evaluation and stability analysis of different load frequency control systems with constant...

International journal of scientific and technical research in engineering (IJSTRE)

www.ijstre.com Volume 1 Issue 1 ǁ April 2016.

Manuscript id. 980039586 www.ijstre.com Page 5

Evaluation and Stability Analysis of Different Load Frequency

Control Systems with Constant Communication Delays

A.SivaPreethi1, V.GnanaThejaRakesh

2 1(EEE Department, SV Engineering College for Women, India) 2(EEE Department, SV Engineering College for Women, India)

[email protected]

[email protected]

ABSTRACT : The extensive usage of open communication networks in power system control analysis causes

inevitable time delays. This paper studies impacts of such delays on the steadiness of multi area load frequency

control (LFC) systems and proposes an analytical methodology to investigate delay margins. The proposed

methodology first eliminate the transcendental terms in characteristic equation of LFC systems without

generating any proximate esteem and transforms the transcendental characteristic equation into a general

polynomial. The key consequences of this elimination procedure makes real roots of the new polynomial compared to imaginary roots of the transcendental characteristic equation. With the help of latest polynomial, it

is additionally possible to determine the delay-dependency of system stability and root tendency with respect to

the time delay. An analytical formula is then developed by using Fuzzy logic controller implementation to figure

delay margins in terms of system parameters. For a large number of controller gains, delay margins of LFC

systems are calculated, for further investigation process the qualitative impact of controller gains on the delay

margin. Finally, simulations studies are carried out to validate the effectiveness of the proposed methodology by

comparing traditional PI controller with Fuzzy logic implementation.

KEYWORDS - Communication time delays, controller, delay margin, delay dependent stability,load frequency

control system.

I. INTRODUCTION

Phasor Measurement units (PMU) and open correspondence networks have been broadly utilized as a

part of the wide Area Measuring systems (WAMS). This causes inescapable time delays which incorporate

estimation and correspondence delays [1], [2]. It is understood that such time delays might diminish the control

systems damping execution and even could bring about unsteadiness on the upper bound limit or delay margin

for dependability [3]–[6]. This paper primarily researches the effect of time postponements on the steadiness

execution of LFC areas.

The primary objectives of LFC areas are to control the frequency and to keep up scheduled power exchange in an interconnected area with one or all the more autonomously controlled regions [7]. Time delays

because of communication connections were mostly disregarded in steady examine of LFC systems since

devoted systems with littler postponements were regularly utilized for exchanging information and control

signals. Be that as it may, bigger measure of time postponements in the scope of 5–15s in LFC frameworks are

seen with the use of open and distributed communication links. There are a few variables that influence size of

delays. These incorporates communication means, for example, fiber-optic-links, power line transporters, phone

line carriers and so forth., phasor bundle size, transmission convention utilized and systems load. Thus,

communication delays might varies randomly in a specific extent. Henceforth, the estimation of the delayed

margin is the prominent issue in strength investigation of time-postponed LFC systems. The knowledge of delay

margins empowers us to design for proper controller guaranteeing steadiness of the system for unreliable delays.

There exist a few techniques for processing delay margins for steadiness of time-deferred dynamical networks. These techniques could be gathered into two fundamental types, in particular frequency domain direct

and time-domain indirect strategies. The fundamental objective of frequency domain methodologies is to

analyse all critical imaginary roots of eigen equation for which the system will be hardly steady. The

accompanying three strategies are the ones usually utilized as a part of delay margin implementation of power

systems:

1) Schur-Cohn technique [8], [9]; 2) Elimination of exponential terms in the characteristic equation [10];

3) Rekasius substitution [11]–[13]. The Schur-Chon approach introduced in [9] has been effectively executed to

measure the delay margin for Automatic generation control (AGC) systems [6]. The Rekasius substitution has

been effectively used to decide delay margins for a basic electrical system [14] and one-region LFC systems. At

last, the disposal technique reported in [10] has been viably utilized to determine delay margin calculation of

generator excitation control systems [16], [17].However, it must be specified here that frequency domain direct

mailto:[email protected]

Evaluation and stability analysis of different load frequency control systems with constant

communication delays


strategies can't be connected to the time delay systems. This is a disadvantage of such strategies despite the fact

that they can acquire exact delay margins for constant delays. The indirect time-space strategy uses Lyapunov

dependability hypothesis and linear matrix inequalities (LMIs) procedures [18]–[20]. Such techniques have been

utilized to estimate delay margins of the wide-range damping controller [21], [22] and LFC networks[23].

Our previous researches concentrates mainly shows the proposed strategy effectively estimate the delay

margins deferral of generator excitation control system with a steady single delay margin and delay margins of

time-delayed DC motor speed control system [16],[17], [25]. With addition to these applications, this frequency

domain direct strategy was effectively applied to research the stability of other time-delayed systems, for

example, mechanical systems [26]–[28], predator-prey systems [29], [30] and a logistic model [31]. Such

fruitful applications together with right estimation of delay margins have spurred us to apply this strategy into

postponement edge calculation of load frequency control system with steady communication delay. The proposed technique first wipes out transcendental terms in characteristic mathematical statement of

LFC systems without making any estimation, and transforms the transcendental characteristic mathematical

statement into a general polynomial. The key effect after the elimination procedure is that, real roots of the

transcendental new polynomial compare to the critical imaginary roots of transcendental characteristic equation

precisely. With the assistance of new polynomial, it is additionally conceivable to determine the delay

dependency of system stability and root tendency concerning the time delay. A logical formula is then created to

figure delay margins regarding to system parameters, which is the principle commitment of this paper. For a

vast arrangement of controller gains, delay margins of LFC systems are processed to examine the qualitative

effect of controller gain on the delay margin. simulation studies using Matlab/Simulink [32] are done to verify

the theoretical delay margin results. Finally, the comparison delay margin results with ones obtained by the

indirect method [23] obviously demonstrate that the proposed technique gives more exact delay margin results.

