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Event-Triggering Load Frequency Control forMulti-Area Power Systems with Communication

DelaysShiping Wen, Xinghuo Yu,Fellow, IEEE, Zhigang Zeng,Senior Member, IEEE,

Jinjian Wang

Abstract—This paper studies the load frequency control (LFC)for power systems with communication delays via an event-triggered control method to reduce the amount of communi-cations required. The effect of the load disturbances on theaugmented output is defined as a robust performance index(RPI) of the augmented LFC scheme. By utilizing a time-delayedsystem design approach, a new model of LFC scheme withdelays is formulated where the communication delays and event-triggered control are integrated for the LFC scheme. Based onthe Lyapunov-Krasovskii functional method, the criteria for theevent-triggered stability analysis and control synthesisof LFCscheme are derived. Finally, the effectiveness of the proposedmethod is verified by simulation studies.

Index Terms—Power system, Load frequency control, Event-triggered control, Time delay, Stability, Robust performanceindex.

I. I NTRODUCTION

T O operate the interconnected power systems successfully,it is essential to make the total generation match the

load demand. The operating points of a power system may bedisturbed by unexpected events and uncertainties, which resultin deviations from nominal system frequencies and scheduledpower exchanges to other areas [1]. The LFC has shownto be very effective in controlling power systems to supplysufficient and reliable electric power with guaranteed quality[2]. By LFC, two variable frequencies and tie line powerexchanges are combined to establish a new variable knownas area control error (ACE), which acts as the control signalin the LFC scheme. For the last decades, LFC has been utilizedeffectively in power systems to maintain frequency and powerinterchanges with neighborhood areas at scheduled values [3],[4], [5], [6].

Large scale interconnected power systems are controlledthrough communication channels. Due to the inevitable com-munication time delays such as packet dropouts, disorderingand ACE signal updating [7], the control performance sufferssevere time delays which may destabilize the power system[8], [9]. The LFC in the presence of communication delays has

S.P. Wen and Z.G. Zeng are with School of Automation, Huazhong Univer-sity of Science and Technology, and Key Laboratory of Image Processing andIntelligent Control of Education Ministry of China, Wuhan,Hubei, 430074,China (Corresponding author: Z.G. Zeng) (email: wenshiping226@126.com,zgzeng527@126.com).

X.H. Yu and J.J. Wang are with Platform Technologies Research In-stitute, RMIT University, VIC 3001, Australia (email: x.yu@rmit.edu.au,s3207417@student.rmit.edu.au).

been studied in recent years [4], [5], [10], [11]. In particular,the delay-dependent stability of the LFC has been investigatedvia calculating delay margins, and the delay margins are usedas a performance index to guide the design and operation ofthe LFC controller [12], [13], [14], [15].

However, for systems with heavy transmission burdens andsmall communication times, it would be desirable to reducethe communication burdens between the sensors and controllernodes [16], [17], [18], [19], [20], [21], [22], [23], [24],[25]. Maximizing the time intervals that the nodes need tosend data to each other is an effective way. Furthermore, toreduce the control costs, the LFC scheme may be embeddedwith micro-processors which are usually run with limitedenergy resources and computing capability. This inspires theresearchers to establish event-triggered control schemesoverdigital-operating platforms for LFC [26], [27], [28], [29]. Onthe other hand, event-triggered control has been investigated inwireless sensor/actuator systems via event-triggered commu-nication [30], [31]. It would be beneficial to develop an event-triggered control for LFC to reduce communication delays.

In this paper, an event-triggered control algorithm is pro-posed for the LFC scheme of multi-area power systems. It isworth pointing out that, the proposed LFC problem is non-trivial because of the difficulties of (i) the LFC scheme underconsideration is comprehensive that comprises time delaysand uncertainties that can reflect more realistic dynamicalbehaviors; (ii) an event-triggered control algorithm is proposedfor the LFC scheme of multi-area power systems; (iii) theconditions for the system stability under the event-triggeredLFC are derived by the free-weighting matrix approach whichis known as the least conservative method.

The rest of the paper is organized as follows. In SectionII, the dynamic model of LFC scheme and the problem tobe investigated are formulated. In Section III, the descriptionof the event-triggered communication scheme is presented.InSection IV, sufficient conditions are derived for the stability ofthe power systems under the LFC scheme with time-varyingdelays, and a controller design method is proposed for theevent-triggered sampled-data output feedback control in theLFC scheme. Case studies are carried out based on three-areatraditional LFC scheme in Section V. Finally, conclusions aredrawn in Section VI.

