CN108988361A - The quick suppressing method of two-shipper interconnected electric power system chaotic oscillation - Google Patents
The quick suppressing method of two-shipper interconnected electric power system chaotic oscillation Download PDFInfo
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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Abstract
The present invention provides a kind of quick suppressing method of two-shipper interconnected electric power system chaotic oscillation, is suitable for electro-engineering field.According to set time Theory of Stability, in conjunction with traditional self adaptive control and quick Terminal sliding-mode control, gamma controller is designed, control amount is reached in finite time in the arbitrarily small neighborhood of its reference value, realizes the global set time in the asymptotically stability of equalization point.It under conditions of not depending on initial value stablizes in finite time system；It can guarantee that stablizing time range has the upper bound, and the upper bound for stablizing the time can be calculated in method provided by the present invention；With very strong robustness and anti-interference ability；It can realize system global consistent asymptotic stability in the given time under arbitrary initial conditions, more effectively inhibit the chaotic oscillation in electric system, improve the stability of electric system.
Description
Technical field
The present invention relates to a kind of quick suppressing methods, are particularly suitable for two-shipper interconnecting electric power used in electrical engineering field
The quick suppressing method of chaotic systems oscillation.
Background technique
Electric system is shown big in the process of running as a kind of nonlinear system of typical multivariable close coupling
Measure non-linear dynamical behavior, such as low-frequency oscillation, bifurcated and chaos phenomenon.Especially when by external disturbance, as long as disturbing
Dynamic amplitude is met certain condition, and electric system just will appear persistently random chaotic oscillation, and this oscillation, which may result in, is
System unstability, collapse of voltage cause large-area power-cuts.Therefore, it is very necessary for studying the suppressing method of electric system chaos.It is double
Machine interconnected electric power system is the system that two isolated power systems containing generator are connected by interconnection, each is individually
System mainly by equivalent generator, equivalent main transformer, breaker and load composition, and be connected by system interconnection.
Chaos controlling is the research hot topic of academia.The control method of chaotic oscillation is divided into targeting chaos and vibrates to expected
Track development and two aspects of generation for inhibiting chaotic systems, currently used control method have parameter perturbation method (OGY), feedback
Control methods, adaptive control law, fuzzy control, it is counter push away control etc..Stabilization of these control methods for guarantee electric system
It is had great theoretical and practical significance with safe operation, but also has certain disadvantage simultaneously.It is for example, counter that push away control be to pass through
Reverse design makes design process systematization, the structuring of the Lyapunov function and controller of system.But structure is very multiple
Miscellaneous, especially when model has uncertainty there are when nonlinear dampling, regression matrix will become increasingly complex.Fuzzy control
It is difficult to adapt to the requirement adjusted on a large scale, needs constantly to adjust control rule and parameter.Also, all above-mentioned control methods can only
Reach asymptotic eventual stability, i.e., cannot distribute convergence time in advance.
From the viewpoint of Operation of Electric Systems, if damped within the limited time, oscillation is only acceptable.And
Finite-time control can then guarantee that t is greater than T (t, x0) after, system is stablized, and with so that system has better robust
Property, anti-external interference and the advantages of faster restrain.Quick Terminal sliding formwork control can also be realized within the limited time
System is stablized, and in addition compares the speciality that traditional sliding formwork control can only make system mode reach sliding-mode surface in finite time, quickly
Terminal sliding formwork control can guarantee system mode on sliding-mode surface in finite time convergence control to equalization point.Both control methods
System can be made to stablize in finite time, but the limitation system stability for being similarly limited to finite time stability is
System Effects of Initial Conditions, and system primary condition is generally difficult to obtain accurate parameter in the power system.
Summary of the invention
The present invention provide it is a kind of by design Chaos Oscillation of Power System system it is adaptive fixed when quick Terminal
Sliding mode controller reaches control amount in the finite time independent of initial value in the arbitrarily small neighborhood of its reference value, thus
System global consistent asymptotic stability is realized, effectively inhibits the chaotic oscillation in electric system, effectively improves power system stability
The quick suppressing method of the two-shipper interconnected electric power system chaotic oscillation of property.
