BE1030275B1 - Method for designing an event-triggered performance-preserved regulator for power systems - Google Patents

Abstract

The present invention lies in the field of automatic production control technology and relates to the Method of designing an event-triggered performance-preserved regulator for power systems, comprising the following steps. construct a closed-loop model for regulating the charging frequency of a power system containing parameter uncertainty; construct a generalized multiple discontinuity Lyapunov-Krasovskii function based on the event-triggered closed-loop model for charging frequency regulation; and determining the stability of the power system containing parameter uncertainty using a stability criterion. The experimental control effect of the invention deviates little from the theoretical value, which not only allows the system to have robust stability and strong anti-disruption ability, but also to make the index optimal. predefined secondary performance.

Description

1 BE2023/5275

Method for designing an event-triggered performance-preserved regulator for power systems

Technical areas

The invention belongs to the field of automatic energy production control technology and more particularly relates to the method of designing an event-triggered performance-preserved regulator for electrical systems.

Core technology

The application of synchronous phase measurement technology and large-scale measurement systems enables the dynamic and synchronous collection of information across the entire network, which lays the foundation for detection, protection and control large-scale networked power systems, but also poses some problems. The frequency of data collection by the large-scale measurement system is very high, and the phase measurement units transmitted at a high frequency without interruption, causing the communication channels to be flooded with massive amounts of data information. status and control, etc. which arrive at the control center even when the electrical system is functioning correctly.

The large amount of redundant data uploaded by the acquisition units not only puts pressure on the system to collect, store and process the information, but also increases the network communication load and prevents the control center functions from working efficiently in real time.

In addition, with the continuous development of the power grid, the uncertainty in the system is increasing day by day, which seriously affects the safe and reliable operation of the power system. When designing a controller for a real power system with uncertain parameters, it is expected not only that the controller can keep the system stable, but also that the system has high robustness.

However, the available research material does not take into account the uncertainty of the actual model, which results in an inaccurate model of the system, and the actual control effect may deviate from the theoretical value and the controller is less resistant to interference.

Content of the invention

With this in mind, the present invention provides a Method for designing an event-triggered performance-preserved regulator for power systems to solve the technical problems mentioned above.

The technical solution of the invention is as follows.

Method for designing an event-triggered performance-preserved regulator for power systems, including the following steps.

2 BE2023/5275

Construction of closed-loop models for load frequency control of power systems containing parameter uncertainties.

Construction of multiple discontinuous generalized Lyapunov-Krasovskii functions based on event-triggered closed-loop models for load frequency control.

Derivation of stability criteria for power systems with parametric uncertainties based on generalized Lyapunov-Krasovskii functions with multiple discontinuities.

Use of stability criteria to determine the stability of power systems containing parameter uncertainties.

A matrix inequality approach is used to derive controller parameters and obtain performance-preserving, event-triggered control laws, assuming that power systems containing uncertain parameters are stable.

Preferably, a closed-loop model for controlling the charging frequency of a power system containing uncertain parameters is constructed according to the following equation (1). x(£) =(A+A4)x(f) +(B + AB)u(t;) + FAP, (t) (0 = Cx) © x(f) is the state vector, X( #) is the derivative of the system state vector, A‚B andC are the matrices of the system parameters, A4 and AB are the changes in the system parameters due to deviations from the nominal values of the inertia time constants of the main motor and speed controller respectively, u(/,) is the system control input, AP,(f) is the load disturbance, V({) is the system output vector, 4, is the trigger time of the event and/is the load disturbance factor.

Preferably, the construction of a multiple discontinuous generalized Lyapunov-Krasovskii function based on an event-triggered closed-loop model for load frequency regulation comprises the following steps

Define ex (f) as the system state deviation, y as the performance indicator Hs , and the time shift functions 7(#) ,7({)=1—1t, ‚and

Of which , is the triggering time of the event and is the present time.

When? e[t, +d,,t,,, +4, ), equation (1) is distorted into

(f)

3 BE2023/5275

By constructing the generic Lyapunov-Krasovskii function with multiple discontinuities according to equation (3) below, we obtain the generic Lyapunov-Krasovskii function with multiple discontinuities.

V(1)= YO © 5 Of which

V(0)=x"(0)Px(t).

Î

‚0 =| x" (S)Ox(o)ds

TM t to ; v,0)=|_ | x"s)Res)dsdv. t-Ty VV t tor t w tor ; v,0)=|_ [ [OS 5)dsvdæw+|" | [X ()S,#(s)dsdvdw tT, SW OV tT, Tag JV _ 2 2 { , T . 1 [ T

V0) = ri], $ YTo)ds LT Ts) -x4)]ds © tr D.

Here are some of the most important features of the new system

Of which

PO ,R ,T ,S, ‚5, are the weight matrices, "M is the upper limit of the time shift function r(t) ‚t. is the trigger time of the event, / is the time current, andX(#) is the state vector.

Preferably, deriving a stability criterion for a power system containing parameter uncertainty based on a generalized multiple discontinuity Lyapunov-Krasovskii function, comprising the following steps

The derivation ofW(#) in equation (3) gives equation (4), the

4 BE2023/5275

V(1)=2x"(0)PX(1)+x"(1)Ox(1)-x"(t-1,,)Qx(t-—T,;) 2-T . { T . ft) 1, . +7,

Ty-T. t tp | t-c(t) pt-rlt) op + OS [ m [ u [ 7 (s)S, X(s)dsdv { .T Sx d Ti, .T Cx { V .T Si dsd

Eu D Sd IOS [ot SA dsdh (to)

ST Sas [RT (Sk (ds + Tik (OTK)

HAS? 7 - 7 KO -x(£-T(1))] T[x()-x(-T())]

By defining a, TO ‚a, €[0,1] , and applying Lemma 1 and Lemma 2 to the terms

TM

5, 6, 10 and 14 of Va) of equation (4), then Lemma 3 for the other terms, we obtain equation (5).

V(t) Ss &(1){Sym(e; PE, +24))+ er Oe, 7 e; Oe, +21 +2 > (TR + 2 2

MS MS, +7), +2) - CT diag2S, 45.25, 45,)G, 2 2 (5) 2 . a A 7 G; diag(2S,,45,,2S,,4S, )G, — Gi HG, —(e - e‚) ze —e, IS; (f)

Including €, = [On I, 0 (8D 0 in } ie {1,2,3,...,9} t) = col{x(t t—T(1 ft a d a A d

Sol HO) A Te) 1 t-z(f) 1 t-7(f)

Of ads Sf Ate OXG)ds,e, (MAP, (0)}

Ty 7 T4) 7th Ty TI) 74 (a)

The following products are some of the most common and popular in the world. 23 = Ae, - BKCe, — BKCe, + Fe, >, = HD(t)E,e — HD(1)E,KCe, — HD(t)F,KCe,

He diag{R,3R,5R} V a VT diag{R,3R,5R}

€76, e +e, —2nd, e —e, —6th

G, — 1 2 5 €, 76; e, +e, —2nd, e, —e, — 6th, e €, e —e, —3rd

G, — 1 4 5 e, € e, — e, — 3rd, e, —e, e, —e, +3e

G, — 2 4 5 €; — 66th, — th, +3rd,

Lemma 1: For any given matrix R ES, , suppose that there exists a matrix X € R"""

R

5 satisfying YR 20 then the following inequality is valid. 1

TR 0 x Rx 1 ll R Vve(0,1) 0 —R 1-v ;

Lemma 2: Given a matrix R €S, , for any differentiable function x: [a,b] + R” , the following inequality is valid. b 1 +" (@)Rèa)du > 57765 diag(R,3R5R)©, a —a

Of which

6 BE2023/5275 x(a)-x(b) 2 b ©, =| x(b)+x(a)=-— | x(u)du b — a“ 6 b x(b)-x(a)-—[_ 4.5 (W)x(u)du b-av * . u—a

At, (u)=2———1 b-a (a) Here are some of the most important features of the new system

Lemma 3: For a positively defined matrix R > O and a differentiable function x) Ive [a, b}}, the following inequalities are valid. brb, T. T 7. [A (W)RéGOdudv > 9j diag(2R,AR)3, b pv [ [ 17 (u)Rè()dudv > 9! diag(2R,4R)3,

Including 1 b x(h) = — | x(u)du

G = b—a

The 1 b 3 b > (B) | x(u)du | At ,(U)x(u)du (b)-— fx) —— [A (x) 1 b x(a)-— [ x(u)du b—a 0 4= x(a)-— [ lada + 3 [205 (ed b—a va bah “*

Define the triggering mechanism for discrete events as in equation (6) below. ben = 1, + Min{((+Dh|[xG,)=x(, JF O[x(1,)=x(4,)]2 âc” (1, )Ox )} (6)

Of which i, =t, +(1+ Dh is the present time. {, is the time when the last event was triggered. h is the sampling period.

