WO2022110290A1 - 一种基于变机械负载转矩的感应电动机转子滑差计算方法 - Google Patents
一种基于变机械负载转矩的感应电动机转子滑差计算方法 Download PDFInfo
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- the invention relates to the field of transient calculation of power systems, and more particularly, to a method for calculating rotor slip of an induction motor based on variable mechanical load torque.
- the receiving end system has a heavy load and a high load ratio of the induction motor.
- the induction motor often absorbs a large amount of transient power due to the decrease of electromagnetic torque and the increase of slip, which makes the transient voltage of the system difficult to recover.
- the academic community has noticed the importance of studying the interaction between the transient characteristics of the motor and the voltage sag, and proposed three types of research methods: the experimental method, the time-domain simulation method and the analytical method.
- the test method uses the voltage sag generator to generate various types of voltage sag waveforms, records the output response of the motor, and then analyzes the interaction between the voltage sag and the motor.
- the time-domain simulation method uses the electromagnetic transient or electromechanical transient simulation program of the power system, considers the more accurate transient model of the motor, and discusses the problem through numerical calculation. The advantage of the two is that the results are real and credible, but in order to reveal the influence of a certain factor, multiple tests or simulations are often required. In addition to being cumbersome and time-consuming, it is beyond the ability to analyze and interpret experimental and simulated phenomena.
- the publication date is February 19, 2019, and the Chinese Patent Publication No. CN109359266A discloses a method for solving the transient response of an induction motor, including: obtaining parameter data of the induction motor; calculating an approximate negative sequence impedance Zin2 according to the parameter data; Combined with the power grid data of the power grid system, the negative sequence electromagnetic torque Te-, the negative sequence active power consumption Pe- and the negative sequence reactive power consumption Qe- are determined; the simultaneous positive sequence power network equation, the induction motor stator voltage equation, the induction motor third-order Electromechanical transient equation, active power consumption calculation equation and reactive power consumption calculation equation, calculate the transient response; the total electromagnetic torque Te in the third-order electromechanical transient equation during the asymmetric fault is included in the negative sequence electromagnetic torque Te-, the active power The consumption calculation equation is included in the negative sequence active power consumption Pe-, and the reactive power consumption calculation equation is included in the negative sequence reactive power consumption Qe-.
- the analytical method is the most fundamental and thorough research method to explicitly solve the interaction between the transient response of the motor and the system voltage with the help of circuit and motor analysis theory, but it is difficult. So far, the analytical method has not made a breakthrough, and cannot accurately obtain all the operation and state variables of the disturbed motor.
- the electromechanical transient simulation because the transient process of the stator winding of the induction motor cannot be simulated in detail, the third-order transient model of the induction motor derived based on the three-phase symmetrical positive sequence voltage is still used as its simulation model. Its calculation accuracy is not high in the case of symmetrical faults.
- the invention provides an induction motor rotor slip calculation method based on variable mechanical load torque, which can quickly calculate the dynamic characteristics of the induction motor load rotor slip under the symmetrical and asymmetrical fault voltage drop of the power system.
- the technical scheme of the present invention is as follows:
- a method for calculating rotor slip of an induction motor based on variable mechanical load torque comprising the following steps:
- a method for calculating rotor slip of an induction motor based on variable mechanical load torque comprising the following steps:
- S1 Construct steady-state equivalent circuit, transient equivalent circuit and equivalent circuit based on negative-sequence power frequency component of induction motor based on positive sequence power frequency component;
- S5 According to S4, the positive and negative sequence voltage components at the stator end of the induction motor are obtained, and the positive and negative sequence stator current and electromagnetic torque equations at the stator end of the induction motor are obtained;
- the steady-state equivalent circuit of the induction motor based on the positive sequence power frequency component includes an externally applied stator terminal voltage.
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the stator reactance X s
- the other end of the stator reactance X s is respectively connected to one end of the excitation reactance X m and the other end of the rotor reactance X r .
- One end is electrically connected, and the other end of the rotor reactance X r is connected to the rotor resistance after considering the rotor slip
- One end is electrically connected, and the rotor resistance after considering the rotor slip
- the other end is respectively connected with the other end of the excitation reactance X m and the externally applied stator terminal voltage The other end of the electrical connection;
- the transient equivalent circuit of the induction motor based on the positive sequence power frequency component includes an externally applied stator terminal voltage.
- Stator resistance R s , rotor short-circuit reactance X' and rotor transient potential E' where:
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the rotor short-circuit reactance X'
- the other end of the rotor short-circuit reactance X' is electrically connected to one end of the rotor transient potential E', The other end of the rotor transient potential E' and the applied stator terminal voltage The other end of the electrical connection;
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the stator reactance X s
- the other end of the stator reactance X s is respectively connected to one end of the excitation reactance X m and the other end of the rotor reactance X r .
