Name: power-system stability in multi-machine networks
Uploaded: 2024-11-21T14:54:38.000Z
Duration: 1 min 25 s
Channel: JoVE

在包含 M 台同步电机的 N 节点电力系统中，负荷被建模为恒定导纳，且网络节点方程被建立。
导纳矩阵被划分为包含负载导纳和逆向发电机阻抗。
在已知电机电压的情况下，通过求解母线电压方程来确定电机电流和电功率。
暂态稳定性分析需要求解发电机的转子运动方程和网络的潮流方程。
首先，运行故障前潮流程序，以确定初始母线电压、电机电流和电气输出。
将机械输出功率设为等于电输出功率，并将频率初始化为同步角频率。
接下来，计算负荷导纳、电机内部电压以及潮流计算节点导纳矩阵。
根据开关操作、负荷变化或故障等事件调整导纳矩阵。
在每个时间步 t 加上 Δt 时，计算电机的电功率、功率角和转速。
重复此过程直至达到预期的时间范围，确保步长在解的精度与计算时间之间取得平衡。

暂态稳定性和系统控制

多机稳定性分析对于理解具有多台同步电机的电力系统的动态和稳定性至关重要。目标是求解连接到 N 母线电力系统的 M 台电机网络的摆动方程。

在分析系统时，节点方程表示母线电压、电机电压和电机电流之间的关系。节点方程由以下公式给出:

<img alt="Equation1" src="/files/ftp_upload/16170/16170_PT_Equation_1.svg" style="height: 44px;">

V 是母线电压的 N 向量，E 是电机电压的 M 向量，I 是电机电流的 M 向量。Y_11、Y_12、Y_21 和 Y_22 分别是 N×N、N×M、M×N 和 M×M 导纳矩阵。这些方程可以分解为:

<img alt="Equation2" src="/files/ftp_upload/16170/16170_PT_Equation_2.svg" style="height: 28px;">

假设 E 已知，则可以使用高斯消元法或高斯-赛德尔消元法等方法迭代求解第一个方程以得到 V。一旦计算出 V，便可以从第二个方程得到 I。

电机 n 的有功电功率输出为:

<img alt="Equation3" src="/files/ftp_upload/16170/16170_PT_Equation_3.svg" style="height: 16px;">

暂态稳定性计算过程涉及迭代求解摆动方程和功率流方程:

<ul style="list-style-type: disc; padding-left: 35px; margin: 0;">
 <li>确定初始母线电压 V_k、电机电流 I_n 和功率输出 pan。</li>
 <li>计算E_n。</li>
 <li>修改Y_11，并计算Y_22和Y_12。</li>
 <li>初始化t=0。</li>
 <li>确定是否存在任何切换操作或负载变化。</li>
 <li>计算时间 t 时的电机电功率。</li>
 <li>使用初始估计值计算时间 t+Δt 时的功率角δ和机器速度ω。</li>
 <li>细化时间 t+Δt 时的电器功率的估计值。</li>
 <li>最终确定时间 t+Δt 时的功率角δ和机器速度ω。</li>
 <li>增加时间并重复这些步骤直到模拟结束。</li>
 <li>准确的时间步长选择对于避免数值积分的不稳定性至关重要。</li>
</ul>

通过遵循这些步骤并使用提供的方程，工程师可以分析多机电力系统的暂态稳定性并确保在各种条件下可靠运行。

在一个包含 N 个母线和 M 台同步电机的电力系统中，负荷被建模为恒定导纳，网络节点方程被建立。  
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Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Transient Stability and System Controls

暂态稳定性与系统控制

KEY TERMS AND CONCEPTS

关键术语与概念

Multimachine Stability

power-system stability in multi-machine networks

Power-system stability in multi-machine networks focuses on how several synchronous machines behave as they stay connected to an N-bus power system. The analysis solves the swing equations for the machines and tracks how the network responds when conditions change. It is used to study transient stability, which is the system’s ability to stay in step after a disturbance.

The network is written with nodal equations that link bus voltages, machine voltages, and machine currents. In this model, V is the N-vector of bus voltages, E is the M-vector of machine voltages, and I is the M-vector of machine currents. The admittance matrices Y11, Y12, Y21, and Y22 describe the connections between buses and machines.