7 BE2023/5275 is the number of cycle intervals between the last trigger and the current time when a sample was taken but did not cause a trigger. ©) is the trigger matrix.

By applying the principle of equation (6) to equation (5), we obtain the following equation (M.

V(1)< OLE (0) = y" (Dy) + 7° APE (MAP (1) - (7) x°(0)ZX(0)=[x(t=1(1))+e , (I (KC) Z,(KC)[x(t=7(1))+e,(1)]

Of which

II, = ee = rele, + Sym(e{ P(Z,, +2, ))+e/ Oe, — ef Oe, 2 2

T T

+ (X; + 3) (TR + 55 + = 5. + Ty TE; + 2) 7

G! diag(2S,,4S,,25,,4S,)G, = GT diag(28, 48, 28,45, )G, - 2 — A

G/ HG, —(e —e,)" 7 "6 —e,)+ô(e, +e,) C OC(e, +e,)- e, CTQCe, + e Ze, +(e , +eg) (KC) Z,(KC)T(e, +e,)

For ll, , use Lemma 4 when the following inequality is respected II, < 0

I, * * * * * * * *

F* Pe, _ y’I * * * * * * *

RE +5) TRE OR HH #4 + 8 2 2

SE +2) us 0 a S, * * * * *

V2 V2 5 PS0 +23) SF 0 0 —S, * * * * |<0 (82)

T,T(E,+E») TIEF 0 0 0 -T #* * * 1

Zie 0 0 0 0 0 —- * * 1

Z}KC(e,+ez) 0 0 0 0 0 0 II *

This, 0 0 0 0 0 0 0 —!

Of which

8 BE2023/5275

T T T T4;

IL, = Sym(e, P(X, +X,,))+e, Oe, —e; Oe, — G, diag(2S,,4S,,2S,,4S,)G, 2

G'diag(2S,,48,,28, 48,)G, —(e —e,) T(e —e,)-G'HG —e; CTQC-G; diag(2S,,4S,,2S,,4S,)G; — (€ — €) 4 (e —e,)—G; 1768 &

TAT

+ô(e, +e,) C OC(e, +5) >, = Ae, — BKCe, — BKCe,

Lemma 4: Schur complement of a matrix For a given symmetric matrix

Su Sr | . . . 2.

S= ss ‚where Si is of dimension r x r, the following three conditions are equivalent. 21 22

DD S<0. (2) Sir <0,8,,-S,,5;18,, <0 (a) Here are some of the most important features of the new version of the new version. (3) S, <0.8,,-S,,$3;$;; <0 (a) Here are some of the most important features of the new version of the new version.

By transforming inequality (8a) into equation (8b), we obtain the following result

HE; * * * * * * * *

F"Pe, VI * * * * * * *

T,RE, T,RF _R * * * * * *

Ds: Not 0 Sj * * kkk

Duss. Drs, 0 0 -S, #* #* # #* +

T,T Es T,,TF 0 00 00 -T * * * 1

Zie, 0 0 0 0 0 — * * (8b) 1

Z2KC(e, +e;) 0 60 0 0 0 0 -/ *

This, 0 60 0 0 0 0 0-1

Sym(F" PZ) & kok # # # # # 0 Q * #* * # # # #

TRX 0 0 * #* * * * * rsz, 0 0 0 #* * * * * 0

Lisa, 0000

T,TE, 00000 * * * 0 000000 * * 0 0000000 * 0 00000000 .

Transform the second term on the left side of inequality (8b) into the following equation (9).

9 BE2023/5275

Sym( FT PZ) & ok ok ok ok * * * 0 0 XK * * * + * t, RE, 0 Q * * * * * *

V2

PR > 0 0 0 * * * * *

V2 = 7 Tu5: > 0 000 * * * *

Tyl Na 0 0000°* * * 0 0 0 0 0 0 O0 ** 0 0000000 * 0 00000000 el PH (9) 0 tu RH 2

DSH

Sym{ V2 D(t) [Fe -F,KCe, —-F,KCe, 0000000 0] — 75H 2 tu IH 0 0 0

After replacing the second term on the left side of inequality (8b) with equation (9), we can finally deform it into equations (10) and (11) by applying Lemma 4 and Lemma 5. diag{ R+S, 3(R+S,) S(R+S,} V 0 . >U(10 y" diag{R+S, AR+S,) 5(R+SH|7 09

HE, * * * * * * * * * *

F"Pe, 7 * * * * x + * * *

TRE, T,RF —R * * * x x x * + rsz, Lose 0 -S, * * * + * * * any ss. Bes 0 0 -S, * tk

TTE, T,TF 0 0 0 -T xxx |<Û

Zèe, 0 0 0 0 001 * + # *

Z?KC(e, +eg) 0 0 0 0 0 07 #* # *# oH" Pe, 0 ot, HR Lords, Lords, OT,HT 0 O0 07 #* *

Eee — E,KCe, — F,KCe, 0 0 0 0 0 0 0 0 -d *

Ce, 0 0 0 0 0 00 0 0-7 where Lemma 5 reads: given a matrix of appropriate dimensions Z =

10 BE2023/5275

ZI, Het E, the following inequalities are satisfied

Z+HD(T)E+E" D'(T)H" <0

For any D(t) satisfying D" (t)D(t) <1, there exists a scalar 6 > 0, such that the following inequality holds.

Z+0 HH +0 E<0 €, = [On I, O0 (8-1) Open ]

IT, =Symfe; PL,)—G, diag{25,,45,,25,,4S, }G, —G; diag{2S,,45,,25,,48,}G, +e Oe, _ ? ; and —e;Qe, —G{ HG, —e; C'OCe, + Ê(e, +e,)"C"OC(e, +e,)—(e —e,)" ZT —e,) >, = Ae, — BKCe, — BKCe, >, = Ae, - BKCe, — BKCe, + Fe, 2. = HD(t)E,e, — HD(t)E, KCe, — HD(t)F, KCe, ñ diag{R,32,5R} V a VT diag{R,3R,5R} €76, summer, —2nd,

G = ee, — 6e, &76; e‚ te, 2nd, e‚—e, — 6th, ee,

G,- e, — e, — 3rd, 6, 64 e, — e, —3rd,

11 BE2023/5275 e‚ —e, e, —e, +3e

G, — 2 4 5 6 e, — e, + 3e, (a) Here are some of the most important features of the new system

Assuming that inequality (10)(11) holds, we have IT, < 0‚ then the first term to the right of the inequality sign of inequality (7) is a value less than 0. Removing the > first term to the right of the inequality sign of inequality (7), the equation still holds, and we have

T 2 DT T

V(OS-y ()y()+7 AP (MAP) x (ZX)

T T xl) +, (I (KC) ZK) T0) te] 9) > [4 + fr + Tp) =[0, ®) V(t)

According to k=0 ‚ x(t) is continuous on t and is continuous on t, which gives the following equation (13)

T 2 DT T

V(00)-V(0)< [-y (y) + AP (MAR (1) x (2x0) 0 (13) —[x(-1())+e, (OT (KC) Z, (KC)[x(t -1(1))+e, (E)Idt

With zero initial conditions, we obtain the following equation.

T 2 DT [EyTOy@ +77 AP! (DAP (Dlt 20 0 (a) Here are some of the most important characteristics of the new system then for any AP (1) € [0,+æ) non-zero and any performance index y, |» Ol, sy | A? GI ‚ when AP;(1)=0 , there exists a constant £ > 0 for *() #0 such that

Va.x()) < cell, ‚ the system is asymptotically stable and has a parametric limit H.

Define the secondary performance indicator for power systems whose parameters are uncertain as in equation (14).

12 BE2023/5275

OT T

J= [ (OZD) +4" (DZ u()]dt (14) where Z1, Z2 is the positive definite matrix representing the energy weights of the system and

J is the sum of state energy accumulation and control energy consumption throughout the control process.

The progressive stabilization of the system results in the following equation (15)

V(o)=0(15)

Substituting equation (15) into equation (13), we obtain equation (16), the -V(0)< fs” (y) + APE MAP MO =x"()ZX(0) 0 (16) xt 70) +e (OJ (KC) Z,(KC)[x( = 7(0))+e, (Ir

By distorting equation (16) to obtain equation (17), we obtain the following result [102 x0) +0" Zu = [ x (07 X0)+[X(@=7())+e; (OT (KC) Z (KOK = 1(1))+e, Bd < © 2

VOR ESTOY APT MAP (Dt <V(0) +7" ]AP, | a

Transform V,(f) into equation (3) as follows, 2 2 ft, T . 7 { T

V0 =|, #(9)7ts)ds <= |, Palo) = GIF Tas) = xt, Yes k k 2 (ff T . T° / T = [ ot TS)" [ EG) = 10 = OT Tas) = x Bones

Since 1(0) = 0, it follows that equation (18), the 2 0 T . at 0 T

V,(0)=Tw Le Xx (s)TX(s)ds = Leo") —x(-T(0))] 7[x(s)— (18) 2 2

J<V(0)+77|AP O)|, = x" (0)Px(0)+ [* x" (S)Ox(s)ds +7, [+ ORidsde | [| [ +" ()SA0S)dschvdw ff (SAG) dscvdw +" AP, Of; = 77 (a) Here are some of the most important features of the new version of the

13 BE2023/5275 new version.

A new time lag-dependent and less conservative stability criterion was obtained as follows.

