Stability of Wide-Area Power System Controls With ...

3494 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 34, NO. 5, SEPTEMBER 2019

Stability of Wide-Area Power System Controls WithIntermittent Information Transmission

Fatima Zohra Taousser , Member, IEEE, M. Ehsan Raoufat, Member, IEEE, Kevin Tomsovic, Fellow, IEEE,and Seddik M. Djouadi, Fellow, IEEE

Abstract—This paper investigates the stability problem of wide-area damping controllers with intermittent information transmis-sion. Due to the interruption in communication links betweenremote measurements and damping controller or from the dampingcontroller to the damping actuators, the closed-loop system mightbecome unstable. The instability is strongly related to the durationof interruption of information transmission. To estimate instabil-ity, this paper formulates the problem as continuous/discrete-timeswitched system and the stability conditions are derived using time-scale theory. This method allows us to handle continuous and dis-crete dynamics as two pieces of the same framework, such that thesystem will switch between a continuous-time subsystem (when thecommunication occurs without any interruption) and a discrete-time subsystem (when the communication fails). The contributionis to estimate the maximum allowable value of the time of interrup-tion of information transmission that does not violate the exponen-tial stability of the closed-loop system. The findings are useful inspecifying the minimum requirements for communication infras-tructure and the time to activate remedial action schemes. Simula-tions are performed based on both linear and nonlinear systems tovalidate the theoretical development.

Index Terms—Low-frequency oscillations, intermittent informa-tion, time scale theory, switched systems.

I. INTRODUCTION

W ITH increasing interconnection complexity, moderngrids are more vulnerable to system-wide disturbances.

These wide-area disturbances require more sophisticated mea-surement systems and coordinated control actions to avoid sys-tem collapse as local responses (delivered based on the localobservations) are not sufficient. This will bring new challengesas coping with instability problems requires wide-area measure-ment and control systems (WAMCS) [1]. Phasor measurementunits (PMUs) can play a crucial role in these applications byproviding the necessary measurement infrastructure.

Manuscript received June 7, 2018; revised December 12, 2018; acceptedMarch 16, 2019. Date of publication April 9, 2019; date of current versionAugust 22, 2019. This work was supported in part by the National ScienceFoundation under Grant CNS-1239366, and in part by the Engineering Re-search Center Program of the National Science Foundation, the Department ofEnergy under NSF Award EEC-1041877, and in part by the CURENT IndustryPartnership Program. Paper no. TPWRS-00881-2018. (Corresponding author:Fatima Zohra Taousser.)

The authors are with the Min H. Kao Department of Electrical Engineeringand Computer Science, The University of Tennessee, Knoxville, TN 37996 USA(e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2019.2908922

Fig. 1. A networked control system with components that are remotelyoperated over a communication network.

Traditionally, PMU data has been used only for off-line postevent analysis. However, with recent advancement in commu-nications (e.g. faster communication channels) and processingpower, it is now possible to use geographically dispersed PMUsfor real-time applications in power systems [2]. PMUs are cur-rently installed in different point in the North American grid,to record and communicate GPS-synchronized, hight samplingrate (60 sample/sec), dynamic power systems data. They canbe used to address the problem of inter-area oscillations whichhappens between several areas and require wide-area supervi-sion and control schemes. In these applications, usually dampingcontrollers (located in control centers, substations), sensors (e.g.PMUs) and actuators (e.g. synchronous generators, FACTS de-vices and energy storage systems) are located remotely and canonly communicate with others over a communication networkas shown in Fig. 1.

Implementation of damping controllers over a network suchthat, the portions of the control system located remotely, mightcreate challenges as the closed loop performance is highly de-pendent on the communication network. In this study, we aim toconsider the effect of the communication network in our analy-sis. The main question is what happens when the communicationis lost during some periods. The network may experience con-stant or time varying delays [3], packet dropout [4] or packetdisordering [5]. Hence, the communication network introducesuncertainty in the operation and the performance of the closed-loop system.

In the power systems literature, communication effects areoften ignored [6], [7]. Reference [8] studied the impact of in-duced network delays using LMIs but only for state feedbackcontrollers. In [9], [10], simple models were considered to cap-ture the effects of communication failure with known lower andupper bounds. The problem of network control system with datapacket dropout and transmission delays, was studied in several

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TAOUSSER et al.: STABILITY OF WIDE-AREA POWER SYSTEM CONTROLS WITH INTERMITTENT INFORMATION TRANSMISSION 3495

ways in the literature [11]–[13]. In fact, existing methods basedon sampling signal output such that, the samples only arrive atthe destination after a (possibly variable) delay which is assumedalways smaller than one sampling interval, but the delays longerthan one sampling interval may result in more than one signalarriving. Other approaches consider the problem as a differentialequations with delay (where the restriction to assuming delayssmaller than one sampling interval is lifted). The Lyapunov–Krasovskii and the Razumikhin theorems are the two main toolsavailable to study the stability of such systems and some LMI-based conditions are derived [14], [15]. However, the requiredcommunication time rate conditions are rather complex to verify.