The notations used through the paper are standard andintroduced here before proceeding.diag· · · denotes a block-diagonal matrix. Matrices, if their dimensions are not explicitly

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stated, are assumed to be compatible for algebraic operations.In is an n dimensional identity matrix.Rn stands for then dimensional Euclidean space.XH denotes the generalizedinverse of X . ||.|| stands the Euclidean norm for vectorsand to the induced 2-norm for matrices.Z+ denotes thenonnegative integers.R+ stands for the positive real numbersandR+

0 = R+⋃0.

II. DYNAMIC MODELS OFLFC

A typical power system model with LFC is shown in Fig.1 [3], which can guarantee no deviation in the ACE andfrequency and result in such that the control areas complywith the North American Electric Reliability Council (NERC)control performance standards, CPS1 and CPS2 [15]. Thereare two types of models: One-area LFC model and multi-areaLFC model, which will be described below.

Fig. 1. Single-line diagram of the two-area four-machine power system.

A. One-Area LFC model

As shown in Fig. 2, the dynamic model of the one-area LFCsystem model can be represented as follows [3], [4]

tsde S

KK- I

p

gsT1

1

chsT1

1

DsM

1

R1

ACEvP mP f

dP

PI Controller

Fig. 2. Dynamic model of one-area LFC scheme.

x(t) = Ax(t) + Bu(t) + F∆Pd,y(t) = Cx(t),

(1)

where

x =[

∆f ∆Pm ∆Pv

]T, y = ACE,

A =

− DM

1M

00 − 1

Tch

1Tch

− 1RTg

0 − 1Tg

, B =[

0 0 1Tg

]T

,

F =[

− 1M

0 0]T

, C =[

β 0 0]

,

where ∆f , ∆Pv, ∆Pm, and ∆Pd denote the deviation offrequency, valve position, generator mechanical output, and

load, respectively.M,Tch, D, Tg, andR are the moments ofinertia of the generator, time constant of the turbine, generatordamping coefficient, time constant of the governor, and speeddrop, respectively.

As there exists no net tie-line power exchange in the one-area LFC scheme, the area control errorACE can be definedas

ACE = β∆f, (2)

whereβ is the frequency bias factor. For technical simplicity,the delays at the transmission of control signals between thecontrol center and the plant, such as sample-induced delays,and fault-induced delays, can be combined as one singledelay in the transmission ofACE and represented by anexponential blocke−sd(t) as shown in Fig. 2. UsingACEas a corresponding control input of load frequency controller,a PI controller is designed as

u(t) = −KPACE −KI

∫

ACE, (3)

whereKP , KI are proportional and integral gains respectivelylLet K = [KP KI ].

To simplify the analysis, the PI control problem is trans-formed into a static output feedback control problem [5].Define the following virtual state and output vectors

x =[

∆f ∆Pm ∆Pv

∫

ACE]T

,

y =[

ACE∫

ACE]T

.

The closed-loop system can be rewritten as

x(t) = Ax(t) +Bu(t) + F∆Pd,y(t) = Cx(t),

(4)

where

A =

− DM

1M

0 00 − 1

Tch

1Tch

0

− 1RTg

0 − 1Tg

0

β 0 0 0

, B =[

0 0 1Tg

0]T

,

F =[

− 1M

0 0 0]T

, C =

[

β 0 0 00 0 0 1

]

,

∫

ACE is the integration of the area control errorACE.Therefore, the practical measurement of outputy(t) is ACE.

B. Multi-Area LFC Model

In each control area of a multi-area LFC scheme, allgeneration units are simplified as an equivalent generationunit.The structure of areap is shown in Fig. 3. The dynamic modelof the multi-area power system is described as above withdifferent parameters [5]

x(t) = Ax(t) +Bu(t) + F∆Pd,y(t) = Cx(t),

(5)

where

xp(t) =[

∆fp ∆Pmp ∆Pvp

∫

ACEp ∆Ptie−p

]T,

yp(t) =[

ACEp

∫

ACEp

]T,

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tsdie S

KK- Ii

pi

gisT1

1

chisT1

1

ii DsM

1

n

ij,j

ijT1

s

2

i iR1

iACEviP miP

if

diP

i-tiePi

n

ij,j

ij fT1

PI Controller

Fig. 3. Dynamic model of theith control area in a multi-area LFC scheme.

x(t) =[

x1(t) x2(t) · · · xn(t)]T

,

y(t) =[

y1(t) y2(t) · · · yn(t)]T

,

u(t) =[

u1(t) u2(t) · · · un(t)]T

,

∆Pd(t) =[

∆Pd1(t) ∆Pd2(t) · · · ∆Pdn(t)]T

,

A =

A11 A12 · · · A1n

A21 A22 · · · A2n

......

. . ....

