CN109378807B - Suppression method of ferromagnetic resonance overvoltage chaotic fixed time sliding mode - Google Patents

Suppression method of ferromagnetic resonance overvoltage chaotic fixed time sliding mode Download PDF

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CN109378807B
CN109378807B CN201811299024.6A CN201811299024A CN109378807B CN 109378807 B CN109378807 B CN 109378807B CN 201811299024 A CN201811299024 A CN 201811299024A CN 109378807 B CN109378807 B CN 109378807B
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ferromagnetic resonance
time
fixed time
sliding mode
resonance overvoltage
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马草原
张勇
赵东艳
刘伟韬
刘鹏娟
李志杰
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China University of Mining and Technology CUMT
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02H—EMERGENCY PROTECTIVE CIRCUIT ARRANGEMENTS
    • H02H9/00—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection
    • H02H9/005—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection avoiding undesired transient conditions
    • H02H9/007—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection avoiding undesired transient conditions avoiding or damping oscillations, e.g. fenoresonance or travelling waves
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02H—EMERGENCY PROTECTIVE CIRCUIT ARRANGEMENTS
    • H02H9/00—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection
    • H02H9/04—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection responsive to excess voltage
    • H02H9/045—Emergency protective circuit arrangements for limiting excess current or voltage without disconnection responsive to excess voltage adapted to a particular application and not provided for elsewhere

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Abstract

The invention discloses a restraining method of a fixed time sliding mode of ferromagnetic resonance overvoltage chaos, which comprises the following steps: (1) establishing a mathematical model of the ferromagnetic resonance system; (2) designing an integral sliding mode surface according to a fixed time stability theory; (3) obtaining a nonlinear control law and an adaptive law of uncertain parameters through theoretical derivation; (4) determining an upper bound of a stable time range according to a fixed time stability theory related theorem and a Lyapunov function stability analysis; (5) the control effect is verified through numerical simulation. The suppression method of ferromagnetic resonance overvoltage disclosed by the invention not only can stabilize the system within a limited time without depending on an initial value, but also can calculate the upper bound of the stable time, has stronger robustness and anti-interference capability, can realize the global consistent gradual stabilization of the system within a preset time under any initial condition, more effectively suppresses the ferromagnetic resonance overvoltage of the power system, and improves the stability of the power system.

