This application claims priority to Chinese Patent Application Ser. No. CN202111246916.1 filed on 26 Oct. 20.
The present invention belongs to the technical field of generators, and relates to a fuzzy finite-time optimal synchronization control method for a fractional-order permanent magnet synchronous generator.
A permanent magnet synchronous generator is one of the most important electric power equipment, and can convert wind energy into mechanical work, and the mechanical work drives magnetic rotors to rotate so as to output alternating current. At present, it is widely used for high-speed power generation, renewable energy power generation, wind power generation, and the like. Permanent magnet synchronous generators in wind power plants are typically operated in harsh environments, and fluctuations from local wind speeds, generator temperatures, magnetic fields, friction and workload are unavoidable. These fluctuations may cause many nonlinear dynamics phenomena such as chaotic oscillation, multi-stability and bifurcation of the permanent magnet synchronous generator. These fluctuations can lead to a degradation of the performance of the permanent magnet synchronous generator and even to a damage to the generator without taking effective measures. In addition, with the improvement of living standard, there are higher and higher requirements on the reliability and safety of the permanent magnet synchronous generator system. Therefore, how to accurately model, reveal stable and unstable boundaries of system motion, and provide a fuzzy finite-time optimal synchronous control scheme between fractional-order driving and driven permanent magnet synchronous generators within a prescribed performance range has important significance and challenge.
Compared with integral-order calculus, fractional-order calculus can establish a more accurate model, describe dynamic characteristics of an actual engineering system more truly, and simultaneously obtain the optimal dynamic response by adjusting fractional-order values of the system. Westerlund and Ekstam experimentally demonstrated fractional-order values of various capacitor media, and found a close fractional-order relationship between current and voltage. Luo et al. experimentally verified the control performance of a fractional-order system using a fractional-order capacitive membrane model. For modeling and dynamic analysis of fractional-order generators, there have been sporadic literature reports over the past few decades. Xu et al. established a fractional-order model of a hydro-generator set system and performed nonlinear dynamic analysis on the model using six typical fractional-order values. Borah and Roy performed dynamic analysis of a fractional-order permanent magnet synchronous generator and proposed a single-state predictive control method. Ardjal et al. established a fractional-order model of a wind energy conversion system and designed a nonlinear cooperative controller of a generator and a grid converter. These important findings have certain inspirations for the current research. However, these models are limited to analyzing a single generator, and do not consider the coupling strength between generators; and the corresponding dynamic analysis does not give stable and unstable boundaries and regions, and does not consider the system control problem such as finite-time, uncertainty, performance constraints and optimization.
The synchronization of coupled systems has attracted much attention in nonlinear science, for example, synchronization control can make multiple generators achieve accurate frequency coincidence in parallel mode in the field of power generation. In order to better address this problem, an enormous amount of valuable work such as robust synchronization, adaptive synchronization and optimal synchronization control are emerging continuously. However, these works are limited to integral-order systems that do not consider the strengths of the bidirectional and unidirectional coupling, and these systems do not generate complex nonlinear dynamics when conditions fluctuate, deviating from the discussed subject of synchronous control. Sadeghi et al. solved the problem of smooth synchronization of brushless doubly-fed induction generators by using machine models. Zhu et al. solved the synchronization problem of chaotic oscillation permanent magnet synchronous generator networks by using a self-adaptive pulse control method. The above literatures do not disclose chaos and periodic intervals in an integer-order generator. At the same time, failure of these schemes without pre-configured performance is inevitable due to perturbation and uncertainty of system modeling parameters.
As an effective control method, backstepping control is widely applied to integer-order uncertain nonlinear systems by fusing a fuzzy logic system or a neural network. Some researchers extend the backstepping control to the field of a fractional-order. In view of the inherent “complexity explosion” problem of backstepping control, whether the backstepping control belongs to an integer-order or a fractional-order, a first-order low-pass filter, a tracking differentiator and an observer are generally adopted to solve the problem. However, the coupled permanent magnet synchronous generator is completely different from a nonlinear mathematical model, the mathematical model can not generate undesirable dynamic behaviors such as chaotic oscillation and multi-stability, and meanwhile, the fuzzy finite-time optimal synchronous control is not mentioned in the above work. Optimal control is an important research topic because it consumes less resources. Some scholars introduce the optical control into the backstepping control of the nonlinear system, thus fully exerting their advantages. The finite-time control of the nonlinear system has the advantages of high response speed, high convergence precision and the like, and is a research topic worthy of attention. In a wind power farm, a unidirectional coupling fractional-order permanent magnet synchronous generator system is a highly complex system with surrounding generators coupled. For a method to fuse effective methods such as optimal control, backstepping control and finite-time control and the like, a fuzzy finite-time optimal synchronous control scheme of a unidirectional coupled permanent magnet synchronous generator system is provided to achieve specified performance, which is still a prominent problem in the field of fractional-order control.
In view of the above, the present invention is intended to provide a fuzzy finite-time optimal synchronization control method for a fractional-order permanent magnet synchronous generator.
For the above purposes, the present invention provides the following technical scheme.
