## Static Stability Analysis of Simple System Based on Small Interference Method

分类：作业 ; 热度：353 ; 最后更新于2022 年 04 月 18 日

Static Stability Analysis of Simple System Based on Small Interference Method

Zhao Fan 17xxxxxx11

***Abstract******* \--The power system will inevitably encounter interference from external factors during normal operation. After these external interferences are generated, no self-oscillation or monotonic out-of-step occurs, and the original operating state can be restored. This article describes the theory and simple application of the small interference analysis method for the static stability of the power system, and makes a detailed deduction and demonstration of the eigenvalue analysis method, and gives an example of the small interference method of the simple system. After the system is subject to small disturbances, it is still stable.\**

***Index Terms******* \-- Small interference stability analysis method; power system; static stability.\**

# I. Introduction

The multiple interconnection of the power system in different regions continuously superimposes, on the one hand, it improves the economics and reliability of transmission, and on the other hand, it increases the instability of the power system. The stable power supply of the power system is a necessary condition to meet the needs of people's daily life, but its stable operation needs to meet some necessary conditions. If unstable, it will bring huge economic and property losses, and even cause personal safety problems. The power system's ability to quickly recover to the operating state before being disturbed by a certain factor in a certain operating mode, or to recover to another stable operating mode, is called the small disturbance stability of the power system . In fact, when the power system is in normal operation, it has always received small disturbances of different degrees. Therefore, in order to improve the stability of the system, it is very necessary to maintain the stability of the small disturbance of the power system.

# II. Small interference in power system

*A.* **Generation and causes of small interference**

After the power system receives the slight interference from the interference source, the state of each generator is that the power angle is gradually pulled apart during synchronization and eventually the out-of-step phenomenon occurs. The reason for this result is that the synchronization torque in the power system is insufficient Large, this kind of out-of-step is called acyclic out-of-step. After the power system receives the small interference from the interference source, the amplitude of the power angle between the units in the system deviates from the synchronization and continuously increases to cause the oscillation. After the oscillation is generated, it continues to increase until it is out of sync, resulting in this result. The reason is that the damping in the power system is not large enough, and this out-of-step is called oscillation out-of-step.

*B.* **Small interference stability analysis scheme**

The study of the transient process and its stabilization process after the power system was affected by small disturbances was proposed by the famous 19th century Russian scholar Lyapunov. Lyapunov believes that any system in the dynamic range can be represented by a multivariate function, Here we make it denoted by , When the system is affected by small disturbances and makes the parameter value different from the original function, we call it function , When the increment of all parameters approaches zero, namely and , the function can be regarded as the system tending to be stable.

At present, there are three kinds of judgment methods applicable to the analysis of power system stability, which are frequency domain method, electrical torque method and eigenvalue analysis method. The theoretical basis of the frequency domain method is the transfer function and the multivariable Nyquist stability criterion. It is mainly used to design various controllers and select installation locations. The disadvantage is that the calculation amount is very large, which will greatly reduce the operation speed. , The key model information provided is also very limited. The physical meaning of the electromagnetic torque method is relatively clear, but the calculation is also relatively complicated. At present, the most commonly used effective method to judge the stability of the system is the eigenvalue analysis method, which not only absorbs the advantages of the frequency domain method, but also can reveal the essence of the stability problem of small interference and seek to solve his countermeasures. Make a comparative analysis table with time domain method and eigenvalue method:

TABLE I

Comparison of advantages and disadvantages of time domain method and eigenvalue method

Method | Time domain analysis | Small interference analysis |
---|---|---|

Means | Apply disturbances and perform integral calculations | Feature solving |

Advantages | Calculate the nonlinearity of all components of the system in detail, which is more reliable | ①Only one feature solution is required; ②Respectively study each oscillation mode; ③Can explain the stable phenomenon; ④Provide important information for controller layout and design |

Disadvantages | ①It is unrealistic to draw all the dynamics of the system; ②Multi-mode coupling of time-domain simulation results; ③Cannot explain the phenomenon; ④Not helpful for the layout and design of the controller | Non-linear features are ignored, model linearization and feature solving are difficult |

It can be known from the above that the eigenvalue analysis method has the superiority and convenience of the argument. Below we will conduct a detailed analysis of the calculus and argumentation process on the eigenvalue analysis method.

# III. Stability Analysis of Small Disturbance in Power System Based on Eigenvalue Analysis

The static stability analysis of small signals can be carried out using the characteristic root judgment method. The stability judgment method is generally completed by the following steps: ① The state equation after the power system is subjected to small interference is appropriately listed according to the situation; ② The listed non- The linear differential equation is transformed into a linearized equation; ③ Solve the root of the characteristic equation, and judge the stability quality of the system according to the change rule of the small increment of the parameter.

*A.* **Linear state equation of system state variable offset**

Since in the transformer, the reactance in its line can be regarded as a part of the leakage reactance of the generator, and the frequency of the system will not change when the capacity of the system is infinite, its state equation can be omitted and not written. Then in the simple micro system of the power system, there is a generator element that needs to list the state equation, and the state equation of the rotor is equation (1).