Motivated by that, in this paper, new stability conditions arederived using time scale theory. Dynamical systems modeledusing time scales theory shows promise as a new approach tosolve this problem. Based on this theory, it will be shown thatthe problem of communication loss can be converted into theasymptotic stabilization problem of a switched system on a par-ticular non-uniform time domain, formed by a union of disjointintervals with variable lengths and variable gaps [16], [17]. In-deed, the closed loop system evolves during some continuoustime intervals when the communication occurs without any inter-mittence in information transmission. When the communicationfails, the control will not evolve, holds its last value and it willbe updated after some periods (considered to be variable). Inthis case, the system acts as if discretized with a variable stepsize and the system will be modeled as a switched system be-tween a continuous-time dynamics with variable intervals lengthand a discrete-time dynamics with variable step size. Thus, it isof interest to mix the continuous-time and discrete-time casesunder a unified framework [18]–[20]. In this paper, new con-ditions are derived to estimate the allowable value of the timeof interruption in information transmission in order to maintainthe exponential stability of the closed loop power system. Thefindings are useful in specifying the minimum requirements forcommunication infrastructure and the time to activate remedialaction schemes to avoid the critical situation [21]. Moreover,we have explored realistic cases involving sensor-to-controllerand/or controller-to-actuator communication failures.

The remainder of this paper is organized as follows. Back-ground on time scale theory is presented in Section II. InSection III, it is shown that the stability problem of linear sys-tem with intermittent information transmission is equivalent tothe stabilization of a switched system consisting of a linearcontinuous-time and discrete-time subsystem. A set of condi-tions on the maximum time of interruption to guarantee the ex-ponential stability of the closed-loop power system is derived inSection III. Numerical results and conclusions are presented inSection IV and Section V, respectively.

II. PRELIMINARIES ON TIME SCALE THEORY

Basic notations and properties of time scales theory [18] arepresented in this section. A time scale, noted T is an arbitrarynonempty closed subset of R. The usual integer sets hZ, N, thereal numbers R, any discrete subset or any combination of dis-crete points with union of closed intervals, are examples of timescales. The forward jump operator σ(t) : T → T is defined by

σ(t) := inf{s ∈ T : s > t}. The mapping μ : T → R+, calledthe graininess function, is defined by μ(t) = σ(t)− t, whichmeasure the distance between two consecutive points. In par-ticular, if T = R, σ(t) = t and μ(t) = 0. If T = hZ, σ(t) =h and μ(t) = h. For T = ∪∞

k=0[k(a+ b), k(a+ b) + a], witha, b ∈ R,

σ(t) =

{t, t ∈ ∪∞

k=0[k(a+ b), k(a+ b) + a[

t+ b, t ∈ ∪∞k=0{k(a+ b) + a}

μ(t) =

{0, t ∈ ∪∞

k=0[k(a+ b), k(a+ b) + a[

b, t ∈ ∪∞k=0{k(a+ b) + a}

Let f : T → R. The Δ-derivative of f at t ∈ T is defined as

fΔ(t) = lims→t

f(σ(t))− f(s)

σ(t)− s(1)

The Δ-derivative, unify the derivative in the continuous senseand the difference operator in the discrete sense. If T = R,σ(t) = t and fΔ(t) = f(t). If T = hZ, σ(t) = t+ h andfΔ(t) = f(t+h)−f(t)

h . In particular, if h = 1, fΔ(t) = f(t+1)− f(t) = Δf(t), the difference operator. Note that the Δ-derivative, generalizes the continuous and discrete derivatives.A function f : T → R is regressive if 1 + μ(t) f(t) �= 0, ∀t ∈T . A matrix A is called regressive, if ∀t ∈ T , the matrix(I + μ(t)A) is invertible, where I is the identity matrix (equiv-alently, (1 + μ(t)λi) �= 0, ∀t ∈ T , for all eigenvalues λi of A[18]). We denote the set of all regressive functions by R andby R+, if they satisfies 1 + μ(t)f(t) > 0, ∀t ∈ T (positivelyregressive). The generalized exponential function of p ∈ R isexpressed by

ep(t, t0) =

⎧⎨⎩ e

∫ tt0

log(1+μ(τ)p(τ))μ(τ)

Δτ , if μ(τ) �= 0

e∫ tt0

p(τ)Δτ , if μ(τ) = 0(2)

where s, t ∈ T , log is the principal logarithm function and thedelta integral is used [18], [22]. Let p ∈ R and t0 ∈ T , forT = R, ep(t, s) = exp(

∫ t

s p(τ) dτ) and for T = hZ, ep(t, s) =∏t−hτ=s(1 + hp(τ)). Notice that the regressivity of p is needed for

the exponential function to be well defined, in particular on dis-crete time scales.

