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Chapter 12

Small-signal Stability and Time Domain Analysis

of a Power System with Feedback Delay

Vahid S. Bokharaie, Rifat Sipahi, and Federico Milano

This chapter describes the impacts that time-delays in feedback have on small-signal

as well as transient stability of power systems. We present a power system model

comprising of Delay Differential Algebraic Equations (DDAEs) and describe general

techniques to compute the spectrum of DDAE and to integrate such equations in time

domain. The focus is on delays arising in measured signals, e.g., remote frequency

measurements for power system stabilizers of synchronous machines. Several ex-

amples based on a benchmark system, the IEEE 14-bus test system, as well as a

real-world system are discussed and analyzed

12.1 Introduction

While time-delays are intrinsic of physical and control systems, these are typically

neglected or approximated with simple lag blocks in the conventional model of

power systems for voltage and transient stability analysis. Most power system de-

vices, like transformers and synchronous machines, are actually not affected by de-

lays, except for very long transmission lines [1]. However, regulators are affected

by delays, and in recent years, the ubiquitous presence of communication systems

and remote measurements, e.g., Phasor Measurement Units (PMUs), has attracted

the attention of researchers in academia and industry to the impact of delays on these

signals and on the stability of the overall power grid.

The focus of the most of the research in this ﬁeld is devoted to the design of

robust controllers that are able to reduce the impact of communication delays. For

example, the work in [2,3] deals with the robust control of wide area control schemes.

The main goal in there is to improve the effect of power system stabilizers to damp

inter-area oscillations. Another emerging area where delays are relevant is the load

frequency control; see [4] for further details.

Existing studies on small-signal stability of power systems with delays develop

model equations and analyze them in two main categories: (i) time-domain methods

and (ii) frequency-domain methods. Relevant contributions to these two approaches

are brieﬂy reviewed below.

1

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2Chapter 12

12.1.1 Time-domain methods

These methods are based on the Lyapunov-Krasovskii’s stability theorem and the

Razumikhin’s theorem, see for example [5–10]. The application of time-domain

methods allow deﬁning robust controllers (e.g., H∞control) and dealing with uncer-

tainties and time-varying delays. While the conditions of the Lyapunov-Krasovskii’s

stability theorem and the Razumikhin’s theorem provide strong tools for the stabil-

ity and control of many problems, including time-varying effects and nonlinearities,

these conditions are only sufﬁcient and hence, in the context of linear stability many

studies prefer frequency domain tools in order to capture necessary and sufﬁcient

conditions of stability. In this context, computation of delay margin - the largest

delay less than which the system is stable - has been one of the main foci research

topic. Another challenge with time-domain methods could be that it is necessary to

ﬁnd a Lyapunov functional or, according to the Razumikhin’s theorem, a Lyapunov

function that bounds the Lyapunov functional. In the context of time-delay systems,

developing these functionals indeed requires deep expertise, and this may pose chal-

lenges in analyzing nonlinear and DAE systems; see [11] for applications to small

scale power systems.

If the DDAE is linear or is linearized around an equilibrium point, ﬁnding the

Lyapunov function, in turn, implies ﬁnding a solution of a Linear Matrix Inequality

(LMI) problem [12]. A drawback of this approach is that the size and the compu-

tational burden of LMI highly increase with the size of the DDAE and it is only in

the last two decades that such calculations have become tractable [12]. Other ap-

proaches are based on the deﬁnition of a Lyapunov function with the well-known

difﬁculties in ﬁnding such a function [11] or on the solution of a LMI problem [12],

whose computational burden, however, is cumbersome but has become tractable for

some applications [13].

12.1.2 Frequency-domain methods

These methods are mainly based on the evaluation of the roots of the characteristic

equation of the corresponding linear time-invariant system [7,14–20]. This approach

in principle follows necessary and sufﬁcient conditions of linear stability however

due to the difﬁculty in determining the roots of the characteristic equation (see Sec-

tion 12.2.1), the analysis is challenging. In the context of power systems, this limits

the analysis to One-Machine Inﬁnite-Bus (OMIB) systems.

While there are attempts to deﬁne an exact analytic solution for oversimpliﬁed

power system models [21], an explicit solution cannot be found in general. Outside

the ﬁeld of power systems, many developments have been published yet majority of

these results are limited to low dimensional problems, see for example [20] and the

references therein. But there exist methods that by providing an approximation of

the system, enable us to analyze the stability of higher dimensional systems.

This chapter considers four different approaches that approximate the solution

of the small-signal stability of DDAEs. These are: (i) a Chebyshev discretization

of a set of Partial Differential Equations (PDEs) that are equivalent to the original

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 3

DDAEs [22]; (ii) a discretization of the Time Integration Operator (TIO) as proposed

in [23]; a Linear Multi-Step (LMS) approximation which has been proposed in [24]

and is implemented in the open-source software tool DD E-BIFTOO L [25]; and the

well-known Pad´

e approximants [26].

We wish to note that the Chebyshev discretization has been successfully applied

to power systems with a single delay [27] and the Pad´

e approximants have been used

in [4] but not considered for the solution of the small-signal stability problem.

A common characteristics of the above techniques is the high computational bur-

den, which, unfortunately, increases more than linearly with the size of the problem.

Hence proper numerical schemes and implementations have to be used. One op-

tion that has proven to be very promising, is to use GPU-based numerical libraries.

For example, all simulation results discussed in this chapter are obtained based on

MAG MA, which provides an efﬁcient GPU-based parallel implementation of LA PACK

functions and QR factorization for solving the linear eigenvalue problems [28].

It is important to note that the above list of techniques is by no means exhaus-

tive. It is aimed only to present a set of options for solving benchmark problems.