II. TIME DELAYED LFC SYSTEM

The dynamics of power systems including LFC systems with or without time delay are generally

described by a set of non-linear differential and/or differential-algebraic equations [16],[33]. When the LFC

system is subjected to a small disturbance, nonlinear equations are linearized around an equilibrium point to

obtain a linear state-space equation model. The linear models suffice to assess the steady-state or small-signal

stability of the system around an equilibrium point [4], [7], [23]. If there exists a time delay, the conventional

LFC model needs to be modified to include delay in LFC system model. Multiple constant or time-varying

delays are generally observed in multi-area LFC systems. As explained in [4] and [5], in an open

communication network, delay can arise during: 1) transmission of area control error (ACE) signals from the

control center to the individual generation units and 2) from a telemetry delay when remote terminal units

(RTUs) send the telemetry signals to the control center. Assuming that the control center waits to receive the telemetered values, the analysis for each delay case is identical. Therefore, all delays are generally aggregated

into a single constant or time-varying delay from the control center. The model of the ith control area of the

multi-area LFC system is illustrated in Fig. 1. Note that all time delays are lumped into a single constant delay

and included into the control loop.

Fig. 1: Block diagram for LFC system (one-Area)




A proportional-integral (PI) controller is used as the load frequency controller in the model. Moreover,

generation units in each area are assumed to be all equivalent. The dynamics of multi-area (N control areas)

LFC system are described by the following state-space equation model:

𝐱 𝐭 = 𝐀𝐱 𝐭 + 𝐁𝐮 𝐭 + 𝐅∆𝐏𝐝 𝐭

𝐲 𝐭 =C 𝐱 𝐭 (1)

Where 𝐱𝐢 𝐭 = [∆𝐟𝐢 ∆𝐏𝐦𝐢 ∆𝐏𝐯𝐢 𝐀𝐂𝐄𝐢 ∆𝐏𝐭𝐢𝐞−𝐢]𝐓

𝐲𝐢(𝐭) = [𝐀𝐂𝐄𝐢 𝐀𝐂𝐄𝐢]𝐓

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=diag [𝐁𝟏 𝐁𝟐 . . .

C=diag [𝐂𝟏 𝐂𝟐 . . .

F=diag [𝐅𝟏 𝐅𝟐 . . .

𝐁𝐢= 𝟎 𝟎𝟏

𝐓𝐠𝐢 𝟎 𝟎

𝐓

, 𝐂𝐢= 𝛃𝐢 𝟎 𝟎𝟎 𝟎 𝟎

𝟎 𝟏𝟏 𝟎

, 𝐅𝐢= −𝟏

𝐌𝐢𝟎 𝟎 𝟎 𝟎

𝐓

𝐀𝐢𝐢 =

−𝐃𝐢

𝐌𝐢

𝟏

𝐌𝐢 𝟎 𝟎 −

𝟏

𝐌𝐢

𝟎 −𝟏

𝐓𝐜𝐡𝐢

𝟏

𝐓𝐜𝐡𝐢 𝟎 𝟎

−𝟏

𝐑𝐢𝐓𝐞𝐢 𝟎 −

𝟏

𝐓𝐠𝐢 𝟎 𝟎

𝛃𝐢 𝟎 𝟎 𝟎 𝟏

𝟐𝛑 𝐓𝐢𝐣𝐧𝐣=𝟏,𝐣≠𝐢 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎

𝐀𝐢𝐣 = 𝟎 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎

-2𝛑𝐓𝐢𝐣 𝟎 𝟎 𝟎 𝟎

𝐓𝐢𝐣 = 𝐓𝐣𝐢




Moreover,∆fi , ∆Pmi , ∆Pvi , ∆Pdi are the deviation in the frequency, the generator mechanical output, the

valve position, and the load of the ith control area, respectively Mi,Di,Tgi , Tchi and Ri denote the generator inertia

constant, damping coefficient, time constant of the governor and turbine, and speed drop of the control area,

respectively.ACEi and ACEi represent the area control error and ith integral.βi is the frequency bias factor.

Finally,Tij denotes the tie-line synchronizing coefficient between the ith and jth control areas. It must be noted

that the linear state-space equation model of (1) is commonly used in various types of analysis in power

systems. These analysis include power system stabilizer design in the presence of time delays [3], the stability

analysis of automatic generation control (AGC) with commensurate delays [6],wide-area damping controller

design for time-delayed power systems [21], [22], stability and delay margin computation of LFC systems [4],

[23], and the region-wise small-signal stability analysis of power systems with time delay [33].The ACE signal

for each control area is the sum of the tie-line power exchange and the frequency deviation weighted by a bias

Factor. 𝐀𝐂𝐄𝐢 = 𝛃𝐢∆𝐟𝐢 + ∆𝐏𝐭𝐢𝐞−𝐢 (2)

