CN114004063B - Method for establishing Hamiltonian model of asynchronous motor - Google Patents
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Abstract
The invention discloses a method for establishing a Hamiltonian model of an asynchronous motor, which is characterized in that from the analysis of the relation between electromagnetic energy and mechanical energy of a basic stator and a basic rotor of the asynchronous motor, a stator Hamiltonian function is established according to electromechanical coupling dynamics and a basic differential equation of the asynchronous motor, a stator Hamiltonian model is further obtained through deduction, the Hamiltonian function and the Hamiltonian equation of the rotor of the asynchronous motor are established by adopting a similar method, and finally the energy functions of the two are synthesized, and the Hamiltonian model of the asynchronous motor is established.
Description
Technical Field
The invention belongs to the technical field of motor modeling and control, in particular to an induction motor modeling method based on a dynamics theory, and more particularly relates to an asynchronous motor Hamiltonian modeling method.
Background
Asynchronous motors are the most widespread consumer, are the main loads in the power grid, often need to model the loads in researching the stable control of the power grid, are regarded as static loads in simplified calculation, can be replaced by equivalent impedance, but under the condition of limited power supply, such as micro-power grid, a large amount of motor loads cannot be regarded as static loads, because the dynamic characteristics of the asynchronous motors have important significance for the stability of the micro-power grid, for example, the starting/stopping and the adjustment of the motors affect the busbar voltage and the frequency, and even cause the voltage and the frequency to be unstable. The model of the asynchronous motor is usually a set of differential equations, and the Hamiltonian modeling method and the Hamiltonian model of the asynchronous motor are researched from the differential equations, so that a foundation is provided for Hamiltonian control based on the model.
Disclosure of Invention
The invention provides a Hamiltonian modeling method of an asynchronous motor, which carries out structural correction control based on the model, the key of the Hamiltonian model establishment is to obtain a Hamiltonian energy function and the deduction evolution of the Hamiltonian model, the invention starts from the basis of the electromagnetic relationship and the motion equation of a stator and a rotor of the asynchronous motor, according to differential equation models of a stator and a rotor of an asynchronous motor, proper variables are selected, firstly, a Hamiltonian energy function of the stator and the rotor is established, and a standard form of the Hamiltonian model is obtained through further deduction and evolution.
A method for establishing a Hamiltonian model of an asynchronous motor includes the steps of respectively establishing Lagrangian functions of a stator loop and rotor rotational kinetic energy, defining generalized speed according to electromechanical coupling dynamics principle, converting the Lagrangian functions of the stator loop and the rotor rotational kinetic energy into Hamiltonian functions, synthesizing the Hamiltonian functions to obtain the Hamiltonian function of the whole asynchronous motor, combining stator and rotor differential equations on the basis, and obtaining the Hamiltonian model of the asynchronous motor through conversion, wherein current is supplied to a stator and a rotor of the asynchronous motor, a stator winding is connected with a three-phase symmetrical power supply, a rotor winding or an internal short circuit is connected with an external passive circuit through a slip ring, and the rotor winding is equivalent by a three-phase winding, and specifically comprises the following steps:
step1, differential equation of asynchronous motor
The asynchronous motor belongs to an induction motor, and the differential equation of the induction motor is as follows:
Wherein, the stator d-axis transient voltage v 'd＝-ωsLmψqr/Lrr, the stator q-axis transient voltage v' q＝ωsLmψdr/Lrr and the stator transient reactance Stator reactance X s＝ωsLss,ψdr、ψqr is the flux linkage of the d and q windings of the rotor respectively, L ss is the self-inductance of the stator, L rr is the self-inductance of the rotor, and L m is the leakage inductance of the stator; i ds、iqs is stator d, q winding current, s= (ω s－ωr)/ωs is motor slip, ω s、ωr is stator and rotor angular velocity per unit value, ω B is angular velocity base value, T' 0＝Xrr/ωBRr is transient open time constant (seconds), X rr、Rr is reactance and resistance per unit value of rotor winding;
according to the motor principle, the rotor current equation is:
wherein i dr、iqr is the current of the d and q windings of the rotor respectively;