An exhaustive study is thus left to future work. Needless to say, the following

techniques, although not covered here, have been widely used and accepted in the

community, namely TRACE-DDE [29], QPMR [30], and Lambert W function ap-

proach [31]. The backbone of TRACE DDE also makes use of Chebyhshev dis-

cretization techniques and has been successfully implemented on problems with var-

ious multiple delay models. Matlab-compatible TRACE-DDE software can be freely

downloaded from https://users.dimi.uniud.it/˜dimitri.breda/.

As for QPMR approach, it aims to start directly with the characteristic equation

of the system. It ﬁrst decomposes the equation into real and imaginary parts, and

then computes the intersection points of the two parts to calculate the system eigen-

values in any region of interest on the complex plane. QPMR can be success-

fully implemented for studying a range of characteristic equations with single and

multiple delays; visit http://www.cak.fs.cvut.cz/vyhlidal for down-

load of Matlab-compatible source ﬁles. Last but not least, Lambert W functions

have been utilized to approximate the rightmost roots of delay differential equa-

tions. This approach carefully maps the inﬁnite dimensional eigenvalue problem

to a Lambert W function representation, for which efﬁcient solvers exist. Matlab-

compatible toolbox can be downloaded from http://www-personal.umich.

edu/˜ulsoy/TDS_Supplement.htm.

12.2 A General Model for Power Systems with Time-Delays

The conventional power system model used for solving voltage and transient stability

analyses consists of a set of Differential Algebraic Equations (DAEs) as follows [32]:

˙x=f(x,y,u)(12.1)

0=g(x,y,u)

where f(f:Rn+m+p7→ Rn) represents the differential equations, g(g:Rn+m+p7→

Rm) represents the algebraic equations, x(x∈Rn) the state variables, y(y∈Rm) the

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4Chapter 12

algebraic variables, and u(u∈Rp) are discrete variables modeling events, e.g., line

outages and faults.

The DDAE formulation is obtained by introducing time delays in (12.1). Con-

sider for now the single delay case, and let

xd=x(t−

τ

),yd=y(t−

τ

)(12.2)

be the delayed state and algebraic variables, respectively, where tis the current simu-

lation time, and

τ

(

τ

>0) is the time delay. In the remainder of this chapter, the main

focus is on small-signal stability analysis. In order to capture the fundamental roles

of time-delays on stability, they are assumed to be constant here, and this will also

allow us to beneﬁt from a number of analysis tools available for linear time-invariant

systems. Interested readers are also referred to the contributions by A. Halanay and

VL. Rasvan, see for example [33].

If some state or algebraic variables in (12.1) are affected by time-delay as repre-

sented in (12.2), one obtains:

˙x=f(x,y,xd,yd,u)(12.3)

0=g(x,y,xd,u)

which is the index-1 Hessenberg form of DDAE given in [34]. Note that gdoes not

depend on yd. This allows us to obtain a closed form expression for the small-signal

stability analysis and, as discussed in [27], (12.3) is adequate to model power system

models without loss of generality. Note that this assumption is usually satisﬁed in

physical systems including in power system since it is quite uncommon that the same

source of time-delay affects several system variables and, in particular, both state and

algebraic ones. Finally, it is straightforward to extend (12.3) to the multiple delay

case: it sufﬁces to deﬁne as many vectors of state and algebraic variables (12.2)

as the number of delays present in the system. For simplicity and without lack of

generality, in Subsections 12.2.1 and 12.3.2 we only consider the single-delay case.

12.2.1 Steady-State DDAE

Small-signal stability analysis deals with power system stability when it is subject to

small disturbances around its equilibrium points. For the model (12.3), assume that

a stationary solution of (12.3) is known and has the form:

0=f(x0,y0,xd0,yd0,u0)(12.4)

0=g(x0,y0,xd0,u0)

Note that in steady-state, xd0=x0and yd0=y0. Moreover, discrete variables u0are

assumed to be constant in the remainder of this chapter. Then, differentiating (12.3)

at the stationary solution yields:

∆˙x=fx∆x+fxd∆xd+fy∆y+fyd∆yd(12.5)

0=gx∆x+gxd∆xd+gy∆y(12.6)

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 5

Without loss of generality, we ignore singularity-induced bifurcation points and it

can be assumed that gyis non-singular. Substituting (12.6) into (12.5), one obtains:1

∆˙x=A0∆x+A1∆x(t−

τ

) + A2∆x(t−2

τ

)(12.7)

which described a retarded type delay differential equation since delay affects only

the states but not the state derivatives. In the above equation we have:

A0=fx−fyg−1

ygx(12.8)

A1=fxd−fyg−1

ygxd−fydg−1

ygx(12.9)

A2=−fydg−1

ygxd(12.10)

The ﬁrst matrix A0is the well-known state matrix that is computed for standard

DAEs of the form (12.1). The interested reader can refer to Appendix 12.6.1 for a

proof of (12.9) and (12.10). The other two matrices are not null matrices since the

system is of retarded type. The matrix A1is found in any delay differential equations,

while A2appears speciﬁcally in DDAEs, although it can be null if either fdoes not

depend on ydor gdoes not depend on xd. If one of the two conditions above are

satisﬁed, then (12.14) becomes:

∆(

λ

) =

λ

In−A0−A1e−

λτ

,(12.11)

which is the case considered in [27].

Equation (12.7) is a particular case of the standard variational form of the linear

delay differential equations:

∆˙x=A0∆x(t) +

ν

∑

i=1

Ai∆x(t−

τ

i),(12.12)

which is studied in various forms; see the references in Section 12.1.2. As we de-

scribed above, equation (12.12) describes a retarded-type system with multiple de-

lays if

ν

>1. In the special case of

ν

=1, this system is known to be of single-delay

type since all the states are affected by the same delay

τ

1. Substituting a sample so-

lution of the form e

λ

t

υ

, with

υ

being a non-trivial possibly complex vector of order

n, the characteristic equation of (12.12) can be stated as follows:

det ∆(

λ

) = 0,(12.13)

where

∆(

λ

) =

λ

In−A0−

ν

∑

i=1

Aie−

λτ

i,(12.14)

is called the characteristic matrix [14, 35]. In (12.14), Inis the identity matrix of

order n. The solutions of (12.14) are called the characteristic roots or spectrum,

similar to the ﬁnite-dimensional case of ordinary differential equations, i.e., the case

for which Ai=0,∀i=1,...,

ν

.