Given & 7>Û and a controller gain K, for a power system with parameter uncertainty, if there exist positive definite matrices P ,O ,R ,T ,S, ,S, ‚and matrices V € R °*"*" such that the following equations (19) and (20) are valid diag{R+S, 3R+S,) S(R+S,)} V 009) > y" diag{R+S, 3R+S,) S(R+S,)}

HE, * * * * * * * * * *

F"Pe, „VI * * * * * * * * *

TR >; TRF —R * * * * * * * *

J2 2 trick, SuSE 0 -S, * * + + + * *| 20)

Ps, se 0 0 -S, * #40 # + +

TTE, TyTF 0 0 0 TB * #* + *|<0

Zie, 0 0 0 0 0 Jot ok # #

ZIKC(e, tes) 0 0 0 0 0 0 0 #* # # oH” Pe, 0 or, HR Lords, Lords, or,HT 0 0 -ol * *

E,e, — F,KCe, - F,KCe, 0 0 0 0 0 00 0 -di *

This, 0 0 0 0 0 00 0 0 +7

The electrical system containing uncertainty on the parameters is then asymptotically stable and the corresponding secondary performance satisfies: J < J”,

Of which x 0 0 po / 0 Foro /

J = x"(0)Px(0) + f x'(s)Ox(s)ds+T, f Í x” (s)Rt(s)dsdv + [ Í Í (SS t(s)dsdvdw 0 w 0. - 2 + f IN Í X(s)S,x(s)dsdvdw +7 |AP, Of,

Preferably, the use of stability criteria to determine the stability of a power system containing uncertain parameters, comprising the following steps.

Define . X, = P"

Multiply inequality (19) backwards and right sides by diag { , XX XX

14 BE2023/5275

Q=X OX, R=

A — A — T —

DS ATX Q= X,C QCX,,

Applying Lemma 4 and Lemma 6 to the two inequalities after the transformation gives equations (21) and (22): diaglPf +5 AR +5) 5(R +5 V auf +5 a D 5(@ + 3 ) 11 A A 1>0 En ÿ diaglf +5, AP +5) 5(R +5)

IT, * * * * * * + * * *

F'e, I * * * * + + * * *

T,X,T,F R —2X, * * * * 0% +# * ok

Lt rr 0 5-24, * * EEE (22)

Ls, rr 0 0 S, —2X, * * *%* * xx

Ty, zu 0 0 0 T-2X, * * * * *|<0

Z?X,e, 0 0 0 0 01 * * #* *#

Z2X,(e, +e;) 0 0 0 0 0 017 * ox * oH"e 0 ot,H'R Lord Lord or,H* 0 0 ol * *

E,X;e, -E,X,e, -E,X,e, 0 0 0 0 0 00 0 ol *

This, 0 0 0 0 0 00 0 0 7

Of which e =|0 I, 0, Ô ie {1,2,3,...,9 ‚Tl ee n° PSN NXMara b (123,29) (a) Here are some of the most important features of the new system

IT, =Sym(e'$,)-G1 diag{25,,45,,2$,,45,}G, -G! diag{25,,48,,25,,4$,}G, + el Oe,

A A A A A? A —e, Qe, -G/ HG, — eg Qe, +Ô(e, +e,)" Ole, +e)-(e 6) ze —e,) >, = AX,e, — BX, e, — BX,e,

An diag{R,3R5R} V

PT diag{R,3R,5R}

Lemma 6: For any positive definite matrix R > 0 and a symmetric matrix X, we have. —XR’X < p’R-2pX

15 BE2023/5275 where p is any positive real number.

Under the event-triggered communication mechanism, given 6, % Ty >O , the positive definite array Z1 , Z2 , the power system containing parameter uncertainty is asymptotically stable when there exist real positive definite matrices

X, Ô ‚RT $, ‚i=L2 and real matrices” and Xb such that equations (21) and (22) are valid.

Preferably, the regulator parameters are derived using a matrix inequality method to obtain a performance-preserving event-triggered control law, comprising the following steps

Define the performance-preserving controller model as follows u=-KCx(t,) (23)

Of which

K=X ‚lt CX 1 y C is the matrix of the system parameters.

The parameters X 1 and X. 2 of the performance-preserved regulator are obtained by solving Eqs. (21) and (22) with the LMI solver, and the performance-preserved regulator is obtained by substituting À 1 and X 2 into Eq. (23), and the upper limit of the preserved performance index of the performance-preserved regulator is J*.

POT (TÔ (ds +, [TATRA Rs dd eu my dv + ATS

The method for designing an event-triggered performance-preserved regulator for power systems provided by the present invention takes into account the uncertainty of system parameters and obtains a system model. The experimental control effect of its controller has a small deviation from the theoretical value, which not only makes the system robust and stable and resistant to interference, but also makes the preset secondary performance index optimal, which can reduce the transmission pressure of the communication network and save the limited resources of network bandwidth.

Illustrations

Figure 1 is a flowchart of the design of the present invention.

16 BE2023/5275

Figure 2 is a model of the control system of Embodiment 1 of the present invention. (a) Figure 3 shows the frequency deviation response of the system in the case where the initial condition of state x(t) is constant in Embodiment 1 of the present invention.

Figure 4 shows the frequency deviation response of the system in the case of the initial condition of state x(t) in Embodiment 1 of the present invention, as a function of time.

Specific implementation

The present invention provides a Method for designing an event-triggered performance-preserved regulator for power systems, to make the objective, technical solutions and advantages of the present application clearer. In order to make the purpose, technical solutions and advantages of the present embodiment more clear, the technical solutions of this embodiment will be described in more detail below, in conjunction with the accompanying drawings. In the accompanying drawings, the same or similar designations from start to finish indicate the same or similar components or components with the same or similar functions. The embodiments described constitute only part of the embodiments of the present application and not their entirety. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present application and should not be construed as limiting the present application. Based on the embodiments described in the present application, all other embodiments obtained without creative work by a person having ordinary skill in the art fall within the scope of protection of the present application. Embodiments of the present application are described in detail below in connection with the accompanying drawings.

The method for designing an event-triggered performance-preserved regulator for power systems, as shown in Figure 1, includes the following steps.

Construction of closed-loop models for load frequency control of power systems containing parameter uncertainties.

Construction of multiple discontinuous generalized Lyapunov-Krasovskii functions based on event-triggered closed-loop models for load frequency control.

17 BE2023/5275

Derivation of stability criteria for power systems with parametric uncertainties based on generalized Lyapunov-Krasovskii functions with multiple discontinuities.

Use of stability criteria to determine the stability of power systems containing parameter uncertainties.

A matrix inequality approach is used to derive controller parameters and obtain performance-preserving, event-triggered control laws, assuming that power systems containing uncertain parameters are stable.

Furthermore, a closed-loop model for controlling the load frequency of a power system containing uncertain parameters is constructed according to equation (1), as follows

X(1)=(A+ AA)x(1)+(B+ AB)u(t, ) + FAP, (1) bi = Cx(1) ©

Where x({) is the state vector, to the deviations from the nominal values of the inertia time constants of the main motor and the speed regulator respectively, u(£,) is the system control input, AP,(f) is the load disturbance, V({) is the system output vector, 4, is the event trigger time and ” is the load disturbance factor.

Furthermore, construction of a multiple discontinuous generalized Lyapunov-Krasovskii function based on a closed-loop model triggered by load frequency control event, comprising the following steps

Define ex (f) as the system state deviation, y as the performance indicator Hs , and the time shift functions 7(#) ,7({)=1—1t, ‚and

Of which , is the triggering time of the event and is the present time.

When e[f, +d,,t,,, +4, ), equation (1) is distorted into

(f)

By constructing the generic Lyapunov-Krasovskii function with multiple discontinuities according to equation (3) below, we obtain the generic Lyapunov-Krasovskii function with multiple discontinuities.