To achieve the goals above, the quick suppressing method of a kind of two-shipper interconnected electric power system chaotic oscillation of the invention,
Its step are as follows:
1) second order for establishing electric system using the traditional two-shipper interconnected electric power system mathematical model of electromechanical engineering field is micro-
Divide equation, the two-shipper interconnected electric power system includes that two isolated power systems containing generator utilize interconnection interconnection composition
System；
2) the quick Terminal for designing two-shipper interconnected electric power system state variable according to set time Theory of Stability is sliding
The adaptive law of die face and uncertain parameter；
3) pass through set time Theory of Stability using the adaptive law of quick Terminal sliding-mode surface and uncertain parameter
It is derived from the Nonlinear control law of two-shipper interconnected electric power system state variable；
4) liapunov function is constructed, using the Nonlinear control law of two-shipper interconnected electric power system state variable to prove
The two-shipper interconnected electric power system set time stablizes；
5) two-shipper interconnecting electric power system is determined according to set time Theory of Stability and liapunov function stability analysis
The stabilization time range upper limit of system, two-shipper interconnected electric power system stablize time maximum and are no more than the upper limit；
6) control effect for verifying the gamma controller u according to nonlinear control law design, is restrained using controller u
The stabilization time of chaotic oscillation is necessarily stablizing within time upper limit.
The two-shipper interconnected electric power system mathematical model second order differential equation is as follows:
In formula, δ, ω are respectively generator amature angle and relative rotation speed, are differentiated to δ, ω；PsAnd PmRespectively generator
Electromagnetic power and input mechanical output；H is equivalent inertia time constant, and D is damped coefficient；PeFor the amplitude for disturbing load, λ
For the frequency for disturbing load；Nondimensionalization processing is carried out to two-shipper interconnected electric power system mathematical model second order differential equation, so that
Controlled electric chaotic systems oscillatory system model equivalency is in being converted to system:
In formula, u is that the control of two-shipper interconnected electric power system inputs, a=Ps/ H, b=D/H, c=Pm/ H, F=Pe/ H, [x1,
x2]=[δ, ω], and f (x)=- asinx1-bx2+ c, a, b, c, d, F are letter character, are the nondimensionalizations to archetype
Processing.
The adaptive law design method of quick Terminal sliding-mode surface and uncertain parameter are as follows:
First according to quick Terminal sliding mode design principle, quick Terminal sliding-mode surface is selected are as follows:
In formula, α0, β0, q0, p0For parameter to be designed, meet α0, β0> 0, q0, p0For positive odd number,
To formula III derivation:
Take global quickly Terminal sliding mode form are as follows:
Thus realize the set time stability of system state variables, in formula:γ, α, β, p, q are system ginseng to be designed
Number meetsγ > 0,0 < α < 1, β > 1, p, q are positive odd number, and q < p；
The adaptive law expression formula of uncertain parameter is designed according to auto-adaptive control theory are as follows:
In formula,For the estimated value of uncertain parameter F, g is any normal number；
To sum up, two-shipper interconnected electric power system designs Nonlinear control law, that is, controller u mathematic(al) representation are as follows:
The calculation method of the stable time range upper limit are as follows:
Consider following nonlinear system:
X ∈ R in formulan, f is system state variables and mission nonlinear function respectively, and t is the time, is managed according to the set time
By if there are continuous positive definite differentiable function V (x), first derivatives by nonlinear system VIIINegative definite, then nonlinear system
VIIILyapunov stablizes, right if existing simultaneously Local Bounded stablizes function of time T (x)As t >=T (x), x (t)=0 is permanent
It sets up, then nonlinear system VIII in origin is global finite time stability at this time；If the convergence time of nonlinear system VIII
There is the upper bound, and dividing value is unrelated with state variable x thereon, i.e., under arbitrary initial conditions,So that
And as t >=T (x), x (t) ≡ 0, it is stable to be referred to as the global set time by nonlinear system VIII at this time；For nonlinear system
VIII, it is assumed that there are function V (x): RnThe continuous positive definite of → R can be micro-, includes balance neighborhood of a point D ∈ R for onen, V's (x) leads
Number meets:
D*V(x)≤-[αVp(x)+βVq(x)]k,
Wherein α, β, p, q, k > 0, and pk < 1；If V (x) is from D ∈ R at this timenAny position starts, in set time T
Interior can to make V (x) ≡ 0, i.e., the system set time stablizes, and its convergence time are as follows:
Determine that designed controller u control two-shipper interconnected electric power system is stablized according to liapunov function stability analysis
The time range upper limit:
Construct Lyapunov function:
The controller u and corresponding tuner parameters designed using quick Terminal sliding formwork control principle, when with fixing
Between Theory of Stability obtained the derivative of system Lyapunov function:
In formula,
It is hereby achieved that system stablizes time upper limit are as follows:
I.e. as t >=t1When, two-shipper interconnected electric power system reaches stable, and chaotic oscillation is inhibited.