Because the prime mover is larger in terms of time constant, the dynamic characteristics of the prime mover can be ignored, and the output power of the prime mover is a constant by default. When the generator is a hidden pole machine, the no-load electromotive force of the generator is a fixed value, which is regarded as a constant, because the excitation adjustment system can adjust the electromotive force parameters. Let be the power expression of the generator's electromotive force as equation (2).

Then, the power expression (2) is substituted into the rotor state equation (1) to obtain the rotor state equation (3).

Since it is known from the rotor state equation (3) that the equation contains a nonlinear function , the simple power system is a nonlinear system. Simplify the state variables of a simple system, the state variables can be expressed as formula (4).

Substitute the state variable formula (4) into the state equation (3) to get the formula (5):

Using the Taylor series to expand in the vicinity of , ignoring the term whose offset is higher than quadratic (including quadratic), the linear relationship between and can be obtained as:

Substituting equation (6) into equation (5), we can get the system state differential equations (7):

*B.* **Linearize equation of state at equilibrium**

Transform the state offset differential equations (7) into a matrix form to obtain the linearized system matrix (8):

Including:

And the general form of matrix (8) can be transformed into formula (10):

In equation (10), is a vector, which is composed of the derivative of the state offset. is also a vector, consisting of the offset of the state variable. A is the coefficient matrix of the system equations of state. Then you can get a simple block diagram of the above system:

Fig. 1. Block diagram of generator operation in a simple system

*C.* **Use feature root to judge system stability**

In order to find the characteristic root of the linearized system matrix (8), the characteristic equation (11) is listed:

According to the law of the second-order system, the characteristic values can be listed by the characteristic equation:

Then the characteristic root can be obtained as (13):

It is easy to get through formula (13) that when is less than zero, produces two real roots with different signs. The monotonicity of and increases with the increase of t, so that for the generator, The aperiodic system of its infinite system loses its synchronization function, that is, the system is unstable. When is greater than zero, a pair of imaginary roots are generated, and and will continue to oscillate with the same amplitude. The expression of the oscillation frequency is as shown in equation (14):

If there are no other special circumstances, the range of is limited to 0.5~1, and is limited to five to ten seconds. At this time, the oscillation frequency f should be around 1Hz, and its oscillation frequency is very small, so it is low frequency oscillation. If there are still damping factors in the system, there will also be the effect of damping oscillation (oscillation factors after small interference attenuation), and finally the synchronization is restored, then the system is stable at this time.

*D.* **The influence of damping link on stability determination**

Add a damping link to the simple system of Fig. 1 to obtain the corresponding block diagram 2.

Fig. 2. Simple system considering damping effects

Similarly, the characteristic equation can be quickly derived from equation (12) as equation (15):

Find its characteristic root as:

Through the analysis of equation (16), when is less than zero, no matter what the value of D is, there will always be a positive real number root, and the system will lose stability. When is greater than zero, if D is greater than zero, you will get a pair of negative roots (both real and imaginary roots may exist, but not both), the system is stable; if D is less than zero, you will get a pair Conjugate complex number, in fact, the positive part is positive, and will produce oscillations with increasing amplitude, and the system is unstable at this time.

# IV. Example analysis

Figure 3 shows a simple power system, the unit value of each component parameter is known, generator G: , , , . Transformer reactance: , . Line L: Double loop wire . The initial state of system operation is , . The generator has no excitation regulator and the damping coefficient is . A simple analysis of its stability is made.

Fig. 3. Simple power system diagram

Analyse as below:

*A.* **Obtain the initial oscillation frequency without considering the damping coefficient of the system**

When the damping coefficient of the system is not considered, the parameters of the initial state of the system are: , ,

Can be obtained from formula (9):

From equation (14), we can see that the initial oscillation frequency of this system is:

*B.* **Judging the stability of the system when considering its damping coefficient**

Judging the stability can be done by analyzing the root locus, which can be obtained from equation (16):

Use Matlab to run the following code to draw the root locus diagram:

x=0:0.005:pi

y1=-0.45*pi+((0.45*pi)^2-4.5576*pi*cos(x)).^0.5

y2=-0.45*pi-((0.45*pi)^2-4.5576*pi*cos(x)).^0.5

plot(y1,'r')

hold on

plot(y2,'b')

The root locus diagram of the characteristic value changing with is as follows:

Fig. 4. Matlab drawing feature root locus map

According to the analysis in Figure 4, when , the system no longer oscillates after small interference. When , the system will have two negative roots, at this time the system will maintain stability. When , the characteristic value of the system is two real numbers with different signs, saddle junction bifurcation occurs, and the system will monotonously lose stability. Then its conclusion is consistent with the conclusion drawn by equation (16).

# V. Conclusion

The various devices of the power system are multiple interconnected and superimposed. The power generation equipment, power consumption equipment and power supply equipment are all inseparable. The operation status of a single device will affect other devices, and even more. The normal operation of the power system. To this end, the system should have a self-adjusting function to enable it to run stably. This article briefly describes the basis for whether the power system can continue to maintain stability after receiving small interference signals during normal operation, that is, whether it will produce out-of-step phenomenon, and gives the conditions for judging the stability of the system, that is, without considering the impact of the damping factor D At this time, the condition for stable operation of the system is ; the condition for stable operation of the system is when considering the influence of the damping factor D.

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