18 BE2023/5275

V(1)= YO i=l (3)

Of which

V(t)= x" (t)Px(t) (a) Here are some of the most important features of the new system

Î

5 V, (ft) = [ x" (s)Ox(s)ds (a) Here are some of the most important features of the new t t system

A) =| [ x” (s)Rx(s)dsdv (a) Here are some of the most

Ty important JV of the new system t tor t w tor; v,0)=|_ [ [OS 5)dsvdæw+|" | [X ()S,#(s)dsdvdw tT, SW OV tT, Tag JV (a) Here are some of the most important features of the new system 2 2 { , T . 1 [ T

V0) = ri], $ YTo)ds LT Ts) -x4)]ds ©

Here are some of the most important features of the new system

Of which

PO ,R ,T ,S, ‚5, are the weight matrices, "M is the upper limit of the time shift function r(t) ‚t. is the trigger time of the event, / is the time current, andX(#) is the state vector.

Furthermore, derivation of a stability criterion for power systems containing parametric uncertainty based on generalized Lyapunov-Krasovskii functions with multiple discontinuities, comprising the following steps

The derivation ofW(#) in equation (3) gives equation (4), the

19 BE2023/5275

V(1)=2x"(0)PX(1)+x"(1)Ox(1)-x"(t-1,,)Qx(t-—T,;) 2-T . { T . ft) 1, . +7,

Ty.T. {CT. t=c(t) pt=r(t).T . + OS [ m [ X'(s)S, X(s)dsdv— | u [ 7 (s)S, X(s)dsdv a, HN TS ENS NE

Ci) OS + EOS AO | ISS) a) [Of Sas [RT SAG + TX (OTK) 2 7

Kx 70) TK) xl -7(0)]

By defining a, TO ‚a €[0,1] , and applying Lemma 1 and Lemma 2 to the terms

You 5, 6, 10 and 14 of VO) of equation (4), then Lemma 3 for the other terms, we obtain equation (5).

V(t) < &(1){Sym(e;P(X;, +Z3 + e Oe, 7 e, Oe, +2 +2 > (TR + 2 2

MS + MS, +77), + Es) - CT diag(2S, 45,.25,,45,)G, 2 2 (5) 2 . = IT 7 G, diag(2S,,4S,,25,,4S,)G, 7 Gi HG, —(e - e, 7 "6 —e,)}5, (4)

Including €, = [On I, Ö (8D 0 in } ie {1,2,3,...,9} 1 and 1 and

EO OE 0 A (NS 1 t-z(f) 1 t-7(f)

ZZ Í u Od | Peer nj SXGS)ds,e, (0), AP} 9 >, = Ae, - BKCe, — BKCe, + Fe, 2, = HD(t)F,e — HD(t)E,KCe, — HD (t)E, KCe,

He diag{R,3R,5R} V a VT diag{R,3R,5R}

20 BE2023/5275 €76, e +e, —2nd, e —e, —6th

G, — 1 2 5 €76 e, +e, —2nd, e, —e, — 6th, ee, e —e, —3rd

G, — 1 4 5 6 e, —e, 3rd, e‚ Ee, e, —e, +3e

G, — 2 4 5 €76 e, —e, + 3rd,

Lemma 1: For any given matrix R ES, , suppose that there exists a matrix X € R"""

R X satisfying YR 20 then the following inequality is valid. 1

TR 0 x Rx 1 “YT RR Vve(0,1) 0 —R 1-v ;

Lemma 2: Given a matrix R €S, , for any differentiable function x: [a,b] > R”, the following inequality is valid. b 1 [ +" (u) R&(u)du > 5776: diag(R,3R5R)©, a —a

Of which

21 BE2023/5275 x(a)-x(b) 2 b ©, =| x(b)+x(a)=-— | x(u)du b — a “a 6 b x(b)-x(a)=-—[" 4.5 (W)x(u)du b-ava * . u—a

At, (u)=2———1 b-a

Lemma 3: For a positively defined matrix R > O and a differentiable function x) Ive [a, b}}, the following inequalities are valid. b cb T . T 1. | [ (w)Ré(/dudy >

Including 1 b x(b)-— [| x(u)du b— aa 4 = .

Bf x)du-—Z [to d 6) wave === [| A (X) 1 b x(a)-— [ x(u)du b—a a 4 = (a)-— [xda + 3 Aaso)xGdu x(a) -— —— — u b— a» b—a % “*

Define the triggering mechanism for discrete events as in equation (6) below. ben = 1, + Min{(+Dh|[xG,)=x(, JT O[x(1,)=x(4,)]2 " (tp) (1, )} (6)

Of which i, =t, +(+ Dh is the present time. {, is the time when the last event was triggered. h is the sampling period. is the number of cycle intervals between the last trigger and the current time when a sample was taken but did not cause a trigger. ©) is the trigger matrix.

22 BE2023/5275

Applying the principle of equation (6) to equation (5), we obtain the following equation (7).

VOE ODILE) y" (Oy) +77AP MAR (D)- (7) x°(0)ZX(0)=[x(t=1(1))+e, (I (KC) Z,(KC )[x(t=7(1))+e,(1)]

Of which

II, = e' C'Ce, = vele, + Sym(e{ P(Z, +2, ))+e/ Oe, — el Oe, 2 2 +(24, + 3) (TR + AS, + As, + TT ME; +33)

G! diag(2S,,4S,,25,,4S,)G, - G! diag(2S,.45,,2S,45,)G, - 2

G/ HG, —(e —e,)" TT —e,)+ô(e, +e,) C'OC(e, +e,)- e, C'ACe, +e"Z,e, +(e, +e,)"(KC)" Z,(KC)T(e, + e,) N! (a) Here are some of the most important features of the new system

For ll, , use Lemma 4 when the following inequality is respected II, < 0

I, * * * * * * * *

F* Pe, — y’I * * * * * * *

URE +5) TRE ORO

SE +2) us 0 a S, * * * * *

DSE, +2) SF 0 0 -5, * * * * |<0 2 2 (8a)

T,T(E,+E») TIEF 0 0 0 -T #* * * 1

Zie 0 0 0 0 0 —- * * 1

Z?KC(e,+e;) 0 0 0 0 0 0 -1 *

This, 0 0 0 0 0 0 0 —!

Of which

IL, = Sym(e; P(E, +2,))+ er Oe, 7 e, Oe, 7 G, diag(2S,,4S,25,,45,)G, 2 ; T — 0 7 G; diag(2S,,4S,,2S,,4S,)G, — (e — e,)" + T(e -e,)-G/ HG, zes C'QCe, ; +ô(e, +65 )" C OC(e, +e,) >, = Ae, — BKCe, — BKC,

23 BE2023/5275

Lemma 4: Schur complement of a matrix For a given symmetric matrix _ Su Sp | . . . _ .

S= ss, WHERE If is of dimension r x r, the following three conditions are equivalent. 21 22

M S<0. (2) S,, <O,S,-SLSIS, <0 (a) Here are some of the most important features of the new system (3) S, <0,8,,-8,,53,$} , <0 (a) Here are some of the most important features of the new system

By transforming inequality (8a) into equation (8b), we obtain the following result

TL, * * * * * * * *

FT Pe, _ y I * * * * * * * 7 RX; TRF -R * + * + + + 2 2 73 745 >, Tus” 0 —S, * * * * * 2 V2

Sas Es Sul 0 0 Ss + 8 + OF

TT, TIF 0 0 60 -T * * * 1

Zie, 0 0 0 0 60 -I * * 1 2} KC(e, +e;) 0 0 0 0 0 0 II *

This, 0 0 0 0 0 0 0 —-/

Svm( FT PE kok # kkk ym( 32) (8b) 0 0 * * # # # # *

T, RE, 0 0 * xx # # #

V2 2 TS, > 0 0 0 * +4 +4 #* * 2 <0 ae >, 0 000% + + *

Tyl La 0000 0 *%* #* * 0 000000 * * 0 000000 0 * 0 0 0000000

Transform the second term on the left side of inequality (8b) into the following equation (9).

24 BE2023/5275

Sym( FT PZ) & ok ok ok ok * * * 0 0 XK * * * + * t, RE, 0 Q * * * * * *

V2

PR > 0 0 0 * * * * *

V2 = 7 Tu5: > 0 000 * * * *

Tyl Na 0 0000°* * * 0 0 0 0 0 0 O0 ** 0 0000000 * 0 00000000 el PH (9) 0 tu RH 2

DSH

Sym{ V2 D(t) [Fe -F,KCe, —-F,KCe, 0000000 0] — 75H 2 tu IH 0 0 0

After replacing the second term on the left side of inequality (8b) with equation (9), we can finally deform it into equations (10) and (11) by applying Lemma 4 and Lemma 5. diag{ R+S, 3(R+S,) S(R+S,} V 0 . >U(10 y" diag{R+S, AR+S,) 5(R+SH|7 09

HE, * * * * * * * * * *

F"Pe, 7 * * * * x + * * *

TRE, T,RF —R * * * x x x * + rsz, Lose 0 -S, * * * + * * * any ss. Bes 0 0 -S, * tk

TTE, T,TF 0 0 0 -T xxx |<Û

Zèe, 0 0 0 0 001 * + # *

Z?KC(e, +eg) 0 0 0 0 0 07 #* # *# oH" Pe, 0 ot, HR Lords, Lords, OT,HT 0 O0 07 #* *

Eee — E,KCe, — F,KCe, 0 0 0 0 0 0 0 0 -d *

Ce, 0 0 0 0 0 00 0 0-7 where Lemma 5 reads: given a matrix of appropriate dimensions Z =

25 BE2023/5275

Z*, Het E, the following inequalities are satisfied

Z+HD(T)E+E" D'(T)H" <0

For any D(t) satisfying D" (t)D(t) <1, there exists a scalar 6 > 0, such that the following inequality holds.