Description

Suppression method of ferromagnetic resonance overvoltage chaotic fixed time sliding mode
Technical Field
The invention belongs to the field of electrical engineering, and particularly relates to a suppression method of a ferromagnetic resonance overvoltage chaotic fixed time sliding mode.
Background
A plurality of iron core inductive elements such as transformers, voltage transformers, generators, arc suppression coils, reactors and the like exist in the power system; there are also many capacitive elements such as the ground and phase to ground capacitances of the conductors, compensation capacitors, stray capacitances of high voltage devices, etc. that form a complex tank circuit in which it is possible to excite long-lasting ferroresonant overvoltages if there is a large disturbance or operation in the power system.
At present, along with the massive progress of the reconstruction and extension projects of power grids in various regions, the structure of the power grid is more complex. Complex power systems also increase the probability of ferroresonance occurring. Ferroresonance frequently occurs in both a neutral point ungrounded system of below 110kV and a neutral point directly grounded system of above 110 kV. During resonance, overvoltage is generated, overcurrent is caused, the duration time is long, even the overvoltage can exist stably, insulation flashover, explosion of a lightning tube and equipment damage can be caused, power failure accidents can be caused in serious cases, and the safety operation of a power grid is seriously threatened. In a neutral-grounded system, a circuit formed by a system power supply via a breaker grading capacitor and an electromagnetic voltage transformer may also induce ferroresonance. The "self-healing" function included in the robust smart grid strategy proposed by the national grid requires isolation of the problematic elements of the grid from the system and rapid restoration of the system to normal operation with little or no human intervention, thereby providing little interruption to the power supply service to the user. Therefore, the strengthening of the research on the early suppression of the ferromagnetic resonance overvoltage has very important significance.
In recent years, many experts and scholars theoretically and deeply analyze the chaotic state of a neutral point grounding system when a ferromagnetic resonance phenomenon occurs and the representation of the chaotic state, and theoretically analyze and inhibit the chaotic state caused by ferromagnetic resonance overvoltage in a power system. In the aspect of ferromagnetic resonance overvoltage suppression, chaos control methods such as constant pulse and neural network based on maximum entropy have also been proposed, but these methods all have certain disadvantages (such as large control energy consumption, complex method, etc.), and do not have strong practical significance in engineering.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a chaos suppression method of ferromagnetic resonance overvoltage, and designs a self-adaptive fixed time sliding mode controller for a ferromagnetic resonance chaotic system according to a fixed time stability theory, so that the ferromagnetic resonance overvoltage can be suppressed within a limited time independent of an initial value.
The technical scheme adopted by the invention is as follows: a restraining method of a ferromagnetic resonance overvoltage chaotic fixed time sliding mode specifically comprises the following steps:
(1) establishing a mathematical model of the ferromagnetic resonance system;
(2) designing an integral sliding mode surface according to a fixed time stability theory;
(3) obtaining a nonlinear control law and an adaptive law of uncertain parameters through theoretical derivation;
(4) determining an upper bound of a stable time range according to a fixed time stability theory related theorem and a Lyapunov function stability analysis;
(5) the control effect is verified through numerical simulation.
Further, for the power system with the neutral point directly grounded, the mathematical model of the ferroresonant system in step (1) has the following equation:
Figure BDA0001850951170000021
in the formula: r is the core loss of the TV; emIs the power supply amplitude; c ═ C1+C2Is the equivalent capacitance in the equivalent circuit;is the magnetic flux in the non-linear inductor; u is the effective value of the voltage at the two ends of the iron core; omega is the system frequency;
Figure BDA0001850951170000023
is a non-linear inductance magnetization characteristic
Figure BDA0001850951170000024
A relationship; u is the control input.
Further, the slip form surface designed in the step (2) is as follows:
firstly, a switching function is designed, and the system outputs a tracking target of
Figure BDA0001850951170000025
UdThe error in defining the controlled quantity is:
in order to expand the robustness of the system to the whole system and eliminate steady-state errors, an integral term is added into a linear sliding mode surface, and the following integral sliding mode surface is constructed:
Figure BDA0001850951170000027
Figure BDA0001850951170000028
further, the nonlinear control law designed for the ferroresonance overvoltage system in the step (3) is as follows:
Figure BDA0001850951170000029
-ls-ksign(s)|s|a-ksign(s)|s|β
and:
Figure BDA0001850951170000038
wherein, 0 is more than α and less than 1, β is more than 1, and g is any normal number.
Further, the nonlinear system designed according to the fixed time stability theory in the step (4) is as follows:
wherein
Figure BDA0001850951170000039
Respectively, a system state variable and a system nonlinear function, the first derivative of which is the derivative of the continuous positive definite differentiable function V (x) if present for the system (7)
Figure BDA0001850951170000031
If a local bounded stable time function T (x) exists, if any t is greater than or equal to T (x), x (t) is equal to 0, and then the system (7) is called as global finite time stable at the origin; if the convergence time of the system (7) is bounded above and its value is independent of the state variable x, i.e. at any initial conditionIn the following, the first and second parts of the material,
Figure BDA0001850951170000032
so thatAnd when t ≧ T (x), x (t) is ≡ 0, at which time the system (7) is said to be globally fixed-time stable; for the nonlinear system (7), it is assumed that there is a function v (x): rn→ R continuous positive definite microminiature, for a neighborhood D epsilon R containing equilibrium pointsnV (x) satisfies:
D*V(x)≤-[αVp(x)+βVq(x)]k
or
Figure BDA0001850951170000034
Wherein α, p, q, k > 0 and pk < 1, when V (x) is selected from D e RnStarting at an arbitrary position, v (x) 0, i.e. the system fixed time, must be stabilized for a fixed time T, and the convergence time is:
further, determining an upper bound of the stability time range according to lyapunov function stability analysis:
constructing a Lyapunov function:
Figure BDA0001850951170000036
a derivative of a Lyapunov function of the system is obtained by using a designed controller u and corresponding tuning parameters and applying a fixed time stability theory:
wherein:
Figure BDA0001850951170000042
Figure BDA0001850951170000043
the upper bound of the system settling time can thus be found to be:
Figure BDA0001850951170000044
i.e. when t ≧ t1When the voltage is high, the ferromagnetic resonance overvoltage is suppressed.
And (5) performing a numerical simulation experiment on an MATLAB simulation platform by using the adaptive law of uncertain parameters and the nonlinear controller designed in the steps (2) and (3) to verify the control effect of the controller.
Compared with the prior art, the invention has the technical effects and advantages that: the method for quickly suppressing the ferromagnetic resonance overvoltage of the power system disclosed by the invention not only can stabilize the system within a limited time without depending on an initial value, but also can calculate the upper bound of the stabilization time, has stronger robustness and anti-interference capability, and most importantly, can realize the global consistent gradual stabilization of the system within a preset time under any initial condition, more effectively suppress the ferromagnetic resonance overvoltage of the power system, and improve the stability of the power system.
Drawings
FIG. 1 is a flow chart of a method for suppressing a fixed-time sliding mode of ferromagnetic resonance overvoltage chaos provided by the invention;
FIG. 2 is a schematic diagram of a typical ferroresonant tank circuit of a neutral point direct-grounded system substation employed in the present invention;
fig. 3 is a ferroresonant circuit diagram of a single phase of a substation and a simplified circuit diagram thereof, as employed in the present invention;
FIG. 4 is a time response graph of a ferroresonant overvoltage chaotic state variable without a controller in an embodiment of the present invention;
FIG. 5 is a phase diagram of a ferroresonant overvoltage chaotic state without a controller in an embodiment of the present invention;
FIG. 6 is a time response graph of a ferroresonant overvoltage chaotic state variable with the addition of a designed controller in an embodiment of the present invention;
figure 7 is a phase diagram of a ferroresonant system with the addition of a designed controller in an embodiment of the invention.
Detailed Description
For the purpose of enhancing the understanding of the present invention, the present invention will be further explained with reference to the accompanying drawings and examples, which are only for the purpose of explaining the present invention and do not limit the scope of the present invention.
As shown in fig. 1 to 7, the chaos fast suppression method for over-voltage of ferromagnetic resonance in power system provided by the present invention comprises establishing a mathematical model for a ferromagnetic resonance system, designing an integral sliding mode surface according to a fixed time stability theory, deriving a nonlinear control law and an adaptive law of uncertain parameters through theory derivation, designing a controller, determining an upper bound of a stable time range according to a fixed time stability theory correlation theorem and a lyapunov function stability analysis, and verifying a control effect through numerical simulation.
The specific process is as follows:
(1) a mathematical model is established from the typical ferromagnetic resonance circuit schematic of the neutral point direct grounding system substation shown in fig. 2 and its simplified equivalent circuit of fig. 3, as follows:
Figure BDA0001850951170000051
in the formula: r is the core loss of the TV; emIs the power supply amplitude; c ═ C1+C2Is the equivalent capacitance in the equivalent circuit;is the magnetic flux in the non-linear inductor; u is the effective value of the voltage at the two ends of the iron core; omega is the system frequency;
Figure BDA0001850951170000053
is notLinear inductance magnetization characteristic
Figure BDA0001850951170000054
A relationship; u is the control input.
When the parameter C is selectedpu＝21.9633，Rpu＝8.6508，
Figure BDA0001850951170000055
ωpu＝1，EmpuWhen the signal is 1, the chaotic response of the voltage waveform is clearly shown as shown in fig. 4(a) and (b), and fig. 5 is a dynamic phase diagram in this case. The phase traces in the diagram are not in an equilibrium state and do not have periodic solutions, but are disordered and have certain random characteristic behaviors, which indicate that the ferroresonance behaviors are in a chaotic state. If no measures are taken, immeasurable damage will be caused to the power system.
(2) In order to realize the control target, an integral sliding mode surface is designed according to a fixed time stability theory.
Firstly, a switching function is designed, and the system outputs a tracking target ofUdThe error in defining the controlled quantity is as follows:
Figure BDA0001850951170000062
in order to expand the robustness of the system to the whole system and eliminate steady-state errors, an integral term is added into a linear sliding mode surface, and the following integral sliding mode surface is constructed:
Figure BDA0001850951170000063
and (5) obtaining a derivative:
(3) according to the formulas (1), (3) and (4), the nonlinear control rate of the chaotic suppression of the ferromagnetic resonance overvoltage can be designed:
wherein k is an uncertain parameter, and the self-adaptation law is as follows:
k＝|s|a+1+|s|β+1-(k-g)a-(k-g)β(6)
where 0 < α < 1, β > 1, and g is any normal number, this example is α -0.5, β -1.5, and g-1.
(4) Constructing a Lyapunov function:
Figure BDA0001850951170000066
a derivative of a Lyapunov function of the system is obtained by using a designed controller u and corresponding tuning parameters and applying a fixed time stability theory:
Figure BDA0001850951170000067
wherein:
Figure BDA0001850951170000068
Figure BDA0001850951170000069
the upper bound of the system settling time can thus be found to be:
i.e. when t ≧ t1When the voltage is high, the ferromagnetic resonance overvoltage is suppressed.
The parameters taken in the embodiment are substituted to obtain t1 ≦ 4.38. Namely, the upper limit of the system stable time is within 4.38s after the controller is applied, in other words, after the controller is applied for 4.38s, the system is always stable, and the ferromagnetic resonance overvoltage is inhibited. The flow of the provided chaotic suppression method for ferromagnetic resonance overvoltage is shown in fig. 1.
(5) And (3) performing data simulation on the MATLAB simulation platform by using the controller designed by the fixed time stability theory, and verifying the control effect of the controller. The initial value of this embodiment is taken as
Figure BDA0001850951170000072
The time response of the state variable of the ferroresonant overvoltage system after the controller designed by the invention is added and the phase diagram of the chaotic power system are respectively shown in fig. 6(a), (b) and fig. 7. It can be seen that the control target has stabilized to the required equilibrium point and the ferroresonance overvoltage is suppressed, thereby verifying the effectiveness of the controller.
The embodiments of the present invention are disclosed as the preferred embodiments, but not limited thereto, and those skilled in the art can easily understand the spirit of the present invention and make various extensions and changes without departing from the spirit of the present invention.