Let A be a regressive matrix. The unique matrix-valuedsolution of

xΔ(t) = A x(t), x(t0) = x0, t ∈ T , (3)

is the generalized exponential function denoted by eA(t, t0)x0.The dynamical system (3) is exponentially stable on an arbi-

trary time scale T , if there exists a constant β ≥ 1 and a constantλ < 0 and λ ∈ R+, such that the corresponding solution satis-fies ‖x(t)‖ ≤ β‖x0‖eλ(t, t0), ∀t ∈ T .

This characterization is a generalization of the definition ofexponential stability for dynamical systems defined on R orhZ. More specifically, the condition that λ < 0 and λ ∈ R+ inthe characterization of exponential stability is reduced to λ < 0for T = R, and to 0 < 1 + μ(t)λ < 1, ∀t ∈ T (an arbitrarydiscrete time scale). Since, the generalized exponential func-tion can be negative, the positive regressivity of λ is needed(see [18]).



Fig. 2. Sample of control signal with intermittent communication transmis-sions when assigning a value a) equal to zero (zero strategy) and b) equal to theprevious value (hold strategy).

III. WIDE AREA CONTROL WITH INTERMITTENT

INFORMATION TRANSMISSION

Damping controllers relying on a communication networkwhere portions of the control system are located remotely, cre-ates challenges. The closed loop performance is highly depen-dent on the communication network. In this study, we considerthe effects of the communication network in the stability analy-sis. To begin, nonlinear power system models can be expressedas the following differential algebraic equations

x(t) = f(x(t), y(t), u(t)

)(4)

0 = g(x(t), y(t)

)(5)

wherex is the state vector, y is a vector of algebraic variables,u isthe vector of control inputs and t is the time variable. Linearizingthe power system model (4) around the operating point leadsto the following generalized form

x(t) = Ax(t) +Bu(t) (6)

y(t) = Cx(t) (7)

where A and B are constant real matrices with appropriate di-mensions such that (A,B) is stabilizable and u ∈ Rm is thecontrol input. The aim in this section is to estimate the timeof interruption of information transmission and analyze whathappens when the communication network is no longer perfectdue to packet loss, delay or any other common communicationfailure.

Two general schemes are generally used when faced withintermittent communication: the zero strategy, in which the in-put/measurement of the plant is set to zero if a packet is dropped,and the hold strategy, in which the latest arrived/measured packetis kept constant until the next packet arrived/measured [23] (seeFig. 2,). In this paper, the hold strategy is used in control andmeasurement loops in which the last value of the control beforecommunication failure is hold and continues to be used whenpacket dropouts happen. However, if the failure time becomeslarge, and since the control will not evolve, the system may be-come unstable. Hence, communication network reliability is akey requirement and the goal is to estimate the maximum timeof interruption of communication that does note violate the sta-bility of the system.

Fig. 3. Time scale T = P{σ(ti),ti+1}.

A. State Feedback Problem Formulation

Consider the particular time scale T = ∪∞i=0[σ(ti), ti+1],

where σ(.) is the forward jump operator, such that, σ(t0) = t0and the graininess functionμ(ti) = σ(ti)− ti, ∀i ∈ N∗ (Fig. 3).To solve the power systems problem under intermittent infor-mation transmission between generators and controllers, thefollowing switched control law is applied

u(t) =

{Kx(t), if t ∈ ∪∞

i=0[σ(ti), ti+1)

Kx(ti+1), if t ∈ ∪∞i=0[ti+1, σ(ti+1))

(8)

where K is an appropriate state feedback controller. The unionof time intervals over which the communication occurs is repre-sented by ∪∞

i=0[σ(ti), ti+1). The remaining intervals representthe time intervals over which the feedback does not evolve (i.e.,maintained constant to its value at the switching times instantsti+1) due to the absence of local information. The time sequence{t1, t2, t3, . . .} characterizes the time when the communicationfailure occurs with no accumulation points. The duration of acommunication failure equal to μ(ti) which is assumed to bevariable and bounded, ∀i ∈ N∗. With the control law (8), thedynamical system (6) is equivalent to

x(t) =

{(A+BK)x(t), if t ∈ ∪∞

i=0[σ(ti), ti+1)

Ax(t) +BKx(ti+1), if t ∈ ∪∞i=0[ti+1, σ(ti+1)).