- One end is electrically connected, and the other end of the rotor reactance X r is connected to the rotor resistance after considering the rotor slip
- One end is electrically connected, and the rotor resistance after considering the rotor slip
- the other end is respectively connected with the other end of the excitation reactance X m and the externally applied stator terminal voltage the other end of the electrical connection.
- the rotor voltage equation described in step S2 is:
- E' d and E' q are the d-axis and q-axis components of the rotor transient potential E', respectively, and I ds and I qs are the stator currents, respectively
- stator current equation described in step S2 is:
- V ds and V qs are the applied stator voltages, respectively The d- and q-axis components of .
- step S3 the rotor motion equation described in step S3 is:
- T j is the motor inertia time constant
- T e is the electromagnetic torque
- T m is the mechanical load torque
- variable mechanical load torque equation of the induction motor in step S3 is:
- T m0 is the initial mechanical torque of the induction motor during stable operation
- ⁇ 0 is the torque coefficient
- step S4 the symmetrical component method is used to obtain the positive and negative sequence voltage components of the stator terminal of the induction motor, and the calculation formula is:
- step S5 the positive and negative sequence voltage components at the stator end of the induction motor are obtained according to S4, and the positive and negative sequence stator current and electromagnetic torque equations at the stator end of the induction motor are obtained, wherein:
- the sending-end generator power P eq +jQ eq and the receiving-end motor power P d +jQ d satisfy the following formula:
- the motor terminal positive and negative sequence voltages are calculated as follows:
- the subscripts "1" and “2" represent the positive and negative sequence components, respectively, is the conjugate value of the positive and negative sequence voltages of the power supply, and the generator power at the sending end is P eq +jQ eq ;
- T e1 and T e2 represent the positive and negative sequence electromagnetic torques, respectively.
- step S6 obtains the first-order inhomogeneous linear differential equation of the rotor slip s with respect to time t during the asymmetric fault voltage drop according to the positive and negative sequence stator currents and the electromagnetic torque equations at the stator terminals of the induction motor:
- the initial rotor slip of the induction motor be s 0 , and it will continue to increase from the time t 0 when the voltage sag occurs, to the time t 1 when the fault is cleared, the rotor slip of the induction motor increases to s 1 , and after the voltage sag is cleared, After a period of time, the rotor slip of the induction motor recovers;
- V s1 is the positive sequence component of the externally applied stator terminal voltage
- V s2 is the negative sequence component of the externally applied stator terminal voltage
- R r is the rotor resistance
- ⁇ s is the motor synchronous speed
- ⁇ 0 is the torque coefficient
- T j is the motor Inertia time constant
- T m0 is the initial mechanical torque of the induction motor during stable operation
- Equation (13) the approximate analytical expression for the rotor slip S dur of the induction motor during the fault period is obtained as:
- T m is the mechanical load torque
- the invention simplifies the complex electromagnetic transient time-domain simulation calculation process, saves time-domain simulation calculation time and workload, and can obtain the dynamic process change characteristics of the rotor slip s of the induction motor after any type of fault occurs.
- the analytical calculation method explicitly gives the calculation expressions of the mechanical and electrical parameters of the induction motor after disturbance. In addition to being used to quickly evaluate the interaction between the rotor slip and the voltage sag of the induction motor, it also has the potential to be applied to power system stability. sex analysis.
- FIG. 1 is a schematic flow chart of the method of the present invention.
- Figure 2 is a schematic diagram of a steady-state equivalent circuit of an induction motor based on a positive sequence power frequency component.
- Figure 3 is a schematic diagram of a transient equivalent circuit of an induction motor based on a positive sequence power frequency component.
- Figure 4 is a schematic diagram of a single machine with a motor power supply system.
- FIG. 5 is a schematic diagram of the steady-state equivalent circuit of the induction motor based on the negative sequence power frequency component.
- FIG. 6 is a graph showing the change trend of the rotational speed of the induction motor when the voltage drops.
- Figure 7 is a schematic diagram of a single machine with an induction motor load power supply system.
- FIG. 8 is a schematic diagram showing the comparison of the A-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1400kW motor.
- FIG. 9 is a schematic diagram showing the comparison of the B-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1400kW motor.
- FIG. 10 is a schematic diagram showing the comparison of the C-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1400kW motor.
- 11 is a schematic diagram showing the comparison of rotor slip obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1400kW motor.
- FIG. 12 is a schematic diagram showing the comparison of the A-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1200kW motor.
- FIG. 13 is a schematic diagram showing the comparison of the B-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1200kW motor.
- FIG. 14 is a schematic diagram showing the comparison of the C-phase voltage obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1200kW motor.
- FIG. 15 is a schematic diagram showing the comparison of rotor slip obtained by using the calculation method of the embodiment and the simulation method of the electromagnetic transient simulation program PSCAD/EMTDC in a 1200kW motor.