Fig.2.Dynamic Model of the 𝐢𝐭𝐡 control area in a multi-area LFC system

As shown in Fig. 2, the delayed ACE signal is the input of the PI controller. For each control area, a PI

controller is chosen as 𝐮𝐢 𝐭 = −𝐊𝐩𝐢𝐀𝐂𝐄𝐢 − 𝐊𝐈𝐢 𝐀𝐂𝐄𝐢

= −𝐊𝐢𝐲𝐢 𝐭 − 𝛕𝐢 = −𝐊𝐢𝐂𝐢𝐱𝐢(𝐭 − 𝛕𝐢) (3)

and the closed-loop system dynamic model is obtained as

𝐱 𝐭 = 𝐀𝐱 𝐭 + 𝐀𝐝𝐢𝐱(𝐭 −𝐧𝐢=𝟏 𝛕𝐢) + 𝐅∆𝐏𝐝(𝐭) (4)

Where 𝐀𝐝𝐢 = 𝐝𝐢𝐚𝐠 𝟎… . −𝐁𝐢𝐊𝐢𝐜𝐢 … . . 𝟎

𝐊𝐢 = [𝐤𝐏𝐢 𝐊𝐈𝐢]

𝐊 = 𝐝𝐢𝐚𝐠[𝐊𝟏 𝐊𝟐 ……𝐊𝐧]

Obviously, the multi-area LFC system will have multiple time delays. τi,i = 1, … . , n. In order to

simplify the delay margin computation, it is assumed that multiple delays are all equal and represented as

constant single delay. With this simplification, it is possible to build a simple model as shown in (5) which

provides an appropriate depiction of a single time delay included in state variable:

𝐱 𝐭 =𝐀𝐱 𝐭 + 𝐀𝐝𝐱 𝐭 − 𝛕 + 𝐅∆𝐏𝐝 𝐭 (5)

Where 𝐀𝐝 = 𝐀𝐝𝐢𝐧𝐢=𝟏

The characteristic equation of multi-area LFC system will have multiple exponential terms with commensurate

delays as follows: ∆ 𝐬, 𝛕 = 𝐝𝐞𝐭[𝐬𝐈 − 𝐀 − 𝐀𝐝𝐞−𝐬𝛕]= 𝐚𝐤 𝐬 𝐞

−𝐤𝛕𝐬𝐧𝐤=𝟎 = 𝟎 (6)

Where ak(s) is a polynomial in s with real coefficients For the one-area LFC system, the characteristic equation

is given as follows: ∆ 𝐬, 𝛕 = 𝐝𝐞𝐭[𝐬𝐈 − 𝐀 − 𝐀𝐝𝐞−𝐬𝛕]=𝐚𝟎 𝐬 + 𝐚𝟏 𝐬 𝐞

−𝐬𝛕 = 𝟎 (7)

Where a0 s and a1 s are polynomials in with following real coefficients

𝐚𝟎 𝐬 = 𝐏𝟒𝐬𝟒 + 𝐏𝟑𝐬

𝟑 + 𝐏𝟐𝐬𝟐 + 𝐏𝟏𝐬 (8)




𝐚𝟏 𝐬 = 𝐪𝟏𝐬 + 𝐪𝟎

𝐏𝟒 = 𝐑𝐓𝐠𝐓𝐜𝐡𝐌,

𝐏𝟑 = 𝐌𝐑𝐓𝐜𝐡 + 𝐑𝐃𝐓𝐠𝐓𝐜𝐡 + 𝐑𝐓𝐠𝐌,

𝐏𝟐 = 𝐌𝐑 + 𝐑𝐃𝐓𝐜𝐡 + 𝐑𝐓𝐠𝐃, 𝐏𝟏 = 𝐑𝐃 + 𝟏,

𝐪𝟏 = 𝛃𝐑𝐊𝐩, 𝐪𝟎 = 𝛃𝐑𝐊𝟏 (9)

From (6), the characteristic polynomial of a two-area LFC system will be

∆ 𝐬, 𝛕 = 𝐚𝟎 𝐬 + 𝐚𝟏(𝐬)𝐞−𝛕𝐬+𝐚𝟐 𝐬 𝐞−𝟐𝛕𝐬=0 (10)

The degree of polynomials a0 s , a1 s and a2 s 9, 6, and 3, respectively. Similar to the one-area

LFC system case, the coefficient of these polynomials clearly depend on system parameters. Those coefficients

are not presented due to insufficient space.

III. DELAY-DEPENDENT STABILITY ANALYSIS The stability studies of time-delayed systems aim to determine whether the system delay-independent

or delay-dependent stable. For the delay-independent stability, the system remains stable for all finite values of

time delays. In a delay-dependent stability case, the system remains stable for τ < τ∗ where τ and τ∗represent

delay and delay margin, respectively. If the delay exceeds the margin τ > τ∗, the system becomes unstable.

The delay margin is the key factor for stability evaluation of LFC systems. The total time delays

observed in the system must be less than the delay margin. The knowledge of delay margins for a large set of system parameters is essential to assess the stability of LFC systems. The following subsections present the

implementation of a frequency domain based direct method [10] to delay margin computation for both one-area

and two-area LFC systems.

3.1. Direct Method

From the general stability theory of dynamical systems, it is well known that all the roots of the

characteristic equation of (7) or (10) must lie in the left half of the complex plane for LFC systems to be

asymptotically stable. These characteristic equations may have infinitely many roots due to exponential type

transcendental terms. Consequently, the stability problem has become a complex task. However, for stability

assessment, the knowledge of all roots is not required. It is sufficient to find delay margin values τ∗at which the characteristic polynomial of (7) or (10) has roots (if any) on the imaginary axis.

