CN110048435B - Electric power wide area time-lag controller design method based on Jensen inequality - Google Patents

Abstract

The invention discloses a design method of an electric power wide area time-lag controller based on Jensen inequality, which is used for designing a time-lag stable controller for a wide area system and improving the time-lag stability margin of a closed loop system, and comprises the following steps: step 1, randomly generating primary controller parameters of a system to be solved; step 2, establishing a time-lag-free closed-loop system state space model; step 3, for a closed loop system stable without time lag, iteratively calling a Jensen inequality time lag criterion by a dichotomy to obtain an allowable time lag stability upper limit; step 4, generating new generation controller parameters by applying crossover, mutation and selection operations of a differential evolution algorithm; step 5, judging whether the iteration times reach the upper limit or whether the difference between the upper limit and the upper limit of the time lag stability allowed by the front and the back optimal times is smaller than a specified value, if so, entering a step 6, otherwise, returning to the step 2; and 7, outputting the optimal controller parameters and the time lag stability upper limit, and ending the operation.

Description

Electric power wide area time-lag controller design method based on Jensen inequality

Technical Field

The invention relates to the technical field of power systems, in particular to a power wide-area time-lag controller design method based on a Jensen inequality.

Background

With the expansion of the interconnection scale of the power system, the risk of serious threat safety and stability such as low-frequency oscillation is generated, and the risk needs to be restrained by adopting wide-area PSS or using wide-area control means such as additional control of high-voltage direct current. However, in practical applications, it is found that the wide area control system needs to collect a plurality of signals separated by thousands of kilometers, and the time lag of signal transmission has an important effect on the control effect. The wide area measurement and control signal transmission of the power system are realized by transmission means such as optical fibers, carriers and satellites. According to the previous researches, the time lag of optical fiber transmission is shortest, generally 60-80 ms, and the time lag of satellite transmission can reach 500ms; the time consumption related to signal collection, calculation and transmission is measured, the maximum time lag of the PMU-like measured signal is more than 600ms, and the time lag of the PMU-like measured signal is controlled to be more than 40 ms. If time lag influence is not considered in the design of the power wide-area controller, the system may be unstable in actual implementation.

Time lag system stability analysis of wide area control systems generally includes two types of methods, namely frequency domain and time domain. The theoretical basis of the frequency domain method is the condition for the stability of the linear system, namely, the roots of the characteristic equation are all positioned on the left half plane of the complex plane. However, the equation of the system after Laplace transformation is an overrun equation, solving is not easy, and when the uncertainty exists in the system and the time delay changes with time, solving is very difficult, the research of the current frequency domain method is concentrated on the analysis and synthesis of the fixed time delay system, the precision is lower, and the practical application has larger limitation.

The time domain method is based on the Lyapunov-Krasovskii, rzumishin stability theorem, and a proper Lyapunov or Lyapunov-Krasovskii functional is constructed, and linear is appliedMatrix Inequality (LMI) techniques solve. Time domain method research is focused on a Lyapunov-Krasovskii functional (L-K for short) structure with time lag related stability, the time lag is considered to be stable when 0, and a time lag upper bound exists in the system

The system is in the time lag zone->

The inner part is stable. The more mature methods in the field are a model transformation method and a free weight matrix method. The various model transformation methods mainly aim at defining cross terms of L-K functional derivatives, and the Park inequality and the Moon inequality are used for defining the cross terms to reduce the conservation of the method. The weight matrix introduced by the model transformation method is fixed and is based on greater conservation. The free weight matrix is for this case, for x (t), in the L-K functional>

The weight matrix of the terms of x (t-h) and the like is replaced by a free weight matrix with adjustable elements, and when the number of the referenced free weight matrices is larger, the obtained time lag upper limit fixing result is more accurate. The best results reported at the time are obtained by the free weight matrix method and the improved class method. The free weight algorithm not only can be used for the performance analysis of the controller, but also can be used for the design of the controller by a cone-patch linearization method, and a dynamic feedback controller is designed for the wide-area control of the electric power by the existing application of the free weight method. However, the conservation of time-lapse stability analysis is greatly reduced by the free weight matrix, the introduced free weight matrix greatly increases the LMI scale of the system, the solution is particularly difficult to apply to a large-scale system, the CCL-based controller design method is a local optimization method, and the design result still has certain conservation. Because of the large number of additional variables introduced by the free weight matrix method, even in a small-sized power system, the analysis by using the free weight matrix is difficult, and when the system order exceeds 40 orders, the solution by using the free weight matrix is basically impossible in practical application.