Step 2, establishing an asynchronous motor stator Hamiltonian model
Step1, stator Hamiltonian establishment:
In the induction motor modeling, stator d and q winding transient states are ignored, so that a stator energy storage part is not included in the Lagrange function, the currents i dr、iqr of the axes d and q of a rotor are taken as generalized speeds, a matrix is expressed as i= [ i dr iqr]T, and the rotor energy storage is taken as the Lagrange function:
Wherein, L rr is rotor self-inductance, L m is stator leakage inductance, i ds、iqs is stator d and q winding currents respectively, and i dr、iqr is rotor d and q winding currents respectively;
From electromechanical coupling dynamics, the generalized momentum is defined: p= [ P dr pqr]T:
The Hamilton function of the system is h=p T i-L, i.e.:
Bringing the rotor current equation (2) into (5) to obtain a Hamiltonian:
Wherein v' d,v′q is the transient voltages of the d and q axes of the rotor; x s、X′s is the stator reactance and transient reactance, ω s is the stator angular velocity nominal value ;v′d＝-ωsLmψqr/Lrr,v′q＝ωsLmψdr/Lrr, X s＝ωsLss;ψdr、ψqr is the flux linkage of the d and q windings of the rotor respectively; l ss is stator self-inductance, L rr is rotor self-inductance, and L m is stator leakage inductance;
equation (6) is stator Hamiltonian;
Step2, stator Hamiltonian model establishment:
The hamiltonian H obtained in equation (6) can be derived from v' d、v′q:
wherein: R s is the stator resistance, v ds、vqs is the stator d and q axis voltages; A. b is determined by asynchronous motor parameters and is a constant;
Solving the formula (7), and expressing the A, B front coefficients in the formula as:
Using the above equation, the differential equation (1) can be transformed into hamiltonian form:
Wherein s= (ω s-ωr)/ωs is motor slip, ω s、ωr is stator and rotor angular velocity nominal value, ω B is angular velocity base value, T' 0 is transient open circuit time constant, sec;
step3, establishing a rotor Hamiltonian
A. Establishing a Lagrangian equation for the rotor
The Lagrange function of the motor rotating part only contains rotational kinetic energy, the rotation angular speed omega r is taken as a generalized speed, and the Lagrange function is as follows:
Wherein: omega r rotor angular velocity per-value, T j motor inertia time constant;
Neglecting friction loss of a shafting, setting corresponding dissipation energy as zero, setting electromagnetic torque (main torque) on a motor shaft as m e and mechanical torque as m t, and then setting Lagrange equation as:
b. derivation of Hamiltonian model
The hamiltonian form of the electromagnetic torque m e and the mechanical torque m m in the following derivative (12);
according to kundur "electric power system stabilization and control", the rotor electromagnetic torque expressed by the stator parameters is:
By combining the formulas (2) and (8), the electromagnetic moment expression can be obtained through deformation and deduction as follows:
Wherein,
The mechanical moment can be expressed as:
wherein: m t0 is the rated torque;
Because of According to the analysis dynamics principle H (2)＝pωωr-L(2), the Hamiltonian is obtained by conversion:
Taking into account that The mechanical moment is expressed in the form of a hamiltonian function:
The equation of motion of the rotor can then be written in hamiltonian:
Step4, building a Hamiltonian model of the asynchronous motor:
Selecting a state variable x= [ x 1,x2,x3]＝[ωr,v′d,v′q ], and adding the formulas (6) and (16) to obtain a Hamiltonian of the asynchronous motor, wherein the Hamiltonian is as follows:
formulas (9) and (18) together form the hamilton version of the asynchronous motor:
wherein:
J (x) is an antisymmetric matrix, called a structural matrix, reflecting the associations between the state variables inside the system; r (x) is a symmetric matrix, called a damping matrix, reflecting the damping characteristics of the system state variables on the ports.
The beneficial effects of the invention are as follows:
1. in the application research of the asynchronous motor, the model is usually processed according to two conditions, one is to simplify the model into a constant, the other is to adopt a differential equation, and the Hamiltonian model and the differential equation model provided by the invention provide a structure and a damping matrix, thus being a new thought for modeling the asynchronous motor and creating conditions for further structure correction, damping injection and other control.
2. The model provided by the invention can be used for system control under a Hamiltonian framework.
3. The invention provides a new method for modeling the load class of the asynchronous motor.
Drawings
FIG. 1 is a diagram of a simulation structure;
FIG. 2 shows the variation of bus frequency and voltage when a motor is put into operation;
FIG. 3 shows the variation of bus frequency and voltage when one motor is cut off;
Fig. 4 shows the change in bus voltage when three motors are put into operation.