Z+0 HH +0 E<0 €, = [On I, O0 (8-1) Open ]

IT, =Symfe; PL,)—G, diag{25,,45,,25,,4S, }G, —G; diag{2S,,45,,25,,48,}G, +e Oe, _ ? ; and —e;Qe, —G{ HG, —e; C'OCe, + Ê(e, +e,)"C"OC(e, +e,)—(e —e,)" ZT —e,) 2, = Ae, — BKCe, — BKCe, >, = Ae, - BKCe, — BKCe, + Fe, 2. = HD(t)E,e, — HD(t)E, KCe, — HD(t)F, KCe,

A diag{R,3R,5R} V a VT diag{R,3R,5R} €76, summer, —2nd,

G = ee, — 6e, &76; e‚ te, 2nd, e‚—e, — 6th, ee,

G,- e, — e, — 3rd, 6 64 e, —e, — 3rd,

26 BE2023/5275 e, —e, e, —e, +3e

G, — 2 4 5 6 e, — e, + 3e, (a) Here are some of the most important features of the new version of the new version.

Assuming that inequality (10)(11) holds, we have IT, < 0‚ then the first term to the right of the inequality sign of inequality (7) is a value less than 0. Removing the > first term to the right of the inequality sign of inequality (7), the equation still holds, and we have

T 2 DT T

V(OS-y ()y()+7 AP (MAP) x (ZX)

T T xt) +, (OT (KO) Z;(KC)x(—r())+e, (01) > [4 + fr + Tp) =[0, ®) V(t)

According to k=0 ‚ x(t) is continuous on t and is continuous on t, which gives the following equation (13)

T 2 DT T

V(00)-V(0)< [-y (y) + AP (MAR (1) x (2x0) 0 (13) —[x(-1())+e, (OT (KC) Z, (KC)[x(t -1(1))+e, (E)Idt

With zero initial conditions, we obtain the following equation.

T 2 DT [EyTOy@ +77 AP! (DAP (Dlt 20 0 (a) Here are some of the most important features of the new version of the new version. then for any non-zero AP (1) e[0,+æ) and any performance index y,

U (Ol, sy |AP; (WHERE, ; when A: ()=0, there exists a constant £ > 0 for *() #0 such that

Va) < elk, ‚ the system is asymptotically stable and has a parametric limit H.<

Define the secondary performance indicator for power systems whose parameters are uncertain as in equation (14).

27 BE2023/5275

OT T

J= [ (OZD) +4" (DZ u()]dt (14) where Z1, Z2 is the positive definite matrix representing the energy weights of the system and

J is the sum of state energy accumulation and control energy consumption throughout the control process.

The progressive stabilization of the system results in the following equation (15)

V(o)=0(15)

Substituting equation (15) into equation (13), we obtain equation (16), the -V(0)< fs” (y) + APE MAP MO =x"()ZX(0) 0 (16) xt 70) +e (OJ (KC) Z,(KC)[x( = 7(0))+e, (Ir

By distorting equation (16) to obtain equation (17), we obtain the following result [102 x0) +0" Zu = [ x (07 X0)+[X(@=7())+e; (OT (KC) Z (KOK = 1(1))+e, Bd < © 2

VOR ESTOY APT MAP (Dt <V(0) +7" ]AP, | a

Transform V,(f) into equation (3) as follows, 2 2 ft, T . 7 {T

V0 =|, #(9)7ts)ds <= |, Palo) = GIF Tas) = xt, Yes k k 2 (ff T . T° / T = [ 9 Tds [ „ATO TLs) x Os t Ti tT

Since T(0) = 0, it follows that equation (18), the 2 0 T . at 0 T

V,(0)=Tw Lot (s)TX(s)ds = Leo") —x(-T(0))] 7[x(s)— x(=7(0))]ds = 0 ( 18) 2 2

J<V(0)+77|AP O)|, = x" (0)Px(0)+ [* x" (S)Ox(s)ds +7, [+ ORidsde | [| [ +" ()SA0S)dschvdw ff (SAG) dscvdw +" AP, Of; = 77 (a) Here are some of the most important features of the new version of the

28 BE2023/5275 new version.

A new time lag-dependent and less conservative stability criterion was obtained as follows.

Given & 7>Û and a controller gain K, for a power system with parameter uncertainty, if there exist positive definite matrices P ,O ,R ,T ,S, ,S, ‚and matrices V € R °*"*" such that the following equations (19) and (20) are valid diag{R+S, 3R+S,) S(R+S,)} V 009) > y" diag{R+S, 3R+S,) S(R+S,)}

HE, * * * * * * * * * *

F"Pe, „VI * * * * * * * * *

TR, TRF —R * * * * * * * *

J2 2 77 Tu TS TE 0 5, + * * # + * * (20)

Bes ET 0 0 -S, * RE

Ty TE, TyTF 0 0 0 -T ot #* # # *|<0

Zie, 0 0 0 0 0 Jot ok # #

ZIKC(e, tes) 0 0 0 0 0 0 0 #* # # oH” Pe, 0 or, HR Lords, Lords, or,HT 0 0 -ol * *

E‚e, - F,KCe, - F,KCe, 0 0 0 0 0 0 0 0 -d *

This, 0 0 0 0 0 0 0 0 0 I

The electrical system containing uncertainty on the parameters is then asymptotically stable and the corresponding secondary performance satisfies: J < J”,

Of which x 0 0 po / 0 Foro /

J = x"(0)Px(0) + f x" (s)Ox(s)ds +1, f Í x” (s)Rt(s)dsdv + [ Í Í (SS t(s)dsdvdw 0 w 0. - 2 + f IN Í X(s)S,x(s)dsdvdw +" |AP, Of,

Furthermore, using the stability criterion to determine the stability of a power system containing parameter uncertainty, includes the following steps

Define . X, =P!

Multiply inequality (19) backwards and right sides by diag { , XX XX

29 BE2023/5275

Q=X OX, R=

A — A — T —

DS ATX Q= X,C QCX,,

Applying Lemma 4 and Lemma 6 to the two inequalities after the transformation gives equations (21) and (22): diaglPf +5 AR +5) 5(R +5 V auf +5 a D 5(@ + 3 ) 11 A A 1>0 En ÿ diaglf +5, AP +5) 5(R +5)

IT, * * * * * * + * * *

F’e, Jy’ * * * * + + * * *

Ty X, T,F R-2X, * * * * + * * *

Lt rr 0 5-24, * * BKK (22)

Ls, rr 0 0 S, —2X, * EEE

Ty, TyF 0 0 0 T-2X, * * #* * *|<0

Z?X,e, 0 0 0 0 01 * * #* #

Z2X,(e, +e;) 0 0 0 0 0 017 * #* * oH'e 0 ot,H'R Lori Lori or,H* 0 0 -ol * *

E Xe, -E,X,e, - FE, X es 0 0 0 0 0 000 0 I *

This 0 0 0 0 0 00 0 0 -

Including €, = [O enn I, O8 Open } ie {1,2,3,...,9}

ÎL, =Sym(e$,)-G{diag{25,,45,,2$, 45,}G, - GT diag{28,,48,,28.,48,}G; + el Oe,

At 2

A Le A A TT A -e; Oe, -G/ HG, — €, Nes +Ô(e, +6)! Ole, +e,)-(e -e,) 7/6 -€,) >, = AX,e, — BX,e, — BX,e,

Be diag{R,3R,5R} V

DT diag{R,3R,5R}

Lemma 6: For any positive definite matrix R > 0 and a symmetric matrix X, we have. — XRX < p'R-2pX where p is any positive real number.

30 BE2023/5275

Under the event-triggered communication mechanism, given 6% Tu >O , the positive definite table Z1 , Z2 , the power system containing parameter uncertainty is asymptotically stable when there exist real positive definite matrices

X, Ô ‚R JT $, ‚i=L2 , and real matrices / and X2 such as equations (21) and (22) are valid.