An1 An2 · · · Ann

,

B = diagB1, B2, · · · , Bn, C = diagC1, C2, · · · , Cn,

F = diagF1, F2, · · · , Fn, Bi =[

0 0 1Tg−i

0 0]T

,

Ci =

[

βi 0 0 0 10 0 0 1 0

]

, Fi =[

− 1Mi

0 0 0 0]T

,

Aii =

−Di

Mi

1Mi

0 0 − 1Mi

0 − 1Tch−i

1Tch−i

0 0

− 1RpTg−i

0 − 1Tg−i

0 0

βi 0 0 0 0

2πn∑

j=1,j 6=i

Tij 0 0 0 0

,

Aij =

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

2πTij 0 0 0 0

, Tij = Tji,

whereTij is the tie-line synchronization coefficient betweenthe ith andjth control areas. TheACE signal for each area isdefined as a linear combination of the tie-line power exchangebetween areas and frequency deviation:

ACEi = βi∆fi +∆Ptie−i, (6)

where∆Ptie−i is the net exchange of tie-line power of thepth control area. Note that theACE signal combines the areafrequency deviation and the net tie-line power exchange as theinput of the PI controller. For the multi-area LFC scheme, thenet tie-line power exchange between the control areas satisfiesthe following equation

n∑

i=1

∆Ptie−i = 0. (7)

Consider a nominal LFC scheme as in [13]

x(t) = Ax(t) +Bu(t) + Fω(t), (8)

wherex(t) ∈ Rn is the state vector,ω(t) ∈ Rm is the systemdisturbance. From (3) and (6), the real-time input control termof the multi-area LFC scheme

u(t) = Ky(t) = KCx(t) (9)

is performed in continuous time [12], [13], which implicitlyimplies that the whole communication network bandwidth isunlimited, where

K = diagK1,K2, · · · ,Kn,Ki =[

KPi KIi

]

.

III. T HE EVENT-TRIGGEREDCOMMUNICATION SCHEME

The measured output of the LFC scheme is sampled beforebeing sent to the controller of the LFC system with data-sampling technique and zero-order hold (ZOH), and presentedas follows

y(t) = Cx(tk), t ∈ [tk, tk+1), (10)

where x(tk) ∈ Rm is the measured output of the LFCscheme (8), andtk represents the sampling instant whichsatisfieslimk→∞ tk =∞. The sampled datax(tk) is directlytransmitted to the event generator which decides whether thecurrent sampled datax(tk+j) needs to be transmitted to thecontroller are based on the following judgement condition:

(

y(tk+j)− y(tk))T

Ω(

y(tk+j)− y(tk))

< yT (tk+j)Ωy(tk+j), (11)

whereΩ ∈ Rm×m is a positive definite matrix,j ∈ 1, 2, · · · , ∈ [0, 1). In other words, the sampled datay(tk+j) will betransmitted when condition (11) is satisfied. In particular, when = 0, it is impossible to satisfy inequality (11) with almostall the sampled datay(tk+j), and the event-triggered controlstrategy reduces to a periodic release method.

Under the event-triggered control (11), it is obvious that thesignals transmitted are the subsequences of the sampled data,and the release instantst0, t1, t2, t3 · · · ⊆ 0, 1, 2, 3, · · · have a relationship with the value of, as well as the systemsampled states.

As there exist time-varying delays over the network com-munication asτk ∈ [0, τ ], k ∈ 1, 2, · · · ,+∞ the sampledstate data sent to the terminal end can be rewritten as

y(t) = Cx(tk), t ∈ [tk + τk, tk+j + τk+j). (12)

Consequently, the control law of the LFC scheme can bepresented as

u(t) = KCx(tk), t ∈ [tk + τk, tk+1 + τk+1). (13)

Substituting the output (13) into condition (11), we have thefollowing inequality

(

x(tk+j)− x(tk))T

CTΩC(

x(tk+j)− x(tk))

< xT (tk+j)CTΩCx(tk+j). (14)

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For convenience, the following two intervals will be con-sidered:

[tk + τk, tk + h+ τ), [tk + lh+ τ, tk + lh+ h+ τ),

where l denotes a positive integer satisfyingl ≥ 1 and hpresents a sampling period.

If tk + h+ τ > tk+1 + τk+1, a functionτ(t) is defined asfollows:

τ(t) = t− tk, t ∈ [tk + τk, tk+1 + τk+1).

Thus

τk ≤ τ(t) ≤ tk+1 − tk + τk+1 ≤ h+ τ.