The utility model has the advantages that the quick suppressing method of two-shipper interconnected electric power system chaotic oscillation of the invention can not only disobey
Relying the upper limit for stablizing system in finite time under conditions of system state variables initial value, and stablizing the time to pass through calculating can
, there is very strong robustness and anti-interference ability, while can realize in the given time in the case where not depending on primary condition double
The globally consistent ultimate boundness of machine interconnected electric power system is stablized, and the chaotic oscillation in two-shipper interconnected electric power system is more effectively inhibited,
Improve the voltage stability of electric system；
Set time stability is a popularization of stability in finite time, it has all excellent of finite-time control
Point, and the above control method is compared, the set time can not only guarantee that stablizing time range has the upper bound, have stronger robust
Property and anti-interference ability, it is most important that the system global consistent asymptotic stability under arbitrary initial conditions；
Quick Teminal sliding mode controller, may be implemented two machine interconnected electric power systems when the present invention is using adaptive fix
Chaotic oscillation it is limited when inhibit, the stable time upper bound can be by being calculated.
This method step is simple, accurately stablizes time upper limit by calculating can be obtained, anti-interference ability is good, robustness
By force, there is wide applicability.
Detailed description of the invention
Fig. 1 is two-shipper interconnected electric power system structural schematic diagram of the present invention；
Fig. 2 is the Lyapunov exponential spectrum of two-shipper interconnected electric power system in the embodiment of the present invention；
Fig. 3 (a) is two-shipper interconnected electric power system chaotic oscillation state variable when not adding controller in the embodiment of the present invention
Time response figure；
Fig. 3 (b) is two-shipper interconnected electric power system chaotic oscillation state variable when not adding controller in the embodiment of the present invention
Time response figure；
Fig. 4 is two-shipper interconnected electric power system phasor when not adding controller in the embodiment of the present invention；
Fig. 5 is the flow chart of the quick suppressing method of two-shipper interconnected electric power system chaotic oscillation of the invention；
Fig. 6 (a) is two-shipper interconnected electric power system chaotic oscillation shape when the controller of design being added in the embodiment of the present invention
The time response of state variable schemes；
Fig. 6 (b) is two-shipper interconnected electric power system chaotic oscillation shape when the controller of design being added in the embodiment of the present invention
The time response of state variable schemes；
Fig. 7 is two-shipper interconnected electric power system phasor when the controller of design being added in the embodiment of the present invention.
Specific embodiment
Below with reference to examples and drawings, the invention will be further described, and embodiment provided by the invention is not used in restriction
Invention.
The quick suppressing method of two-shipper interconnected electric power system chaotic oscillation of the invention, first to two-shipper interconnected electric power system
Carry out mathematical model, then according to set time Theory of Stability design quick Terminal sliding-mode surface and uncertain parameter from
Rule is adapted to, Nonlinear control law is obtained secondly by theory deduction and designs controller, is managed again according to set time stability
It is determined by affiliated lemmas and liapunov function stability analysis and stablizes the time range upper bound, verified finally by numerical simulation
Its control effect；
As shown in fig. 6, specific steps are as follows:
1) second order for establishing electric system using the traditional two-shipper interconnected electric power system mathematical model of electromechanical engineering field is micro-
Divide equation, the two-shipper interconnected electric power system includes that two isolated power systems containing generator utilize interconnection interconnection composition
System；
2) the quick Terminal for designing two-shipper interconnected electric power system state variable according to set time Theory of Stability is sliding
The adaptive law of die face and uncertain parameter；
3) pass through set time Theory of Stability using the adaptive law of quick Terminal sliding-mode surface and uncertain parameter
It is derived from the Nonlinear control law of two-shipper interconnected electric power system state variable；
4) liapunov function is constructed, using the Nonlinear control law of two-shipper interconnected electric power system state variable to prove
The two-shipper interconnected electric power system set time stablizes；
5) two-shipper interconnecting electric power system is determined according to set time Theory of Stability and liapunov function stability analysis
The stabilization time range upper limit of system, two-shipper interconnected electric power system stablize time maximum and are no more than the upper limit；
6) control effect for verifying the gamma controller u according to nonlinear control law design, is restrained using controller u
The stabilization time of chaotic oscillation is necessarily stablizing within time upper limit.