(9)Since the feedback does not evolve when local information is notavailable, the study of system (9) is not trivial. There exist previ-ous works dealing with the stabilization of linear systems undervariable sampling periods or by considering a differential equa-tions with delay. The approaches are usually based on LMIs andderived using Lyapunov-Razumikhin stability conditions, whichare rather complex to verify [24], [14], [15]. To reduce the con-servatism and facilitate the analysis, the problem (9) is convertedto a switched system on time scale T = ∪∞

i=0[σ(ti), ti+1] suchthat, the communication fails at ti+1 and only the behavior of thesolution of the second equation in (9) at the discrete times {ti+1}and {σ(ti+1)} is considered. The descritization is as follows (formore details see [25]):

For t ∈ [ti+1, σ(ti+1)), i ∈ N, we have

x = Ax(t) +Bu(ti+1) (10)

such that u(ti+1) = Kx(ti+1) is constant on the time interval[ti+1, σ(ti+1)). The solution of (10) is given by

x(t)=eA(t−ti+1)[x(ti+1)+A−1Bkx(ti+1)

]−A−1BKx(ti+1)

= eA(t−ti+1)[I +A−1BK

]x(ti+1)−A−1BKx(ti+1)



At time t = ti+1, the Δ-derivative of x(t) is given by

xΔ(ti+1) =x(σ(ti+1))− x(ti+1)

σ(ti+1)− ti+1

=

(eAμ(ti) − I

μ(ti)

)[I +A−1BK

]x(ti+1).

By using the above development, the closed-loop system (9) ismodelled as the following switched linear system

xΔ(t) =

⎧⎨⎩

(A+BK)x(t), t ∈ ∪∞i=0[σ(ti), ti+1)(

eAμ(t)−Iμ(t)

) (I+A−1BK

)x(t), t ∈ ∪∞

i=0{ti+1}(11)

on T = ∪∞i=0[σ(ti), ti+1]. The derived system commutes be-

tween a stable continuous-time linear subsystem (on contin-uous intervals with variable length) and may be an unstablediscrete-time linear subsystem with variable discrete-step sizeμ(t), which corresponds to the interruption time of the controlevolution. Note that, the stability and instability of the discrete-time subsystem is strongly related to μ(t) [17], [25], [26]. Itis known that switching between stable and unstable (or evenbetween stable) systems may make the overall system unstableif we will not put some restriction on the dwell time of eachsubsystem [27], [28].

B. Stability Criteria

In this section, sufficient conditions are derived to guaranteethe stability of system (11).

Proposition 1: Consider the switched system (11), and sup-pose that the following assumptions are fulfilled:

i) (A,B) is stabilizable and the matrix control law K isdetermined such that (A+BK) is stable.

ii) Suppose that μ(t) is bounded and the discrete subsystemis regressive and can be stable or unstable.

iii) Let τ(ti) = ti+1 − σ(ti) be the duration of eachcontinuous-time subsystem, such that ∀i ∈ N, we have∥∥∥e(A+BK)τ(ti)

[I+

(eAμ(ti)−I

)(I +A−1BK)

]∥∥∥ <1.

(12)Then the switched system (11) is exponentially stable.Proof: For the proof see Appendix B �Remark 1: Notice that, if A is not invertible, we can always

determine the discrete matrix via the convergence power series

E(Aμ(t)) =∞∑

n=1

(Aμ(t))n−1

n!, (13)

and the matrix of the discrete subsystem in (11) is equal to

E(Aμ(t))(A+BK).

Condition (12) will be: ∀i ∈ N,∥∥∥e(A+BK)τ(ti) [I + μ(ti)E(Aμ(ti))(A+BK)]∥∥∥ < 1. (14)

C. Extension to Dynamic Output-feedback

In practical applications for power systems, the full state vec-tor is not available. Consequently, it is desirable to adopt the dy-namic output-feedback controller to directly use the measured

Fig. 4. Failure of communication in the output-controller.

output signals for damping the oscillations. This type of con-troller can be defined as

xc(t) = Akxc(t) +Bky(t) (15)

u(t) = Ckxc(t) +Dky(t) (16)

where xc ∈ Rn is the controller states, Ak, Bk, Ck, Dk are anappropriate matrices to be designed, u and y are the controllerand system outputs, respectively. This controller yields with (6)and (7) the following system.

˙x(t) =

[A 0

BkC Ak

]x(t) +

[B

0

]u(t) (17)

u(t) =[DkC Ck

]x(t) (18)

where xT = [xTxTc ] is the augmented system state vector, and

the closed loop matrix is

Acl =

[A+BDkC BCk

BkC Ak

].