Further, obtain the performance-preserving event-triggered control law using a matrix inequality method to derive the controller parameters, including the following steps

Define the performance-preserving controller model as follows u=-KCx(t,) (23)

Of which

K=X ‚lt CX 1 y C is the matrix of the system parameters.

The parameters X 1 and X. 2 of the performance-preserved regulator are obtained by solving Eqs. (21) and (22) with the LMI solver, and the performance-preserved regulator is obtained by substituting À 1 and X. 2 into Eq. (23), and the upper limit of the preserved performance index of the performance-preserved regulator is J*.

POT (TÔ el) +, [HT ()X RXT Rs dd eu my dv + ATS [37 (XS, X 3(s)dsdvdwe + 7 |A, (Of, rjg dw dv or Jr dv

Example 1

The event-triggered guaranteed performance monitoring method for power networks with uncertain parameters, described in this application, is described in more detail below, in conjunction with the attached Figures 2 to 4. 1, As an example, the single-domain load-frequency control system model is shown in Figure 2. Let us first consider changes in system parameters caused by deviations from the nominal values of the constants of inertia time of the prime mover and speed regulator in the electrical system, assuming that

Tja € [0 -0%)T 4, 1 +0%T,,]

T„e[0-107,,0+1%)T, | ©

Of which

31 BE2023/5275

Lana !,a Te represents the actual inertia time constant of the main motor, represents the nominal inertia time constant of the main motor, 7, 7, represents the nominal inertia time constant of the regulator eto: represents the percentage deviation from the inertia time constant of the main motor and the governor respectively.

Equation (1) can be re-expressed as a time-varying function in equation (2) as follows. 1 O O _ 4 2

TT ft) T. cha ch ch l 1 1 @ _ 1 2

T. 7 T +f,(6) T ga g g

Including 1

Oo = . (1=0%)(1 + 0%) 0% oa on: (1 =0%)(1 + 0%) felt] 1 1, = ————

A=1%)(1+1%)

The 1% 27 (1-2%)1+1%) felt]

Based on Figure 2 and the description of equation (2) above, the following equation can be constructed for the LFC model of a single-domain parametric uncertainty power system.

X(1)= (A+ AA)x(1)+(B+ AB)u(t,)+ FAP,(1), te[t,+d,,t,,, des) where AA = HD(t )F, ,AB = HD(t)E, , H, El and E2 are constant matrices with appropriate dimensions, D(f) = diag{0 f() fl) 0} . AP,(f)is the load disturbance, and tk denotes the time when the measurement data is released by the event detector, i.e. the time when the event is triggered.

M ap, ap, face]

32 BE2023/5275

D 1 —_ — 0 0

M M

0 LA 4 9

A= the To 4 0 49

RT; [,

B 0 0 0

T

4

B=|0 0 — O0]. 1, 0 0 0 0 9 > LL 9 52 [, Ta 1 L L -—— 0 —-— 0

RT, I, 0 0 0 0

T

1,

E,=10 0 — O0|. 1,

For the above single parameter uncertain power system LFC model, the relevant parameters are as follows:7,, = 0.3.7, =0.1,R= 0.05, D= 1.0, M = 10.8 =21, =o1 = 10, H = 0.1/la (Ja designates a four-dimensional unitary matrix), D(f) = diag{0, sin(t), sin(t), 0}. 2, Stability criterion and method for designing a performance-preserving controller using the stability criterion obtained in the case where the initial conditions of state x(t) are constant.

Set the initial condition of state x(t) to? €[-7,,,0] as a continuous function of , on lt) p(tf) te[-7,,,0] . Assuming thatg(f) =[0.01 0 0.1 OJ! , an upper limit on the time offset 7, = 0.2s, the event trigger parameters ô = 0.01, 0 = 0.06 and Zi =l4, Z2 = 1, based on the stability criterion ci (above, the controller gain matrix K and the trigger matrix) can be solved using the matrix inequality method in the following way.

K =|0.3108 0.7055]; o- 208.4932 1241966 “ |124.1966 156.3325 |’

33 BE2023/5275

According to the above derivation of the performance-preserving controller design method, the performance-preserving control law can be obtained as u'(f)=-[0.3108 0.7055]y(t,) , an upper bound for the minimum value of the secondary performance index is J* =5.1336. By selecting the time offset z(f) = 0.2sint | , a load disturbance of 0.1pu. occurs at t=2s for a duration of 2s, and the response of the system is shown in Figure 3, which shows that the system is stable at this time. 3, A stability criterion with a performance-preserving controller design method using the stability criterion obtained in the case where the initial conditions of state x(t) vary with time. t

Assuming that g(1) =[0 e' 0 e?]” ‚the event trigger parameters 6 = 0.01,0 = 0.06 and Zi =l , Z2 = 1, the gain matrix of the controller K and the trigger matrix © can be solved using the matrix inequality method based on the aforementioned stability criterion as follows

K =|0.4079 0.7207]; 5 a bone sos 30.0294 50.8403

According to the above derivation of the performance-preserving controller design method, the performance-preserving control law can be obtained as u'(f)=—[0.4079 0.7207]y(t,) , an upper bound for the minimum value of the secondary performance index is J” = 4.9897. By selecting the time offset 7(#) =|0.2sint | , a load disturbance of 0.1pu. occurs at t=5s for a duration of 5s, without disturbance the rest of the time, the response of the system is shown in Figure 4, which shows that the system is stable at that time.

The method for designing an event-triggered performance-preserved regulator for power systems provided by the present invention takes into account the uncertainty of system parameters and obtains a system model. Reasonable, the experimental control effect of its controller deviates little from the theoretical value, which not only makes the system robust and stable and resistant to interference, but also makes the preset secondary performance index optimal, which can reduce the transmission pressure of the communication network and save the limited resources of network bandwidth.

The above discussion only constitutes a specific preferred embodiment of the invention; however, the embodiments of the invention are not limited and any variation which may be considered by a person skilled in the art will fall within the scope of protection of the invention.

Claims (7)