1The interested reader can ﬁnd in [27] the details on how to determine (12.8)-(12.10) from (12.5) and

(12.6).

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6Chapter 12

Similar to the ﬁnite-dimensional case (i.e.,

ν

=0), the stability of (12.12) can

be deﬁned based on the location of the roots of (12.14) on the complex plane; the

equilibrium point is stable if and only if all the roots have negative real parts, and

unstable otherwise [14, 15].

Equation (12.14) is transcendental and, hence, has inﬁnitely many roots. In gen-

eral, the explicit solution of (12.14) is not known and only a subset of this solution

can be approximated numerically, as will be discussed in Section 12.3.3. What is

critical in this approximation is to make sure that one approximates the solutions

that are relevant from stability and system performance points-of-view; that is, one

must approximate the dominant/rightmost roots of the system characteristic equa-

tion. Needless to say, this problem does not have a trivial solution; see also Section

12.1.2.

12.3 Numerical Techniques for DDAEs

This section introduces a number of techniques for the numerical analysis of DDAE.

The section begins with the well-known Pad´

e approximants, which allow reducing an

inﬁnite dimensinoal DDAE into a ﬁnite dimensional DAE. Then, the implementation

of the time domain integration schemes of DDAEs are brieﬂy discussed. Finally, the

section introduces three discretization techniques to approximate the spectrum of

DDAEs with inclusion of multiple delays.

12.3.1 Pad´

e Approximants

Pad´

e approximants are the most common and simplest implementation of time delays

for the numerical analysis of dynamical systems. Roughly speaking, Pad´

e approx-

imants allow representing time-delays through a set of linear ordinary differential

equations. The higher the order of such equations, the more precise the represen-

tation. Hence, DDAEs can be rewritten as a set of higher dimensional DAEs. The

rationale behind Pad´

e approximants is brieﬂy discussed below.

Firstly, let us recall the well-known time shifting property of the Laplace trans-

form:

f(t−

τ

)u(t−

τ

)L

−→ e−

τ

sF(s)(12.15)

where sis the Laplace variable obtained via the Laplace transform L, or it can be de-

ﬁned as the complex frequency;u(t)is the unit step function; and F(s)is the Laplace

transform of the function f(t). The approach based on Pad´

e approximants consists

of deﬁning a rational polynomial transfer function, say P(s), that approximates e−

τ

s.

Then, the inverse Laplace transform L−1allows obtaining the approximated time

domain function

ϕ

(t)that leads to an approximated solution of the DAE, such as the

one in (12.1):

e−

τ

sF(s)≈P(s)F(s)L−1

−→

ϕ

(t)(12.16)

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 7

Such an approximation can be obtained using the Taylor’s expansion of e−

τ

saround

τ

=0:

e−

τ

s=1−

τ

s+(

τ

s)2

2! −(

τ

s)3

3! +··· ≈ b0+b1

τ

s+···+bq(

τ

s)q

a0+a1

τ

s+···+ap(

τ

s)p(12.17)

where coefﬁcients a1,...,apand b1,...,bqare obtained by dividing the polynomials

of the right hand side of (12.17) and imposing that the ﬁrst p+qcoefﬁcients are the

same as those of the Taylor’s expansion [26]. Note that shas a different meaning

than

λ

in (12.14). In fact,

λ

takes an inﬁnite number of discrete values that solve

(12.14). Note that sis the continuous independent variable of the Laplace transform.

Generally, p≥qis imposed in (12.17). If p=q, the coefﬁcients aiand biare

obtained by the following iterative formula:

a0=1,ai=ai−1

p−i+1

i·(2p−i+1),bi= (−1)i·ai(12.18)

The case p=qis noteworthy as the amplitude of the frequency response of the Pad´

e

approximant is exact, only the phase is affected by an error. p=q=6 is a common

choice in numerical simulations.

The higher the order of the Pad´

e approximant, the lower the phase error (see, for

example the discussion on Pad´

e approximants in [4]). However, for small delays,

i.e. time-delays of the order of milliseconds (which are common in power systems),

there is no point in considering high order Pad´

e approximants. For example, let

p=9 and

τ

=10−3s. Then, one obtains a9=−b9=5.6679 ·10−11 and

τ

9=

10−27, which leads to a9·

τ

9=5.6679 ·10−38. This number is critically close to

the minimum positive value that can be represented by the single-precision binary

ﬂoating-point deﬁned by the IEEE 754 standard, i.e., 2−126 ≈1.18 ·10−38 . High

order Pad´

e approximants may also show unstable poles of defects (i.e., a pair of a

pole and a zero that are very close but not equal, see [26]). Hence, the ﬂoating point

representation binds the maximum value of pas pmax =qmax =10, which is the most

commonly used upper limit.

As an example on how to use Pad´

e in practice, let us obtain the Pad´

e approximant

for a unit step function u(t). For the sake of simplicity, we consider the case with

p=q. The approximant udof order p, in time domain, of u(t−

τ

)given by (12.18)

is as follows:

ud=˜x1+b1

τ

˜x2+···+bp−1

τ

p−1˜xp+bp

τ

p˙

˜xp(12.19)

where:

˙

˜xi=˜xi+1,i=1,2,...,p−1 (12.20)

and

ap

τ

p˙

˜xp=u−(a0˜x1+a1

τ

˜x2+···+ap−1

τ

p−1˜xp)(12.21)

Knowing these coefﬁcients, we can easily obtain the time-domain function of (12.17).