Consider the case where a communication failure happens inthe control signal (as is shown in Fig. 4). The system can berewritten on the time scale T = ∪∞

i=0[σ(ti), ti+1] as follows

˙x(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

[A+BDkC BCk

BkC Ak

]x(t), t ∈ ∪∞

i=0[σ(ti), ti+1)

[A 0

BkC Ak

]x(t) +

[B

0

]u(ti+1),

t ∈ ∪∞i=0[ti+1, σ(ti+1)),

with u(ti+1) =[DkC Ck

]x(ti+1) is maintained constant on

[ti+1, σ(ti+1)]. So we get

˙x(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

[A+BDkC BCk

BkC Ak

]x(t), t ∈ ∪∞

i=0[σ(ti), ti+1)[A 0

BkC Ak

]x(t) +

[B0

][DkC Ck

]x(ti+1),

t ∈ ∪∞i=0[ti+1, σ(ti+1)).

(19)



Fig. 5. Failure of communication in the output measurement.

Similarly to the above analysis, the system can be rewritten asfollows:

xΔ(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[A+BDkC BCk

BkC Ak

]x(t), t ∈ ∪∞

i=0[σ(ti), ti+1)

⎛⎜⎜⎝e

⎡

⎣A 0

BkC Ak

⎤

⎦μ(t)

− I

⎞⎟⎟⎠

μ(t)×⎡

⎣I +[

A 0

BkC Ak

]−1[BDkC BCk

0 0

]⎤⎦ x(t),

if t ∈ ∪∞i=0{ti+1}

(20)The stability criteria (21), shown at the bottom of this page,can be formulated for the augmented system with output-feedback controller and communication failures in the controlsignal.

Similarly, for the case where a communication failure hap-pens in the measurement signal (Fig. 5), the output feedbackcontroller (15) and (16) with (6) and (7) yields the followingsystem:

˙x(t) =

[A BCk

0 Ak

]x(t) +

[BDk

Bk

]y(t) (22)

y(t) =[C 0

]x(t) (23)

such that y(ti+1) = [C 0 ] x(ti+1) is constant on [ti+1,σ(ti+1)) when communication fails at ti+1. The switched

system on the time scale T = ∪∞i=0[σ(ti), ti+1] will be

xΔ(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[A+BDkC BCk

BkC Ak

]x(t),

t ∈ ∪∞i=0[σ(ti), ti+1)⎛

⎜⎜⎝e

⎡

⎣A BCk

0 Ak

⎤

⎦μ(t)

− I

⎞⎟⎟⎠

μ(t)×[

I +

[A BCk

0 Ak

]−1 [BDkC 0

BkC 0

]]x(t),

t ∈ ∪∞i=0{ti+1}

(24)

The stability criteria (25), shown at the bottom of this page, isdeduced for the augmented system with output-feedback con-troller and communication failure in the measurement signal.

Remark 2: Note that, the matrix[

A 0BkC Ak

]is invertible if

both A and Ak are invertible. If not, we can always use theconvergence power series as in (13).

IV. APPLICATION TO POWER SYSTEMS

In this section, stability conditions provided above, will be ap-plied to the Single-Machine Infinite Bus (SMIB) and Kundur’stwo-area power systems. Both systems are modified to have un-damped inter-area modes and the accuracy of the stability con-ditions will be verified. In case of two-area system, the dynamicoutput feedback controller has been designed based on a re-duced model and the results has been tested for a large system.In practice, use of the reduced model avoid feasibility problemand realize practical lower-order controller. The reduced modelis based on the balanced model truncation method, which re-tains the most important states variables for control purposes.The appropriate order of the reduced model can be determinedby comparing the accuracy of frequency response of the full or-der and the reduced order system which has a closer response tothe full-order system.

A. Case Study I: SMIB Power System

In this subsection, a SMIB power system model is consid-ered. As shown in Fig. 6, this system consists of a synchronous

∥∥∥∥∥∥∥∥e

⎡

⎣A+BDkC BCk

BkC Ak

⎤

⎦τ(ti)

⎡⎢⎢⎣I +

⎛⎜⎜⎝e

⎡

⎣A 0

BkC Ak

⎤

⎦μ(ti)

− I

⎞⎟⎟⎠(I +

[A 0

BkC Ak

]−1[BDkC BCk

0 0

])⎤⎥⎥⎦∥∥∥∥∥∥∥∥< 1, ∀i ∈ N (21)

∥∥∥∥∥∥∥∥e

⎡

⎣A+BDkC BCk

BkC Ak

⎤

⎦τ(ti)

⎡⎢⎢⎣I +

⎛⎜⎜⎝e

⎡

⎣A BCk

0 Ak

⎤

⎦μ(ti)