34 BE2023/5275 Claims 1. Method for designing an event-triggered performance-preserved regulator for electrical systems, characterized in that it comprises the following steps Construction of closed-loop models for the control of the charging frequency of electrical systems containing parameter uncertainties; Construction of multiple discontinuous generalized Lyapunov-Krasovskii functions based on event-triggered closed-loop models for load frequency control; Derivation of stability criteria for power systems with parametric uncertainties based on generalized Lyapunov-Krasovskii functions with multiple discontinuities; Use of stability criteria to determine the stability of power systems containing parameter uncertainties; A matrix inequality approach is used to derive controller parameters and obtain performance-preserving, event-triggered control laws, assuming that power systems containing uncertain parameters are stable. 2. Method for designing an event-triggered performance-preserved regulator for electrical systems according to claim 1, characterized in that the following equation (1) is used to construct a closed-loop model containing a loop model closed loop for controlling the charging frequency of an electrical system with parameter uncertainty, the closed loop model containing the following elements X(f) = (A+ A4)x(1) + (B + AB)u(t ,) + FAP,() fs = Cx) © Oux(f) is the state vector, A4 and AB are the changes in system parameters due to deviations from the nominal values of the inertia time constants of the main motor and speed controller respectively, u(£,) is the system control input, AP ,(f) is the load disturbance, V({) is the system output vector, 4, is the event trigger time and ” is the load disturbance factor. 3. Method for designing an event-triggered performance-preserved regulator for electrical systems according to claim 2, characterized in that the closed-loop model of load frequency regulation based on the triggering of an event allows one to construct generalized Lyapunov-Krasovskii functions with discontinuities 35 BE2023/5275 multiples. closed-loop load frequency regulation model based on event triggering to construct generalized Lyapounov-Krasovskii functions with multiple discontinuities, including the following steps Define ex(f) as the state deviation of the system , y as the performance indicator Hs , and the time shift functions (#) ,7({)=1{—1, ‚and Whose {, is the trigger time of the event andt is the time here. When? e[f, +d,,t,., +4, ) , equation (1) is deformed into X(f) = (A+ A4)x(1)+(B+AB)KC[e,( t)+x(t-7()]+FAP, © By constructing the generic Lyapunov-Krasovskii function with multiple discontinuities according to equation (3) below, we obtain the generic Lyapunov-Krasovskii function with multiple discontinuities . 5 V(1)= VO 5 G) Of which _vT Vı(t)= x (£)Px(f) Î VO =] x"(s)Ox(s)ds M t tr V.(t)= [ [ [x 0)S,#(s)dsdvdw tT, JW eV tT, Tag JV 2 ff 7 . T° | T Vs = ri, (Ti) [ ets) a] Ts) x Nds k k Of which PO ,R ,T ,S, ,S, are the weight matrices, "M is the upper limit of the time shift function r( t) ‚t. is the trigger time of the event, f is the current time and X(#) is the state vector. 4. The method of designing a performance-preserved regulator triggered by an event for electrical systems according to claim 3, characterized in that, on the 36 BE2023/5275 basis of multiple discontinuities, the generalized Lyapunov-Krasovskii functions derive stability criteria for electrical systems containing parametric uncertainties. generalized Lyapounov-Krasovskii functions, the derivation of stability criteria for power systems containing parametric uncertainties, comprising the following steps The derivation of} (#) in equation (3) gives equation (4), the V (1)=2x"(1)PR(0)+x"()Ox(1)-x"(t-T,,)Qx(t—T,) . . f . . <<). . +74 (H)RX(1)- ru] | < (s)RX(s)ds — ru] c(t) 7 J +3" (SX) [ m [ X'(s)S #(s)dsdv | [ v T Si dsd Eu AD Sd IE OSEO = pd OS Adsdy a É-T (1) pv . . {-T({) . . . — [ [ 1) | (SS, (-7())] By defining a, 10) ‚a €[0,1] , and applying Lemma 1 and Lemma 2 to terms 5, 6, Tu and 14 of VW) of equation (4 ), then Lemma 3 for the other terms, we obtain the equation (5). ; T T T T T (2 V(1)<E, (O){Sym(e, PZ, +Z4,))+ e, Oe — e, Oe, +(E4, +24) (Ty R+ 2 2 TM TM 2 T 7. 10 5 3 + EE + T (23 +247) C, diag(2S ,45,,25,45, CG, x? Toy: TH T -G, diag(2S,,4S,,2S,,4S , )G, -G, HG, —(e — e,) 4 76e — 6, )} 63 (1) (5) Including €, = [O enn I, 0 8 nn Open } ie {1,2, 3,....9} 1 a as, | to d Ezel 0) Ad do ME et (CS 1 t-7(1) 1 t-7(1) —_ x(s)ds, —_— A (S)x(s)ds,e, (1), AP (t 7 [6 ee TOO ep (AP) 23 = Ae, - BKCe, — BKCe, + Fe, 37 BE2023/5275 >, = HD(f)E,e, — HD(t)F,KCe, — HD(t)E, KCe, ñ diag{R,3R,5R} V a VT diag{R,3R ,5R} €76, e te, — 2nd, e —e, —6th G, — 1 2 5 €, 76; e, te, —2nd, e, —e, — 6th, e — €, G, = e, — €, —3e, 6 76 e, —e, —3e, e, —e, e, —e, +3e G, — 2 4 5 €76 e, — e, +3e, Lemma 1: For any given matrix R ES, , suppose that there exists a matrix X € R""" R following inequality is valid; 1 —R 0 Rx Zl p|Ye(01) o LR X“ R 1-v; Lemma 2: Given a matrix R € S, , for any differentiable function x: [a,b] > R" , the following inequality is valid; b 1 [ +" (@)Rèar)du > 57765 diag(R,3R5R)©, a —a Of which 38 BE2023/5275 x(a)-x(b) 2 eb ©, =| x(b)+x(a)=-— | x(u)du b — a “a 6 b x(b)-x(a)-—[ 4.5 (W)x(u)du b-ava * u—a À, (u)=2—— —1 b-a Lemma 3: For a positively defined matrix R > O and a differentiable function x) Ive [a, b}}, the following inequalities are valid; b eb, T . T 7. f(x" (WRi(u)dudv > X diag(2R,4R)3 a JV b pv [ [ 1 b x(b)-—f x(u)du b—a** á 5 1 b 3 b > a)-—f x(u)du b—a 4 = (a)=-—[" x(arjdu +" 2, (0)<(er)du x(a)=—— | x(u — u b —a b-ava 0 Define the triggering mechanism for discrete events as in equation (6) below: ben = 1, + Min{(+Dh|[xG,)=x(, JT O[x( 1,)=x(4,)]2 x” (1, )OQx(f,)} (6) Of which i, = 1, +(J+Dh is the present moment; ff, is the moment when the last event was triggered: h is the sampling period; is the number of cycle intervals between the last trigger and the current time when a sample was taken but did not trigger a trigger, © is the matrix of trigger; Applying the principle of equation (6) to equation (5), we obtain the following equation (7); 39 BE2023/5275 V()<E DILg (0) -y "(yn +77AP (AP, ()- (7) (Zx) [kt 701) +e (I (KC) Z(KO)[K( E-7(1)) + e(1)] Of which II, = e C'Ce, — vee, + Sym(e; P(E,, +E,))+e; Oe, — ef Oe, 2 2 +2 +2 y (TR + GS, + AS, + TT (Es +2)" GT diag(2S,,4S,,2S,,4S,)G, - GT diag(2S,,4S,,25 ,,4S,)G, - 2 Gi HG, —(e, —e,)" TTC —e,)+6(e, +4) C'QC(e, +e,)- e, C'ACe , + e Ze, +(e, +e,) (KC) Z,(KC)T(e, +e;) For II; , use Lemma 4 when the following inequality is respected IT, < 0 II, * * * * * * * * F* Pe, _ y'I * * * * * * * Tu RZ; +) TURF O-R *#* #* * #* #* *# 2 2 2 SE +2) ns 0 a S, * * * * * V2 V2 Tus (2s +23) 7 us 0 0 —S, * * * * < 0 (8a) Ty T(E;+Es) Ty TF 0 0 0 -T * * * 1 Zie, 0 0 0 0 0 -1 * * 1 2; KC(e, +e;) 0 0 0 0 0 0 II * Ce, 0 0 0 0 0 0 0 —! Including II, = Sym(e , P(Z, +2,))+ e Oe 7 e, Qe, 7 G, diag(2S,,4S,,25,,45,)G, 2 —G, diag(2S,,45,,2S ,,45,)G, —(e —e,)" T T(e —e,)-G/ HG, —e, C'QCe, ; +ô(e, +65)" C OC(e, +e,) >, = Ae, — BKCe, — BKCe, . Lemma 4: Schur complement of a matrix For a given symmetric matrix 40 BE2023/5275 _ Su Sp | . . . _ . ee S= Ss Ss ‚where Si is of dimension r x r, the following three conditions are equivalent; 21 22 MD S<0; (2) Sj <0.8,,-S1.5;S, <O (a) Here are some of the most important features of the new version of the new version; (3) S, <0.5,,-8.83:$}> <O (a) Here are some of the most important features of the new version of the new version; By transforming inequality (8a) into equation (8b), we obtain the following result II, * * * * * * * * FT Pe, _ , I * * * * * * * RX, T,RF _R * * * * * * V2 V2 7 79 >, 5 Tus 0 —S, * * * * * V2 V2 72 Tus? >, 7 Tu52F 0 0 —S, * * * * |+ TT >, Ty TE 0 0 0 -T * #* * 1 Zie 0 0 0 0 0 -I * * 1 Z} KC(e, +e; ) 0 0 0 0 0 0 —I * Ce, 0 0 0 0 0 0 0 -/ Sym(F" PZ) & kk # # + Æ x* (8b) 0 Ok * * x # x * t, RE, 0 Q * * * * * * 2 Drs >, 0.0.0 xxx