Provided is a fuzzy finite-time optimal synchronization control method for the fractional-order permanent magnet synchronous generator, which comprises the following steps:
S1: modeling a system;
S2: designing a numerical method and applying the numerical method to solve a nonlinear fractional-order system; and
S3: establishing a hierarchical type-2 fuzzy neural network and designing a controller that provides an ordered and coordinated motion for the fractional-order permanent magnet synchronous generator.
Optionally, the S1 specifically comprises that: a wind energy conversion system consists of a wind turbine, a permanent magnet synchronous generator and three converters, and the three converters are sequentially a diode bridge rectifier, a DC/DC boost converter and an inverter in the order; the electric energy generated by the permanent magnet synchronous generator is transmitted to a grid through the converters;
in combination with local aerodynamic characteristics, the power generated by the wind turbine is expressed as:
wherein ρ, R, ωr, β, ωr and νw represent air density, turbine radius, rotation speed, propeller blade angle, rotation speed and wind speed, respectively, and Cp(ωrR/νw,β) represents turbine power coefficient;
a mechanical fractional-order model of the permanent magnet synchronous generator is provided according to rotation law:
wherein α, J, Tt, Tg, {tilde over (t)} and b represent fractional-order coefficient, system inertia, turbine torque, generator torque, time and viscous friction coefficient, respectively;
the torque of an electromagnetic generator is expressed as:
wherein Ld and Lq represent d-axis inductance and q-axis inductance, respectively, id and iq represent d-axis stator current and q-axis stator current, respectively, and p and ϕ represent the number of pole pairs and magnetic flux of permanent magnets in a three-phase permanent magnet synchronous generator, respectively;
the fractional-order model of the permanent magnet synchronous generator in a synchronous rotating d-q reference frame is expressed as:
wherein Rs, Vd and Vq represent stator resistance, d-axis stator voltage and q-axis stator current, respectively;
in the Laplace domain, a fractional-order integrator in the form of linear approximation with a zero pole pair is represented by a transfer function with a slope of −20m dB/decade in a Bode diagram:
wherein
and Pƒ, ωmax and dƒ represent difference values between angular frequency, bandwidth and actual line and approximate line, respectively; Qi and Pi represent zero and pole of singularity function, respectively;
L=Ld=Lq is obtained due to symmetrical stator windings; a fractional-order model of the permanent magnet synchronous generator is defined by equation (2) and equation (4) as follows:
wherein C D0,tα represents Caputo fractional-order derivative with α>0 and starting point at origin;
by introducing new variables x1=Lωr/Rs, x2=pLϕiq/bRs and x3=pLϕid/bRs, a normalized fractional-order model of a permanent magnet synchronous main motor is expressed as:
wherein t={tilde over (t)}Rs/L, uq=pLϕVq/bRs2, ud=pLϕVd/bRs2, TL=−L2Tt/JRs2, μ=−pϕ2/bRs, σ=3Lb/2JRs and ρ=bL/JRs, and x1, x2, x3, t, uq, ud and TL represent normalized angular velocity, q-axis current, d-axis current, time, q-axis voltage, d-axis voltage and load torque, respectively; σ, ρ and μ represent system parameters;
a fractional-order model of the driven permanent magnet synchronous generator is established by using a Heaviside function H(t−Tg):
wherein tg, κ1, and κ2 represent primary synchronous time, the capacitance coupling and the resistance coupling, respectively, and u2 and u3 represent control inputs;
synchronization errors are defined as e1=y1−x1, e2=y2−x2 and e3=y3−x3; the following equation is obtained by subtracting equation (7) from equation (8):
Definition 1: for a sufficiently differentiable function F(t), the Caputo fractional-order derivative is expressed as:
wherein
represents Euler gamma function, n−1<α<n, and n∈+;
a Laplace transform is performed on equation (10) to obtain:
for any continuous function, when 0<α<1 and F1(t) and F2(t) are within an interval [0,tζ], the following equation is obtained:
when F1(t)=F2(t), the following inequation is derived as:
C D0,tα (F2(t))≤2F(t)·C D0,tαF(t) (13)
Lamma 1: for any xs, ys∈, ϑ(xs, ys) is any positive real function and then the following inequation holds if cs>0 and ds>0.
Definition 2: a minimum performance cost function is as follows:
wherein
Optionally, the S2 specifically comprises that:
for a fractional-order differential equation C D0,tαyƒ(t)=g(t, yƒ(t)), t≥0 with a given function g(⋅,⋅), the fractional-order derivative quotient of yƒ(t) is defined as an infinite series, i.e.,
wherein h>0 and ωj(α) represent step size and coefficient, respectively, and a condition based on the Euler-Gamma function is satisfied as follows:
an equidistant grid tn=nh, n=0, 1, ⋅ ⋅ ⋅ , N is defined in an interval [0 T], wherein N=T/h; an integral expression is then obtained when t=tn:
a vector field g(τ, yƒ(τ)) is approximated by a constant g(tj, yƒ
system parameters of the fractional-order model of the driving/driven permanent magnet synchronous generator are set to be three working conditions:
working condition 1: α=0.99, σ=3, ρ=4, μ=25, TL=0, and the initial conditions are x1(0)=0.1, x2(0)=0.9, and x3(0)=20;
working condition 2: α=0.99, σ=17, ρ=16, μ=25, TL=0, and the initial conditions are x1(0)=1.5, x2(0)=0.5, and x3(0)=20; and
working condition 3: α=0.99, σ=5.5, ρ=5.5, μ=20, TL=0, and the initial conditions are x1(0)=0.1, x2(0)=0.1, and x3 (0)=3;
approximate transfer functions of different fractional-orders are given by introducing equation (5):
error values in the above equations are ≤0.1 dB, ≤0.2 dB and ≤0.4 dB, respectively.