Meanwhile, if tk + h + τ < tk+1 + τk+1, we have thefollowing inequality

tk +mh+ τ < tk+1 + τk+1 < tk +mh+ h+ τ,

wherem ∈ 1, 2, · · · , l − 1.DefineI1k = [tk + τk, tk +h+ τ), I2k =

∪l−1m=1 [tk +mh+

τ, tk+mh+h+τ)

, I3k = [tk+lh+τ, tk+1+τk+1). Then, the

range[tk + τk, tk+1, τk+1) can be divided into the followingl + 1 sub-ranges

[tk + τk, tk+1, τk+1) = I1k ∪ I2k ∪ I3k .

Therefore, the functionτ(t) can be defined in the range[tk + τk, tk+1 + τk+1) as follows:

τ(t) =

t− tk, t ∈ I1k ;t− tk − lh, t ∈ I2k ;t− tk − lh, t ∈ I3k .

Consequently, we have

0 ≤ τk ≤ τ(t) < h+ τ, t ∈ I1k ;0 ≤ τk ≤ τ ≤ τ(t) < h+ τ, t ∈ I2k ;0 ≤ τk ≤ τ ≤ τ(t) < h+ τ, t ∈ I3k .

As a result,0 ≤ τ(t) < τM , where τM = h + τ . Then,∀t ∈ [tk + τk, tk+1 + τk+1), we can define

xk(t) =

0, t ∈ I1k ;x(tk + lh)− x(tk), t ∈ I2k ;x(tk + lh)− x(tk), t ∈ I3k .

Based on the definitions of functionsτ(t) and xk(t), theevent-triggered control (11) can be rewritten as

xTk (t)C

TΩCxk(t) < xT (t− τ(t))CTΩCx(t − τ(t)). (15)

Through transforming (14) to (15), the event-triggered systemis transferring to the time-delay systems, this will be conve-nient for the theoretical analysis.

Based on the available sampled state measurementy(t), theLFC dynamics can be derived from

x(t) = Ax(t) +BKCx(t− τ(t)) −BKCxk(t) + Fω(t).(16)

The initial condition ofx(t) on [−τ, 0] as

x(t) = φ(t), t ∈ [τ , 0]

whereφ(t) is a continuous function on[−τ , 0].Under the proposed event-triggered scheme, we aim to

design a PI controller (9) such that

• The closed-loop LFC dynamics (16) with time-varyingdelays is asymptotically stable;

• For any nonzeroω(t) ∈ L2[0,+∞) and a givenγ > 0,the RPI performance||y(t)|| < γ||ω(t)|| is guaranteedunder the zero initial state condition.

IV. D ESIGN OF EVENT-TRIGGEREDLFC

In this section, we will design an event-triggered controllerfor LFC scheme (16). First, sufficient conditions are derivedfor the augmented LFC scheme (16) with time-varying delaysto be asymptotically stable. Then, a design method is proposedfor the event-triggered sampled-data output feedback controlof LFC scheme (16) with a novel event-triggering algorithm(11). Correspondingly, two derivations are obtained by thederived results to design such systems with constant timedelays. To reduce the conservatism of the obtained results,The technique of free-weighting matrix is employed to obtainthe delay-dependent asymptotic stability of augmented LFCschemes with event-triggered controller in the case of under-going disturbances and time delays.

The LFC system (16) with even-triggering condition (11)can be asymptotically stable provided certain conditions aresatisfied. Now we derive the conditions. Choose the Lyapunov-Krasovskii functional candidate to be

V (t) = xT (t)Px(t) +

∫ t

t−τM

x(s)Qx(s)ds

+

∫ t

t−τM

∫ t

η

xT (s)Rx(s)dsdη,

+ (τ − τ(t))[

x(t) − x(t− τ(t))]T

W

×[

x(t) − x(t− τ(t))]

+

∫ t

sk

xT(s)Zx(s)ds

(17)

whereP > 0, Q > 0, R > 0,W > 0 andZ > 0 are matriceswith appropriate dimensions to be determined,sk = tk + τk.

Using the Newton-Leibnitz formula,

ζT(t)L[

x(t) − x(t− τ(t)) −∫ t

t−τ(t)

x(s)ds]

= 0, (18)

ζT(t)M[

x(t− τ(t)) − x(t − τ)−∫ t−τ(t)

t−τ

x(s)ds]

= 0,

(19)

ζT(t)N[

x(t) − x(sk)−∫ t

sk

x(s)ds]

= 0, (20)

where

ζ(t) = [xT(t) xT(t− τ(t)) xT(t− τ ) xTk (t) x

T(sk) ωT(t)]T,

and matricesL,M,N are with appropriate dimensions.