Detailed Description
Further analysis was performed in connection with specific examples in order to further illustrate the present invention.
Example 1
A method for establishing a Hamiltonian model of an asynchronous motor comprises the steps of respectively establishing Lagrangian functions of a stator loop and rotor rotational kinetic energy, defining generalized speed according to an electromechanical coupling dynamics principle, converting the Lagrangian functions of the stator loop and the rotor rotational kinetic energy into Hamiltonian functions, obtaining the Hamiltonian function of the whole asynchronous motor through synthesis of the Hamiltonian functions, combining stator differential equations and rotor differential equations on the basis, and obtaining the Hamiltonian model of the asynchronous motor through conversion; the stator and the rotor of the asynchronous motor are both provided with current, the stator winding is connected with a three-phase symmetrical power supply, the rotor winding or the inside is short-circuited, or the rotor winding is connected with an external passive circuit through a slip ring, the rotor winding is equivalent by a three-phase winding, the asynchronous motor belongs to an induction motor, and the differential equation of the induction motor can be expressed by the following formulas (1) and (2):
Wherein, the stator d-axis transient voltage v 'd＝-ωsLmψqr/Lrr, the stator q-axis transient voltage v' q＝ωsLmψdr/Lrr and the stator transient reactance Stator reactance X s＝ωsLss,ψdr、ψqr is the flux linkage of the d and q windings of the rotor respectively, L ss is the self-inductance of the stator, L rr is the self-inductance of the rotor, and L m is the leakage inductance of the stator; i ds、iqs is stator d, q winding current, s= (ω s－ωr)/ωs is motor slip, ω s、ωr is stator and rotor angular velocity per unit value, ω B is angular velocity base value, T' 0＝Xrr/ωBRr is transient open time constant (seconds), X rr、Rr is reactance and resistance per unit value of rotor winding;
according to the motor principle, the rotor current equation is:
Where i dr、iqr is the rotor d, q winding current, respectively.
The building process of the Hamiltonian model of the asynchronous motor based on the conditions specifically comprises the following steps:
Step 1: establishing an asynchronous motor stator Hamiltonian model
A. Establishing a stator Hamiltonian:
In the induction motor modeling, stator d and q winding transient states are ignored, so that a stator energy storage part is not included in the Lagrange function, the currents i dr、iqr of the axes d and q of a rotor are taken as generalized speeds, a matrix is expressed as i= [ i dr iqr]T, and the rotor energy storage is taken as the Lagrange function:
Wherein, L rr is rotor self-inductance, L m is stator leakage inductance, i ds、iqs is stator d and q winding currents respectively, and i dr、iqr is rotor d and q winding currents respectively;
From electromechanical coupling dynamics, the generalized momentum is defined: p= [ P dr pqr]T:
The Hamilton function of the system is h=p T i-L, i.e.:
Bringing the rotor current formula (2) into formula (5) to obtain a hamilton function:
Wherein v' d,v′q is the transient voltages of the d and q axes of the rotor; x s、X′s is the stator reactance and transient reactance, ω s is the stator angular velocity nominal value ;v′d＝-ωsLmψqr/Lrr,v′q＝ωsLmψdr/Lrr, X s＝ωsLss;ψdr、ψqr is the flux linkage of the d and q windings of the rotor respectively; l ss is stator self-inductance, L rr is rotor self-inductance, and L m is stator leakage inductance;
equation (6) is stator Hamiltonian;
b. Establishing a stator Hamiltonian model
The hamiltonian H obtained in equation (6) can be derived from v' d、v′q:
wherein: R s is the stator resistance, v ds、vqs is the stator d and q axis voltages; A. b is determined by asynchronous motor parameters and is a constant;
Solving the formula (7), and expressing the A, B front coefficients in the formula as:
Using the above equation, the differential equation (1) can be transformed into hamiltonian form:
Wherein s= (ω s-ωr)/ωs is motor slip, ω s、ωr is stator and rotor angular velocity nominal value, ω B is angular velocity base value, T' 0 is transient open circuit time constant, sec;
Step 2: establishing a rotor Hamiltonian
A. Establishing a Lagrangian equation for the rotor
The Lagrange function of the motor rotating part only contains rotational kinetic energy, the rotation angular speed omega r is taken as a generalized speed, and the Lagrange function is as follows:
Wherein: omega r is the value of the rotor angular velocity per unit; t j is the inertial time constant;