Optionally, the S3 specifically comprises that:
the hierarchical type-2 fuzzy neural network consists of an input layer, a membership layer, a rule layer, a type-reduction layer and an output layer;
at the second level, upper and lower membership levels are written as:
wherein mÀ
in the rule layer, an upper/down trigger rule is calculated as:
in the fourth layer, YR and YL are expressed in the form of a set type-reduction center as:
wherein wRj and wLj are referred as weight, Tj and
in the output layer, the fuzzy output in the form of vector is derived as:
Y=wTξ=wRTξR+wLTξL (26)
wherein
by calling equation (26), the hierarchical type-2 fuzzy neural network realizes high-precision approximation to any unknown but bounded function on a compact set, then
wherein X≡[x1 ⋅ ⋅ ⋅ xi]T∈, i=1 ⋅ ⋅ ⋅ N, N represents the number of inputs, ε(X)>0 represents approximation errors, and Ωw and DX are compact sets of appropriate bounds of w and X, respectively; an optimal parameter w* is introduced, and the parameter meets the condition:
ĥ represents approximation values of h; {tilde over (w)}=w−W* is defined, wherein w* represents an amount of labor for analysis;
in order to avoid exponential increase of the number of rules, two criteria of tracking error reduction rate and fuzzy rule ε completeness are adopted to execute the structural adjustment of the hierarchical type-2 fuzzy neural network. The tracking error reduction rate is equivalent to the derivative of the square of tracking errors between a drive system output and a response system. The completeness of the fuzzy rule is defined as that “at least one fuzzy rule ensures that the trigger strength within an operation range is not less than ε”. If the membership grade is equal to or greater than 0.5 in the hierarchy structure, the membership grade is saved. Otherwise, these unimportant rules in a self-structure algorithm is deleted.
In order to improve solving speed and simplify the structure, the hierarchical type-2 fuzzy neural network is transformed as follows:
wTξ(X)≤ζξT(X)ξ(X)/2b2+b2/2 (28)
wherein ζ=∥w∥2, and b>0; {tilde over (ζ)}=ζ−{circumflex over (ζ)}, wherein {circumflex over (ζ)} represents estimate values of ζ, and C D0,tα{tilde over (ζ)}=−C D0,tα{circumflex over (ζ)};
in order to avoid system performance degradation and suppress the convergence characteristics of error variables, a positive and strictly monotonically decreasing finite-time prescribed performance function is designed:
wherein ai, i=0, ⋅ ⋅ ⋅ , 3 represents design parameters, and T0 and βT
wherein β0 represents the initial value of the finite-time prescribed performance function;
when a constraint signal is matched with the convergence rate of the prescribed performance function, a constraint condition suppresses a large overshoot of the control output at an initial stage; for the pre-given parameters β0, T0 and βT
in order to achieve a faster response, a fractional-order finite-time command filter based on a first order Levant differentiator is provided:
wherein Zi,1∈ and Zi,2∈ represent states of the command filter, αr∈ represents input signals of the command filter, ci,1 and ci,2 represent positive design constants, and positive constants Γi,1 and Γi,2 satisfy conditions |Zi,1−αr|≤Γi,1 and |
obviously, by appropriate selection of ci,1 and ci,2, Zi,1=αr and
error variables are introduced:
z2=e2−α2c, z3=e3−α3c (32)
wherein αic, i=2, 3 and Zi,1, i=2, 3 are the same, representing the output of the fractional-order command filter;
the compensated tracking error is represented as:
υi=zi−θi, i=1,2,3 (33)
wherein θi represents a compensation signal between virtual control and filtered signals;
the error variables have the following inequation:
−ρβ(t)<e1<
wherein 0<ρ,
a smooth and invertible function S1(z1) is defined, and the above inequation is rewritable to an unconstrained form:
e1=β(t)S1(z1) (35)
it is noted that (
then the inverse transform of equation (35) is written as:
z1=S1−1(ϕ(t)) (36)
wherein ϕ1(t)=e1(t)/β(t);
the fractional-order derivative of equation (36) is derived as:
C D0,tαz1=β−1(t)·C Dϕ(t)αS1−1(⋅)(C D0,tαe1(t)−ϕ1(t)CD_hd 0,tαβ(t))=η1(C D0,tαe1−r1), (37)
wherein η1=β−1(t)·C Dϕ(t)αS1−1(ϕ(t)), and r1=ϕ1(t)C D0,tαβ(t).