Note that the equations (12.19)-(12.21) are linear and introduces pstate variables

per each delay. Clearly, there is no limitation to the number of delays that can be

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8Chapter 12

included in the systems, and there is no structural difference between the single-delay

and the multiple-delay case. Moreover, since the system is approximated through a

DAE, conventional time domain integration and the small-signal stability analysis

can be used.

12.3.2 Numerical Integration of DDAEs

While Pad´

e approximants avoid the need to implement numerical methods that deal

with time delays, they also introduce numerical issues. A well-known issue is that

they may cause the birth of spurious high-frequency oscillations. Speciﬁc methods

to deal with time delays can thus be desirable. However, integrating DDAEs is not

an easy task. For example, despite being A-stable for standard DAE equations, the

implicit trapezoidal method may show numerical issues in case of DDAEs. Thus,

higher-order time integration methods for DDAE have been developed. Interested

readers can ﬁnd an excellent discussion on this topic in [36].

Although the general case can show interesting numerical issues, in this chapter

we focus only on the index-1 Hessenberg form (12.3). Furthermore, we only con-

sider implicit integration schemes, for two reasons: (i) they are the most adequate to

deal with stiff DAEs in general and power system models in particular; and (ii) most

state-of-the-art power system software tools implement implicit integration meth-

ods. We thus provide the modiﬁcations that are required to adapt a general implicit

integration scheme up to the second order to integrate (12.3).

We start with DAEs and then we will show how the method can be extended

to DDAEs. While using implicit methods, each step of the numerical integration is

obtained as the solution of a set of nonlinear equations. At a generic time t, and

assuming a step length h, one has to solve:

0=p(x(t+h),y(t+h),u(t+h),h)(12.22)

0=q(x(t+h),y(t+h),u(t+h),h)

where p, (p:Rn+m+p7→ Rn) and q, (q:Rn+m+p7→ Rm) are nonlinear functions that

depend on the DAE and on the implicit numerical method. In particular, paccounts

for differential equations, while qfor algebraic ones.

Since (12.22) are nonlinear, their solution is generally obtained using a direct

solver, e.g., Newton method, which in turn, consists of iteratively computing the

increments ∆x(i)and ∆y(i)and updating state and algebraic variables:

∆x(i)

∆y(i)=p(i)

xp(i)

y

q(i)

xq(i)

yp(i)

q(i)=−[A(i)]−1p(i)

q(i)(12.23)

x(i+1)(t+h)

y(i+1)(t+h)=x(i)(t+h)

y(i)(t+h)+∆x(i)

∆y(i)

For simplicity, we ignore the functional dependence on variables and iteration

indexes. Assuming that, a general expression for pand qthat is able to represent

the Backward Euler Method (BEM), the Implicit Trapezoidal Method (ITM) and

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 9

Backward Differentiation Formula (BDF) is as follows:

p=

ξ

−

β

h(f+

κ

ft)(12.24)

q=−g

where ftis known vector of differential equations at time tand

ξ

=x−

ν

∑

ℓ=1

γ

ℓx(t−(ℓ−1)h)(12.25)

and without a substantial loss of generality, we assume a constant step length hfor

x(t−(ℓ−1)h)values. The Jacobian matrix of (12.24) is given by:

A=In−

β

h f x−

β

h f y

−gx−gy(12.26)

The coefﬁcients

γ

ℓand

β

are computed according to a straightforward procedure

given in [37]. Table 12.1 summarizes the coefﬁcients for the BEM, ITM and order-2

BDF. BEM and order-2 BDF are L-stable, where BEM can be, in occasions, hy-

perstable, while ITM is A-stable. Implicit methods of order higher than two are

not considered in this chapter. A comprehensive discussion on implicit integration

schemes and their properties can be found in [38].

Table 12.1: Coefﬁcients of the order 1 and order 2 BDF and ITM

Scheme Order

γ

1

γ

2

β κ

BEM 1 1 - 1 0

BDF 2 4/3−1/3 2/3 0

ITM 2 1 - 0.5 1

To account for delays, we need to expand the set of pand q. Let us deﬁne two

general functional expressions:

0=

ϕ

(x,xd,t) = ˆx(

α

(x,t)) −xd(12.27)

0=

ψ

(y,yd,t) = ˆy(

β

(y,t)) −yd(12.28)

where

α

(x,t)and

β

(y,t)represent the functional dependence of state and algebraic

variables on the delays. For the constant time-delay (12.2), we have:

α

(x,t) = t−

τ

,

β

(y,t) = t−

τ

(12.29)

but, of course, more complex expressions can be considered [36].

Applying the same rule to (12.3) and using the functional equations (12.24),

(12.27) and (12.28), one obtains:

A=

In−

β

h f x−

β

h f y−

β

h f xd−

β

h f yd

gxgygxd0

ϕ

x0

ϕ

xd0

0

ψ

y0

ψ

yd

(12.30)

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10 Chapter 12

where the superscript ihas been omitted to simplify the notation. From (12.27) and

(12.28),

ϕ

xd=−Indand

ψ

yd=−Imdare negative identity matrices, while

ϕ

xand

ψ

y

can be obtained using the chain rule:

ϕ

x=diag˙

ˆx(x,t)

α

x(12.31)

ψ

y=diag˙

ˆy(y,t)

β

y(12.32)

where ˙

ˆx(x,t)and ˙

ˆy(y,t)are the rate of change of xand yat time

α

(x,t) = t−

τ

and

β

(y,t) = t−

τ

, respectively. While ˙

ˆx(x,t)is easy to obtain by simply storing ˙xduring

the time domain integration, ˙

ˆy(y,t)requires an extra computation, i.e., solving the

following equation at each time t:

0=gxf+gy˙y+gxd˙

ˆx

α

t(12.33)

from which ˙ycan be obtained, if gyis not singular, and stored. Observe that ˙ycan be

discontinuous.