In order to reduce the conservation of the time lag stability criterion and the decision variable number, the time lag stability criterion based on the Jensen inequality and the Wirtinger inequality is intensively studied. The Jensen method can account for interval time lag and time-varying time lag, the introduced decision variables are relatively less, the obtained result has lower conservation, and the method has advantages in calculation efficiency compared with the free weight matrix method, but the result is equivalent to the free weight matrix method, so that the method receives more attention.

The needed decision variables based on the Jensen inequality are greatly reduced, meanwhile, higher precision is reserved, but when the design of the output feedback controller is carried out, as the parameters of the controller are unknown, the matrix inequality to be solved relates to product terms of 2-3 unknown matrixes, so that the inequality of the nonlinear double matrix is the NP difficulty problem, the time-lag controller synthesis based on the Jensen inequality method is very difficult, and compared with the effective cone-complement linearization design method of the free weight method, no document currently discusses the time-lag controller design method based on the Jensen inequality.

Disclosure of Invention

In view of the difficulty of designing an output feedback controller for a time-lag system based on a Jensen inequality directly, the invention provides a 'generation-inspection' thought based on a random method, namely, the output feedback controller is randomly generated first, and then the Jensen inequality criterion is applied to iteratively solve the upper time-lag stability limit of a closed-loop system, so that the output feedback controller meeting the requirement can be designed. In order to improve the design efficiency and the controller performance, the invention also introduces a high-efficiency differential evolution algorithm to provide an optimization direction for the controller parameters. Based on the thought, the Jensen inequality method is popularized to the design of the wide area controller of the power system, and the DOF controller is solved by applying the differential evolution-LMI hybrid algorithm, so that the conservation is reduced, and the calculation efficiency is improved.

Specifically, the invention provides a power wide-area time-lag controller design method based on a Jensen inequality, which comprises the following steps:

step 1: input power system model state space model parameters A, B, C and D, input signal dimension n _u Output signal dimension n _y The controller order n is to be calculated _k Decision variable number n _v ＝(n _u +n _k )×(n _y +n _k ) Initializing differential evolution parameters: scaling factor C _F Probability of crossover C _R And population size N _P Maximum number of iterations g _max Iterative error limit epsilon, let g=1;

step 2: random generation controller K _s Is a starting population of (a)

Its ith row vector K _i Representing an ith controller individual;

step 3: for N _P Individual controllers K _i Conversion to a controller state space matrix K _{m_i} And generating a closed loop system matrix A without time delay _ci ，B _ci ，C _ci ，D _ci ；

Step 4: check A one by one _ci If the rightmost eigenvalue of (a) is positioned on the left half plane, the step 5 is entered; such as N _P If the closed loop systems corresponding to the individuals are not established, returning to the step 2;

step 5: based on Jensen inequality criterion, a dichotomy search is applied to calculate the time lag stability upper limit T of the closed loop system corresponding to each body _{dmax_i} Obtaining the optimal individual of the present generation

Is->

Step 6: for N in this generation _P The individuals perform crossover, mutation and selection operations to generate a new generation of individuals

Let g=g+1;

step 7: if g is less than or equal to g _max Or (b)

Returning to the step 3; otherwise, go intoStep 8, entering;

step 8: outputting the optimal solution

And corresponding controller matrix->

Ending the operation.

Further, in step 5, the Jensen inequality criterion is:

time lag system described below

Wherein x (t) ∈R ⁿ A and A are state variables _d The constant matrix with proper dimension is characterized in that h and mu are constant scalar quantities, and respectively represent a time lag upper limit and a time lag change rate upper limit;

for the time-lag system defined by equation (1), given scalar h > 0 and μ, if there is a symmetric matrix p=p ^T ＞0，Q＝Q ^T ≥0，S＝S ^T >0，V＝V ^T > 0 holds the following linear matrix inequality:

Ψ ₁ ＝Ψ-[I -I 0] ^T V[I -I 0]＜0 (2)

Ψ ₂ ＝Ψ-[0 I -I] ^T V[0 I -I]＜0 (3)

wherein:

Ψ _o ＝PA+A ^T P+Q+S-V

the linear system with time-varying lag described by equation (1) is asymptotically stable.