− I

⎞⎟⎟⎠(I +

[Ak 0

BCk A

]−1[BDkC 0

BkC 0

])⎤⎥⎥⎦∥∥∥∥∥∥∥∥< 1, ∀i ∈ N (25)



Fig. 6. A single-machine infinite-bus (SMIB) power system.

generator connected through two transmission lines to an infi-nite bus that represents an approximation of a large system. Aflux-decay model of the synchronous generator equipped witha fast excitation system can be represented by the following setof dynamic equations

δ = ωs(ωr − 1) (26)

ωr =1

2H

[TM−(E ′

qIq + (Xq −X ′d)IdIq+Dωs(ωr − 1)

)](27)

E ′q = − 1

T ′d0

[E ′

q + (Xd −X ′d)Id − Efd

](28)

Efd = −Efd

TA+

KA

TA

[Vref − Vt + sat(Vs)

](29)

while satisfying the following algebraic equations

ReIq +XeId − Vq + V∞ cos (δ) = 0 (30)

ReId −XeIq − Vd + V∞ sin (δ) = 0 (31)

Vt =√

V 2d + V 2

q (32)

where Re and Xe = Xt +12Xl are the total external resistance

and reactance respectively. The SMIB power system is con-sidered to demonstrate the idea and verify the resulting im-provement. The parameters of the machine, excitation system,transformer and transmission lines are listed as follows

Xt = 0.1, Xl = 0.8, Re = 0, V∞ = 1.05∠0◦,Xd = 2.5, Xq = 2.1, X ′

d = 0.39, Vt = 1∠15◦,T ′d0 = 9.6, H = 3.2, D = 0, ωs = 377,

TA = 0.02,KA = 100, V maxs = −V min

s = 0.05,

where V maxs = −V min

s = 0.05 are the saturation limit consid-ered for the control signal. In case of generator supplementarydamping controller (SDCs), saturation limit should be consid-ered in the supplementary control input signal and are usuallyin the range of ±0.05 to ±0.1 per unit. These limits allowan acceptable control rang in the excitation system to preventundesirable tripping of the equipments protection initiated byover-excitation or under-excitation of generators. The abovenonlinear model can be linearized around the nominal oper-ating point and expressed in the following fourth order state-space representation, such that x = [Δδ Δωr ΔE ′

q ΔE ′fd]

Fig. 7. System performance of the closed loop and open loop SMIB powersystem with ideal network communication.

Fig. 8. Stability criteria for SMIB power system.

and x(t) = Ax(t) +Bsat(Vs), with Vs = Kx(t).

A =

⎡⎢⎢⎢⎢⎣

0 ωs 0 0

−K1

2H −Dωs

2H −K2

2H 0

− K4

T ′d0

0 − 1K3T ′

d0

1T ′d0

−KAK5

TA0 −KAK6

TA− 1

TA

⎤⎥⎥⎥⎥⎦, B =

⎡⎢⎢⎢⎣

0

0

0KA

TA

⎤⎥⎥⎥⎦(33)

and K1 −K6 are the well-known linearization constants basedon the system parameters [29]. Eigenvalue analysis shows thatthe open loop system has unstable complex eigenvalues of+0.2423± 7.6064i with frequency of 1.21 Hz and damping of−3.18%. Using LQR control design method, the following state-feedback damping controller is designed in [30] to enhance thedamping performance by regulating the exciter of SMIB system.

K = [−0.22 7.75 − 0.28 − 0.0006]. (34)

The controller is designed such that, a domain of attraction (DA)is estimated to guarantee a safety region and the state trajectoriesmust remain inside the (DA) to guarantee the stability. In prac-tice, these controllers can be implemented based on dynamicstate estimation using PMUs measurement. For ideal networkcommunications, the performance of the closed-loop system isshown in Fig. 7. Consider now the case where communicationfails, for some time variable durationμ and assume that the dura-tion of the ideal communication is τ = 0.2 s. Using the stabilitycriteria (21), two intervals can be found analytically (withoutany simulations) to determine the time of interruptions to be re-spected in order to maintain the stability of the system, as shownin Fig. 8. In Table I, these intervals are compared with the realvalues found using trial and error simulations. Compared to the



TABLE ICOMMUNICATION FAILURE DURATION FOR SMIB SYSTEM

Fig. 9. Speed deviation of the closed loop SMIB power system in case of idealand non-ideal (τ = 0.2 and μ = 0.23) communication network which is stable.

Fig. 10. Speed deviation of the closed loop SMIB power system in case ofideal and non-ideal (τ = 0.2 and μ = 0.35) communication which is instable.

developed stability criteria, excessive effort is needed to identifythe unstable regions.