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Notice that there exist matricesR andZ, such that

− ζT(t)(

L

∫ t

t−τ(t)

x(s)ds+M

∫ t−τ(t)

t−τ

x(s)ds

+N

∫ t

sk

x(s)ds)

≤ τ(t)

2ζT(t)(LR−1LT +NZNT)ζ(t)

+τ − τ(t)

2ζTMR−1MTζ(t) +

1

2

∫ t

t−τ

xT(s)Rx(s)ds

+1

2

∫ t

sk

xT(s)Zx(s)ds.

Taking the Lyapunov-Krasovskii functional (17) along (16)with x(sk) = 0 and τ(t) = 1, we can get

V (t) ≤ ζT(t)Φζ(t) − yT(t)y(t) + γ2ωT(t)ω(t) (21)

with

Φ = k+ (τ − τ(t))Σ1 + τ(t)Σ2 + τCTRC

+ ATA+∆+∆T,

Σ1 = MR−1MT + CTZC+ 2BWC,

Σ2 = LR−1LT +NZ−1NT,

whereA,B,C and k are defined in Theorem 1. Using theLyapunov-Krasovskii functional (17) and following matrixinequalities

Ψκ =

[

Ξ + (2− κ)Υ ∗Πκ −Π

]

< 0, κ = 1, 2 (22)

where

Ξ = k+∆+∆T + ATA,

k = diagQ, Ω,−Q, 0,−Ω,−γ2I+ PC+ CTP − BWB

T,

∆ =[

L+N M − L −M −N 0 0]

,

Υ = τBWC+ τCTW

TBT,Π1 = colτRC, τZC, τMT,

Π2 = colτRC, τNT, τLT,Π = diagτR, τR, τZ,A = colI, 0, 0,−I, 0, 0,B = colA,BKC, 0, 0,−BKC,F,C = colC, 0, 0, 0, 0, 0,we can derive that system (16) withω(t) = 0 is asymptoticallystable under the zero initial condition. Next, we need toguarantee that system (16) is asymptotically stable with anH∞

norm boundγ, such that||y(t)|| < γ||ω(t)|| for any nonzeroω(t) ∈ L2[0,+∞). Since condition (22) holds, which ensuresΦ < 0 in (21), then

V (t) < −yT(t)y(t) + γ2ωT(t)ω(t). (23)

Integrating both sides of (23) fromt0 to t where t → +∞under the zero initial condition, we have||y(t)|| < γ||ω(t)||,the asymptotic stability of LFC scheme (16) is guaranteed.

Now we can conclude that if there exist matricesP > 0,Q > 0, R > 0, W > 0, andZ > 0, and matricesL,M,Nwith appropriate dimensions, such that LMIs (22) hold, then,for given positive parameters, τ , γ, and feedback gainK, LFC system (16) with event-triggering condition (11) is

asymptotically stable with an RPIγ.DefineX = P−1, XQXT = Q, XRXT = R, XWXT =

W , XZXT = Z, XLXT = L, XMXT = M , XNXT = N ,XΩXT = Ω, andY = KCXT. For givenǫ > 0,

τBXWA+ τATWTXTB

≤ τ2BWX−1ǫX−1WBT + ATǫ−1A. (24)

Then, pre- and post-multiplying both sides ofΨ1 in(22) with diagX,X,X, I,X,X, I, I,X and Ψ2 withdiagX,X,X, I,X,X, I,X,X respectively. Based on (24)and Schur complement, we can get the following LMIs from(22)

Ψκ < 0, κ = 1, 2 (25)

where

Ψ1 =

[

Ψ11 ∗Ψ21 Ψ22

]

, Ψ2 =

[

Ψ11 ∗Ξ21 Ξ22

]

,

with

Ψ11 = k+ ∆ + ∆T ,

k = diagQ, Ω,−Q, 0,−Ω,−γ2I+ C + CT − BWBT,

∆ =[

L+ N M − L −M −N 0 0]

,

Ψ21 = col√τ C,√τ C, A, C, τWB

T,Ψ22 = −diagXR−1X,XZ−1X, R, I, ǫ,Xǫ−1X,Ξ21 = col

√τ C,√τ N ,√τ L, A,

Ξ22 = −diagXR−1X, R, I, Z, I,B = colAX,BY, 0, 0,−BY, F,C = colCX, 0, 0, 0, 0, 0,

the other parameters are defined as above.Now we can conclude that if there exist matricesX > 0,

Q > 0, R > 0, W > 0, andZ > 0, and matricesY, L, M, Nwith appropriate dimensions, such that LMIs (25) hold, then,for given parameters, γ andǫ, LFC scheme (16) with event-triggering condition (11) is asymptotically stable with RPI γand controller feedback gainsK = Y (CXT)H .