By using equations (9) and (37), then:
C D0,tαz1=η1(ƒ1+σe2−r1), C D0,tαe2=ƒ2+u2, C D0,tαe3=ƒ3+u3 (38)
ƒ1=−ρe1, ƒ2=−e2−y1y3+x1x3−(κ1e1+κ2e2)H(t−tg)+μe1 and ƒ3=−e3−y1y2−x1x2 are all regarded as unknown nonlinear functions due to perturbations from wind speeds, generator temperatures, stator resistance, friction coefficient, workload, and the like;
the design of the controller comprises three steps based on a fractional-order backstepping control principle;
step 1: the above-described hierarchical type-2 fuzzy neural network is adopted to perform estimation on a compact set for facilitating controller design considering the uncertainty of ƒ1, i.e.,
ƒ1=w1Tξ1(⋅)+ε1(⋅) (39)
wherein (⋅) represents abbreviations of (x1, x2, x3),
a first Lyapunov-candidate-function is selected as:
the fractional-order derivative of V1 is taken to obtain:
C D0,tαV1≤υ1η1[ƒ1+
wherein α2ν and
the following cost function is designed to achieve its minimum value by definition 2;
wherein uoi, δi2, and κi represent an optimal control input, a normal constant and a design parameter, respectively; the following inequation ξi2(⋅)≤δi2υi2, i=1, 2, 3 is used in order to compensate estimation errors of the hierarchical type-2 fuzzy neural network;
the optimal control input is designed as uoi=−Piυi/κi2, i=1, 2, 3, wherein Pi is the solution of an algebraic Riccati equation Pi2+2kiPi−κi2−1=0, ki>0;
the optimal control input is derived as:
uoi=−(√{square root over (1+ki2+δi2)}−ki)υi/κi2, i=1,2,3 (43)
virtual control, and an adaptive law and a compensation signal thereof are selected:
α2ν=(−k1z1−υ1η1{circumflex over (ζ)}1ξ1T(⋅)ξ1(⋅)/2b12+r1−s1υ1γ+uoi)/
C D0,tα{circumflex over (ζ)}1=υ12η12ξ1T(⋅)ξ1(⋅)/2b12−γ1{circumflex over (ζ)}1 (45)
C D0,tαθ1=−k1η1θ1+
wherein k1>0, l1>0, γ1>0, s1>0, and 0<γ<1;
equations (44)-(46) are substituted into inequation (41) by inequation (14) to obtain:
C D0,tαV1≤−k1η1υ12−k1η1θ12+
wherein b1=s1η1−l1ϑ1/(γ+1), and d1=l1(1−γϑ1−1/γ/(γ+1));
step 2: a hierarchical type-2 fuzzy neural network is used to approximate the aforementioned unknown nonlinear function ƒ2 with high precision in the following form to solve the unknown nonlinear function:
ƒ2=w2Tξ2(⋅)+ε2(⋅) (48)
a second Lyapunov-candidate-function is selected as:
the fractional-order derivative of V2 is written as:
wherein ζ2=∥w2∥2, and b2>0;
the q-axis control input and the adaptive law and the compensation signal are designed as follows:
wherein k2>0, l2>0, γ2>0 and s2>0;
by equations (51)-(53), inequation (50) is further simplified as:
wherein b1=s2−l2ϑ2/(γ+1), and d2=l2(1−γϑ2−1/γ/(γ+1));
step 3: a hierarchical type-2 fuzzy neural network with very high precision and repeatability is used for estimation to process the unknown nonlinear function ƒ3:
ƒ3=w3Tξ3(⋅)+ε3(⋅) (55)
the magnetic field orientation control is adopted, so that α3c is equal to zero; a last Lyapunov candidate function is defined as:
the fractional-order derivative of V3 is calculated as:
wherein ζ3=∥w3∥2, and b3>0;
the d-axis control input, the adaptive law and the compensation signal are designed as follows:
u3=−k3z3−{circumflex over (ζ)}3υ3ξ3T(⋅)ξ3(⋅)/2b32−s3υ3γ+uo3 (58)
C D0,tα{circumflex over (ζ)}3=υ32ξ3T(⋅)ξ3(⋅)/2b32−γ3{circumflex over (ζ)}3 (59)
C D0,tαθ3=−k3θ3−l3θ3γ (60)
wherein k3>0, l3>0, γ3>0 and s3>0;
obtaining:
wherein b3=s3−l3ϑ3/(γ+1), and d3=l3(1−γϑ3−1/γ/(γ+1)).
The present invention has the following beneficial effects:
Firstly, unlike an integer-order model of a single isolated permanent magnet synchronous generator, the present invention establishes a unidirectional coupling fractional-order permanent magnet synchronous generator synchronization model which can realize ordered and coordinated motion. The model can accurately describe the dynamic characteristics of the system, increasing the degree of freedom of design. Meanwhile, under the designed numerical method, dynamic analysis reveals the stable and unstable boundaries of the system along with the change of time, and illustrates the chaos and regular motion trend of the system along with the change of parameters.
Secondly, the present invention solves the problem of “complexity explosion” of the traditional backstepping control of a nonlinear system (even in a high-order system), processes the problem of infinite-time convergence in an exponential form and the problem of non-adjustable convergence speed between an initial stage and a steady-state stage, and simultaneously provides greater flexibility and better functional characteristics compared with a common fuzzy logic system and a neural network.