- This embodiment provides a method for calculating rotor slip of an induction motor based on variable mechanical load torque, as shown in Figure 1, including the following steps:
- S1 Construct steady-state equivalent circuit, transient equivalent circuit and equivalent circuit based on negative-sequence power frequency component of induction motor based on positive sequence power frequency component;
- S5 According to S4, the positive and negative sequence voltage components at the stator end of the induction motor are obtained, and the positive and negative sequence stator current and electromagnetic torque equations at the stator end of the induction motor are obtained;
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the stator reactance X s
- the other end of the stator reactance X s is respectively connected to one end of the excitation reactance X m and the other end of the rotor reactance X r .
- One end is electrically connected, and the other end of the rotor reactance X r is connected to the rotor resistance after considering the rotor slip
- One end is electrically connected, and the rotor resistance after considering the rotor slip
- the other end is respectively connected with the other end of the excitation reactance X m and the externally applied stator terminal voltage The other end of the electrical connection;
- transient equivalent circuit of the induction motor based on the positive sequence power frequency component is shown in Figure 3, including the applied stator terminal voltage Stator resistance R s , rotor short-circuit reactance X' and rotor transient potential E', where:
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the rotor short-circuit reactance X'
- the other end of the rotor short-circuit reactance X' is electrically connected to one end of the rotor transient potential E', The other end of the rotor transient potential E' and the applied stator terminal voltage The other end of the electrical connection;
- stator resistance R s is electrically connected to one end of the stator resistance R s
- the other end of the stator resistance R s is electrically connected to one end of the stator reactance X s
- the other end of the stator reactance X s is respectively connected to one end of the excitation reactance X m and the other end of the rotor reactance X r .
- One end is electrically connected, and the other end of the rotor reactance X r is connected to the rotor resistance after considering the rotor slip
- One end is electrically connected, and the rotor resistance after considering the rotor slip
- the other end is respectively connected with the other end of the excitation reactance X m and the externally applied stator terminal voltage the other end of the electrical connection.
- step S2 The rotor voltage equation described in step S2 is:
- E' d and E' q are the d-axis and q-axis components of the rotor transient potential E', respectively, and I ds and I qs are the stator currents, respectively
- step S2 The stator current equation described in step S2 is:
- V ds and V qs are the applied stator voltages, respectively The d- and q-axis components of .
- step S3 The rotor motion equation described in step S3 is:
- T j is the motor inertia time constant
- T e is the electromagnetic torque
- T m is the mechanical load torque
- variable mechanical load torque equation of the induction motor in step S3 is:
- T m0 is the initial mechanical torque of the induction motor during stable operation
- ⁇ 0 is the torque coefficient
- step S4 the symmetrical component method is used to obtain the positive and negative sequence voltage components of the stator terminal of the induction motor, and the calculation formula is:
- step S5 the positive and negative sequence voltage components at the stator end of the induction motor are obtained according to S4, and the positive and negative sequence stator current and electromagnetic torque equations at the stator end of the induction motor are obtained, wherein:
- the sending-end generator power P eq +jQ eq and the receiving-end motor power P d +jQ d satisfy the following formula:
- the positive and negative sequence voltages at the motor terminals are calculated as follows:
- the subscripts "1" and “2" represent the positive and negative sequence components, respectively, is the conjugate value of the positive and negative sequence voltages of the power supply, and the generator power at the sending end is P eq +jQ eq ;
- T e1 and T e2 represent the positive and negative sequence electromagnetic torques, respectively.
- Step S6 According to the positive and negative sequence stator currents and the electromagnetic torque equations at the stator terminals of the induction motor, the first-order inhomogeneous linear differential equation of the rotor slip s with respect to the time t is obtained during the asymmetric fault voltage drop:
- Figure 6 shows the dynamic change trend of the rotor slip of the induction motor when the voltage drops due to the failure of the upper-level power supply network.
- the initial rotor slip of the induction motor be s 0 , and it will continue to increase from the moment t 0 of the voltage drop occurs, to the time t 1 when the fault is cleared, the rotor slip of the induction motor increases to s 1 , after the voltage drop is cleared, after a period of time , the rotor slip of the induction motor is restored;
- V s1 is the positive sequence component of the externally applied stator terminal voltage
- V s2 is the negative sequence component of the externally applied stator terminal voltage
- R r is the rotor resistance
- ⁇ s is the motor synchronous speed
- ⁇ 0 is the torque coefficient
- T j is the motor Inertia time constant
- T m0 is the initial mechanical torque of the induction motor during stable operation
- Equation (13) the approximate analytical expression for the rotor slip S dur of the induction motor during the fault period is obtained as:
- T m is the mechanical load torque
- the motor parameters in Table 1 are used to build the 10kV power supply system shown in Figure 7 in the electromagnetic transient simulation program PSCAD/EMTDC.