Further, in step 5, the method for calculating the upper time lag stability limit is as follows:

step 5-1: setting a step length delta h and a maximum iteration number Iter _max And an error limit epsilon _h An upper desired closed loop system time lag limit T _dexpect Let the time lag viable value h _f Time lag range h=0 _max ＝0,h _min ＝0，Iter＝0；

Step 5-2: individual parameters of the ith controller according to the given generation

Conversion into matrix K _m ＝[D _k C _k ；B _k A _k ]Computing A of closed loop System _c ，A _dc ；

Step 5-3: order the

According to a given A _c ，A _dc ，h _test Solving the inequality (2) and (3) of the linear matrix, and if feasible, turning to the step 5-4; otherwise, executing the step 5-5;

step 5-4: h is a _f ＝h _test ，h _min ＝h _test ，h _text ＝2×h _f Turning to step 5-6;

step 5-5: h is a _max ＝h _test ，h _test ＝(h _max +h _min )/2；

Step 5-6: iter = Iter +1; if Iter is less than or equal to Iter _max Or h _gap ＝|h _max -h _min |≥ε _h Or h _f ≤T _dexpect Returning to the step 5-3; otherwise, turning to the step 5-7;

step 5-7: upper limit T of output time lag stability _dmax ＝h _f And (5) ending the calculation.

The invention has the beneficial effects that:

(1) The invention can directly design a controller with any appointed order for the electric power wide area system with input time lag, and comprises a static output feedback controller with the order of 0, wherein the controller has the characteristics of high efficiency and high time lag stability margin, and the system has less conservation;

(2) The invention is based on the method of generating-checking, firstly randomly generating the controller, and then checking whether the controller meets the Jensen inequality or not, and the product term of two or more unknown matrixes does not exist, so that the controller can be directly solved by applying the Jensen inequality criterion based on the linear matrix inequality framework, the difficulty of solving the non-linear matrix inequality is avoided, and the optimized controller can be obtained by a differential evolution search method;

(3) The invention randomly generates the controller, then checks whether the rightmost eigenvalue of the closed loop system matrix is positioned on the left half plane one by one, and obtains the time lag stability upper limit for the closed loop system meeting the condition through the Jensen inequality, thereby avoiding a large amount of invalid operations, improving the calculation efficiency and being applicable to the controller synthesis of a larger-scale time lag system.

Drawings

FIG. 1 is a flow chart of a design method of the present invention;

FIG. 2 is a single machine to infinity system wiring diagram;

FIG. 3 is a time domain simulation result of a static output feedback controller based on the Jensen method;

FIG. 4 is a 4-order dynamic output feedback controller time domain simulation result based on the Jensen method;

FIG. 5 is a diagram of the result of a static output feedback controller time domain simulation based on the free weight method;

fig. 6 is a 4-order dynamic output controller time domain simulation result based on the free weight method.

Detailed Description

For a clearer understanding of technical features, objects, and effects of the present invention, a specific embodiment of the present invention will be described with reference to the accompanying drawings. It should be understood that the particular embodiments described herein are illustrative only and are not intended to limit the invention, i.e., the embodiments described are merely some, but not all, of the embodiments of the invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the invention, as presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by a person skilled in the art without making any inventive effort, are intended to be within the scope of the present invention.

1. Theoretical basis of the invention

Consider the following time-lag state equation:

x(t)＝φ(t)t∈[-h,0] (2)

0＜d(t)＜h (3)

wherein x (t) ∈R ⁿ A and A are state variables _d And h and mu are constant scalar quantities which are constant matrixes with proper dimensions, and respectively represent a time lag upper limit and a time lag change rate upper limit.