Wherein as shown in Figure 1, two-shipper interconnected electric power system carries out mathematical modeling, in figure: E1, E2 and T1, T2 shows respectively is
The equivalent generator and main transformer electric appliance of system, P, Q indicate active power and reactive power, and σ 1 and σ 2 indicate the rotor of equivalent generator
Angle；
It is as follows:
Wherein, δ, ω are respectively generator amature angle and relative rotation speed, are differentiated to δ, ω；PsAnd PmRespectively generator
Electromagnetic power and input mechanical output；H is equivalent inertia time constant, and D is damped coefficient；PeFor the amplitude for disturbing load, λ
For the frequency for disturbing load；Nondimensionalization processing is carried out to two-shipper interconnected electric power system mathematical model second order differential equation: enabling a=
Ps/ H, b=D/H, c=Pm/ H, F=Pe/ H, [x1, x2]=[δ, ω], then enable f (x)=- asinx1-bx2+ c, then controlled electric
Chaotic systems oscillatory system model can simplify are as follows:
In formula, u is control input, and a, b, c, d, F is letter character, is the nondimensionalization processing to archetype；
As selected parameter Ps/ H=1, D/H=0.02, Pm/ H=0.2, PeWhen/H=0.2593, such as Fig. 2 system
Shown in Lyapunov exponential spectrum, Lyapunov index LE1> 0, LE2=0, LE3< 0, and LE1<-LE3, illustrate that system is chaos
, and there are chaos attractors.It, can be with if Fig. 3 (a), Fig. 3 (b) and Fig. 4 clearly demonstrate the chaotic behavior of electric system
The time response for finding out state variable is irregular and aperiodic oscillatory regime, and their track is in a very long time
It is inside uncertain.If taken measures not in time, electric system unstability will be caused, collapse of voltage occurs.
Target is controlled to realize, designs the adaptive law of quick Terminal sliding-mode surface and uncertain parameter:
Quick Terminal sliding-mode surface is selected first are as follows:
In formula: α0, β0, q0, p0For parameter to be designed, α0, β0> 0, q0, p0For positive odd number.This embodiment takes parameter alpha0=2,
β0=1, q0=5, p0=9；
To formula III derivation, can obtain:
In order to realize the set time stability of system variable, global quickly Terminal sliding mode form is taken are as follows:
Thus realize the set time stability of system state variables, in formula:γ, α, β, p, q are system ginseng to be designed
Number,0,0 < α < 1 of γ >, β > 1, p, q are positive odd number, and q < p, this embodiment take parameterγ=10, α=0.5, β
=1.5, p=3, q=1.
Design the adaptive law of uncertain parameter are as follows:
Wherein,It is the estimated value of uncertain parameter F, g is any normal number, this embodiment takes g=0.3.