- the calculation results are shown in Figure 8 to Figure 15. It can be seen from the figure that the dynamic characteristics of the rotor slip of the motor calculated by the present invention are in good agreement with the PSCAD/EMTDC simulation results, indicating the effectiveness and accuracy of the algorithm of the present invention. It is worth mentioning that at the moment when the voltage drop occurs and clears, the calculation results of the analytical method will change abruptly, while PSCAD/EMTDC will not. The main reason for this difference is that the analytical method does not take into account the electromagnetic transient process of the stator winding, that is, it is considered that the electrical quantities of the stator winding can be abruptly changed.
Abstract
一种基于变机械负载转矩的感应电动机转子滑差计算方法，快速计算电力系统对称和不对称故障电压跌落下感应电动机负荷转子滑差动态特性，与PSCAD/ EMTDC电磁暂态仿真结果的比较，验证了算法的有效性。该解析计算方法给出了受扰后感应电动机各机械、电气参量的计算表达式，除可用于快速评估感应电动机转子滑差与电压跌落的相互影响外，也有潜力应用于电力系统稳定性分析中，可以求出任意类型故障的电压跌落发生后感应电动机转子滑差s的动态过程变化特性。
Description
本发明涉及电力系统暂态计算领域，更具体地，涉及一种基于变机械负载转矩的感应电动机转子滑差计算方法。
随着“西电东送”战略的实施和深化，我国“长三角”和“珠三角”等负荷中心均已形成典型的受端系统结构。受端系统负荷重、感应电动机负荷比例高，电网发生故障及恢复过程中，感应电动机由于电磁转矩下降，滑差增大，常会吸收大量的暂态功率，导致系统暂态电压难以恢复。学术界注意到研究电动机暂态特性与电压跌落相互影响的重要性，提出试验法、时域仿真法与解析法等三类研究方法。
试验法借助电压跌落发生器产生各种类别的电压跌落波形，记录电动机的输出响应，进而分析电压跌落与电动机的相互影响。时域仿真法借助电力系统电磁暂态或机电暂态仿真程序，考虑较为精确的电动机暂态模型，通过数值计算对问题进行探讨。二者的优点在于结果真实可信，但为揭示某一因素的影响，常需进行多次试验或仿真。除了繁琐、费时之外，在分析和解释试验和仿真现象时，都显得力所不及。
公开日为2019年02月19日，公开号为CN109359266A的中国专利公开了一种感应电动机暂态响应的求解方法，包括：获取感应电动机的参数数据；根据参数数据计算出近似负序阻抗Zin2；再结合电网系统的电网数据，确定出负序电磁转矩Te-，负序有功消耗Pe-及负序无功消耗Qe-；联立正序电力网络方程，感应电动机定子电压方程，感应电动机三阶机电暂态方程，有功消耗计算方程以及无功消耗计算方程，计算出暂态响应；不对称故障期间三阶机电暂态方程中的总电磁转矩Te计入负序电磁转矩Te-，有功消耗计算方程计入负序有功消耗Pe-，无功消耗计算方程计入负序无功消耗Qe-。
现有技术中，解析法借助电路和电机分析理论，显式求解电动机暂态响应和系统电压间的交互作用，是最根本、最透彻的研究方法，但难度较大。到目前为止，解析法尚未取得突破性进展，还不能准确求得受扰后电动机的全部运行和状态变量。在机电暂态仿真中，由于无法详细模拟感应电动机定子绕组的暂态过程， 截止目前仍采用基于三相对称正序电压推导出的感应电动机三阶暂态模型作为其仿真模型，因而在发生不对称故障时其计算精度不高。
发明内容
本发明提供一种基于变机械负载转矩的感应电动机转子滑差计算方法，快速计算电力系统对称和不对称故障电压跌落下感应电动机负荷转子滑差动态特性。
为解决上述技术问题，本发明的技术方案如下：
一种基于变机械负载转矩的感应电动机转子滑差计算方法，包括以下步骤：
一种基于变机械负载转矩的感应电动机转子滑差计算方法，包括以下步骤：
S1：构建感应电动机基于正序工频分量的稳态等值电路、暂态等值电路和基于负序工频分量的等值电路；
S2：根据S1构建的等值电路，采用考虑机电暂态过程，得到感应电动机的转子电压方程、定子电流方程，所述转子电压方程、定子电流方程与转子滑差关联；
S3：构建转子运动方程，采用变机械负载转矩对转子滑差进行解析计算，得到感应电动机的变机械负载转矩方程；
S4：应用对称分量法求得感应电动机定子端的正、负序电压分量；
S5：根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程；
S6：根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程；