Lemma 1 (Jensen integral inequality): for arbitrary symmetry positive definite constant matrix M E R ^n×n Scalar r ₁ ,r ₂ And satisfy r ₁ ＜r ₂ Vector function omega [ r ] ₁ ,r ₂ ]→R ⁿ The following integral inequality holds

Lemma 2 (dead time stability discrimination based on Jensen integral inequality): given scalar h > 0 and μ, if there is a symmetric matrix p=p ^T ＞0，Q＝Q ^T ≥0，S＝S ^T >0，V＝V ^T > 0 holds the following Linear Matrix Inequality (LMI):

Ψ ₁ ＝Ψ-[I -I 0] ^T V[I -I 0]＜0 (6)

Ψ ₂ ＝Ψ-[0 I -I] ^T V[0 I -I]＜0 (7)

wherein:

Ψ _o ＝PA+A ^T P+Q+S-V

then the linear system with time-varying lags described by equations (1) - (4) is asymptotically stable.

And (3) proving: defining the Lyapunov function as:

/>

in the above, x _t =x (t+θ), -2h+.θ+.0, the derivative of Lyapunov function of the above formula can be obtained:

wherein, the liquid crystal display device comprises a liquid crystal display device,

and (3) making:

β＝d(t)/h (12)

there is a case where the number of the group,

and is also provided with

The primer 1 comprises:

from (10) to (15)

Above type ψ, ψ ₁ ,Ψ ₂ The definition of (c) is described in the theorem,

ζ(t)＝[x(t) ^T x(t-d(t)) ^T x(t-h) ^T ] (17)

since 0.ltoreq.β.ltoreq.1, (1-. Beta.) ψ in the derivative of the energy function ₁ +βΨ ₂ Is convex and combined when the vertex ψ ₁ ,Ψ ₂ For negative timing, the system is asymptotically stable.

2. Main aspects of the invention

Considering signal transmission time lags, assume that the state equation of the power wide area system is:

wherein:

and n is _x 、n _u 、n _y The state matrix, the input matrix and the output matrix dimensions are respectively defined.

For the wide area system described by equation (18), the following dynamic output feedback control law is selected:

in the above formula:

n _k is the controller order. Where K is _s Can be selected as a reduced order controller (n _k ＜n _x ) Or a full-order controller (n) _k ≥n _x ) For example, n is selected _k =0, then becomes a static output feedback controller.

Order the

The closed loop system can be written as:

obtaining A of closed-loop System _c ，A _dc Then, the primer 2 cannot be directly applied to solve the dynamic output feedback controller, and the following 3 problems exist:

(1) In a closed loop system, the PA of formulas (6), (7) will be present _dc ，h ² A _c RA _dc The product term containing two or three unknown matrixes is called a nonlinear matrix inequality, and cannot be solved by a linear matrix method; (2) The quotation 2 only gives out the judgment of whether the time lag system with given time lag and the upper limit of the time lag change rate is stable, and the upper limit of the time lag stability for a given feedback control law is required to be given in order to realize the design of the time lag stability controller; (3) The initially given controller time lag stability upper limit index does not necessarily meet the requirements, and optimization of the controller is required.

In view of the above problems, the present invention first randomly generates a series of controllers K _s So that at most one unknown matrix appears in each item of formulas (6) and (7), and the algorithm solution of the linear matrix inequality can be applied; for a randomly generated controller K _s Solving the upper time lag stability limit by iteratively calling the lemma 2 by a half search method; in order to optimize the performance of the controller, a differential evolution algorithm is applied to generate a new generation of controllers to be tested, and the optimized controllers are obtained through loop iteration. The method comprises the following steps:

(1) Random generation method of controller

For controller matrix K _m Hypothesis elementThe element is

Since the differential evolution algorithm can only optimize one-dimensional vectors, K _m The following one-dimensional row vector K is converted:

the vectors can be expressed as

Let the vector K element K _i Value range [ k ] _imin ,k _imax ]Randomly generating a random pattern with N using the following rule _p Controller initial population of individuals:

in the method, in the process of the invention,

a j-th dimensional component representing an i-th individual of the g-th generation controller; n (N) _p The number of the population, namely the number of controllers to be inspected in the present generation; the rand () function produces a value of 0,1]Random numbers of intervals. The following uses->

Representing the g generation controller ith individual.