One-Area LFC System: The characteristic equation of one-area LFC system is given in (7). The characteristic

equation ∆ s, τ = 0 clearly shows that it is an implicit function of s and τ . For simplicity, it is assumed that the

delay free system is stable. In other words, all the roots of ∆ s, 0 = 0 are in the left half-plane. This is a

realistic assumption since a delay-free one-area LFC system is stable for practical values of parameters. Suppose

that the characteristic equation ∆ s, τ = 0 has a root on the imaginary axis at s = jωc (where subscript c refers

to “crossing” the imaginary axis), for some finite value of the time delay τ . Because of the complex conjugate

symmetry of complex roots, the equation Δ −s, τ = 0 will also have the same root at for the same value of the

time delay. Consequently, the problem now reduces to finding values of time delay τ such that both ∆ s, τ = 0

and ∆ −s, τ = 0 have a common root at s = jωc .This result could be stated as follows:

𝐚𝟎(𝐣𝛚𝐜)+ 𝐚𝟏(𝐣𝛚𝐜)𝐞−𝐣𝛚𝐜𝛕=0

𝐚𝟎(−𝐣𝛚𝐜)+ 𝐚𝟏(−𝐣𝛚𝐜) 𝐞−𝐣𝛚𝐜𝛕=0 (11)

The following augmented characteristic equation in ωc2 is obtained by eliminating exponential terms in (11):

𝐖 𝛚𝐜𝟐 = 𝐚𝟎(𝐣𝛚𝐜)𝐚𝟎(−𝐣𝛚𝐜) − 𝐚𝟏(𝐣𝛚𝐜)𝐚𝟏(−𝐣𝛚𝐜)=0

𝐦𝟖 =𝐦𝟖𝛚𝐜𝟖 + 𝐦𝟔𝛚𝐜

𝟔 + 𝐦𝟒𝛚𝐜𝟒 + 𝐦𝟐𝛚𝐜

𝟒 + 𝐦𝟎 = 𝟎 (12)

Where 𝐦𝟖 = 𝐩𝟒𝟐, 𝐦𝟔 = 𝐩𝟑

𝟐 − 𝟐𝐏𝟒𝐏𝟐, 𝒎𝟒 = 𝑷𝟐𝟐 − 𝟐𝑷𝟑𝑷𝟏, 𝐦𝟐 =𝐩𝟏

𝟐−(𝛃𝐑𝐊𝐩)𝟐, 𝐦𝟎 = −(𝛃𝐑𝐊𝐈)𝟐.

It should be emphasized here that the characteristic equation with a transcendental term given in (7) is

now transformed into a regular polynomial without transcendentality presented in (12).The positive real roots of

(12) ωc correspond to the magnitude of purely imaginary roots of (7), s = ±jωc exactly. The computation of




positive real roots of (12) is much easier than computation of purely imaginary roots of (7). Depending on the

nature of roots of (12), the following two different stability phenomena may be observed:

1) The one-area LFC system is delay-independent stable if the augmented characteristic equation of (12) does

not have any positive real roots for all finite delays τ ≥ 0.The non-existence of such roots implies that the roots

of (7) remain in the left-half stable plane for all finite delays τ ≥ 0.

2) The one-area LFC system is delay-dependent stable if the augmented characteristic equation of (12) has at

least one positive real root. The existence of such roots implies that the roots of (7) cross the imaginary axis

at s = ±jωc for a finite delay τ∗.

The time delay value, delay margin, for which the roots of (7) cross the imaginary axis, is determined using (7) as [10]

𝛕∗ =𝟏

𝛚𝐜𝐓𝐚𝐧−𝟏

𝐭𝟓𝛚𝐜𝟓+𝐭𝟑𝛚𝐜

𝟓+𝐭𝟏𝛚𝐜

𝐭𝟒𝛚𝐜𝟒+𝐭𝟐𝛚𝐜

𝟒 +𝟐𝐫𝛑

𝛚𝐜;

𝐫 = 𝟎, 𝟏,𝟐,… …… , ∞ (13)

where the corresponding coefficients are given as

𝐭𝟓 = −𝐩𝟒𝛃𝐑𝐊𝐏, 𝐭𝟒 = 𝐩𝟑𝛃𝐑𝐊𝐩−𝐩𝟒𝛃𝐑𝐊𝐈,

𝐭𝟑 = (𝐩𝟐𝛃𝐑𝐊𝐏− 𝐩𝟑𝛃𝐑𝐊𝐈), 𝐭𝟐 = (𝐩𝟐𝛃𝐑𝐊𝐈 − 𝐩𝟐𝛃𝐑𝐊𝐏)

𝐭𝟏 = 𝛃𝐑𝐊𝐈𝐩𝟏

For a positive root of (12), we should investigate if at s = ±jωc, the root of (7) crosses the imaginary

axis with increasing τ. The necessary condition for the existence of roots crossing the imaginary axis is that

roots cross the imaginary axis with non-zero velocity as given in the following:

𝐑𝐞 𝐝𝐬

𝐝𝛕 𝐬=𝐣𝛚𝐜

≠ 𝟎 (14)

Where Re (.) represents the real part of a complex variable. The sign of root sensitivity is defined as root

tendency (RT) [10], [12]

𝐑𝐓 𝐬=𝐣𝛚𝐜 = 𝐬𝐠𝐧 𝐑𝐞 𝐝𝐬

𝐝𝛕 𝐬=𝐣𝛚𝐜

= 𝐬𝐠𝐧[𝐖𝐈 𝛚𝐜𝟐 ] (15)

Where the prime denotes the derivative of (12) with respect to ωc2. The derivation of (15) could be

found in [16]. The RT expression given in (15) gives a practical tool to evaluate the direction of transition of the

roots at as increases from τ1 = τ − ∆τ to τ2 = τ∗ + ∆τ ,0< ∆τ ≪ 1. The root crosses the imaginary axis

either to unstable right half plane when , or to stable left half plane when.