S7：通过常数变易法，求解转子滑差s关于时间t的一阶非齐次线性微分方程，得到故障期间及恢复过程中转子滑差s动态变化的解析计算表达式。
优选地，所述感应电动机基于正序工频分量的稳态等值电路包括外施定子端电压
定子电阻R
s、定子电抗X
s、励磁电抗X
m、转子电抗X
r和考虑转子滑差后的转子电阻
其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与定子电抗X
s的一端电连接，定子电抗X
s的另一端分别与励磁电抗X
m的一端、转子电抗X
r的一端电连接，转子电抗X
r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X
m的另一端、外施定子端电压
的另一端电连接；
所述感应电动机基于正序工频分量的暂态等值电路包括外施定子端电压
定子电阻R
s、转子短路电抗X'和转子暂态电势E'，其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与转子短路电抗X'的一端电连接，转子短路电抗X'的另一端与转子暂态电势E'的一端电连接，转子暂态电势E'的另一端与外施定子端电压
的另一端电连接；
当电源通过等值阻抗为Z
eq＝R
eq+jX
eq的供电网络向感应电动机IM供电时，感应电动机的负序等值电路包括外施电压
定子电阻R
s、定子电抗X
s、励磁电抗X
m、转子电抗X
r和考虑转子滑差后的转子电阻
其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与定子电抗X
s的一端电连接，定子电抗X
s的另一端分别与励磁电抗X
m的一端、转子电抗X
r的一端电连接，转子电抗X
r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X
m的另一端、外施定子端电压
的另一端电连接。
优选地，步骤S2所述的转子电压方程为：
式中，
为外施定子端电压，E'
d和E'
q分别为转子暂态电势E'的d轴和q轴分量，I
ds和I
qs分别为定子电流
的d轴和q轴分量，X＝X
s+X
m为转子开路电抗；
为转子短路电抗；
为定子开路、转子回路暂态时间常数，ω
s为电动机同步转速，ω
m为电动机转速，
为转子滑差，R
s和X
s为定子电阻、电抗，R
r和X
r为转子电阻、电抗，X
m为励磁电抗。
优选地，步骤S2所述的定子电流方程为：
式中，V
ds和V
qs分别为外施定子电压
的d轴和q轴分量。
优选地，步骤S3所述的转子运动方程为：
式中，T
j为电动机惯性时间常数，T
e为电磁转矩，T
m为机械负载转矩。
优选地，步骤S3中的感应电动机的变机械负载转矩方程为：
T
m＝T
m0+β
0s (4)
式中，T
m0为稳定运行时感应电动机的初始机械转矩，β
0为转矩系数。
优选地，步骤S4中应用对称分量法求得感应电动机定子端的正、负序电压分量，其计算式为：
式中，a＝1∠120°，
分别为电源相电压，
分别为电源正、负、零序电压。
优选地，步骤S5中根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程，其中：
送端发电机功率P
eq+jQ
eq与受端电动机功率P
d+jQ
d满足下式：
电动机端正、负序电压计算如下：
式中，下标“1”、“2”分别表示正、负序分量，
为电源正、负序电压共轭值，送端发电机功率为P
eq+jQ
eq；
考虑到
2-s≈2，有：
式中，下标“1”、“2”分别表示正、负序分量，Z
rs为中间变量，
不对称故障电压跌落期间，电动机的转子运动方程为：
式中，T
e1、T
e2分别表示正、负序电磁转矩。
优选地，步骤S6根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：
联合(6)、(10)、(11)和(12)，得不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：
电压跌落清除后，定子电压中不含负序电压分量，转子运动方程也不再含有 负序电磁转矩T
e2，有：
联立式(6)、(10)、(11)和(14)，可得故障清除后转子滑差s关于时间t的一阶非齐次线性微分方程：
优选地，令感应电动机的初始转子滑差为s
0，从电压跌落发生时刻t
0开始不断增大，到故障清除时刻t
1，感应电动机转子滑差增大到s
1，电压跌落清除后，经过一段时间，感应电动机的转子滑差才恢复；
通过常数变易法令：
式中，V
s1为外施定子端电压正序分量，V
s2外施定子端电压负序分量，R
r为转子电阻，ω
s为电动机同步转速，β
0为转矩系数，T
j为电动机惯性时间常数，T
m0为稳定运行时感应电动机的初始机械转矩；
代入式(13)，求得故障期间感应电动机转子滑差S
dur的近似解析计算表达式为：
通过常数变易法令：
式中，T
m为机械负载转矩；
代入式(15)求得故障清除后感应电动机转子滑差S
after的近似解析计算表达式为：
与现有技术相比，本发明技术方案的有益效果是：
本发明使复杂的电磁暂态时域仿真计算过程简单化,节约了时域仿真计算时 间和工作量，可以求出任意类型故障发生后感应电动机转子滑差s的动态过程变化特性。同时该解析计算方法显式给出了受扰后感应电动机各机械、电气参量的计算表达式，除可用于快速评估感应电动机转子滑差与电压跌落的相互影响外，也有潜力应用于电力系统稳定性分析中。
图1为本发明的方法流程示意图。
图2为感应电动机基于正序工频分量的稳态等值电路示意图。
图3为感应电动机基于正序工频分量的暂态等值电路示意图。
图4为单机带电动机供电系统示意图。