Thirdly, under a fractional-order backstepping framework, fuzzy finite-time optimal synchronous control of the unidirectional coupling fractional-order permanent magnet synchronous generator system is achieved, and the problems of finite-time convergence, unknown system nonlinear function, prescribed constraint conditions, minimum cost function and the like are solved.
Fourthly, additional advantages, objectives, and features of the present invention will be described below in the specification to some extent, and will be apparent to those skilled in the art upon the study of the following or may be learned from the practice of the present invention. The objectives and other advantages of the present invention will be realized and acquired through the following description of the specification.
For a better understanding of the objectives, technical schemes and advantages of the present invention, details will be described below with reference to the accompanying drawings.
The embodiments of the present invention are illustrated by specific examples below, and additional advantages and functions of the present invention will be readily apparent to those skilled in the art from the disclosure herein. The present invention can further be implemented or applied through different specific embodiments, and various details in this specification can be modified or changed based on different views and applications without departing from the spirit of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and the embodiments of the present invention may be combined with each other without conflict.
The drawings are only for exemplary illustration, showing only schematic diagrams, not physical drawings, and cannot be understood as limiting the present invention. In order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, which does not represent the actual size of products. It is understood by those skilled in the art that some well-known structures in the drawings and descriptions thereof may be omitted.
1. System Modeling
As shown in
In combination with local aerodynamic characteristics, the power generated by the wind turbine is expressed as:
wherein ρ, R, ωr, β, ωr and νw represent air density, turbine radius, rotation speed, propeller blade angle, rotation speed and wind speed, respectively, and Cp(ωrR/νw,β) represents turbine power coefficient.
A mechanical fractional-order model of the permanent magnet synchronous generator is provided according to rotation law:
wherein, α, J, Tt, Tg, {tilde over (t)} and b represent fractional-order coefficient, system inertia, turbine torque, generator torque, time and viscous friction coefficient, respectively.
The torque of an electromagnetic generator is expressed as:
wherein Ld and Lq represent d-axis inductance and q-axis inductance, respectively, id and iq represent d-axis stator current and q-axis stator current, respectively, and p and ϕ represent the number of pole pairs and magnetic flux of permanent magnets in a three-phase permanent magnet synchronous generator, respectively.
The fractional-order model of the permanent magnet synchronous generator in a synchronous rotating d-q reference frame is expressed as:
wherein Rs, Vd and Vq represent stator resistance, d-axis stator voltage and q-axis stator current, respectively.
In the Laplace domain, a fractional-order integrator in the form of linear approximation with a zero pole pair is represented by a transfer function with a slope of −20m dB/decade in a Bode diagram:
wherein
and Pƒ, ωmax and dƒ represent difference values between angular frequency, bandwidth and actual line and approximate line, respectively; Qi and Pi represent zero and pole of singularity function, respectively.
L=Ld=Lq is obtained due to symmetrical stator windings. A fractional-order model of the permanent magnet synchronous generator is defined by equation (2) and equation (4) as follows:
wherein C D0,tα represents Caputo fractional-order derivative with α>0 and starting point at origin.
By introducing new variables x1=Lωr/Rs, x2=pLϕiq/bRs and x3=pLϕid/bRs, a normalized fractional-order model of a permanent magnet synchronous main motor may be expressed as:
wherein t={tilde over (t)}Rs/L, uq=pLϕVq/bRs2, ud=pLϕVd/bRs2, TL=−L2Tt/JRs2, μ=−pϕ2/bRs, σ=3Lb/2JRs and ρ=bL/JRs, and x1, x2, x3, t, uq, ud and TL represent normalized angular velocity, q-axis current, d-axis current, time, q-axis voltage, d-axis voltage and load torque, respectively; σ, ρ and μ represent system parameters.
a fractional-order model of the driven permanent magnet synchronous generator is established by using a Heaviside function H(t−Tg):
wherein tg, κ1 and κ2 represent primary synchronous time, the capacitance coupling and the resistance coupling, respectively, and u2 and u3 represent control inputs.
Synchronization errors are defined as e1=y1−x1, e2=y2−x2 and e3=y3−x3. The following equation is obtained by subtracting equation (7) from equation (8):
Definition 1: for a sufficiently differentiable function F(t), the Caputo fractional-order derivative is expressed as:
wherein
represents Euler gamma function, n−1<α<n, and n∈+.
A Laplace transform is performed on equation (10) to obtain:
for any continuous function, when 0<α<1 and F1(t) and F2(t) are within an interval [0,tζ], the following equation is obtained:
when F1(t)=F2(t), the following inequation is derived as:
C D0,tα (F2(t))≤2F(t)·C D0,tαF (t) (13)
Lamma 1: for any xs, ys∈, ϑ(xs, ys) is any positive real function and then the following inequation holds if cs>0 and ds>0.
Definition 2: a minimum performance cost function is as follows:
wherein
2. Dynamic Analysis and Problem Raising
This section performs a dynamic analysis on the fractional-order driving/driven permanent magnet synchronous generator to reveal its dynamic characteristics. At present, it is difficult to obtain an explicit solution to the nonlinear fractional-order differential equation. The present invention, by using an integration rule, designs a numerical method and applies the numerical method to solve a nonlinear fractional-order system.