By formula IV, Formula V and Formula IV, nonlinear control law design is designed for second order two-shipper interconnected electric power system are as follows:
The calculation method of the stable time range upper limit are as follows:
Consider following nonlinear system:
X ∈ R in formulan, f is system state variables and mission nonlinear function respectively, and t is the time, is managed according to the set time
By if there are continuous positive definite differentiable function V (x), first derivatives by nonlinear system VIIINegative definite, then nonlinear system
VIIILyapunov stablizes, right if existing simultaneously Local Bounded stablizes function of time T (x)As t >=T (x), x (t)=0 is permanent
It sets up, then nonlinear system VIII in origin is global finite time stability at this time；If the convergence time of nonlinear system VIII
There is the upper bound, and dividing value is unrelated with state variable x thereon, i.e., under arbitrary initial conditions,So that
And as t >=T (x), x (t) ≡ 0, it is stable to be referred to as the global set time by nonlinear system VIII at this time；For nonlinear system
VIII, it is assumed that there are function V (x): RnThe continuous positive definite of → R can be micro-, includes balance neighborhood of a point D ∈ R for onen, V's (x) leads
Number meets:
D*V(x)≤-[αVp(x)+βVq(x)]k,
Wherein α, β, p, q, k > 0, and pk < 1；If V (x) is from D ∈ R at this timenAny position starts, in set time T
Interior can to make V (x) ≡ 0, i.e., the system set time stablizes, and its convergence time are as follows:
Determine that designed controller u control two-shipper interconnected electric power system is stablized according to liapunov function stability analysis
The time range upper limit:
Construct Lyapunov function:
Using the controller u and corresponding tuner parameters of design, system has been obtained with set time Theory of Stability
The derivative of Lyapunov function:
Wherein,
It is hereby achieved that system stablizes the time upper bound are as follows:
I.e. as t >=t1When, two-shipper interconnected electric power system reaches stable, and chaotic oscillation is inhibited.
Each parameter taken in the present embodiment is brought into wherein, obtains t1≤30.35.That is system was stablized on the time
Boundary is in the 30.35s after applying controller, and in other words, after applying controller 30.35s, system centainly reaches stable, chaos
Oscillation is inhibited.
Using controller designed by set time Theory of Stability, it is imitative that data are carried out on MATLAB emulation platform
Very, the control effect of access control device.The present embodiment initial value is taken as S (δ0, ω0)=(0.43,0.003).Two-shipper interconnecting electric power
The system time response of state variable and chaos electric system phasor after the controller that the present invention designs is added are shown in Fig. 6
(a), Fig. 6 (b) and Fig. 7, as shown in Fig. 6 (a), Fig. 6 (b) and Fig. 7, control target has been stabilized to required equalization point, mixes
Ignorant oscillation is suppressed, to demonstrate the validity of controller, is had in two-shipper interconnected electric power system chaotic oscillation good
Inhibitory effect.
Claims (4)
1. a kind of quick suppressing method of two-shipper interconnected electric power system chaotic oscillation, it is characterised in that steps are as follows:
1) the second-order differential side of electric system is established using the traditional two-shipper interconnected electric power system mathematical model of electromechanical engineering field
Journey, the two-shipper interconnected electric power system includes two isolated power systems containing generator is using interconnection interconnection composition
System；
2) the quick Terminal sliding-mode surface of two-shipper interconnected electric power system state variable is designed according to set time Theory of Stability
And the adaptive law of uncertain parameter；
3) it is derived using the adaptive law of quick Terminal sliding-mode surface and uncertain parameter by set time Theory of Stability
Obtain the Nonlinear control law of two-shipper interconnected electric power system state variable；
4) liapunov function is constructed, using the Nonlinear control law of two-shipper interconnected electric power system state variable to prove two-shipper
The interconnected electric power system set time stablizes；
5) two-shipper interconnected electric power system is determined according to set time Theory of Stability and liapunov function stability analysis
Stablize the time range upper limit, two-shipper interconnected electric power system stablizes time maximum and is no more than the upper limit；
6) control effect for verifying the gamma controller u according to nonlinear control law design, restrains chaos using controller u
The stabilization time of oscillation is necessarily stablizing within time upper limit.
2. the quick suppressing method of two-shipper interconnected electric power system chaotic oscillation according to claim 1, it is characterised in that: institute
It is as follows to state two-shipper interconnected electric power system mathematical model second order differential equation:
In formula, δ, ω are respectively generator amature angle and relative rotation speed, are differentiated to δ, ω；PsAnd PmThe respectively electricity of generator
Magnetic power and input mechanical output；H is equivalent inertia time constant, and D is damped coefficient；PeFor the amplitude for disturbing load, λ is to disturb
The frequency of dynamic load；Nondimensionalization processing is carried out to two-shipper interconnected electric power system mathematical model second order differential equation, so that controlled
Chaos Oscillation of Power System system model is equivalent to be converted to following system:
In formula, u is that the control of two-shipper interconnected electric power system inputs, a=Ps/ H, b=D/H, c=Pm/ H, F=Pe/ H, [x1, x2]=
[δ, ω], and f (x)=- asinx1-bx2+ c, a, b, c, d, F are letter character, are the nondimensionalization processing to archetype.