xk 2 <0 Ls, >, 0 0 0 0 * * * * Tyl La 0 0 0 0 O0 *** 0 0 0 0 0 0 O0 * * 0 0000000 * 0 0 0000000 Transform the second term on the left side of the inequality (8b) into the following equation (9); 41 BE2023/5275 Sym( F7 PZ) & ok ok ok ok * * * 0 0 XK * * * + * t, RE, 0 Q * * * * * * 2 Los >, 0 0 0 xx xxx V2 = 7 Tu5 : > 0 000 + * #* * TT Na 00000*°** 0 0 0 0 0 0 O0 ** 0 0000000 * 0 00000000 el PH (9) 0 tu RH 2 DSH Sym{ V2 D(t) [Fe - F,KCe, —-F,KCe, 0000000 0] — 75H 2 tu IH 0 0 0 After replacing the second term on the left side of the inequality (8b) by equation (9), we can finally deform it in equations (10) and (11) by applying Lemma 4 and Lemma 5; diag{R+S, 3(R+S,) S(R+S,} V 0 ; > U(10 y" diag{R+S, AR+S,) 5(R+SH|7 09 IL, * * * * * * * * * * F"Pe, yv * * * * kk * * * TRE, T,RF —R * * * * %* x kk rs, Loose 0 -S, * * * + * * * any ss. Bes 0 0 -S, * kkk “+ TTE, T,TF 0 0 0 -T #* * #* # #|<0 Zie, 0 0 0 0 0 Jk # # # Z2KC(e, +e;) 0 0 0 0 0 017 #* #* * oH" Pe, 0 ot, HR Lords, Lords, OT,HT 0 0 od #* * Eee, - E‚KCe, - F,KCe, 0 0 0 0 0 0 0 0 dq * Ce, 0 0 0 0 0 0 0 0 0 -I where Lemma 5 reads: given a matrix of appropriate dimensions Z = Z', H 42 BE2023/5275 and E, the following inequalities are satisfied Z+HD(T)E+E" D'(T)H" <0. For any D(t) satisfying D" (t)D(t) <1, there exists a scalar 6 > 0, such that the following inequality holds. Z+o HH" +0E E<0. €, = [On I, N Open ] . IL, = Syme; PZ;)- G; diag(2S, 45,25, 45,}G, 7 G, diag{25,,4S,,25,,45, jG, +e Oe 2 — el Oe, -G/ HG, —e, C"OQCe , + S(e, teg) C OC(e, +e,)—(e —e,)" zT —e,) ‚X, = Ae, — BKCe, — BKCe, 2. = Ae, - BKCe, — BKCe, + Fe, 103 = HD(1)E,e, - HD(1)E,KCe, - HD(1)E,KCe, diaë{R.3R.5R} V y" diag{R,3R,5R} €76, e te, — 2e, G = e — e, — 6e, €, 76, e, te, — 2e, e, — e, — 6e, ee, e —e, —3e G, 1614 5 6, 6 e, —e, — 3e, 43 BE2023/5275 e, Ee, G = e, — e, +3e, ; = é3 7 C6 e, — e, +3e, (a) Here are some of the most important features of the new version of the new version: Assuming that the inequality (10)(11) holds, we have IT , < 0 ‚ then the first term to the right of the inequality sign of inequality (7) is a value less than 0. By removing the first > term to the right of the inequality sign of inequality (7), the equation still holds, and we have T 2A DT T V(OS-y ()y() +7 AP (MAP) x (ZX) T T RE 7) te (I (KC) Z,(KOIXG-76)) +e,;(01 (2) > [4 + Tr bg + T1) =[0, ) V(t) According to k=0 ‚ x(t) is continuous on t and is continuous on t, which gives the following equation (13) T 2 DT T V(00)-V(0)< [-y ()y() + AP (MAR (1) x (2x0) 0 (13) [xt -7(0 )+e (OP (KC) Z,(KC)[x( = 7(1))+e, ()Jdr With zero initial conditions, we obtain the following equation: T 2 DT [Ey Oy)+7 "APT (WAP (Dlt 20 0 (a) Here are some of the most important features of the new version of the new version; then for any non-zero AP (1) € [0,+æ) and any performance index y, b Ol sy |A? Ol; when AF,() =0 ‚there exists a constant £ > 0 for x(1) #0 such that” & x) < —E| Ol, “the system is asymptotically stable and has a parametric limit H…” Define the secondary performance indicator for power systems whose parameters are uncertain as in equation (14); OT T =| OZ +0" (OZ u(D]dt (14) 44 BE2023/5275 where Z1, Z2 is the positive definite matrix representing the energy weights of the system and J is the sum of the accumulation of the state energy and the consumption of the control energy throughout the process control; The progressive stabilization of the system results in the following equation (15) V(w)=0(15) By substituting equation (15) for equation (13), we obtain equation (16), the T 2A DT T —V(0)< Ey Oy +7 APT (DAP (0) -xT(DZ;x() 0 (16) T T BR -7() + eN] (KC) Z,(KO)[ X(-7())+e,(0)]}at By distorting equation (16) to obtain equation (17), we obtain the following result OZ) +0" Zu) = [ (x OZXOHKE - 7) +e (OT (KC) Z,(KO)[X( -7())+e, (dr < © 2 VO + [EY OO +77 APT MAR, (Old <V(0)+ 7 [AP]; a Transform V(f) in equation (3) as follows, 2 2 ff .T . MT ft T (=| Tias Ef tsx] Tets) Nds k k 2 2 7 T . T 1 T = [ 9 THIS [ „ATO TLs) x Os tT(t T0 Since 1(0) = 0, it follows that equation (18), the 2 (0 LT T° 0 r Of A Ode A [6-0 TIXG) -x(-7(0)lds = 0018) J <V(0)+7°"|AP, 9; = OPO [* x" ()Ox(s)ds +74 | [x )R/ Asv+ | [ [4 ()S,x(s)dschvdw + 98, 4(s)dsdvdw + [APO =" Ty 9 dv (a) Here are some of the most important features of the new version of the new version ; A new stability criterion dependent on the time lag and less conservative has been 45 BE2023/5275 obtained as follows; Given &, # >Ü and a controller gain K, for a power system with parameter uncertainty, if there exist positive definite matrices P ‚OQ ,R ,T ‚Si ‚S, , and matrices V € R °""* such that the following equations (19) and (20) are valid, [diagtR+S, 3R+5)) 5(R+S;}} V 009) > y" diag{R+S, 3R +S,) 5(R+S,)} IL, * * * * * * * * * * F"Pe, „VI * * * * * * * * * TR, TRF —R * * * * * * * * Ps, Loose 0 5, + * * + * * * (20) ss Los, 0 0 -S, * #40 # + + TTE, T4TE 0 0 0 To * #* + *|<0 Zie, 0 0 0 0 0 JO #* # # # Z?KC(e, +e,) 0 0 0 0 0 0 1 * # # oH" Pe, 0 gold, HR Pol's Lords, OT,HTT 0 0 -ol * * E,e, — F,KCe, - F,KCe, 0 0 0 0 0 00 0 -di * Ce, 0 0 0 0 0 00 0 0 +7 The electrical system containing an uncertainty on the parameters is then asymptotically stable and the corresponding secondary performance satisfies: J < J” , Of which x 0 0 po / 0 Foro / J* = x" (0)Px(0) + f x" (s)Ox(s)ds +1, f |" ERES)Asdv+ |" | [JS E)dsdvew M M UV Ty dw dv 0 w 0. - 2 + [Sf x" SS, x(s)dsdvdw +77 |AP, © 5. The method of designing a performance-preserved regulator triggered by an event for electrical systems according to claim 4, characterized in that, using a stability criterion, determining the stability of a system electrical containing parameter uncertainty, including the following steps Define. X, =P Multiply inequality (19) backwards and forwards by diag { and place by diag XX X ,0= XC'OCX,, X, = KCX, NE (a) Here are some of the 46 BE2023/5275 most important features of the new version of the new version. Applying Lemma 4 and Lemma 6 to the two inequalities after the transformation gives equations (21) and (22): diag{Â +5, AR + 5) 5(R +5) ÿ so ©) pi diag{Â + $, 3815) 5(R + 5) { IL, * * * * * * + * * * F'e, Jy' * * * * + + * * * T, X, T,F R —2X, * * * * 0% +# * ok Lt rr 0 5-24, * * EEE (22) Ls, rr 0 0 S, —2X, * EEE Ty, TyF 0 0 0 T-2X, * * #* * *|<0 1 Z?X,e, 0 0 0 0 01 * * #* # 1 Z2X,(e, +e;) 0 0 0 0 0 017 * #* * oH'e 0 ot, H'R Lori Lori or,H* 0 0 -ol * * E Xe, -E,X,e, - FE, €, = [O enn I, 0 8 nn Open } LE {1,2,3,....9} IT, =Sym(e'$,)-G1 diag{25,,45,,25,, 45,}G, -GTdiag{28,,48,,28,,48,}G; + el Oe, A 2 TA TH TÉ T TA + —e, Oe, -G, HG, — e, Oes + ô(e, teg) Me, tes) (ee) ze —e,) e, — BX,e, - BX,e, A diag{R,3R,5R} V H= A CAL A LA y" diag{R,3R,5R} Lemma 6: For any positive definite matrix R > 0 and a symmetric matrix Under the event-triggered communication mechanism, given &%T, >0‚the positive definite array Z1, Z2, the power system containing parameter uncertainty is asymptotically stable when there exist real positive definite matrices X, Ô ‚R JT $, ‚i=L2 and real matrices” and X such that equations (21) and (22) are valid; 47 BE2023/5275 6. Method for designing a preserved performance regulator triggered by an event for electrical systems according to claim 5, characterized in that it uses a matrix inequality Method for designing a preserved performance regulator triggered by an event for electrical systems, including the following steps; Define the performance-preserving controller model as follows: u=-KCx(t,) (3) Whose + K=X (CX 1) C is the system parameter matrix; The parameters X 1 and X. 2 of the performance-preserved regulator are obtained by solving the Egs. (21) and (22) with the LMI solver, and the performance-preserved regulator is obtained by substituting À 1 and X 2 into Eq. (23), and the upper limit of the preserved performance index of the performance-preserved regulator is J*; * = x" (0x7 1 x (S)XTÔX7x(s)d 0 [47 (OXTRXT(5)dsd J =x (0)X, x(0)+ | x" (s)X OX, x(s ) +7, [ [+ (s)X, ; 7 AP, ( Ty dw dv TTM VTM VV