For a general fractional-order differential equation C D0,tαyƒ(t)=g(t, yƒ(t)), t≥0 with a given function g(⋅,⋅), the fractional-order derivative quotient of yƒ(t) is defined as an infinite series, i.e.,
wherein h>0 and ωj(α) represent step size and coefficient, respectively, and a condition based on the Euler-Gamma function is satisfied as follows:
An equidistant grid tn=nh, n=0, 1, ⋅ ⋅ ⋅ , N is defined in an interval [0 T], wherein N=T/h. An integral expression is then obtained when t=tn.
A vector field g(τ, yƒ(τ)) is approximated by a constant g(tj, yƒ
System parameters of the fractional-order model of the driving/driven permanent magnet synchronous generator are set to be three working conditions:
working condition 1: α=0.99, σ=3, ρ=4, μ=25, TL=0, and the initial conditions are x1(0)=0.1, x2(0)=0.9, and x3(0)=20;
working condition 2: α=0.99, σ=17, ρ=16, μ=25, TL=0, and the initial conditions are x1(0)=1.5, x2(0)=0.5, and x3(0)=20; and
working condition 3: α=0.99, σ=5.5, ρ=5.5, μ=20, TL=0, and the initial conditions are x1(0)=0.1, x2(0)=0.1, and x3(0)=3.
Approximate transfer functions of different fractional-orders are given by introducing equation (5):
error values in the above equations are ≤6.1 dB, ≤0.2 dB and ≤0.4 dB, respectively.
The permanent magnet synchronous generator generates chaotic oscillation with two attractors. In order to demonstrate the advantages of the fractional-order model, and reveal the dynamic characteristics of the permanent magnet synchronous generator-related to the fractional-order value, the fractional-order value is set as 0.96, and the periodic motion of the permanent magnet synchronous generator is immediately switched to a chaotic track with an attractor. In the working condition 1, the system has two motion states of T-time periodic behavior and chaotic attractors when α=0.97. In the working condition 2, the motion behavior of the permanent magnet synchronous generator becomes chaotic oscillation with two attractors when α=0.98. Finally, whether in the working condition 1 or in the working condition 2, once a is increased to 0.992 or 0.998, the system motion is maintained in the chaotic oscillation state with two attractors.
3. Fuzzy Finite-Time Optimal Controller Design
3.1. Hierarchical Type-2 Fuzzy Neural Network
The hierarchical type-2 fuzzy neural network is a derivative of a type-1 fuzzy neural network and has strong learning capability, function approximation capability and fault-tolerant capability. The hierarchical type-2 fuzzy neural network consists of an input layer, a membership layer, a rule layer, a type-reduction layer and an output layer. At the second level, upper and lower membership levels are written as:
μÀ
wherein mÀ
In the rule layer, an upper/down trigger rule is calculated as:
In the fourth layer, YR and YL are expressed in the form of a set type-reduction center as:
wherein wRj and wLj are referred as weight, Tj and
In the output layer, the fuzzy output in the form of vector is derived as:
Y=wTξ=wRTξR+wLTξL (26)
wherein,
By calling equation (26), the hierarchical type-2 fuzzy neural network realizes high-precision approximation to any unknown but bounded function on a compact set, then
wherein X≡[x1 ⋅ ⋅ ⋅ xi]T∈i, i=1 ⋅ ⋅ ⋅ N, N represents the number of inputs, ε(X)>0 represents approximation errors, and Ωw and DX are compact sets of appropriate bounds of w and X, respectively. An optimal parameter w* is introduced, and the parameter meets the condition:
and ĥ represents approximation values of h·{tilde over (w)}=w−w* is defined, wherein w* represents an amount of labor for analysis.
In order to avoid exponential increase of the number of rules, two criteria of tracking error reduction rate and fuzzy rule a completeness are adopted to execute the structural adjustment of the hierarchical type-2 fuzzy neural network. The tracking error reduction rate is equivalent to the derivative of the square of tracking errors between a drive system output and a response system. The completeness of the fuzzy rule is defined as that “at least one fuzzy rule ensures that the trigger strength within an operation range is not less than ε”. If the membership grade is equal to or greater than 0.5 in the hierarchy structure, the membership grade is saved. Otherwise, these unimportant rules in a self-structure algorithm is deleted.
In order to improve solving speed and simplify the structure, the hierarchical type-2 fuzzy neural network is transformed as follows:
wTξ(X)≤ζξT(X)ξ(X)/2b2+b2/2 (28)
wherein ζ=∥w∥2, and b>0, {tilde over (ζ)}=ζ−{circumflex over (ζ)}, wherein {circumflex over (ζ)} represents estimate values of ζ.
Note 1: compared to Grunwald-Letnikov and Riemann-Liouville derivatives, the initial conditions of Caputo derivatives of the fractional-order differential equation have the same form as the initial conditions of the integer-order differential equation. When the Caputo derivative is applied to a constant, its value is zero. Thus, C D0,tα{tilde over (ζ)}=−C D0,tα{circumflex over (ζ)} is obtained directly.