Neglecting friction loss of a shafting, setting corresponding dissipation energy as zero, setting electromagnetic torque (main torque) on a motor shaft as m e and mechanical torque as m m, and then setting Lagrange equation as:
b. Building Hamiltonian model
According to kundur "electric power system stabilization and control", the rotor electromagnetic torque expressed by the stator parameters is:
By combining the formulas (2) and (8), the electromagnetic moment expression can be obtained through deformation and deduction as follows:
Wherein,
The mechanical moment can be expressed as:
wherein: m t0 is the rated torque;
Because of According to the analysis dynamics principle H (2)＝pωωr-L(2), the Hamiltonian is obtained by conversion:
Because of The mechanical moment is expressed in the form of a hamiltonian function:
The equation of motion of the rotor can then be written in hamiltonian:
step 3: building Hamiltonian model of asynchronous motor
Selecting a state variable x= [ x 1,x2,x3]＝[ωr,v′d,v′q ], and adding the formulas (6) and (16) to obtain a Hamiltonian of the asynchronous motor, wherein the Hamiltonian is as follows:
Equations (9) and (18) constitute the hamilton version of the asynchronous motor:
wherein:
J (x) is an antisymmetric matrix, called a structural matrix, reflecting the associations between the state variables inside the system; r (x) is a symmetric matrix, called a damping matrix, reflecting the damping characteristics of the system state variables on the ports.
Example 2
The application of the damping correction control to the hamiltonian model type (20) obtained in example 1 is specifically as follows:
(1) Hamiltonian theory
Let R a (x) and J a (x) be damping and structural correction matrices, and J T a(x)＝-Ja(x),RT a(x)＝Ra (x), let R d(x)＝R(x)+Ra(x),Jd(x)＝J(x)+Ja (x) be the equivalent transformation of formula (20):
wherein:
The above two sides are multiplied by g T (x), if g (x) is full of rank, g T (x) g (x) is reversible, and there are:
Where w is a new control and α is an additional control generated by hamilton structure correction, to be used to equivalent the structural changes generated by R a (x) and J a (x);
(2) Damping matrix correction application
Let us say that the damping matrix R a is modified as follows:
the additional control is calculated according to equation (23):
the new control is calculated according to equation (22):
(3) Simulation test
The simulation structure is as shown in fig. 1, simulation parameters: main parameters of the diesel generator are as follows: rated power 1250kW, rated rotating speed n=1500r/min, moment of inertia j=71.8232 kg.m 2, equivalent damping coefficient d= 2.1753, diesel generator adopts traditional PID control, speed regulator PID parameter: k P＝2.0,KI＝2.0,KD =0.1, excitation PID parameters: k P＝2.0,KI＝2.0,KD =0.1, static load and asynchronous motor parameters are converted into standard values (pu) by taking the capacity of a diesel engine as a basic value, each motor has an active power of 0.1pu and a reactive power of 0.047pu, and dynamic load change is simulated by the different input quantity of the motors in simulation.
Initial working conditions: the diesel generator has the active power of 0.8 (pu), the reactive power of 0.6 (pu), 7 motors are put into operation, the total absorption active power of 0.7 (pu), the reactive power of 0.3291 (pu) and the rest active and reactive power are static loads, and the active and reactive power on the bus reach balance.
Disturbance condition 1: let t=0.5 s time, put into 1 motor, at this time two control methods are used: (1) Taking r 12 =0.0005, and calculating additional control according to a formula (23); (2) conventional PID control (no additional control) of diesel engine; the busbar frequency and voltage obtained by two control effects in the input process of the asynchronous motor are shown in fig. 2, and the result analysis is that: the damping structure is adopted to correct additional control, so that the influence on the bus frequency is not obvious, but the disturbance amplitude of the bus voltage can be effectively improved.
Disturbance condition 2: one motor was cut off at t=0.5 s, the other conditions were the same as for disturbance 1, the busbar frequency and voltage variations were as shown in fig. 3, and the results were analyzed: the value of r 12 has little influence on the bus frequency, similar to the situation of input of a motor, but the influence of r 12 on the bus voltage is larger, and the positive and negative values of the value are opposite to the situation of input of the motor, namely, r 12 = -0.0005 plays a role in inhibiting the bus voltage fluctuation.