The simple structure of the Jacobian matrices of

ϕ

and

ψ

allows rewriting (12.30)

as:

A=In−

β

h(fx+fxd

ϕ

x)−

β

h(fy+fyd

ψ

y)

gx+gxd

ϕ

xgy(12.34)

Equation (12.34) is general and can be used for various types of time-varying delays.

In the case of constant time delays, i.e., (12.2), it is straightforward to observe that

α

x=0 and

β

y=0 and, hence,

ϕ

x=0 and

ψ

y=0. Therefore, for constant delays,

(12.26) and (12.34) match. This result was expected since, at a given time t, both xd

and yd, i.e., state and algebraic variables delayed by

τ

, are constants.

12.3.3 Methods to Approximate the Characteristic Roots of DDAEs

As discussed above, a common approach to deﬁne the small-signal stability of the

Delay Differential-Algebraic Equations (DDAEs) is to use Pad´

e approximants, which

in turn lead to the study of stability of standard DAEs. In this section, we discuss

three other commonly-used numerical methods to approximate the eigenvalues of a

DDAE2. These are (i) a Chebyshev discretization scheme of equivalent partial dif-

ferential equations that resemble the original DDAEs; (ii) an approximation of the

time integration operator; and (iii) a linear multi-step discretization of the DDAEs

based on a high-order implicit time-integration scheme.

12.3.3.1 Chebyshev discretization scheme

This approach consists in transforming the original problem of computing the roots

of a retarded functional differential equation into a matrix eigenvalue problem of a

PDE system of inﬁnite dimension. No loss of information is involved in this step.

Then the dimension of the PDE is made tractable using a discretization based on a

ﬁnite element method.

2See also Section 12.1.2 for a list of techniques not reviewed here.

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 11

Consider the single-delay case ﬁrst. Let DNbe the Chebyshev differentiation

matrix of order N(see the Appendix 12.6.2 for details) and deﬁne

M=ˆ

C⊗In

A10... 0A0,(12.35)

where ⊗indicate the tensor product or Kronecker product (see Appendix 12.6.3 for

details), Inis the identity matrix of order n; and ˆ

Cis a matrix composed of the ﬁrst

N−1 rows of Cdeﬁned as follows:

C=−2DN/

τ

.(12.36)

Then, the eigenvalues of Mare an approximated spectrum of (12.11). As it can be

expected, the number of points Nof the grid affects the precision and the computa-

tional burden of the method, as it is further discussed in the case study.

The matrix Mis the discretization of a set of PDEs where the continuum is rep-

resented by the interval

ξ

∈[−

τ

,0]. The continuum is discretized along a grid of

Npoints and the position of such points are deﬁned by the Chebyshev polynomial

interpolation. The last nrows of Mimpose the boundary conditions

ξ

=−

τ

(i.e.,

A1) and

ξ

=0 (i.e., A0), respectively.

Figure 12.1 illustrates matrix (12.35) through a graphical representation. Each

element of the grid is an n×nmatrix and there are N2elements. Light gray blocks

are deﬁned by the Chebyshev discretization and are very sparse. Dark gray blocks

represent the state matrix A0and delayed matrix A1that appear in (12.11). Finally,

white blocks indicate zero matrices.

Figure 12.1: Representation of the matrix Mfor a system with a single delay

τ

and character-

istic equation (12.11).

Let us now consider the general multi-delay case of the characteristic equation

(12.14) and, thus, let us assume that there are

ν

delays, with

τ

1<

τ

2<···<

τν

−1<

τν

. Each point of the Chebyshev grid corresponds to a delay

θ

k= (N−k)∆

τ

, with

k=1,2,...,Nand ∆

τ

=

τν

/(N−1). Hence, k=1 corresponds to the state matrix A

ν

,

which corresponds to the maximum delay

τν

; and k=Nis taken by the non-delayed

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12 Chapter 12

state matrix A0. If a delay

τ

i=

θ

kfor some k=2,...,N−1, then the correspondent

matrix Aitakes the position kin the grid. Of course, in general, the delays of the

system will not match the points of the grid. For such cases, a linear interpolation

is considered in this chapter, as follows. Let the time-delay

τ

i,i̸=k, satisfy the

condition:

θ

k<

τ

i<

θ

k+1.(12.37)

Then, the matrices that will be added to the positions kand k+1 are respectively:

Ak,i=

τ

i−

θ

k

∆

τ

Ai,Ak+1,i=

θ

k+1−

τ

i

∆

τ

Ai.(12.38)

Next, the resulting matrix of each point kof the grid is computed as the sum of the

contributions of each delay that overlaps that point:

Ak=∑

i∈Ωi

Ak,i,(12.39)

where Ωiis the set of delays

τ

ithat satisﬁes (12.37). Other more sophisticated inter-

polation schemes can be used. For example, a Lagrange polynomial interpolation is

implemented in [39]. Figure 12.2 illustrates the Chebyshev discretization approach

for the multiple-delay case.

Figure 12.2: Representation of the Chebyshev discretization for a system with

ν

delays

τ

1<

τ

2<··· <

τν

−1<

τν

. In the general case, the delays do not exactly match the grid. Hence, an

interpolation between consecutive points of the grid is required.

12.3.4 Discretization of the Time Integration Operator (TIO)

The discretization of the time integration operator that is proposed in [39] is similar

to the approach above, but instead of deﬁning the discretization of a PDE, it directly

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 13

discretizes the set of original DDE equations. In the interest of clarity, ﬁrst consider

the single-delay case and the following system:

∆˙x(t) = A0∆x(t) + A1∆x(t−

τ

),(12.40)

which is obtained from (12.7) by assuming that A2=0. The algorithm includes

the following steps: (i) dividing the interval [−

τ

,0]into a mesh of Nintervals with

constant step size h=

τ

/N; and (ii) applying an integration scheme (e.g., a RK

method) to the mesh that approximate the continuous solution of (12.40). Then the

discrete counterpart of (12.40) is given by:

zi+1=SNzi,(12.41)

where z∈Rn·r·N, and SNis the following n·r·N×n·r·Nmatrix:

SN=

B00... 0B1

Inr 0... 0 0

0Inr ... 0 0

.