3.2. Controller Design
In order to avoid system performance degradation and suppress the convergence characteristics of error variables, a positive and strictly monotonically decreasing finite-time prescribed performance function is designed:
wherein ai, i=0, ⋅ ⋅ ⋅ , 3 represents design parameters, and T0 and βT
wherein β0 represents the initial value of the finite-time prescribed performance function.
Note 2: when a constraint signal is matched with the convergence rate of the prescribed performance function, a constraint condition suppresses a large overshoot of the control output at an initial stage. For the pre-given parameters such as β0, T0 and βT
In order to achieve a faster response, a fractional-order finite-time command filter based on a first order Levant differentiator is provided:
wherein Zi,1∈ and Zi,2∈ represent states of the command filter, αr∈ represents input signals of the command filter, ci,1 and ci,2 represent positive design constants, and positive constants Γi,1 and ri,2 satisfy conditions |Zi,1−αr|≤Γi,1 and |
Obviously, by appropriate selection of ci,1 and ci,2, Zi,1=αr and
Error variables are introduced:
z2=e2−α2c, z3=e3−α3c (32)
wherein αic, i=2, 3 and Zi,1, i=2, 3 are the same, representing the output of the fractional-order command filter.
The compensated tracking error is represented as:
υi=zi−θi, i=1,2,3 (33)
wherein θi represents a compensation signal between virtual control and filtered signals.
The error variables have the following inequation:
−ρβ(t)<e1<
wherein 0<p,
A smooth and invertible function S1(z1) is defined, and the above inequation is rewritable to an unconstrained form:
e1=β(t)S1(z1) (35)
It is noted that (
Then the inverse transform of equation (35) is written as:
z1=S1−1(ϕ(t)) (36)
wherein ϕ1(t)=e1(t)/β(t).
The fractional-order derivative of equation (36) is derived as:
C D0,tαz1=β−1(t)·C Dϕ(t)αS1−1(⋅)(C D0,tαe1(t)−ϕ1(t)CS0,tαβ(t))=η1(C D0,tαe1−r1), (37)
wherein η1=β−1(t)·C Dϕ(t)αS1−1(ϕ(t)), and r1=ϕ1(t)C D0,tαβ(t).
By using equations (9) and (37), then:
C D0,tαz1=η1(ƒ1+σe2−r1), C D0,tαe2=ƒ2+u2, C D0,tαe3=ƒ3+u3 (38)
ƒ1=−ρe1, ƒ2=−e2−y1y3+x1x3−(κ1e1+κ2e2)H(t−tg)+μe1 and ƒ3=−e3−y1y2−x1x2 are all regarded as unknown nonlinear functions due to perturbations from wind speeds, generator temperatures, stator resistance, friction coefficient, workload, and the like.
The design of the controller comprises the following three steps based on a fractional-order backstepping control principle.
Step 1: the above-described hierarchical type-2 fuzzy neural network is adopted to perform estimation on a compact set for facilitating controller design considering the uncertainty of ƒ1, i.e.,
ƒ1=w1Tξ1(⋅)+ε1(⋅) (39)
wherein (⋅) represents abbreviations of (x1, x2, x3).
A first Lyapunov-candidate-function is selected as:
the fractional-order derivative of v1 is taken to obtain:
C D0,tαV1≤υ1η1[ƒ1+
wherein α2ν and
The following cost function is designed to achieve its minimum value by definition 2.
In the above equation, uoi, δi2 and κi represent an optimal control input, a normal constant and a design parameter, respectively. The following inequation ξi2(⋅)≤δi2υi2, i=1, 2, 3 is used in order to compensate estimation errors of the hierarchical type-2 fuzzy neural network.
The optimal control input is designed as uoi=−Piυi/κi2, i=1, 2, 3, wherein Pi is the solution of an algebraic Riccati equation Pi2+2kiPi−κi2−1=0, ki>0.
The optimal control input is then further derived as:
uoi=−(√{square root over (1+ki2+δi2)}−ki)υi/κi2, i=1,2,3 (43)
Virtual control, and an adaptive law and a compensation signal thereof are selected:
α2ν=(−k1z1−υ1η1{circumflex over (ζ)}1ξ1T(⋅)ξ1(⋅)/2b12+r1−s1υ1γ+uoi)/
C D0,tα{circumflex over (ζ)}1=υ12η12ξ1T(⋅)ξ1(⋅)/2b12−γ1{circumflex over (ζ)}1 (45)
C D0,tαθ1=−k1η1θ1+
wherein k1>0, l1>0, γ1>0, s1>0, and 0<γ<1.
Equations (44)-(46) are substituted into inequation (41) by inequation (14) to obtain:
C D0,tαV1≤−k1η1υ12−k1η1θ12+
wherein b1=s1η1−l1ϑ1/(γ+1), and d1=l1(1−γϑ1−1/γ/(γ+1)).