(2) Search solving method for time-lag stability upper limit

After obtaining the controller by the random generation method, the formulas (6) and (7) still exist h ² A _c RA _dc The product term of such an unknown variable and an unknown matrix is still a nonlinear matrix inequality. However, since the unknown variable is not one-dimensional, the maximum allowable time lag upper limit can be obtained by a search method. In order to improve the searching efficiency, a dichotomy method is adopted, and the method comprises the following steps:

step 1: setting a step length delta h and a maximum iteration number Iter _max Error limit ε _h An upper desired closed loop system time lag limit T _dexpect Let the time lag viable value h _f Time lag range h=0 _max ＝0,h _min ＝0，Iter＝0；

Step 2: individual parameters of the ith controller according to the given generation

Conversion into matrix K _m _ _i ＝[D _k C _k ；B _k A _k ]Computing A of closed loop System _c ，A _dc ；

Step 3: order the

According to a given A _c ，A _dc ，h _test Solving the linear matrix inequalities (6) and (7), if feasible, turning to step 4; otherwise, executing the step 5;

step 4: h is a _f ＝h _test ，h _min ＝h _test ，h _text ＝2×h _f Turning to step 6;

step 5: h is a _max ＝h _test ，h _test ＝(h _max +h _min )/2；

Step 6: iter = Iter +1; if Iter is less than or equal to Iter _max Or h _gap ＝|h _max -h _min |≥ε _h Or h _f ≤T _dexpect Returning to the step 3; otherwise, turning to the step 7;

step 7: upper limit T of output time lag stability _dmax ＝h _f And (5) ending the calculation.

(3) Determination of optimization targets

To search for more optimal controllers using the differential evolution algorithm, an objective function Obj (K) needs to be defined to evaluate the performance of each candidate controller K. The time lag control system is optimized to be the highest time lag limit which can be born by the closed loop systemIt is reasonable to take the maximum allowable time lag upper limit of the system as an optimization target. However, the process of determining the upper time lag limit for the control system is time consuming, and if the upper time lag limit is determined for each controller to be selected, the solving process will be quite long and unnecessary, because a part of the randomly generated controllers are likely to be unstable in the time lag-free closed loop system, and the calculation of the upper time lag stability limit is unnecessary for the part of the controllers. Judging whether the closed loop system is stable or not without time lag, and obtaining the rightmost characteristic value real part sigma of the closed loop system _max Realization of when sigma _max The system is stable when < 0. The eigenvalue of the closed loop system is obtained, the eig and eigs functions of matlab can be utilized, the maturation is reliable, the calculation efficiency is high, and the system can be used for a large-scale system. Based on this idea, one can only target sigma _max The controllers smaller than 0 can only find the upper time lag limit, so that the number of controllers of the upper time lag limit to be tested can be greatly reduced, and the solving process is accelerated.

The objective function is obtained according to the following steps:

step 1: closed loop system without time lag and optimal characteristic value sigma thereof _max ；

Step 2: if sigma _max If less than 0, iteratively calling the Jensen inequality criterion to obtain T by using the binary search method _dmax Let obj= -T _dmax The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, obj=σ _max 。

(4) Optimized search direction based on differential evolution

The invention applies the mutation, crossing and selection operation of the differential evolution algorithm to generate a new generation of controller individuals to be checked, and provides a search direction for optimizing the performance of the controller.

1) Mutation operation

The mutation operation is to select three different individuals from the present generation controller individuals to conduct differential operation to generate a new generation target individual. Set the target controller individual for carrying out mutation operation on the generation as

(g generation) random selection from the population of present generation controllersThree different individuals->

Generating a candidate individual of a new generation controller based on the following operations +.>

Wherein r1, r2, r3 ε {1,2, …, N _P The r1, r2, r3 are different from the current target vector index i, thus the population size N _P And is more than or equal to 4.F is a scaling factor, and the value range is [0,2 ]]To control the degree of scaling of the differential vector.