2) Two-Area LFC System: The characteristic equation is given by (10). If the characteristic equation of (10) has

a solution of s = jωc then ∆ −s, τ = 0 will have the same solution:

∆ −𝐬, 𝛕 = 𝐚𝟎 −𝐬 + 𝐚𝟏 −𝐬 𝐞𝛕𝐬 + 𝐚𝟐(−𝐬)𝐞𝟐𝛕𝐬 =0 (16)

Similar to the one-area LFC system, the exponential terms should be eliminated to obtain a new

characteristic polynomial without transcendentality. This could be easily achieved using a recursive procedure

as described below. Let us define a new characteristic equation as [10]

∆ 𝟏 𝐬, 𝛕 = 𝐚𝟎 −𝐬 ∆ 𝐬,𝛕 − 𝐚𝟐 𝐬 𝐞−𝟐𝛕𝐬∆(−𝐬, 𝛕)

∆ 𝟏 𝐬,𝛕 = [𝐚𝟎 −𝐬 𝐚𝟎 𝐬 − 𝐚𝟐 𝐬 𝐚𝟐 −𝐬 ] + [𝐚𝟎 −𝐬 𝐚𝟏 𝐬 − 𝐚𝟐 𝐬 𝐚𝟏 −𝐬 ] 𝐞−𝛕𝐬 (17)

Then, we have ∆ 𝟏 −𝐬,𝛕 = 𝐚𝟎 𝐬 ∆ −𝐬, 𝛕 − 𝐚𝟐 −𝐬 𝐞𝟐𝛕𝐬∆ 𝐬,𝛕

∆ 𝟏 −𝐬,𝛕 = [𝐚𝟎 𝐬 𝐚𝟎 −𝐬 − 𝐚𝟐 −𝐬 𝐚𝟐 𝐬 + [𝐚𝟎 𝐬 𝐚𝟏 −𝐬 − 𝐚𝟐 −𝐬 𝐚𝟏 𝐬 ]𝐞𝛕𝐬 (18)




It is clear from (17) and (18) that the root s = jωc of (10) and (16) is also a root of the following new

characteristic equations:

∆ 𝟏 𝐬, 𝛕 = 𝐚𝟎 𝟏 𝐬 + 𝐚𝟏

𝟏 𝐬 𝐞−𝛕𝐬 = 𝟎

∆ 𝟏 −𝐬,𝛕 = 𝐚𝟎 𝟏 −𝐬 + 𝐚𝟏

𝟏 −𝐬 𝐞−𝛕𝐬 = 𝟎 (19)

Where 𝐚𝟎𝟏 𝐬 = 𝐚𝟎 −𝐬 𝐚𝟎 𝐬 − 𝐚𝟐 𝐬 𝐚𝟐 −𝐬

𝐚𝟏𝟏 𝐬 = 𝐚𝟎 −𝐬 𝐚𝟏 𝐬 − 𝐚𝟐 𝐬 𝐚𝟏 −𝐬 (20)

Note that the new characteristic equations in (19) contain only a single e−τs or eτs , indicating that the

degree of commensuracy is reduced from 2 to 1. On the other hand, the degrees of polynomials a0 1

(s) and

a1 1

(s) now become 18 and 15, respectively after eliminating the term e−2τs of in (10). This procedure could be

easily repeated to eliminate exponential terms, e−τs and eτs in (19) and the following augmented characteristic

equation not containing any exponential terms could be obtained:

∆ 𝟐 𝐬 = 𝐚𝟎 𝟐 𝐬 = 𝟎 (21)

Where 𝐚𝟎(𝟐) 𝐬 = 𝐚𝟎

(𝟏) 𝐬 𝐚𝟎(𝟏) −𝐬 − 𝐚𝟏

(𝟏) 𝐬 𝐚𝟏(𝟏) −𝐬 (22)

It should be noted here that the root (10) for some is also a root (10) for some τ is also a root of (21)

since the elimination procedure preserves the imaginary roots of the original characteristic equation of (10). The

substitution of s = jωc into (21) yields the following polynomial in ωc2:

𝐖 𝛚𝐜𝟐 = 𝐚𝟎

𝟏 𝐣𝛚𝐜 𝐚𝟎 𝟏 −𝐣𝛚𝐜 − 𝐚𝟏

𝟏 𝐣𝛚𝐜 𝐚𝟏 𝟏 −𝐣𝛚𝐜 = 𝟎 (23)

Note that the new characteristic equation in (23) has the degree of 36. The time delay value, delay margin, for

which the roots of (10) cross the imaginary axis, is computed by [10]

𝛕∗ =𝟏

𝛚𝐜𝐭𝐚𝐧−𝟏

𝐈𝐦 𝐚𝟎 𝟏

(𝐣𝛚𝐜)

𝐚𝟏 𝟏

(𝐣𝛚𝐜)

𝐑𝐞 −𝐚𝟎 𝟏

(𝐣𝛚𝐜)

𝐚𝟏 𝟏

(𝐣𝛚𝐜)

+𝟐𝐫𝛑

𝛚𝐜;

𝐫 = 𝟎, 𝟏,𝟐, … . . , ∞. (24)

Finally, we should derive an expression similar to one given by (15) to find the R T s . In [10], it was shown that

if the following condition is satisfied at the crossing root s = jωc:

𝐚𝟎 𝟏 𝐬 = 𝐚𝟎 −𝐣𝛚𝐜 𝐚𝟎 𝐣𝛚𝐜 −𝐚𝟐(𝐣𝛚𝐜)𝐚𝟐(−𝐣𝛚𝐜) > 0 (25)

Then, the RT of a root of ∆ s, τ = 0 is the same as that for the corresponding root of

∆ 2 s, τ = 0.The RT is then determined by

𝐑𝐓 𝐬=𝐣𝛚𝐜 = 𝐬𝐠𝐧[𝛂𝐖𝟏 𝛚𝐜𝟐 ] (26)

Where 𝛂 = 𝐚𝟎 𝟏

(𝐣𝛚𝐜) (27)

Summary of Analysis Steps:

The delay margin computation process includes the following steps: Step 1) Select LFC system parameters and obtain the time delayed state-space equation model of the one-area or

two-area LFC systems.

Step 2) Select a set of values for PI controller gains.




Step 3) For each PI controller gain, determine the characteristic equation using (10) for the multi-area LFC

system and.

Step 4) Obtain the augmented characteristic equations of the multi area LFC systems using (12) and (23),

respectively and compute their real positive roots, namely 𝜔𝑐 = 𝜔𝑐1 , 𝜔𝑐2 , … . , 𝜔𝑐𝑞 . Step 5) For all real positive roots, determine the corresponding root tendencies using (15) or (25).

Step 6) Compute the corresponding delay margins using (13) or (24) for those real positive roots found in Step

(4) that have positive root tendency only,𝑅𝑇 = ±1. Step 7) Choose the minimum of delay margins computed in Step 6) as the system delay margin.

Step 8) Verify the accuracy of the theoretical delay margins using time-domain simulations based on the

dynamic models for the one-area and two-area LFC systems, respectively.

IV. FUZZY SYSTEM The fuzzy interface system basically consists of a methodology of the mathematical mapping from a

given input set to an output set using Fuzzy logic. The mapping process provides the basis from which the

interference or conclusion can be made.

Fuzzy interface process consists of following steps Step 1: Fuzzification of input variables.

Step 2: Application of Fuzzy operator.(AND, OR, NOT) In the IF (antecedent) part of the rule.

Step 3: Implication from the antecedent to the consequent (Then part of the rule).

Step 4: Aggregation of the consequents across the rules.

Step 5: Defuzzification.

Fig.3.Fuzzy logic implementation of stability analysis

IV. CASE STUDIES

Case studies are performed for both one-area, two-area LFC systems. Based on that we design a three area LFC system parameters given in [23] are used for comparison purposes. The delay margins are compared

for both PI and Fuzzy logic controllers. Delay margins are obtained for various values of PI controller gains to

investigate the quantitative effect of the controller. Finally, the accuracy of delay margin results computed by

the proposed fuzzy mamdani method is verified utilizing simulation studies.

A. One-Area LFC System

1) Theoretical Delay Margin Results: Delay margin are computed using (13) for a large set of PI

controller gains and are presented in Table I. Results show that a fixed 𝐾𝑝 ,𝜏∗ decreases as 𝐾𝐼 increases. This

indicates that the increase of 𝐾𝐼 causes a less stable LFC system. The impact of 𝐾𝑝 on 𝜏∗ has two different

patterns when 𝐾𝐼 is fixed. For all values of 𝐾𝐼 , 𝜏∗ increases as 𝐾𝑝 increases when 𝐾𝑝 lies in an interval of

𝐾𝑝=0-0.4. However, 𝜏∗decreases with the increase in 𝐾𝑝 for 𝐾𝑝 ≥ 0.6.Such an effect 𝐾𝑝 on 𝜏∗ has also been

noted in the time-delayed excitation control systems [17], in delay margin results of one-area LFC system

obtained by Rekasius substitution [15], and in delay margin results for LFC systems reported in [23]. Moreover,

delay margin results indicate that a small increase in 𝐾𝑝 for relatively larger 𝐾𝑝 values may cause a significant

decrease in delay margins when 𝐾𝐼 is fixed.

2) Computating Theoretical Delay Margins via Simulation:




In order to verify the accuracy of theoretical delay margin results, time-domain simulation are

performed using Matlab/Simulink for the following PI controller gains:(𝐾𝑃 = 0, 𝐾𝐼 = 0.4) and (𝐾𝑃 = 0.6, 𝐾𝐼 =0.6).Frequency response of one-area LFC system for a positive load disturbance of ∆𝑃𝑑 = 0.1 pu at t=0 s is

obtained. For 𝐾𝑃 = 0, 𝐾𝐼 = 0.4, the delay margin is found to be 𝜏∗ = 3.382𝑠 by the proposed method as shown

in Table I and 𝜏∗ = 3.124𝑠 by the method of [23]. However, the simulation result presented in Fig.4 indicates

that the system is marginally stable 𝜏∗ = 3.384𝑠 due to the sustained oscillations. Fig.4 also shows the

frequency deviation for τ = 3.3 s and τ = 3.3 s . Please note that system is stable with decaying oscillations for

τ = 3.3s < τ∗ = 3.384s and it is unstable for τ = 3.5 s > τ∗ = 3.384s with growing oscillations. Moreover,

the comparison of delay margins with the one obtained by the simulation shows that the relative percentage

errors are 0.391% for the presented method and 7.670% for the method of [23].