For the closed-loop LFC scheme with a preset event-triggered controller with a delay upper bound, the minimal RPIcan be derived by the high-effective binary search approach[12] as in the following algorithm

As described in [12], the obtained results can only providesufficient conditions for the stability of LFC scheme (16)with time delays and cannot be used directly to find thedelay margin. However, the MATLAB/LMI Control Toolboxprovides the functionfeasp to check the existence of the delaymargin based on LMI constraints given in above results. Fora given RPIγ, by manually increasing the value ofτ andchecking the feasibility, the allowable upper delay marginτcan be found.

The implementation of the proposed method is detained asfollows:

S1 Model derivation: Obtain the state-space model of theclosed-loop LFC scheme equipped with a PI controller.All types of turbines, such as reheat turbine, non-reheat

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Algorithm 1 Calculateγmin

Initialization:Setγa → 0; ε→ 10−5;γb large enough to satisfy the LMI condition (25)

Iteration:1: while |γa − γb| > ε do2: γt ← 0.5 ∗ (γa + γb);3: if the LMI condition (25) holds withγt then4: γa → 0.5 ∗ (γa + γb)5: else6: γb → 0.5 ∗ (γa + γb)7: end if8: end while

Setγmin → γa.

turbine, and hydro turbine, can be modeled as shown inSection II.

S2 Controller design: Based on the obtained model, designthe controller gains for a given delay upper bound using(13) with the parameters in LMIs (25) and using thefeasp andmincx solvers in MATLAB Robust ControlToolbox. [32].

S3 Robust performance analysis: For given different timedelays, calculate the RPI of the closed-loop system.

Remark 1. Notice that we can only obtain sufficient con-ditions which are conservative. Therefore, the value of thecalculated RPI is larger than its true value, and the controllerderived can tolerate a delay larger than the preselect upperbound. Meanwhile, the obtained results are derived from thefree-weighting matrix approach which is known as the leastconservative method [33], [34] as utilizing in this paper.

Remark 2. For a preset triggered parameter, the corre-sponding allowable delay upper boundτ , controller gainK,and triggered matrixΩ can be obtained from the derivedresults. This means that the chosen triggered parametersdetermine the performance of the controlled LFC system.Meanwhile, if the RPI parameterγ is guaranteed with presettriggered parameter and sampling periodh, the larger means that fewer numbers of packets need to be transmittedwith less delay upper boundτ .

V. CASE STUDIES

For simulation, a three-area power system is depicted asin Fig. 4, in which, each area represents the electric powersystem with the LFC scheme, and each blue line representsthe communication channel. Meanwhile, The LFC will be sus-pended or stopped when the counted steps of the fault counterreach predefined upper bound. In general, this upper boundis selected by the operational experience with a conservativesmall value [3]. As the time delay will degrade the dynamicperformance and cause instability of augmented power system.Instability of the LFC scheme denotes the deviation far awayfrom zero which exists in the ACE and frequency. To analyzethe stability of multi-area LFC schemes with time delays, nominal available parameters defined in equation (5) are

Area 1

Area 2 Area 3

Fig. 4. Schematic diagram for a three-area interconnected power system.

0 1 2 3 4 5 6−15

−10

−5

0

5

10

15

Times(s)

Sys

tem

sta

tes

(a)

0 1 2 3 4 5 6−15

−10

−5

0

5

10

15

Times(s)

Sys

tem

sta

tes

(b)

Fig. 5. (a) State responses of LFC scheme with = 0.1; (b) State responsesof LFC scheme with = 0.6.

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Times(s)

even

t int

erva

ls

(a)

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Times(s)

even

t int

erva

ls

(b)

Fig. 6. (a) The release instants and release intervals with = 0.1; (b) Therelease instants and release intervals with = 0.6.

tabulated in [3], [10] as follows.1) Area-1

Tch−1 = 0.3s, Tg1 = 0.1s,R1 = 0.05, D1 = 1,M1 = 10,

β1 =2

R1+D1.

2) Area-2 (equivalent unit of four generators)

Tch−2 = 0.17s, Tg2 = 0.4s,R2 = 0.05, D2 = 1.5,M2 = 12,

β2 =4

R2+D2.

3) Area-3 (equivalent unit of three generators)

Tch−3 = 0.20s, Tg3 = 0.35s,R3 = 0.05, D3 = 1.8,M3 = 12,

β3 =3

R3+D3,

and T12 = 0.2p.u./rad, T23 = 0.12p.u./rad and T31 =0.25p.u./rad.

