由潮流计算程序求解。
输电线路用等效π电路表示。输入数据包括串联阻抗、并联导纳、连接的母线以及最大兆伏安额定值。
这些数据构建了节点导纳矩阵，进而建立网络的节点方程，并确定输送至各节点的复功率。
高斯-赛德尔潮流解法利用这些节点方程来计算输送的复功率。通过对推导出的方程取实部和虚部，为每个母线建立功率平衡方程。
牛顿-拉夫逊潮流计算利用这些非线性潮流方程，具有更高的精度和更快的收敛速度。

功率流问题及解决方案

功率流问题分析对于确定网络组件（例如输电线、变压器和负载）中的有功和无功功率流至关重要。电力系统的单线图提供有关母线、输电线和变压器的数据。系统中的每个母线 k 都具有四个关键变量:电压幅度 V_k​、相位角 δ_k、有功功率 P_k 和无功功率 Q_k。这四个变量中的两个是输入，而功率流程序计算其余变量。输送到母线 k 的功率可以用发电机和负载组件来表示:

<img alt="Equation1" src="/files/ftp_upload/16149/16149_PT_equation_1.svg" style="height: 14px;">
<img alt="Equation2" src="/files/ftp_upload/16149/16149_PT_equation_2.svg" style="height: 14px;">

根据运行特性，电力系统中的母线可分为三类:平衡母线、负荷（PQ）母线和电压控制母线。平衡母线的电压幅值接近 1.0/单位，相位角为零度。在负荷（PQ）母线处，有功功率和无功功率是指定的，而电压幅值和相位角未知。对于电压控制母线，有功功率和电压幅值是给定的。
网络的电流方程以导纳矩阵表示:
<img alt="Equation3" src="/files/ftp_upload/16149/16149_PT_equation_3.svg" style="height: 13px;">


其中 I 是注入的源电流矢量，V 是母线电压矢量。对于每个母线 k，电流和复数功率为:
<img alt="Equation4" src="/files/ftp_upload/16149/16149_PT_equation_4.svg" style="height: 28px;">
<img alt="Equation5" src="/files/ftp_upload/16149/16149_PT_equation_5.svg" style="height: 16px;">
解决功率流问题的两种主要迭代方法是高斯-赛德尔法和牛顿-拉夫森法。高斯-赛德尔法迭代求解节点方程，使用已知功率值重新计算负载母线的电流，并调整电压控制母线的无功功率，直到收敛。牛顿-拉夫森法将功率流方程线性化，并使用雅可比矩阵进行电压校正，通常收敛速度更快，更适合大型系统。这些迭代方法对于确保电力系统在规定参数范围内运行、维持整个网络的稳定性和效率至关重要。

功率流问题用于计算平衡三相稳态电力系统中的电压幅值、相角以及有功和无功功率流动。

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Steady-State Transmission Lines and Power Flows

稳态传输线与功率流

steady-state-transmission-lines-and-power-flows

KEY TERMS AND CONCEPTS

关键术语与概念

The Power Flow Problem and Solution

power flow analysis in electric networks

Power flow analysis helps determine how real and reactive power move through an electric network. It uses the system’s single-line diagram, which shows the buses, transmission lines, transformers, and loads. The analysis focuses on balanced three-phase steady-state operation.

Each bus in the system has four key variables: voltage magnitude V k, phase angle δ k, real power P k, and reactive power Q k. Two of these four variables are treated as inputs. The power flow program solves for the remaining values.

Buses are grouped by operating role. A swing bus usually has a voltage magnitude close to 1.0 per unit and a phase angle of zero degrees. A load or PQ bus has specified real and reactive power, while its voltage magnitude and phase angle are unknown. A voltage-controlled bus has given real power and voltage magnitude.

The network equations are written with an admittance matrix, where I is the vector of injected source currents and V is the vector of bus voltages. For each bus k, current and complex power are related through these equations. The power flow problem is then solved with iterative methods.

Two main methods are used: Gauss-Seidel and Newton-Raphson. Gauss-Seidel repeatedly solves the nodal equations, updates current for load buses from known power values, and adjusts reactive power for voltage-controlled buses until the results converge. Newton-Raphson linearizes the power flow equations and uses the Jacobian matrix to correct voltages, so it usually converges faster and is better for large systems. These methods help keep the power system within its specified limits and support stable, efficient network operation.

Power Flow Analysis in Electric Networks

transmission-line differential equations

Modeling Voltage and Current on Transmission Lines

Transmission lines carry electric power, and their resistance, inductance, and capacitance are spread out along the line. Because these properties are distributed per unit length, voltage and current change as they move from one end of the line to the other. Differential equations are used to describe that variation. A line-section model helps show how this works. In the model, a small length, Δx, is used to represent the line. Voltage V(x) and current i(x) are measured from the receiving end.

lossless lines

Lossless Lines and Voltage Behavior

Lossless transmission lines in electrical engineering are described by a purely imaginary propagation constant and a resistive characteristic impedance. Their ABCD parameters show an equivalent pi circuit with an imaginary series impedance and a shunt admittance. When the product of beta and line length is less than pi, the line has inductive series impedance and capacitive shunt admittance, which keeps it lossless. Wavelength is the physical length over which voltage or current phase changes...

maximum power flow and line loadability

Transmission Line Power Limits and Loadability

Transmission line power limits and loadability depend on voltage, angle, and line impedance. For lossy transmission lines, ABCD parameters in phasor form are used to relate sending-end and receiving-end voltages and currents. This matrix relation makes it possible to find the receiving-end current and then calculate the complex power delivered to the load. From the complex power, the real and reactive power components can be determined. For a lossless line, the equations become simpler, and...

the power flow problem and solution

Power flow analysis helps determine how real and reactive power move through an electric network. It uses the system’s single-line diagram, which shows the buses, transmission lines, transformers, and loads. The analysis focuses on balanced three-phase steady-state operation. Each bus in the system has four key variables: voltage magnitude V k, phase angle δ k, real power P k, and reactive power Q k. Two of these four variables are treated as inputs. The power flow program solves for the...

control of power flow

Voltage and Reactive Power Control

Voltage and reactive power control help keep a power system stable and efficient. In this topic, generator settings, switched equipment, and transformer taps work together to guide how power moves through the network. A generator can be modeled with a Thevenin equivalent circuit under balanced steady-state conditions. The main terms are terminal voltage Vt, excitation voltage Eg, power angle δ, and positive-sequence synchronous reactance Xg. From this model, the generator current and the...

fast decoupled and dc powerflow

Power Flow Shortcuts for Grid Contingencies

Fast decoupled power flow and DC power flow give engineers faster ways to study a power system during contingencies such as generator outages or transmission line failures. These methods help operators get quick power flow solutions, which is important for real-time system adjustments. Fast decoupled power flow simplifies the Jacobian matrix, the set of equations used in power flow calculations, by neglecting some elements. The method drops J2 and J3 and assumes voltage magnitudes stay close...

key terms and concepts

Learn how power flow analysis tracks voltage, real power, and reactive power in electric networks using bus types, admittance matrices, and iterative methods.

Power Flow Analysis in Electric Networks | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Power Flow Analysis in Electric Networks

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