B. Three-Area LFC System

1) Theoretical Delay Margin Results:

Delay margins are computed using different sets of controller gains that are equal in each control area

and presented in Table I. The effects of controller gains are quite similar to the one-area case. For example, an

increase in KI for fixed KP values causes a decrease in τ∗ . The effect of on the delay margin has two trends for

fixed KI . The delay margin increases as KP increases in an interval of KP = 0 − 0.4 . On the other hand, the

delay margin decreases as increases for KP ≥ 0.6. . Additionally, Table II indicates that delay margins for the

two-area LFC system are slightly smaller than those of the one-area case for all PI controller gains.

2) Computing Theoretical Delay Margins via Simulation:

For two different sets of PI controller gains, simulation studies are performed to investigate the

accuracy of delay margins. These are (KP = 0.2, KI = 0.4)and (KP = 0.4, KI = 0.6)The delay margins for this

set of gains are found to be τ∗ = 3.631s and τ∗ = 2.184s, respectively. For verification purpose, the same

positive load disturbance,∆Pd1 = ∆Pd2 = 0.1 pu at t = 10s is considered. The frequency deviations for the

selected gains are shown in Figs. 4 and 5. Figs. 4 and 5 clearly indicate that the two-area LFC system becomes

marginally stable at τ∗ = 3.632s for KP = 0.2, and KI = 0.4 and τ∗ = 2.189s for, KP = 0.4 , KI = 0.6 .

Observe that theoretical delay margin results are again in close agreement with those obtained by simulations,

verifying the accuracy and effectiveness of the proposed method.

Table I:

Delay Margin Results Obtained By the Proposed Method for Various Values Of And (Three-Area LFC)

𝝉∗(𝒔) 𝑲𝑰

𝑲𝑷 0.05 0.1 0.15 0.2 0.4 0.6 1.0

0 30.812 15.090 9.842 7.211 3.225 1.843 0.591

0.05 31.772 15.570 10.162 7.450 3.345 1.922 0.638

0.1 32.647 16.008 10.453 7.669 3.453 1.993 0.676

0.2 34.122 16.744 10.943 8.035 3.631 2.106 0.725

0.4 35.728 17.542 11.469 8.424 3.802 2.184 0.684

0.6 34.809 17.068 11.136 8.155 3.588 1.881 0.480

1.0 0.510 0.498 0.485 0.472 0.416 0.357 0.243




Fig.4. Frequency response for different time delays for 𝐊𝐏 = 𝟎.𝟒, 𝐊𝐈 = 𝟎. 𝟔: Three areas

LFC system.

SIMULATION RESULTS:

Fig.5: Simulation circuit of a load frequency control with multi-area system using PI controller.

Fig.6: load frequency control with single-area system using PI controller.

Fig.7: load frequency control with multi-area system using PI controller.




Fig.8: load frequency control with multi-area system using PI controller

FLC IMPLEMENTATION SIMULATION CIRCUIT:

Fig.9: Simulation circuit of a load frequency control with multi-area system using

FUZZY logic controller

Fig.10: Simulation circuit of a load frequency control with single-area system using


Fig.11: Simulation circuit of a load frequency control with multi-area system using





Fig.12: comparison of PI and FUZZY CONTROLLER for load frequency control with

Multi-area system

CONCLUSION: This paper has studied the delay-dependent stability of LFC systems with communication delays. A

frequency domain based analytical method that does not use any approximation has been FOR VARIOUS VALUES OF MULTI-AREA (3-AREA) LFC presented to investigate the delay-dependency of stability and to

compute the delay margin for one-area and two-area LFC systems. The method eliminates exponential terms in

the characteristic equation such that the positive real roots of the resulting augmented polynomial give the finite

values of crossing frequencies at which stability feature of the system change. With the help of this polynomial,

a simple root tendency test has been developed to determine the direction of the root transition. The stability of

one-area and three area LFC systems with constant delays have been analyzed and delay margins have been

computed for a wide range controller gains. Simulation studies have been carried out to verify delay margin

results. The following observations and comments can be made from the results:

1) The PI controller gains have significant impact on delay margins. When the proportional controller gain

is kept constant, the delay margin decreases as the integral controller gain is increased, indicating a smaller

stability margin for LFC systems. The delay margin increases at first and decreases with the increase of the

proportional gain when integral controller gain remains unchanged. For relatively larger controller gains, a small increase in controller gains may result in a sharp decrease in delay margins.

2) By implementing fuzzy logic controller mamdani method, Delay margins results obtained by the

proposed method are almost the same as the ones determined by simulations, proving that the proposed

method accurately estimate delay margins of LFC systems.

For all values of controller gains, the proposed method gives relatively larger delay margin results when

compared with methods based on Lyapunov stability theory and LMI techniques with mamdani fuzzification.

With the help of the results presented, the controller gains could be properly selected such that LFC system will

be stable and will have a desired damping performance even if certain amount of communication delays exist.

As future work, the proposed method will extended to the application of the fractional- order controllers into

time-delayed LFC systems and to the computation of stability regions in the controller parameter space.

Moreover, another frequency-domain direct method, Rekasius substitution [11], [34], [35] or ANN techniques will be implemented into three-area LFC scheme with both equal and multiple independent time delays

(incommensurate delays) in each control area since the proposed method cannot be applied to the stability

analysis of such multiple independent delay cases.

REFERENCES: [1] Şahin Sönmez, Saffet Ayasun, and Chika O. Nwankpa, “An Exact Method for Computing Delay Margin for Stability of Load

Frequency Control Systems With Constant Communication Delays”, in IEEE TRANSACTIONS ON POWER SYSTEMS

[2] Xia.X, Y. Xin, J. Xiao, J. Wu, and Y. Han, “WAMS applications in Chinese power systems,” IEEE Power Energy Mag., vol. 4, no.