Step 2: a hierarchical type-2 fuzzy neural network is used to approximate the aforementioned unknown nonlinear function ƒ2 with high precision in the following form to solve the unknown nonlinear function:
ƒ2=w2Tξ2(⋅)+ε2(⋅) (48)
A second Lyapunov-candidate-function is selected as:
The fractional-order derivative of V2 is written as:
wherein ζ2=∥w2∥2, and b2>0.
The q-axis control input and the adaptive law and the compensation signal are designed as follows:
wherein k2>0, l2>0, γ2>0 and S2>0.
By inequation (14) and equations (51)-(53), inequation (50) is further simplified as:
wherein b1=s2−l2ϑ2/(γ+1), and d2=l2(1−γϑ2−1/γ/(γ+1)).
Step 3: also, a hierarchical type-2 fuzzy neural network with very high precision and repeatability is used for estimation to process the unknown nonlinear function ƒ3:
ƒ3=w3Tξ3(⋅)+ε3(⋅) (55)
The magnetic field orientation control is adopted, so that α3c is equal to zero. A last Lyapunov candidate function is defined as:
The fractional-order derivative of V3 is calculated as:
wherein ζ3=∥w3∥2, and b3>0.
The d-axis control input, the adaptive law and the compensation signal are designed as follows:
u3=−k3z3−{circumflex over (ζ)}3υ3ξ3T(⋅)ξ3(⋅)/2b32−s3υ3γ+uo3 (58)
C D0,tα{circumflex over (ζ)}3=υ32ξ3T(⋅)ξ3(⋅)/2b32−γ3{circumflex over (ζ)}3 (59)
C D0,tαθ3=−k3θ3−l3θ3γ (60)
wherein k3>0, l3>0, γ3>0 and s3>0.
The following can be obtained:
wherein b3=s3−l3ϑ3/(γ+1), and d3=l3(1−γϑ3−1/γ/(γ+1)).
Note 3: when the order of a complex nonlinear system increases, the filtering error of the fractional-order finite-time command filter becomes difficult to control. In this case, it is difficult to obtain a minimum tracking error. Therefore, it is necessary and meaningful to solve this problem and achieve finite-time convergence by constructing compensation signals.
4. Stability Analysis
Principle 1: for the problem of fuzzy finite-time optimal synchronous control of the unidirectional coupling fractional-order permanent magnet synchronous generator with chaotic oscillation, if q-axis control input (51) and d-axis control input (58) with adaptive laws (52), (59) and compensation signals (53), (60) are designed, all signals of a closed-loop system are bounded, and synchronization errors meet a prescribed performance requirement. Meanwhile, the purpose of fuzzy finite-time optimal synchronous control is achieved, and the cost function is minimized.
Proof: the whole Lyapunov-candidate-function is defined as:
The fractional-order derivative of equation (62) is obtained:
wherein
The stability of υi, {tilde over (ζ)}i and θi are realized within the compact set Ωe in the finite-time T0, i.e.,
Ωe:={υi∈,θi∈,{circumflex over (ζ)}i∈,i=1,2,3|υi2+{circumflex over (ζ)}i2+θi2≤De} (64)
wherein De=min{ce/(1−λe)ae, (ce/(1−λe)be)2/(γ+1)}, and 0<λe<1.
The finite-time T0 is:
T0≤max{t1,t2} (65)
wherein t1=ln((λeaeVe1−(1+λ
The proof is finished.
5. Results and Analysis
A numerical method for solving the nonlinear fractional-order system based on an integral rule is constructed and realized by programming. The coupling parameters of the driving and driven permanent magnet synchronous generators are κ1=0.1 and κ2=−0.1. The parameters of the finite-time prescribed performance function are selected as follows: a0:=0.4, a1=0, a2=−4.9, a3=6.3, and βT
Real-time changes in the grid caused by the ambient environment through the transmission buses and feeders of the wind farm can have an unavoidable effect on the working conditions of each generator, which can lead to inaccurate modeling and system parameter perturbations. As can be seen from
The size of the coupling coefficient directly determines synchronous/asynchronous motion and harmonic oscillation/periodic motion when the permanent magnet synchronous generator operates in parallel. Three combinations are given, such as, I: κ1=0.1, and κ2=−0.1; II: κ1=0.2, and κ2=−0.15; III: κ1=0.15, and κ2=−0.2. Under all three working conditions, the permanent magnet synchronous generator will generate an oscillation behavior before the proposed method is applied.
The hierarchical type-2 fuzzy neural network with conversion performs compensation and function approximation on the flutter caused by wind speeds, generator temperatures, stator resistance, friction coefficient, workload, and the like. Three sets of system perturbation parameters are defined: A1: ρ=5, and μ=19; A2: ρ=5.5, and μ=20; A3: ρ=6, and μ=21. It can be seen from
The fractional-order finite-time command filter matched with the compensation signal achieves the purposes of high response speed, high estimation precision and finite-time convergence.
As can be seen from
Finally, the above embodiments are only used to illustrate the technical scheme of the present invention without limitation. Although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions of the technical scheme of the present invention may be made within the scope of the appended claims herein without departing from the spirit and scope of the technical scheme.
Number | Date | Country | Kind |
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202111246916.1 | Oct 2021 | CN | national |
Number | Name | Date | Kind |
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8860236 | Nasiri | Oct 2014 | B2 |