Claims (3)

1. A restraining method of a fixed time sliding mode of ferromagnetic resonance overvoltage chaos is characterized by comprising the following steps:
(1) establishing a mathematical model of the ferromagnetic resonance system;
(2) designing an integral sliding mode surface for a controller according to a fixed time stability theory;
(3) obtaining a nonlinear control law and an adaptive law of uncertain parameters through theoretical derivation;
(4) determining an upper bound of a stable time range according to a fixed time stability theory related theorem and a Lyapunov function stability analysis;
(5) verifying the control effect of the control system through numerical simulation;
the mathematical model of the power system ferroresonance in the step (1) is given by the following equation:
Figure FDA0002277151220000011
in the formula: r is the core loss of a voltage Transformer (TV); emIs the power supply amplitude; c ═ C1+C2Is the equivalent capacitance in the equivalent circuit;
Figure FDA0002277151220000012
is the magnetic flux in the non-linear inductor; u is the effective value of the voltage at the two ends of the iron core; omega is the system frequency;
Figure FDA0002277151220000013
is a non-linear inductance magnetization characteristic
Figure FDA0002277151220000014
A relationship; u is a control input;
the design method of the integral sliding mode surface in the step (2) comprises the following steps:
firstly, designing a switching function, and setting a system output tracking target as
Figure FDA0002277151220000015
UdThe error in defining the controlled quantity is as follows:
Figure FDA0002277151220000016
in order to expand the robustness of the system to the whole system and eliminate steady-state errors, an integral term is added into a linear sliding mode surface, and the following integral sliding mode surface is constructed:
the formula (3) is derived:
2. the suppression method of the ferromagnetic resonance overvoltage chaotic fixed time sliding mode according to claim 1, wherein the nonlinear control law and the uncertain parameter adaptation law in the step (3) are as follows:
according to the formulas (1), (3) and (4), a nonlinear control law of the chaotic suppression of the ferromagnetic resonance overvoltage can be designed:
wherein k is an adaptive parameter, and the adaptive law is as follows:
Figure FDA0002277151220000022
wherein, 0 is more than α and less than 1, β is more than 1, and g is any normal number.
3. The suppression method of the ferromagnetic resonance overvoltage chaotic fixed time sliding mode according to claim 1, wherein the upper bound of the stable time range in the step (4) is as follows:
the nonlinear system is designed according to the fixed time stability theory as follows:
Figure FDA0002277151220000023
wherein
Figure FDA0002277151220000024
Respectively, a system state variable and a system nonlinear function, the first derivative of which is the derivative of the continuous positive definite differentiable function V (x) if present for the system (7)
Figure FDA0002277151220000025
If a local bounded stable time function T (x) exists, if any t is greater than or equal to T (x), x (t) is equal to 0, and then the system (7) is called as global finite time stable at the origin; if system (A)7) Has an upper bound on its convergence time and its upper bound value is independent of the state variable x, i.e. under any initial conditions,so that
Figure FDA0002277151220000027
And when t ≧ T (x), x (t) is ≡ 0, at which time the system (7) is said to be globally fixed-time stable; for the nonlinear system (7), it is assumed that there is a function v (x): rn→ R continuous positive definite microminiature, for a neighborhood D epsilon R containing equilibrium pointsnV (x) satisfies:
D*V(x)≤-[αVp(x)+βVq(x)]k
or
Figure FDA0002277151220000028
Wherein α, p, q, k > 0 and pk < 1, when V (x) is selected from D e RnStarting at an arbitrary position, v (x) 0, i.e. the system fixed time, must be stabilized for a fixed time T, and the convergence time is:
Figure FDA0002277151220000031
determining the upper bound of the stability time range of the ferromagnetic resonance overvoltage of the designed controller control power system according to the stability analysis of the Lyapunov function:
constructing a Lyapunov function:
Figure FDA0002277151220000032
a derivative of a Lyapunov function of the system is obtained by using a designed controller u and corresponding tuning parameters and applying a fixed time stability theory:
Figure FDA0002277151220000033
wherein:
Figure FDA0002277151220000034
Figure FDA0002277151220000035
the upper bound of the system settling time can thus be found to be:
i.e. when t ≧ t1When the voltage is high, the ferromagnetic resonance overvoltage is suppressed.
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