3. the quick suppressing method of two-shipper interconnected electric power system chaotic oscillation according to claim 1, it is characterised in that fast
The adaptive law design method of fast Terminal sliding-mode surface and uncertain parameter are as follows:
First according to quick Terminal sliding mode design principle, quick Terminal sliding-mode surface is selected are as follows:
In formula, α0, β0, q0, p0For parameter to be designed, meet α0, β0> 0, q0, p0For positive odd number,
To formula III derivation:
Take global quickly Terminal sliding mode form are as follows:
Thus realize the set time stability of system state variables, in formula:γ, α, β, p, q are system parameter to be designed, full
Footγ > 0,0 < α < 1, β > 1, p, q are positive odd number, and q < p；
The adaptive law expression formula of uncertain parameter is designed according to auto-adaptive control theory are as follows:
In formula,For the estimated value of uncertain parameter F, g is any normal number；
To sum up, two-shipper interconnected electric power system designs Nonlinear control law, that is, controller u mathematic(al) representation are as follows:
4. the quick suppressing method of two-shipper interconnected electric power system chaotic oscillation according to claim 3, it is characterised in that institute
State the calculation method of the stable time range upper limit are as follows:
Consider following nonlinear system:
X ∈ R in formulan, f is system state variables and mission nonlinear function respectively, and t is the time, it is theoretical according to the set time, such as
There are continuous positive definite differentiable function V (x), first derivatives by fruit nonlinear system VIIINegative definite, then nonlinear system
VIIILyapunov stablizes, right if existing simultaneously Local Bounded stablizes function of time T (x)The x (t)=0 as t >=T (x)
Perseverance is set up, then nonlinear system VIII in origin is global finite time stability at this time；If when the convergence of nonlinear system VIII
Between have the upper bound, and dividing value is unrelated with state variable x thereon, i.e., under arbitrary initial conditions,So thatAnd as t >=T (x), x (t) ≡ 0, it is stable to be referred to as the global set time by nonlinear system VIII at this time；For
Nonlinear system VIII, it is assumed that there are function V (x): RnThe continuous positive definite of → R can be micro-, includes balance neighborhood of a point D ∈ for one
Rn, the derivative satisfaction of V (x):
D*V(x)≤-[αVp(x)+βVq(x)]k,
Wherein α, β, p, q, k > 0, and pk < 1；If V (x) is from D ∈ R at this timenAny position starts, must in set time T
It can make V (x) ≡ 0, i.e., the system set time stablizes, and its convergence time are as follows:
Determine that designed controller u control two-shipper interconnected electric power system stablizes the time according to liapunov function stability analysis
Range limit:
Lyapunov function is constructed first:
The controller u and corresponding tuner parameters designed using quick Terminal sliding formwork control principle is steady with the set time
Qualitative theory has obtained the derivative of system Lyapunov function:
In formula,
It is hereby achieved that system stablizes time upper limit are as follows:
I.e. as t >=t1When, two-shipper interconnected electric power system reaches stable, and chaotic oscillation is inhibited.
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| CN111130104A (en) * | 2020-01-07 | 2020-05-08 | 西安理工大学 | Passivity sliding mode control method for chaotic oscillation of power system |
| CN111564841A (en) * | 2020-05-29 | 2020-08-21 | 南京师范大学 | Input quantization finite time fault-tolerant anti-interference control method of multi-machine power system |
| CN112147900A (en) * | 2020-09-30 | 2020-12-29 | 苏州科技大学 | Finite time self-adaptive fuzzy tracking control method of full-state constraint power system |
| CN113867141A (en) * | 2021-09-06 | 2021-12-31 | 国网辽宁省电力有限公司鞍山供电公司 | Multi-machine electric power system fixed time dynamic surface high-order sliding mode controller with SVC |
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| CN113867141B (en) * | 2021-09-06 | 2024-02-06 | 国网辽宁省电力有限公司鞍山供电公司 | Multi-machine power system fixed time dynamic surface high-order sliding mode controller with SVC |
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