图5为感应电动机基于负序工频分量的稳态等值电路示意图。
图6为电压跌落时感应电动机转速变化趋势图。
图7为单机带感应电动机负荷供电系统示意图。
图8为在1400kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的A相电压的比较示意图。
图9为在1400kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的B相电压的比较示意图。
图10为在1400kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的C相电压的比较示意图。
图11为在1400kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的转子滑差的比较示意图。
图12为在1200kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的A相电压的比较示意图。
图13为在1200kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的B相电压的比较示意图。
图14为在1200kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的C相电压的比较示意图。
图15为在1200kW电动机中运用实施例的计算方法与电磁暂态仿真程序PSCAD/EMTDC仿真方法得到的转子滑差的比较示意图。
附图仅用于示例性说明，不能理解为对本专利的限制；
为了更好说明本实施例，附图某些部件会有省略、放大或缩小，并不代表实际产品的尺寸；
对于本领域技术人员来说，附图中某些公知结构及其说明可能省略是可以理解的。
下面结合附图和实施例对本发明的技术方案做进一步的说明。
实施例1
本实施例提供一种基于变机械负载转矩的感应电动机转子滑差计算方法，如图1，包括以下步骤：
S1：构建感应电动机基于正序工频分量的稳态等值电路、暂态等值电路和基于负序工频分量的等值电路；
S2：根据S1构建的等值电路，采用考虑机电暂态过程，得到感应电动机的转子电压方程、定子电流方程，所述转子电压方程、定子电流方程与转子滑差关联；
S3：构建转子运动方程，采用变机械负载转矩对转子滑差进行解析计算，得到感应电动机的变机械负载转矩方程；
S4：应用对称分量法求得感应电动机定子端的正、负序电压分量；
S5：根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程；
S6：根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程；
S7：通过常数变易法，求解转子滑差s关于时间t的一阶非齐次线性微分方程，得到故障期间及恢复过程中转子滑差s动态变化的解析计算表达式。
所述感应电动机基于正序工频分量的稳态等值电路如图2，包括外施定子端电压
定子电阻R
s、定子电抗X
s、励磁电抗X
m、转子电抗X
r和考虑转子滑差后的转子电阻
其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与定子电抗X
s的一端电连接，定子电抗X
s的另一端分别与励磁电抗X
m的一端、转子电抗X
r的一端电连接，转子电抗X
r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X
m的另一端、外施定子端电压
的另一端电连接；
所述感应电动机基于正序工频分量的暂态等值电路如图3，包括外施定子端电压
定子电阻R
s、转子短路电抗X'和转子暂态电势E'，其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与转子短路电抗X'的一端电连接，转子短路电抗X'的另一端与转子暂态电势E'的一端电连接，转子暂态电势E'的另一端与外施定子端电压
的另一端电连接；
当电源通过如图4所示的等值阻抗为Z
eq＝R
eq+jX
eq的供电网络向感应电动机IM供电时，感应电动机的负序等值电路如图5，包括外施电压
定子电阻R
s、定子电抗X
s、励磁电抗X
m、转子电抗X
r和考虑转子滑差后的转子电阻
其中：
外施定子端电压
的一端与定子电阻R
s的一端电连接，定子电阻R
s的另一端与定子电抗X
s的一端电连接，定子电抗X
s的另一端分别与励磁电抗X
m的一端、转子电抗X
r的一端电连接，转子电抗X
r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X
m的另一端、外施定子端电压
的另一端电连接。
步骤S2所述的转子电压方程为：
式中，
为外施定子端电压，E'
d和E'
q分别为转子暂态电势E'的d轴和q轴分量，I
ds和I
qs分别为定子电流
的d轴和q轴分量，X＝X
s+X
m为转子开路电抗；
为转子短路电抗；
为定子开路、转子回路暂态时间常数，ω
s为电动机同步转速，ω
m为电动机转速，
为转子滑差，R
s和X
s为定子电阻、电抗，R
r和X
r为转子电阻、电抗，X
m为励磁电抗。
步骤S2所述的定子电流方程为：
式中，V
ds和V
qs分别为外施定子电压
的d轴和q轴分量。
步骤S3所述的转子运动方程为：
式中，T
j为电动机惯性时间常数，T
e为电磁转矩，T
m为机械负载转矩。
步骤S3中的感应电动机的变机械负载转矩方程为：
T
m＝T
m0+β
0s (4)
式中，T
m0为稳定运行时感应电动机的初始机械转矩，β
0为转矩系数。
步骤S4中应用对称分量法求得感应电动机定子端的正、负序电压分量，其计算式为：
式中，a＝1∠120°，