.

..

.

.....

.

..

.

.

0 0 ... Inr 0

,(12.42)

where

B0=R·(1reT

r⊗In),

B1=hR ·(A⊗A1),(12.43)

and

R= (Inr −hA⊗A0)−1,

1r= (1,...,1)T,

er= (0,...,0,1)T,

(12.44)

and Ais the matrix of the Butcher’s tableau that deﬁnes the integration scheme, as

follows:

C A

B=

c1a11 a12 .. . a1r

c2a21 a22 .. . a2r

.

.

..

.

..

.

.....

.

.

crar1ar2.. . arr

b1b2.. . br

(12.45)

and Inr is the identity matrix of order n·r. Note that Amust be invertible,

which means that an implicit scheme has to be used (e.g., BDF formula and Radau

methods).

The single-delay case can be extended to the multi-delay one by modifying the

ﬁrst row of the matrix SNin (12.42). Assume that there are

ν

delays, with

τ

1<

τ

2<

···<

τν

−1<

τν

. Then, the ﬁrst and the last elements of the ﬁrst row of (12.42) are

occupied by B0and B

ν

, where B0is deﬁned as in (12.43) and B

ν

is:

B

ν

=hR(hA0)(A⊗A

ν

).(12.46)

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14 Chapter 12

The state matrices associated with the remaining

ν

−1 delays are ﬁtted to the grid

through a linear interpolation similar to that described in Subsection 12.3.3.1. The

interested reader can ﬁnd in [39] a more general interpolation approach based on

Lagrange polynomials and a detailed discussion on the convergence properties of

this LMS discretization approach.

12.3.5 Linear Multi-Step (LMS) Approximation

Another possible discretization based on a linear multi-step approximation is the one

proposed in [24] and implemented in the software tool DDE-BIFTOOL. The time inte-

gration operator is discretized using a LMS method with polynomial interpolation to

evaluate the delayed terms. Applying a k-step LMS method to (12.12), one obtains:

k

∑

j=0

α

jxL+j=h

k

∑

j=0

β

jA0xL+j+

ν

∑

i=0

(Ai˜x(tL+j−

τ

i))(12.47)

where

α

jand

β

jare the coefﬁcients of the LMS method and ˜x(tL+j−

τ

i)are approx-

imations of the values of the state variables in past. These are computed using the

Nordsieck interpolation, as follows:

˜x(tp−

ε

h) =

σ

∑

ℓ=−

ρ

P

ℓ(

ε

)xp+ℓ,

ε

∈[0,1)(12.48)

where

P

ℓ=

σ

∏

k=−

ρ

,k̸=ℓ

ε

−k

ℓ−k(12.49)

The resulting method is explicit whenever

β

0=0 and min{

τ

i}>

σ

h. Further details

on this technique can be found in the DD E-BI FT OO L documentation and source code

[25].

The LMS-method forms an approximation of the time integration operator over

the time step h, hence the eigenvalues

µ

of the Jacobian matrix of (12.47) are an

approximation of the exponential transforms of the roots

λ

of (12.14):

µ

=exp(h

λ

)(12.50)

The size of the resulting eigenvalue problem is inversely proportional to the step

length hused in the discretization. The choice of his heuristic and is a critical aspect

of this technique. If the step length is too small, the size Kof the problem can be

huge, i.e., K≫n. If his too large the approximation of the roots of (12.14) might

not be accurate. The heuristic method for estimating hdescribed in [24] leads to

precise results although it might be conservative. Larger values of hcan be obtained

using the approach given in more recent works, e.g., [40]. A root is discarded if the

following condition is satisﬁed:

abs(

µ

j)>exp(h·max{

τ

i}),j=1,2,...,K(12.51)

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 15

12.4 Impact of Delays on Power System Control

Time-delays are commonly considered to have destabilizing effects on dynamical

systems. This is actually the most common effect in most power system applica-

tions and the case studies discussed in Section 12.5 largely conﬁrm this expectation.

However, many studies also demonstrated that delays do not necessarily destabilize

the system, see for example, [41, Chapter 11] [14, 20, 42]. This “duality” charac-

teristic of delay [7] has inspired numerous studies, whereby conditions under which

stabilization can be achieved were investigated.

Inspired by this duality characteristic, here the focus is on the selection of partic-

ular controllers, namely the Power System Stabilizers (PSS), affected by time-delays,

and to demonstrate how proper design of PSS despite the delay can stabilize the con-

trol loop, even if the delay is relatively large. This is in line with the above cited

studies, yet it requires one to take several steps to reveal the controller parameters in

the presence of delays. If carefully engineered, the stable closed loop can even pro-

duce desirable performance in the presence of delays. While this section deals with

a particular case on the equilibrium dynamics through linear stability analysis, the

discussion below allows drawing general conclusions, the most important of which

is, in our opinion, that nonlinear systems can always show unexpected behaviors.