From Table I it can be seen that the stability condition isconservative but reasonably characterizes the limits. The sys-tem response for the case of ideal communication time durationτ = 0.2 s and communication failure μ = 0.23 s is also shownin Fig. 9, where the conditions of stability is respected and thesystem is stable. It is shown in Fig. 10, that if the controller isblocked for duration μ = 0.35 s, then the system will be unsta-ble. It can be seen that the performance of damping controllerwith non-ideal communication network has been degraded sig-nificantly. In practice, in case these intervals are violated (e.g.having a longer communication failure), they can be used asthresholds to activate remedial action schemes [21].

B. Case Study II: Kundur’s Two-Area System

In this subsection, the developed stability condition is appliedto a modified Kundur two-area system [31], shown in Fig. 11.

Fig. 11. A two-area Kundur power system.

TABLE IICRITICAL MODES OF TWO-AREA SYSTEM

Area 1 is transferring 550 MW of active power to area 2. Genera-tors are represented by a fourth-order model and equipped with ahigh-gain excitation system. Generator G1 and G3 are equippedwith IEEE standard speed-based PSS to damp the local modes.More details of the parameters can be found in [31].

The modal analysis summarized in Table II, show that thesystem without a controller has a negatively damped inter-areamode at 0.666 Hz with damping ratio of −1.68% and twodamped local modes. The generator supplementary excitationcontrol and speed deviation are chosen as candidates for the ac-tuator and measurement signals of WADC system, respectively.G1 is chosen as the nominal actuator and measurement signals ofWADC system, respectively. G1 is chosen as the nominal actua-tor for damping controller. Based on the controllability measure,speed deviation ofG3 is identified as the best candidate measure-ment signals for the controller, as it has the highest geometricobservability over the first critical mode [32].

Hankel norm approximation [33] can be used to obtain thereduced-order model where the order of the model reduction canbe determined by examining the Hankel singular values. The lin-ear model is reduced to a second-order model and the followingoutput feedback controller, shown as follows, is designed usingmulti-objective optimization to meet or exceed 11% dampingover all inter-area and local mode, and optimize the H2/H∞performance to limit the control efforts and avoid high gains inthe WADC, since, large gain can lead the system to saturation,more details can be found in [32]. For the ideal communicationnetwork, the performance of the closed loop system is shown inFig. 12. Assuming the maximum time duration of perfect com-munication as τ = 0.2 s and using stability criteria (25), twointervals can be found analytically (without simulations) for thetime of interruption in the control signal as shown in Fig. 13.In Table III, these intervals are compared with the real valuesusing try and error simulations. It can be seen that the stabilitycondition is again conservative but reasonably characterizes thelimits. Compared to the analytical stability condition (25), ex-cessive efforts are needed to explore huge numbers thresholdsthrough simulation. The system response for the cases of perfectcommunication duration τ = 0.2 s and communication failureμ = 0.16 s is shown in Fig. 14, where the condition of stabilityis respected. For μ = 0.32 s is shown in Fig. 15. It is shown in



Fig. 12. Speed deviation of the closed loop and open loop two-area Kundurpower system in case of ideal communication network.

Fig. 13. Stability criteria for two-area Kundur power system.

TABLE IIICOMMUNICATION FAILURE DURATION FOR KUNDUR’S TWO-AREA SYSTEM

Fig. 14. Speed deviation of the closed loop two-area Kundur power system incase of ideal and non ideal (τ = 0.2 s and μ = 0.16 s) communication networkwhich is stable.

Fig. 16, that if the controller is blocked for duration μ = 0.35 s,the system will become unstable. It can be seen that the perfor-mance of the damping controller with non-ideal communicationnetwork has been degraded significantly. In Table III, these in-tervals are compared with the real values using try and errorsimulations. It can be seen that the stability condition is againconservative but reasonably characterizes the limits.

Fig. 15. Speed deviation of the closed loop two-area Kundur power system incase of ideal and non ideal (τ = 0.2 s and μ = 0.32 s) communication networkwhich is stable.

Fig. 16. Speed deviation of the closed loop two-area Kundur power system incase of ideal and non ideal (τ = 0.2 s and μ = 0.35) communication which isunstable.

V. CONCLUSION

In this paper, the problem for power systems with intermittentinformation transmissions is analyzed using time scale theory.This problem is proposed as a particular problem of switchedlinear system which consists of a set of linear continuous-timeand linear discrete-time subsystem on a specific time scale. Us-ing the derived stability criteria, bounds of the communicationloss duration, which guarantees the stability of the system, hasbeen computed in case of state-feedback and output-feedbackcontrollers. Numerical results show the effectiveness of the pro-posed scheme. It is also found that the results based on the linearmodel are reasonably accurate for the nonlinear system. Furtherresearch is needed to model randomness of intermittent trans-mission and random packet losses.