Based on the method in LMIs (25), we will calculate delaymargins of the three-area LFC scheme under event-triggeringto achieve the main goals of LFC to make sure that the

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 7

zero steady states of the errors for frequency deviations areachieved. To simplify the analysis, all generators are assumedto be equipped with a non-reheat turbine. Other types ofturbines, such as reheat turbine and hydro turbine, can bemodeled in a similar way and are not presented.

For fixed KP = 0.2,KI = 0.3, γ = 0.1, more detailedcalculation results for different values of are given in Table I.The larger the triggered parameter, the smaller the allowabletime-delay upper boundτ . This phenomenon is derived fromfact that the larger the triggered parameter in the multi-area LFC scheme, the longer the sampled-data stay whicheventually leads to performance deterioration of the LFCscheme and lowers the allowable time delay upper boundτ .

Table II shows the relationship between the triggered pa-rameter and the maximal allowable sampling periodhmax,andτ = γ = 0.4. For giving sampling periodh = 0.01, TableIII shows the relationship between the triggered parameter,triggered times, average release periods and the percentage ofdata transmissions. It is obvious the average release periodand percentage of the data transmission have an inverse pro-portional relationship, which is reasonable for the LFC schemebecause the larger the triggered parameter in the multi-areaLFC scheme, the longer the sampled-data stay. Therefore, theaverage release period becomes larger if the triggered timebecomes smaller; the average release period becomes largerifthe percentage of the data transmission becomes smaller.

Furthermore, since the sensor with event-triggered controltransmits only48.6%( = 0.1), 37.2%( = 0.2), 18.9%( =0.5) of sampled data produced by time-triggering scheme = 0, it is obvious to reduce the resource utilization via event-triggered control by51.4%, 62.8%, 81.1%, respectively. Bythe event-triggered control, the communication cost reductioncan be achieved through the increasing of the triggered pa-rameter . However, too large may make the LMI condition(25) infeasible. In other words, it is impossible to guaranteethe desired control performance of the controlled system withtoo large triggered parameter. To illustrate the improvedperformances, Figs. 5 and 6 are presented to show the dataderived by different event-triggered parameters.

0 0.01 0.02 0.1 0.2 0.3 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ,hm

ax

τhmax

Fig. 7. The upper bounds of time delayτ and sampling periodh for different.

The simulation results are shown in Figs. 7 and 8, whichillustrate that the proposed event-triggered method can greatlyreduce the communication burden by increasing the value oftriggered parameter which guarantees the stability of the

0 0.1 0.2 0.60

100

200

300

400

500

600

700

800

900

1000

1100

ct

(a)

0 0.1 0.2 0.60

0.01

0.02

0.03

0.04

0.05

0.06

t r

(b)

0 0.1 0.2 0.60

10

20

30

40

50

60

70

80

90

100

110

(%

)

(c)

Fig. 8. For fixedh = 0.01, t = 10, τ = 0.1, triggered timesct, averageperiodstr and data transmission rates(%) for different .

power system under the augmented LFC scheme.

Remark 3. When faults occur in different coupling controlareas of the multi-area power system, the power system canbe modelled as a system without full interconnection as in Fig.9. In this case, the obtained results can also be utilized witha quite different matrixA defined in(5).

Area 1

Area 2 Area 3

(a)

Area 1

Area 2 Area 3

(b)

Area 1

Area 2 Area 3

(c)

Fig. 9. Schematic diagrams for three-area power system without fullinterconnection.

Remark 4. Event-triggered control for the LFC scheme withtime delays is based on the Lebesgue sampling [35], [36]. Thismakes the event-triggered control challenging to develop dueto the non-periodic, non-Gaussian and nonlinear propertiesof the Lebesgue sampling. Meanwhile, the Lebesgue samplingis an event function and the occurrence of the events maybe stochastic in nature. Therefore, event-triggered control inthe LFC scheme should be treated as a nonlinear system withsome hybrid nature in analysis.

Remark 5. A delay system approach was firstly proposed forthe sampling issue by Gao, Chen and Lam [37]. This providesthe hint to deal with the event-triggering problems. Meanwhile,a novel two-layer structure was proposed to solve the setpoints compensation problem for industrial processes undernetwork-based environment [16]. The proposed method hasbeen successfully applied to a rougher flotation process, thisprovides possibility to extend the obtained result in this paperfor network-based performance tracking control of complexindustrial processes through transferring the control problemof time-delay systems into a static feedback control (SOFC)problem [38], [39]. Meanwhile, free-weighting matrix method,as a way to reduce the conservativeness of the obtained fortime-delay systems, has been utilized widely such as filterdesign [40]. Recently, there have been some novel methodsto deal with time-varying delay such as the input-outputapproach [41], or combining multiobjective optimization withdifferential evolution [42], [43], [44], these methods will be

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 8

TABLE ITHE UPPER BOUNDS OFτ DERIVED, FOR DIFFERENT .