Consider the well-known simpliﬁed electromechanical model of a synchronous

machine in steady state [43]:

2H˙

ω

=pm−pe(

δ

),(12.52)

where

ω

is the rotor speed, His the machine inertia constant, pmis the mechanical

power, and peis the electromagnetic power deﬁned as:

pe(

δ

) = e′

qv

x′

d

sin(

δ

−

θ

),(12.53)

where

δ

is the rotor angle, vand

θ

are the machine bus terminal voltage magnitude

and phase angle, respectively, e′

qis the internal fem, and x′

dis the d-axis transient

reactance. Differentiating (12.52) leads to:

2H∆˙

ω

=−

∂

pe

∂δ

∆

δ

−

∂

pe

∂

e′

q

∆e′

q−

∂

pe

∂

v∆v,(12.54)

which can be further simpliﬁed as follows. Without the PSS and assuming an integral

automatic voltage regulator, e′

qand vare constant, the above equation in Laplace

domain with Laplace variable sbecomes:

2Hs∆

ω

=−

∂

pe

∂δ

∆

δ

:=−K∆

δ

,(12.55)

where

K=e′

qv

x′

d

cos(

δ

0−

θ

0),(12.56)

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16 Chapter 12

and we denote with

δ

0and

θ

0the rotor and bus voltage phase angles, respectively, at

the equilibrium point. Since ∆

ω

=s∆

δ

, we obtain the characteristic equation of the

system as:

f(s) = s2+K

2H=0,(12.57)

which corresponds to an oscillator with roots on the imaginary axis of the complex

plane.

As it is well-known [43], the presence of a PSS control loop leads to a right-hand

side term in (12.57) proportional to e′

q∝s∆

δ

. Assuming that the feedback is affected

by a delay term

τ

, then the system characteristic equation becomes:

f(s,e−

τ

s) = s2+Ase−

τ

s+˜

K=0.(12.58)

where ˜

K=K/(2H),Ais proportional to the rotor-speed feedback-controller gain of

the PSS, and

τ

≥0 is the constant delay. Equation (12.58) can be interpreted as the

characteristic equation of a feedback control system where an open loop oscillator

dynamics with natural frequency √˜

Kis controlled only by a derivative controller

constructed based on delayed measurements of the output.

One can utilize the approaches surveyed in [20] to reveal the stability map of

(12.58) in the parameter space of Aversus

τ

. To summarize, this mapping is obtained

based on the following principles [14] [15]:

(a) the system poles move on the complex plane continuously with respect to

system parameters;

(b) the system stability is preserved as delay

τ

transitions from zero to 0+;

(c) the system may lose/recover stability only if at least one of its poles crosses

over the imaginary axis of the complex plane.

In light of the above items (a)-(c), the ﬁrst step is to calculate the “critical” val-

ues of Aand

τ

such that equation (12.58) produces imaginary roots s=j

ω

on the

complex plane, where one can assume

ω

>0 without loss of generality. These roots

are the eigenvalues of the system dynamics at hand, “possibly” corresponding to the

system transition from stable to unstable behavior. These critical parameter settings

can be solved from (12.58) by substituting s=j

ω

, which then reads

f(j

ω

,e−j

τω

) = −

ω

2+jA

ω

e−j

τω

+˜

K=0.(12.59)

Once the critical values of Aand

τ

satisfying (12.59) are solved, along with the

critical values of

ω

, one plots these critical points on

τ

versus Aplane, on which

countably many “regions” will form. That is, the critical values will bisect the pa-

rameter space

τ

−Ainto regions, where inside of each of these regions any choice of

τ

−Apairs will create a system dynamics with a “ﬁxed” number of unstable poles.

Inside of regions where this number is zero, the system will be labeled as stable,

otherwise unstable.

Identiﬁcation of stable and unstable regions requires a sensitivity analysis. This

is done mainly to compute how the eigenvalue s=j

ω

tends to change as a critical

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 17

parameter is slightly increased, while all the remaining parameters are kept ﬁxed

[20]. In some sense, this analysis will reveal whether or not the imaginary eigenvalue

tend to moves in the stabilizing direction toward the left-half complex plane, or in

the destabilizing direction toward the right-half complex plane. This information

will then help one reveal how the number of unstable eigenvalues change between

neighboring regions described above, sharing boundaries determined by the critical

parameters on the

τ

−Aplane.

The above described process to identify the number of unstable roots of a system

requires a starting point, which is often taken at the origin of the parameter space.

One thus calculates ﬁrst the number of unstable poles of the system for A=0 and

τ

=

0, then uses this information to calculate how many unstable poles the neighboring

regions have, in light of the sensitivity computations. Readers are also directed to

[20] for an overview of the state of the art in stability analysis.

In Figure 12.3, we present the stability map of the dynamics represented by

(12.59). The boundaries of this picture were obtained by decomposing (12.59) into

real and imaginary parts, and then solving for the common roots of these two parts.

The sensitivity of the imaginary roots ds/d

τ

is calculated simply by using the chain

rule on (12.59) at the critical points. The regions on

τ

−Aplane where the system

has no unstable poles are shaded for convenience, representing the stable operation

options of the system. This outcome is also validated using the TRACE-DDE tool-

box [29].

The parameter values to generate Figure 12.3 are: e′

q=1.8 pu, x′

d=0.8 pu,

vh=1.0 pu, H=2.0 s, and pm=1.0 pu. These parameters lead to K=2.0156 ≈2.0.

As expected, the delay-free system (

τ

=0) is stable for A>0, as well as for small

positive values of

τ

. Moreover, Figure 12.3 clearly shows that larger delays do not

necessarily destabilize the system as long as the corresponding gain Ais properly

adjusted3. These results are consistent with the famous results in [42] where authors

demonstrated how positive feedback with delay can stabilize an oscillator, which

cannot otherwise stabilize in the absence of delay.

12.5 Case Studies

In this section, we consider two examples. The well known IEEE 14-bus system is

presented ﬁrst to provide proof of concepts of the numerical techniques illustrated

above. The chapter is then completed by a comparative study on a large real-world

power system of the techniques to approximate the spectrum of DDAEs.