2) Crossover operation

Crossover operations are performed by individuals after mutation

Substitute for the present substitution individual->

The partial components thereby produce a new generation of operations of the candidate individual. To ensure individual->

By first randomly selecting, so that +.>

At least one bit is composed of->

Contribution, while for other components, a cross probability factor CR is used to determine +.>

The component is from->

Or->

The method of cross operation is that

Where rand () ∈0,1]For uniformly distributed random numbers, j represents the j-th variable, CR is a crossover probability constant, and the value range is [0,1]The size is predetermined. randi (n) _v )∈[1,2,…,n _v ]Indexing the randomly selected dimension variables.

3) Selection operation

The selection operation determines whether the controller individuals generated by the mutation and crossover can enter a new generation group. Test individuals generated after mutation and crossover operation

And->

Competing only when->

Adaptation value and->

Equal or better are selected as new generation individuals +.>

Otherwise, directly will->

As a child. />

3. Detailed description of the preferred embodiments

In the following, an embodiment of the present invention is described, where the implementation object is a stand-alone to infinity system, as shown in fig. 2, and the system can be described by a fourth-order state space equation, and the system matrix is:

when no control is applied, there is a weakly damped oscillation mode in the system: -0.0207 + -4.7609 i, damping ratio of about 0.4350% and oscillation frequency of 0.7577Hz. The controller design goal is to have the maximum upper bound of the skew stability for the closed loop system when there is a signal transfer skew at the output (assuming a skew rate of change μ=0.001).

The Jensen method and the free weight matrix method provided by the invention are respectively adopted to design a static output feedback controller and a 4-order dynamic output feedback controller. The Jensen method in the test uses the following parameters: differential evolution algorithm population size np=40, decision variable number nv=2, scaling factor f=0.85, crossover probability cr=1.0, maximum population number g _max =20, the number of dichotomy iterations iter_max=10, and the error limit epsilon=0.001. The maximum number of iterations of the free weight method is also set to 20.

The results of the controller performance analysis are shown in table 1.

Table 1 controller performance

From the calculation time, the Jensen method takes 4.6494 seconds and 148.4590 seconds for the static and 4-order dynamic output feedback controllers respectively, and 139.4592 seconds and 830.3688 seconds for the free weight matrix method respectively, and the time required by the Jensen method is only 3.3% -17.9% of that of the free weight matrix method, so that the calculation efficiency is high. In the aspect of allowing time lag upper limit estimation, the static controller and the dynamic controller of the Jensen method are 0.1406 seconds and 0.1563 seconds respectively, and the free weight matrix rule is 0.1563 seconds and 0.1875 seconds respectively, namely the estimation accuracy of the free weight matrix method is superior to that of the Jensen method. In order to verify the actual allowable time lag upper limit of the controller, the system model is built on a simulink simulation platform, and the allowable time lag upper limit of the closed loop system is checked by using a time lag scanning method through time domain simulation, and the results are shown in fig. 3-6.

It can also be seen from fig. 3 to 6 that after the controller is applied, when the input signal has no transmission time lag or the transmission time lag is smaller (Tdelay is less than or equal to 0.1 s), the system oscillation can be quickly subsided, and the system damping is greatly improved.

The closed-loop system output of the controller designed based on the Jensen method and shown in fig. 3 and 4 is output at different input time lags, for example, a static output controller is adopted, the upper limit of the time lag of the closed-loop system estimated by the Jensen method is 0.1406ms, the upper limit of the time lag given by time domain simulation is 0.2050ms, and the estimation accuracy is about 68.58%. If a dynamic output controller of 4 th order is adopted, the upper time lag limit given by the Jensen method is 0.1667ms, the upper time lag limit given by the time domain simulation is 0.2560ms, and the estimation accuracy is about 65.12%.

The controller designed based on the free weight matrix method shown in fig. 5 and 6 outputs a closed loop system with different input time lags, if a static output controller is adopted, the upper time lag limit given by the free weight method is 0.1563ms, the upper time lag limit given by the time domain simulation is 0.2010ms, and the estimation accuracy is about 77.76%. If a 4-order dynamic output controller is adopted, the upper time lag limit given by the OXDE method is 0.1875ms, the upper time lag limit given by the time domain simulation is 0.2460ms, and the estimation accuracy is about 76.22%.