分别为电源相电压，
分别为电源正、负、零序电压。
步骤S5中根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程，其中：
送端发电机功率P
eq+jQ
eq与受端电动机功率P
d+jQ
d满足下式：
电动机端正、负序电压计算如下：
式中，下标“1”、“2”分别表示正、负序分量，
为电源正、负序电压共轭值，送端发电机功率为P
eq+jQ
eq；
考虑到
2-s≈2，有：
式中，下标“1”、“2”分别表示正、负序分量，Z
rs为中间变量，
不对称故障电压跌落期间，电动机的转子运动方程为：
式中，T
e1、T
e2分别表示正、负序电磁转矩。
步骤S6根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：
联合(6)、(10)、(11)和(12)，得不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：
电压跌落清除后，定子电压中不含负序电压分量，转子运动方程也不再含有负序电磁转矩T
e2，有：
联立式(6)、(10)、(11)和(14)，可得故障清除后转子滑差s关于时间t的一阶非齐次线性微分方程：
上级供电网络发生故障导致电压跌落时，感应电动机的转子滑差动态变化趋势如图6所示。
令感应电动机的初始转子滑差为s
0，从电压跌落发生时刻t
0开始不断增大，到故障清除时刻t
1，感应电动机转子滑差增大到s
1，电压跌落清除后，经过一段时间，感应电动机的转子滑差才恢复；
通过常数变易法令：
式中，V
s1为外施定子端电压正序分量，V
s2外施定子端电压负序分量，R
r为转子电阻，ω
s为电动机同步转速，β
0为转矩系数，T
j为电动机惯性时间常数，T
m0为稳定运行时感应电动机的初始机械转矩；
代入式(13)，求得故障期间感应电动机转子滑差S
dur的近似解析计算表达式为：
通过常数变易法令：
式中，T
m为机械负载转矩；
代入式(15)求得故障清除后感应电动机转子滑差S
after的近似解析计算表达式为：
为了验证本文提出的感应电动机动态特性解析算法对于不同故障类型和不 同电动机参数的有效性和适应性，采用表1中电动机参数，在电磁暂态仿真程序PSCAD/EMTDC搭建图7所示10kV供电系统的仿真模型，进行了以下2种情形的比较：1)设置电源在0.2s时发生持续时间为0.2s的不对称故障电压跌落(电压跌落前：E
eqa,pre＝1.0∠0°pu，E
eqb,pre＝1.0∠-120°pu，E
eqc,pre＝1.0∠120°pu；电压跌落期间：E
eqa,dur＝0.8∠0°pu，E
eqb,dur＝0.6∠-120°pu，E
eqc,dur＝0.4∠120°pu)，采用表1中容量为1400kW的电动机；2)设置电源在0.2s时发生持续时间为0.2s的对称故障电压跌落(电压跌落前：E
eqa,pre＝1.0∠0°pu，E
eqb,pre＝1.0∠-120°pu，E
eqc,pre＝1.0∠120°pu；电压跌落期间：E
eqa,dur＝0.6∠0°pu，E
eqb,dur＝0.6∠-120°pu，E
eqc,dur＝0.6∠120°pu)，采用表1中容量为1000kW的电动机。算例中电动机均带额定负载运行，机械负载转矩系数β
0＝0.85，通过初始化计算得到参数T
m0，系统等值阻抗Z
eq＝(1.5+j4.0)Ω，PSCAD/EMTDC仿真步长为100us，解析算法计算步长为0.01s，计算结果见图8至图15。从图中可知，本发明计算的电动机转子滑差动态特性均与PSCAD/EMTDC仿真结果吻合良好，表明了本发明算法的有效性和准确性。值得一提的是在电压跌落发生和清除的瞬间，解析法的计算结果会发生突变，而PSCAD/EMTDC不会。产生这种差异的主要原因在于解析法没有计及定子绕组的电磁暂态过程，即认为定子绕组的各电气量是可以突变的。
表1
相同或相似的标号对应相同或相似的部件；
附图中描述位置关系的用语仅用于示例性说明，不能理解为对本专利的限制；
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Claims (10)
- 一种基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，包括以下步骤：S1：构建感应电动机基于正序工频分量的稳态等值电路、暂态等值电路和基于负序工频分量的等值电路；S2：根据S1构建的等值电路，采用考虑机电暂态过程，得到感应电动机的转子电压方程、定子电流方程，所述转子电压方程、定子电流方程与转子滑差关联；S3：构建转子运动方程，采用变机械负载转矩对转子滑差进行解析计算，得到感应电动机的变机械负载转矩方程；S4：应用对称分量法求得感应电动机定子端的正、负序电压分量；S5：根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程；S6：根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程；S7：通过常数变易法，求解转子滑差s关于时间t的一阶非齐次线性微分方程，得到故障期间及恢复过程中转子滑差s动态变化的解析计算表达式。
- 根据权利要求1所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，所述感应电动机基于正序工频分量的稳态等值电路包括外施定子端电压定子电阻R s、定子电抗X s、励磁电抗X m、转子电抗X r和考虑转子滑差后的转子电阻
其中：