These equations can be separated into two parts. If E is known, the first equation can be solved iteratively for V using Gauss elimination or Gauss-Seidel methods. After V is found, I can be calculated from the second equation. The real electrical power output of machine n is then determined from these values.

Transient stability is computed by stepping through time and solving the swing equations together with the power flow equations. The procedure starts with initial bus voltages Vk, machine currents In, and power outputs pan. Next, En is computed, Y11 is modified, and Y22 and Y12 are calculated. Time begins at t = 0, and the model checks for switching operations or load changes.

At each time step, the machine electrical powers are calculated at time t. Initial estimates are then used to find the power angles δ and machine speeds ω at time t + Δt. The electrical powers are refined at t + Δt, and the final values of δ and ω are obtained. The process repeats until the simulation ends.

Choosing a proper time step is important because it helps prevent numerical instability in the integration. With these equations and steps, engineers can analyze transient stability in multimachine power systems and judge how reliably the network will operate under changing conditions.

Power-System Stability in Multi-Machine Networks

power-system-stability-in-multi-machine-networks

the swing equation

Generator Rotor Motion and the Swing Equation

Generator rotor motion is described by the Swing Equation in power system dynamics. It is used to study generating units such as three-phase synchronous generators. The equation comes from Newton’s second law applied to the generator rotor. It links inertia, angular acceleration, and the balance between mechanical and electrical torques. In steady-state operation, the mechanical torque supplied to the generator is equal to the electrical torque produced by the generator. When these torques...

simplified synchronous machine model

Transient Stability in Synchronous Generators

The synchronous machine model helps students analyze transient stability in power systems. It represents a synchronous generator under balanced three-phase positive-sequence conditions. The model assumes constant excitation and ignores losses and saturation. These simplifications make it easier to study how a generator behaves during a disturbance. In this setup, the generator is connected to a network of transmission lines, transformers, loads, and other machines. That larger system is often...

multimachine stability

Power-system stability in multi-machine networks focuses on how several synchronous machines behave as they stay connected to an N-bus power system. The analysis solves the swing equations for the machines and tracks how the network responds when conditions change. It is used to study transient stability, which is the system’s ability to stay in step after a disturbance. The network is written with nodal equations that link bus voltages, machine voltages, and machine currents. In this model, V...

wind turbine machine models

Wind Turbine Types and Generator Control

Wind turbine machine models are used to study how wind turbines behave in transient stability analysis. In this setting, induction machines and synchronous machines are the main model types. Induction machines are common because they are simple and reliable. Induction machines work through the rotating magnetic field made by the stator and rotor. A key term is slip, which is the difference between synchronous speed and rotor speed relative to synchronous speed. Slip is zero at synchronous...

generator voltage control

Voltage Regulation in Generators and Wind Turbines

Generator voltage regulation helps keep synchronous generators and wind turbines working stably. Older systems use a DC generator driven by the rotor to supply DC power to the rotor field winding. That power is passed through slip rings and brushes. Newer systems use static exciters or brushless exciters. A static exciter rectifies AC power from the generator terminals and sends DC power directly to the rotor. A brushless exciter uses an inverted synchronous generator, takes AC power from...

turbine-governor control

How Turbine Governors Stabilize Power Output

Turbine-governor control helps keep generator frequency and rotor speed stable when electrical load changes. It does this by matching turbine mechanical power with the power demand on the system. In a power grid, that balance is important for steady operation. Turbine-generator units also store kinetic energy because their parts are rotating. When load increases, that stored energy is released to help meet the demand. At first, electrical torque rises while mechanical torque stays nearly...

load-frequency control

Load-frequency control in power grids

Load-frequency control keeps a power grid stable when electrical demand changes. It helps keep system frequency and power flow within acceptable limits. Turbine-governor control reduces rotor speed changes after a load shift, but a steady-state frequency error can still remain when the turbine-governor reference setting does not change. In an interconnected power system, each area agrees to export or import a scheduled amount of power through tie-lines, which are the transmission lines that...

key terms and concepts

Learn how multi-machine power systems use nodal equations, swing equations, and time-step simulation to test transient stability after switching or load changes.

Power-System Stability in Multi-Machine 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-System Stability in Multi-Machine Networks

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