1, pp. 54–63, Feb. 2006.

[3] H. Wu, K. Tsakalis, and G. T. Heydt, “Evaluation of time delay effects to wide-area power system stabilizer design,” IEEE Trans.

Power Syst.,vol. 19, no. 4, pp. 1935–1941, Nov. 2004.

[4] Kundur P (1994) Power system stability and control. McGraw-Hill Inc, New York.

[5] S. Bhowmik, K. Tomsovic, and A. Bose, “Communication model for third party load frequency control,” IEEE Trans. Power Syst.,

vol. 19,no. 1, pp. 543–548, Feb. 2004.

[6] M. Liu, L. Yang, D. Gan, D. Wang, F. Gao, and Y. Chen, “The stability of AGC systems with commensurate delays,” Eur. Trans.

Elect. Power,vol. 17, no. 6, pp. 615–627, Nov./Dec. 2007.

[7] Saadat H (1999) Power system analysis. McGraw-Hill Inc, New York 490.

[8] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time Delay Systems.Boston, MA, USA: Birkhauser, 2003.

[9] J. Chen, G. Gu, and C. N. Nett, “A new method for computing delay margins for stability of linear delay systems,” Syst. Control

Lett., vol.26, no. 2, pp. 107–117, Sep. 1995.

[10] K. E. Walton and J. E. Marshall, “Direct Method for TDS Stability Analysis,” IEE Proc. Part D, vol. 134, no. 2, pp. 101–107, Mar.

1987.

[11] Z. V. Rekasius, “A stability test for systems with delays,” in Proc. Joint Automatic Control Conf., San Francisco, CA, USA, Aug.

13–15, 1980.

[12] N. Olgac and R. Sipahi, “An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems,” IEEE

Trans. Autom.Control, vol. 47, no. 5, pp. 793–797, May 2002.




[13] N. Olgac and R. Sipahi, “A practical method for analyzing the stability of neutral type LTI-time delayed systems,” Automatica, vol.

40, no. 5,pp. 847–853, May 2004.

[14] H. J. Jia, X. D. Cao, X. D. Yu, and P. Zhang, “A simple approach to determine power system delay margin,” in Proc. 2007 IEEE

Power Engineering Soc. General Meeting, Tampa, FL, USA, Jun. 24–28, 2007,pp. 1–7.

[15] S. Sonmez, S. Ayasun, and U. Eminoglu, “Computation of time delay margin for stability of a single-area load frequency control

system with communication delays,”WSEAS Trans. Power Syst., vol. 9, pp. 67–76,2014.

[16] S. Ayasun, “Computation of time delay margin for power system small signal stability,” Eur. Trans. Elect. Power, vol. 19, no. 7,

pp. 949–968,Oct. 2009.

[17] S. Ayasun and A. Gelen, “Stability analysis of a generator excitation control system with time delays,” Elect. Eng, vol. 91, no. 6,

pp.347–355, Jan. 2010.

[18] Y. He, Q. G. Wang, L. H. Xie, and C. Lin, “Further improvement of free-weighting matrices technique for systems with time-

varying delay,” IEEE Trans. Autom. Control, vol. 52, no. 2, pp. 293–299, Feb.2007.

[19] M. Wu, Y. He, J. H. She, and G. P. Liu, “Delay-dependent criterion for robust stability of time-varying delay systems,” Automatica,

vol. 40,no. 8, pp. 1435–1439, Aug. 2004.

[20] S. Y. Xu and J. Lam, “On equivalence and efficiency of certain stability criteria for time-delay systems,” IEEE Trans. Autom.

Control, vol. 52,no. 1, pp. 95–101, Jan. 2007.

[21] W. Yao, L. Jiang, Q. H. Wu, J. Y. Wen, and S. J. Cheng, “Delay-dependent stability analysis of the power system with a wide-area

damping controller embedded,” IEEE Trans. Power Syst., vol. 26, no. 1, pp.233–240, Feb. 2011.

[22] W. Yao, L. Jiang, Q. H. Wu, J. Y. Wen, Q. H. Wu, and S. J. Cheng,“Wide-area damping controller of FACTS devices for inter-area

oscillations considering communication time delays,” IEEE Trans. Power Syst., vol. 29, no. 1, pp. 318–329, Jan. 2014.

[23] B. Naduvathuparambil, Valenti.MC, and A. Feliachi, “Communication delays in wide area measurement systems,” in Proc. 34th

Southeastern Symp. System Theory (SSST2002), Huntsville, AL, USA, Mar.18–19, 2002, pp. 118–122.

Authors profile:

Ms.A.sivapreethi has completed her B.tech in EEE Department from SWETHA

INSTITUTE OF TECHNOLOGY AND SCIENCE FOR WOMEN, Tirupati, Affiliated to

JNTU Anantapur .Presently she is pursuing her Masters in Power Systems in SV

ENGINEERING COLLEGE FOR WOMEN, Karakambadi Road, Tirupati, Andhra Pradesh,

(INDIA).

Mr.V.Gnana Theja Rakesh is currently working as an Assistant Professor in EEE department,

SV ENGINEERING COLLEGE FOR WOMEN, Tirupati. He has received his bachelor of

Technology (B.tech) from Sri Venkateshwara University in Electronic Instrumentation and

Control and M.Tech in Sri Venkateshwara University, Tirupati. Specialized in

Instrumentation Control systems. His areas of interest include Control Systems,

instrumentation and Power systems.