0 0.01 0.02 0.1 0.2 0.3 0.4 0.6τ (second) 0.9146 0.8617 0.7329 0.6552 0.5648 0.4141 0.3876 0.3267

TABLE IIFOR FIXED τ = 0.4, THE UPPER BOUNDS OF SAMPLING PERIODh DERIVED, FOR DIFFERENT .

0 0.01 0.02 0.1 0.2 0.3 0.4 0.6hmax(second)0.4535 0.3724 0.3511 0.3167 0.2598 0.2269 0.1820 0.1647

TABLE IIIFOR FIXEDh = 0.01, t = 10, AND τ = 0.1, TRIGGER TIMESct, AVERAGE RELEASE PERIODStr , AND DATA TRANSMISSION RATES, FOR DIFFERENT .

0 0.1 0.2 0.6ct(time) 1000 486 372 189tr(second) 0.0100 0.0206 0.0269 0.0529(percent) 100% 48.6% 37.2% 18.9%

benefit for the research of event-triggering load frequencycon-trol of multi-area power systems with communication delays.

VI. CONCLUSIONS

In this paper, an event-triggered approach has been pro-posed to update the PI control to stabilize the power systemwith communication delay under the LFC scheme. It hasbeen shown that the controller is updated only at its eventtime, which reduces the communication burdens and lowersthe control updating frequency. Simulation results have beenshown to validate the control design. In the future research,several challenging tasks, such as limitations on computationalprocessing power, cost per actuation, sensor quantization, costof communication signals and so on, should be addressed inevent-triggered control of the LFC scheme.

Acknowledgement

This work was supported by the Natural Science Foundationof China under Grants 61125303, 61403152 and 61402218,National Basic Research Program of China (973 Program) un-der Grant 2011CB710606, the Program for Science and Tech-nology in Wuhan of China under Grant 2014010101010004,the Program for Changjiang Scholars and Innovative ResearchTeam in University of China under Grant IRT1245, theAustralian Research Council (ARC) Discovery Scheme underGrant No. 140100544.

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Shiping Wen received the M.S degreein Control Science and Engineering, from AutomationSchool, Wuhan University of Technology, Wuhan, China, in2010, and received the Ph.D degree in Control Science andEngineering, from Automation School, Huazhong Universityof Science and Technology, Wuhan, China, in 2013. Heis currently working at School of Automation, HuazhongUniversity of Science and Technology, and also in the KeyLaboratory of Image Processing and Intelligent Control ofEducation Ministry of China, Wuhan, China. His currentresearch interests include design and analysis of memristor-based circuits and systems,H∞ control of networked systems.

Xinghuo Yu Xinghuo Yu (M’92-SM’98-F’08) received BEng and MEng degrees from the Univer-sity of Science and Technology of China, Hefei, China, in1982 and 1984, and PhD degree from Southeast University,Nanjing, China in 1988, respectively. He is currently withRMIT University (Royal Melbourne Institute of Technology),Melbourne, Australia, where he is the Founding Director ofRMIT Platform Technologies Research Institute. ProfessorYu’s research interests include variable structure and nonlinearcontrol, complex and intelligent systems and applications. Heis an IEEE Distinguished Lecturer and Vice-President (Publi-cations) of IEEE Industrial Electronics Society. Professor Yureceived a number of awards and honors for his contribu-tions, including 2013 Dr.-Ing. Eugene Mittlemann Achieve-ment Award of IEEE Industrial Electronics Society and 2012IEEE Industrial Electronics Magazine Best Paper Award. Healso holds Fellowship of the Institution of Engineering andTechnology (UK), International Energy Foundation, EngineersAustralia, Australian Computer Society, and Australian Insti-tute of Company Directors.

Zhigang Zeng received his B.S. degree fromHubei Normal University, Huangshi, China, and his M.S.degree from Hubei University, Wuhan, China, in 1993 and1996, respectively, and his Ph.D. degree from HuazhongUniversity of Science and Technology, Wuhan, China, in 2003.

He is a professor in School of Automation, HuazhongUniversity of Science and Technology, Wuhan, China,and also in the Key Laboratory of Image Processingand Intelligent Control of Education Ministry of China,

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 10

Wuhan, China. His current research interests include neuralnetworks, switched systems, computational intelligence,stability analysis of dynamic systems, pattern recognition andassociative memories.

Jinjian Wang received his B.S. degree inelectrical engineering from RMIT University, Melbourne, VIC,Australia, in 2012. He is currently working toward the Ph.D.degree in electrical engineering, RMIT University.

His research interests include the security and reliability ofpower network and complex networks.