Name: transmission line power limits and loadability
Uploaded: 2024-11-21T14:54:26.000Z
Duration: 1 min 23 s
Channel: JoVE

使用相量形式的ABCD参数推导有损线路的最大功率传输。
发送端与接收端电压和电流之间的矩阵关系决定了接收端电流，进而可计算输送至接收端的复功率。
由此可推导出输送至受电端的有功功率和无功功率。
将无损耗线路的参数代入可简化有功功率方程。
当相位角等于总串联阻抗角时，出现理论上的最大有功功率或稳态稳定极限。
然而，当串联阻抗大于电抗时，会降低最大有功功率。
实际上，输电线路并不会根据额定端电压和90度的相位角偏移来输送其理论上的最大功率。
实际线路负载能力曲线绘制在理论稳态稳定极限下方，基于电压降限值为0.95或更低，以及最大角位移为30至35度。
对于25公里以下的短线路，其负载能力受限于导体或终端设备的热容量，而非电压降或稳定性。

最大功率流和线路负载能力

有损传输线的最大功率流是从使用相量形式的 ABCD 参数推导出的。这些参数在发送端和接收端电压和电流之间建立了矩阵关系，从而可以确定接收端电流。矩阵关系帮助计算传输到接收端的复功率，从中推导出有功功率和无功功率分量。
<img alt="Equation1" src="/files/ftp_upload/16148/16148_Equation%201.svg" style="height: 36px;">
<img alt="Equation2" src="/files/ftp_upload/16148/16148_Equation%202.svg" style="height: 36px;">

对于无损线路，简化有助于有功功率的计算。
<img alt="Equation3" src="/files/ftp_upload/16148/16148_Equation%203.svg" style="height: 36px;">
当发送端和接收端电压之间的相位角等于总串联阻抗角时，就会出现理论上的最大有功功率，或稳态时的稳定性极限。
<img alt="Equation4" src="/files/ftp_upload/16148/16148_Equation%204.svg" style="height: 36px;">
然而，如果串联阻抗超过电抗，则输送的最大有功功率就会降低。
在实际情况下，由于额定端电压和 90 度角位移的限制，输电线无法输送理论上的最大功率。相反，实际线路负载能力曲线（绘制在理论稳态时的稳定性极限以下）要考虑 0.95 或更低的电压降极限和 30 到 35 度的最大角位移。这些实际限制通过保持可接受的电压水平和限制角位移以避免不稳定，进而确保电力系统的可靠运行。
对于短距离输电线（通常长度在 25 公里以下），负载能力不受电压降或稳定性问题的限制，而是受导体或终端设备的热容量评级的限制。这些热限制可确保设备在高电流条件下不会出现导致损坏或故障的过热。
总之，理论和实际考虑因素（包括阻抗特性、电压限制和热容量评级）会影响输电线的最大功率流和线路负载能力。管理这些因素对于电力系统的安全高效运行至关重要。

利用相量形式的ABCD参数推导有损线路的最大功率传输。
发送端与接收端之间的矩阵关系……

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

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 real power calculation is easier to handle. The theoretical maximum real power, also called the steady-state stability limit, occurs when the phase angle between the sending-end and receiving-end voltages equals the total series impedance angle.

If the series impedance is larger than the reactance, the maximum real power that can be delivered drops. In real power systems, lines usually do not operate at this theoretical limit. The practical line loadability curve stays below the stability limit because it must respect rated terminal voltages and a 90-degree angular displacement limit.

This practical curve also uses a voltage-drop limit of 0.95 or less and a maximum angular displacement of about 30 to 35 degrees. These limits help keep voltage levels acceptable and reduce the risk of instability. For short transmission lines, usually under 25 kilometers, loadability is not set by voltage drop or stability. Instead, the main limit is thermal rating, which keeps the conductor and terminal equipment from overheating under high current.

Transmission Line Power Limits and Loadability

transmission-line-power-limits-and-loadability

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 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 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...

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 ABCD parameters, voltage limits, stability, and thermal ratings set maximum power flow and loadability in transmission lines.

Transmission Line Power Limits and Loadability | 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

Transmission Line Power Limits and Loadability

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