It can be obtained from this embodiment that, although the Jensen method has lower accuracy in estimating the upper time lag limit than the free weight method, the method requires fewer decision variables, the solving speed is five times faster than the free weight method, and the finally obtained upper time lag limit is better than the free weight method because more controller upper time lag limits can be checked in a short time.

The foregoing is merely a preferred embodiment of the invention, and it is to be understood that the invention is not limited to the form disclosed herein but is not to be construed as excluding other embodiments, but is capable of numerous other combinations, modifications and environments and is capable of modifications within the scope of the inventive concept, either as taught or as a matter of routine skill or knowledge in the relevant art. And that modifications and variations which do not depart from the spirit and scope of the invention are intended to be within the scope of the appended claims.

Claims (3)

1. A design method of a power wide area time-lag controller based on a Jensen inequality comprises the following steps:

step 1: inputting power system model state space model parametersA、B、CAndDdimension of input signal

Output signal dimension +.>

The controller order to be solved>

Decision variable number->

Initializing differential evolution parameters: scaling factor C _F Probability of crossover C _R And population size N _P Maximum number of iterationsg _max Iterative error Limit->

Order-makingg=1; a, B, C and D represent a state matrix, an input matrix, an output matrix, and a direct feed matrix, respectively; considering signal transmission time lag and taking a direct feed matrix D=0, and the state equation of the power wide area system is as follows:

wherein:

，/>

，/>

and->

、/>

、/>

The dimension of the state matrix, the input matrix and the output matrix are respectively;

step 2: random generation controller

Initial population->

Its first oneiIndividual row vectorsK _i Represent the firstiIndividual controllers;

step 3: for N _P Individual controllersK _i Conversion to a controller state space matrixK _{m_i} And generating a closed loop system matrix without time delay

，/>

，/>

，/>

；

Step 4: checking one by one

If the rightmost eigenvalue of (a) is positioned on the left half plane, the step 5 is entered; such as N _P If the closed loop systems corresponding to the individuals are not established, returning to the step 2;

step 5: based on Jensen inequality criterion, a dichotomy search is applied to calculate the upper time lag stability limit of the closed loop system corresponding to each bodyT _{dmax_i} Obtaining the optimal individual of the present generation

Is->

；

Step 6: for N in this generation _P The individuals perform crossover, mutation and selection operations to generate a new generation of individuals

Order-makingg=g+1；

Step 7: if it is

Or->

Returning to the step 3; otherwise, enter step 8;

step 8: outputting the optimal solution

And corresponding controller matrix->

The operation is ended.

2. The method for designing a wide-area power time-lag controller based on Jensen inequality according to claim 1, wherein in step 5, the Jensen inequality criterion is:

time lag system described below

，/>

，/>

，/>

（1）

In the method, in the process of the invention,

is a state variable +.>

And->

Is a constant matrix, d (t) is a time-lag function varying with time, h and +.>

A constant scalar is used for respectively representing a time lag upper limit and a time lag change rate upper limit;

for the time-lag system defined by equation (1), a scalar is given

And->

If there is a symmetry matrix->

，

，/>

，/>

The following linear matrix inequality is made true:

(2)

(3)/>

wherein:

(4)

the linear system with time-varying lag described by equation (1) is asymptotically stable.

3. The method for designing a wide-area power time-lag controller based on Jensen inequality according to claim 2, wherein in step 5, the method for calculating the time-lag stability upper limit is as follows:

step 5-1: setting step length

Maximum number of iterationsIter _max Error Limit->

An upper desired closed loop system time lag limit

Let the time lag feasible value +.>

Time lag range =0->

,/>

，Iter=0；

Step 5-2: according to the given generationiIndividual parameters of individual controllers

Conversion into a matrixK _m =[D _k C _k ; B _k A _k ]Calculating the state matrix of the closed-loop system>

Time-lag state matrix->

;

Step 5-3: order the

According to a given->

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，/>

Solving the inequality (2) and (3) of the linear matrix, and if feasible, turning to the step 5-4; otherwise, executing the step 5-5;

step 5-4:

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turning to step 5-6;

step 5-5:

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；

step 5-6:Iter=Iter+1；if it is

Or->

Or->

Returning to the step 5-3; otherwise, turning to the step 5-7;

step 5-7: upper limit of output time lag stability

And (5) ending the calculation. />

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