APPENDIX A

Proof: Consider the switched system (11). Let Ac = A+BK and Ad = [

(eAμ−I

μ

)(I+A−1BK)]. Using the generalized

exponential function in time scale theory, the solution ofthe switched system (11), for σ(tk) ≤ t ≤ tk+1, is given by(see [25])

x(t) = eAc(t−σ(tk))(I + μ(tk)Ad)eAc(tk−σ(tk−1))

× · · · (I + μ(t1)Ad)eAct1 x0. (35)



So, for t = tk+1, we have

x(tk+1) =k∏

i=0

eAc(tk+1−i−σ(tk−i))(I + μ(tk−i)Ad) x0

=

k∏i=0

e(A+BK)(tk+1−i−σ(tk−i))

×[I + μ(tk−i)

(eAμ(tk−i)−I

μ(tk−i)

)(I+A−1BK)

]x0

=

k∏i=0

e(A+BK)(tk+1−i−σ(tk−i))

×[I +

(eAμ(tk−i) − I

)(I +A−1BK)

]x0. (36)

Let 0 < a < 1 such that, ∀0 ≤ i ≤ k and τi = ti − σ(ti−1),∥∥∥e(A+BK)τi[I +

(eAμ(ti) − I

)(I +A−1BK)

]∥∥∥ ≤ a.

(37)So, the upper bound of x(tk+1) is given by

‖x(tk+1)‖

≤k∏

i=0

∥∥∥e(A+BK)τi[I +

(eAμ(ti) − I

)(I +A−1BK)

]∥∥∥ ‖x0‖

≤ ak+1 ‖x0‖ = e(k+1) log(a) ‖x0‖. (38)

Since log(a) < 0, so the solution x(t) of (11) converges expo-nentially to zero when t → ∞ (i.e., k → ∞). �

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Fatima Zohra Taousser received the B.S. and MS.degrees in mathematics from Sidi Bel Abbes Univer-sity, Sidi Bel Abbes, Algeria, and the Ph.D. degree inapplied mathematics from the Polytechnic Universityof Hauts-de-France, Valenciennes, France, in 2015.She is currently a Postdoctoral Researcher with theDepartment of Electrical Engineering and ComputerScience, The University of Tennessee, Knoxville, TN,USA. Her main research interests include stabilityand control of switched systems on non-uniform timedomain, by introducing time-scale theory, with their

application to communication network systems.



M. Ehsan Raoufat received the B.S. and MS.degrees from Shiraz University, Shiraz, Iran, in2009 and 2011, respectively. He is currently work-ing toward the Ph.D. degree at the Department ofElectrical Engineering and Computer Science, TheUniversity of Tennessee, Knoxville, TN, USA. Hismain research interests include distributed, decentral-ized, and stochastic control with their applications topower systems.

Kevin Tomsovic received the B.S. degree in electri-cal engineering from Michigan Technological Uni-versity, Houghton, MI, USA, in 1982, and the MS.and Ph.D. degrees in electrical engineering fromthe University of Washington, Seattle, WA, USA, in1984 and 1987, respectively. He is currently the CTIProfessor with the Department of Electrical Engineer-ing and Computer Science, The University of Ten-nessee, Knoxville, TN, USA, where he also directsthe NSF/DOE-sponsored ERC CURENT. He was thefaculty with Washington State University from 1992

to 2008. He was the Advanced Technology for Electrical Energy Chair forKumamoto University, Kumamoto, Japan, from 1999 to 2000, and was the NSFProgram Director in the ECS division of the Engineering directorate from 2004to 2006.

Seddik M. Djouadi received B.S. (Hons.) degreein electrical engineering from Ecole Nationale Poly-technique, El Harrach, Algiers, the M.Sc. degree elec-trical engineering from the University of Montreal,Montreal, QC, Canada, and the Ph.D. degree in elec-trical engineering from McGill University, Montreal,QC, Canada. He is currently a Full Professor withthe Electrical Engineering and Computer Science De-partment, The University of Tennessee, Knoxville,TN, USA. He authored or coauthored more than 100journal and conferences papers, some of which are

invited papers. His research interests include filtering and control of systemsunder communication constraints, modeling and control of wireless networks,control systems and applications to autonomous sensor platforms, electrome-chanical and mobile communication systems, in particular smart grid and powersystems, control systems through communication links, networked control sys-tems, and model reduction for aerodynamic feedback flow control. He was therecipient of the Best Paper Award in the 1st Conference on Intelligent Systemsand Automation 2008, the Ralph E. Power Junior Faculty Enhancement Awardin 2005, the Tibbet Award with AFS, Inc., in 1999, and the American ControlConference Best Student Paper Certificate (best five in competition) in 1998.


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