Disturbance condition 3: in order to increase the disturbance quantity, when setting at t=0.5s, 3 motors are put into simultaneously, two control modes for comparison are the same as the disturbance working condition 1, two control effects are as shown in fig. 4, and the result analysis is that: under the condition that the disturbance quantity is increased, the effect of the damping structure correction is more obvious.
Claims (1)
1. A method for establishing a Hamiltonian model of an asynchronous motor is characterized in that Lagrangian functions of a stator loop and a rotor rotational kinetic energy are respectively established, generalized speeds are defined according to an electromechanical coupling dynamics principle, the Lagrangian functions of the stator loop and the rotor rotational kinetic energy are converted into Hamiltonian functions, the Hamiltonian functions of the whole asynchronous motor are obtained through the synthesis of the Hamiltonian functions, and a Hamiltonian model of the asynchronous motor is obtained through the conversion by combining stator differential equations and rotor differential equations on the basis;
the method specifically comprises the following steps:
step1, stator Hamiltonian establishment:
Taking the rotor d, q-axis currents i dr、iqr as the generalized speed, the matrix is expressed as i= [ i dr iqr]T, and the lagrangian function is written as:
Wherein, L rr is rotor self-inductance, L m is stator leakage inductance, i ds、iqs is stator d and q winding currents respectively, and i dr、iqr is rotor d and q winding currents respectively;
From electromechanical coupling dynamics, the generalized momentum is defined: p= [ P dr pqr]T:
The Hamilton function of the system is h=p T i-L, i.e.:
Equation of rotor current Substituting the above formula to obtain a Hamiltonian:
Wherein v d′,vq' is the transient voltages of the d and q axes of the rotor; x s、Xs' is the stator reactance and transient reactance, respectively, ω s is the stator angular velocity nominal value ;v′d＝-ωsLmψqr/Lrr,v′q＝ωsLmψdr/Lrr, X s＝ωsLss;ψdr、ψqr is the flux linkage of the d and q windings of the rotor respectively; l ss is stator self-inductance, L rr is rotor self-inductance, and L m is stator leakage inductance;
Step2, stator Hamiltonian model establishment:
and (3) performing bias derivation on v d′、vq' by using the Hamiltonian H obtained in the formula (1) to obtain:
wherein: R s is the stator resistance, v ds、vqs is the stator d and q axis voltages; A. b is a constant;
the coefficient solution for A, B in the above equation is expressed as:
by using the above formula, the differential equation of the asynchronous motor is calculated Transforming into Hamiltonian form:
Where s= (ω s-ωr)/ωs is motor slip, ω s、ωr is stator and rotor angular speed per-unit value, ω B is angular speed base value, T 0' is transient open time constant, seconds;
step3, building a rotor Hamiltonian model:
taking the rotation angular velocity omega r as a generalized velocity, the rotor Lagrangian function is:
Wherein: omega r is the value of the rotor angular velocity per unit; t j is the motor inertia time constant;
neglecting friction loss of the shafting, and enabling corresponding dissipation energy to be zero, the Lagrange equation is as follows:
Namely:
Wherein: m e is the electromagnetic torque on the motor shaft, and m t is the mechanical torque;
(1) m e establishment of hamiltonian form:
the rotor electromagnetic torque expressed by the stator parameters is as follows:
substituting the formula (2) into the above formula and finishing to obtain an electromagnetic moment expression as follows:
Wherein,
(2) M t establishment of hamiltonian form:
Mechanical moment:
wherein: m t0 is the rated torque;
Because of According to the analysis dynamics principle H (2)＝pωω-L(2), the Hamiltonian is obtained by conversion:
Taking into account that The mechanical moment is expressed in the form of a hamiltonian function:
The rotor equation of motion is written in hamiltonian:
Step4, building a Hamiltonian model of the asynchronous motor:
Selecting a state variable x= [ x 1,x2,x3]＝[ωr,vd′,vq' ], and adding the formulas (1) and (4) to obtain a Hamiltonian of the asynchronous motor, wherein the Hamiltonian is as follows:
the equation (3) and (5) together form a Hamiltonian model of the asynchronous motor, and the Hamiltonian model is written into a unified form:
wherein:
wherein: R s is the stator resistance; A. b is determined by asynchronous motor parameters and is constant.
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