外施定子端电压的一端与定子电阻R s的一端电连接，定子电阻R s的另一端与定子电抗X s的一端电连接，定子电抗X s的另一端分别与励磁电抗X m的一端、转子电抗X r的一端电连接，转子电抗X r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X m的另一端、外施定子端电压
的另一端电连接；
所述感应电动机基于正序工频分量的暂态等值电路包括外施定子端电压定子电阻R s、转子短路电抗X'和转子暂态电势E'，其中：
外施定子端电压的一端与定子电阻R s的一端电连接，定子电阻R s的另一端与转子短路电抗X'的一端电连接，转子短路电抗X'的另一端与转子暂态电势E'的一端电连接，转子暂态电势E'的另一端与外施定子端电压
的另一端电连接；
当电源通过等值阻抗为Z eq＝R eq+jX eq的供电网络向感应电动机IM供电时，感应电动机的负序等值电路包括外施电压定子电阻R s、定子电抗X s、励磁电抗X m、转子电抗X r和考虑转子滑差后的转子电阻
其中：
外施定子端电压的一端与定子电阻R s的一端电连接，定子电阻R s的另一端与定子电抗X s的一端电连接，定子电抗X s的另一端分别与励磁电抗X m的一端、转子电抗X r的一端电连接，转子电抗X r的另一端与考虑转子滑差后的转子电阻
的一端电连接，考虑转子滑差后的转子电阻
的另一端分别与励磁电抗X m的另一端、外施定子端电压
的另一端电连接。
- 根据权利要求2所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S2所述的转子电压方程为：
式中，为外施定子端电压，E' d和E' q分别为转子暂态电势E'的d轴和q轴分量，I ds和I qs分别为定子电流
的d轴和q轴分量，X＝X s+X m为转子开路电抗；
为转子短路电抗；
为定子开路、转子回路暂态时间常数，ω s为电动机同步转速，ω m为电动机转速，
为转子滑差，R s和X s为定子电阻、电抗，R r和X r为转子电阻、电抗，X m为励磁电抗。
- 根据权利要求3所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S2所述的定子电流方程为：
式中，V ds和V qs分别为外施定子电压的d轴和q轴分量。
- 根据权利要求4所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S3所述的转子运动方程为：
式中，T j为电动机惯性时间常数，T e为电磁转矩，T m为机械负载转矩。 - 根据权利要求5所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S3中的感应电动机的变机械负载转矩方程为：T m＝T m0+β 0s (4)式中，T m0为稳定运行时感应电动机的初始机械转矩，β 0为转矩系数。
- 根据权利要求7所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S4中应用对称分量法求得感应电动机定子端的正、负序电压分量，其计算式为：
式中，a＝1∠120°，分别为电源相电压，
分别为电源正、负、零序电压。
- 根据权利要求7所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S5中根据S4求得感应电动机定子端的正、负序电压分量，求得感应电动机定子端的正、负序定子电流和电磁转矩方程，其中：送端发电机功率P eq+jQ eq与受端电动机功率P d+jQ d满足下式：
电动机端正、负序电压计算如下：
式中，下标“1”、“2”分别表示正、负序分量，为电源正、负序电压共轭值，送端发电机功率为P eq+jQ eq；
考虑到2-s≈2，有：
式中，下标“1”、“2”分别表示正、负序分量，Z rs为中间变量，
不对称故障电压跌落期间，电动机的转子运动方程为：
式中，T e1、T e2分别表示正、负序电磁转矩。 - 根据权利要求8所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，步骤S6根据感应电动机定子端的正、负序定子电流和电磁转矩方程，得到不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：联合(6)、(10)、(11)和(12)，得不对称故障电压跌落期间转子滑差s关于时间t的一阶非齐次线性微分方程：
电压跌落清除后，定子电压中不含负序电压分量，转子运动方程也不再含有负序电磁转矩T e2，有：
联立式(6)、(10)、(11)和(14)，可得故障清除后转子滑差s关于时间t的一阶非齐次线性微分方程：
- 根据权利要求9所述的基于变机械负载转矩的感应电动机转子滑差计算方法，其特征在于，令感应电动机的初始转子滑差为s 0，从电压跌落发生时刻t 0开始不断增大，到故障清除时刻t 1，感应电动机转子滑差增大到s 1，电压跌落清除后，经过一段时间，感应电动机的转子滑差才恢复；通过常数变易法令：
式中，V s1为外施定子端电压正序分量，V s2外施定子端电压负序分量，R r为转子电阻，ω s为电动机同步转速，β 0为转矩系数，T j为电动机惯性时间常数，T m0为稳定运行时感应电动机的初始机械转矩；代入式(13)，求得故障期间感应电动机转子滑差S dur的近似解析计算表达式为：
通过常数变易法令：
式中，T m为机械负载转矩；代入式(15)求得故障清除后感应电动机转子滑差S after的近似解析计算表 达式为：
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