12.5.1 IEEE 14-bus System

The IEEE 14-bus system consists of two generators, three synchronous compen-

sators, two two-winding and one three-winding transformers, ﬁfteen transmission

lines, eleven loads and one shunt capacitor (see Fig. 12.4). The system also includes

generator controllers, such as the primary voltage regulators. All dynamic data of

3With this ﬁgure, we also make a correction in the stability map presented in [44].

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18 Chapter 12

Figure 12.3: Stability map on A-

τ

plane for ˜

K=2.0. The power system represented by (12.58)

is stable in the shaded regions.

this system as well as a detailed discussion of its transient behavior can be found

in [43].

In the numerical analyses presented below, we assume that the IEEE 14-bus sys-

tem includes a PSS connected to the generator 1. In typical PSSs, the input signal is

the synchronous machine rotor speed

ω

, which, in our formulation is a state variable.

Of course, in most cases, the rotor speed is measured locally, i.e., it is the rotor speed

of the machine where the PSS is installed. However, there exist Wide-Area Measure-

ment Systems (WAMS) where remote signals are used, e.g., the frequency of a pilot

bus [45]. Local measurements have at most a few milliseconds of delay while remote

measurements can be affected by a delay of up to 100 ms or more [45]. Hence, for

the sake of example, we consider that the input signal of the PSS of generator 1 is

obtained through a WAMS. A typical PSS control scheme includes a washout ﬁlter

and two lead-lag blocks. The resulting control scheme diagram of the PSS is shown

in Fig. 12.5. Observe that the DDAE that describes the PSS satisﬁes the index-1

Hessenberg form (12.3) where xd=

ω

(t−

τω

).

Thus the retarded measure of

ω

“propagates” into the PSS equations, as follows:

˙v1=−(Kw

ω

(t−

τω

) + v1)/Tw(12.60)

˙v2= ((1−T1

T2

)(Kw

ω

(t−

τω

) + v1)−v2)/T2

˙v3= ((1−T3

T4

)(v2+ ( T1

T2

(Kw

ω

(t−

τω

) + v1))) −v3)/T4

0=v3+T3

T4

(v2+T1

T2

(Kw

ω

(t−

τω

) + v1)) −vs

where v1,v2and v3are state variables introduced by the PSS washout ﬁlter and

by lead-lag blocks, and other parameters are illustrated in Fig. 12.5. Observe that

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 19

THREE WINDING

C

C

G

G

C

9

TRANSFORMER EQUIVALENT

8

7

4

9

C

GENERATORS

2

1

5

78

4

10

11

14

13

12

6

SYNCHRONOUS

COMPENSATORS

G

3

C

Figure 12.4: IEEE 14-bus test system. The system includes a PSS device connected to gener-

ator 1.

equations in (12.60) are in the form of (12.3) with x= (v1,v2,v3),xd=

ω

(t−

τω

),

and y=vs.

12.5.1.1 Steady-State Analysis and Delay Margin

From [46], it is known that the IEEE 14-bus system shows undamped oscillations

if the loading level is increased by 20% with respect to the base case and line 2-4

outage occurs.4It is also well-known that such oscillations can be properly damped

through the PSS shown in Fig. 12.5 in the excitation control scheme of the machine

connected to bus 1.

4See also Chapter 13 of this book for a comprehensive discussion on the limit cycles originated by

perturbing the IEEE 14-bus system.

eq4

eq3 eq5

eq6

Vsmax

Vsmin

Kw eq2

eq1 VsVSI

PSfrag replacements

Kw

ω

(t−

τω

)

vs

vmax

s

vmin

s

Tws

Tws+1

T1s+1

T2s+1

T3s+1

T4s+1

1

T

ε

s+1

Figure 12.5: Power system stabilizer control diagram [43].

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20 Chapter 12

Table 12.2: Critical eigenvalue of the IEEE 14-bus system for different values of

τω

τω

[s]

λ

c

τω

[s]

λ

c

0−1.76765 ±12.35628i 0.060 −0.25208 ±12.07820i

0.010 −1.49003 ±12.37513i 0.070 −0.05721 ±11.96773i

0.020 −1.21555 ±12.36373i 0.073 −0.00251 ±11.93290i

0.030 −0.95056 ±12.32437i 0.074 0.01534 ±11.92114i

0.040 −0.69982 ±12.26074i 0.075 0.03300 ±11.90932i

0.050 −0.46645 ±12.17721i 0.080 0.11851 ±11.84932i

In this subsection, we consider the bifurcation analysis for the IEEE 14-bus sys-

tem using as bifurcation parameter the time delay

τω

of the frequency signal that

enters into the PSS; and (ii) the loading level of the system. The parametrization

based on the system delay was proposed for the ﬁrst time in [47].

Table 12.2 shows the critical eigenvalue

λ

cof the IEEE 14-bus system as a func-

tion of the PSS frequency measure time delay

τω

, which is varied in the interval

[0,80]ms. Results shown in Table 12.2 are obtained using the Chebyshev discretiza-

tion discussed in Subsection 12.3.3.1. However, due to the small size of the IEEE

14-bus system, same results can be obtained with many different techniques, includ-

ing those discussed in Sections 12.3.1 and 12.3.3. Table 12.2 indicates that, as the

delay increases, the difference between the non-delayed DAE and the DDAE be-

comes quite evident. A Hopf Bifurcation (HB) occurs for

τω

≈73 ms, which is thus

the delay margin of the PSS. In other words, if the PSS is fed by a remote frequency

measurement signal, the communication system has a margin of 73 ms before the

PSS becomes unstable.

12.5.1.2 Time Domain Analysis

As discussed above, for a 20% increase of the loading level with respect to the base

case, a HB occurs for

τω

>73 ms. Repeating the same analysis for line 2-4 outage,

we observe that the HB occurs for

τω

≈68.6 ms. Thus, setting 73 >

τω

>69 ms, it

has to be expected that the transient following line 2-4 outage is unstable, while the

initial equilibrium point without contingency is stable, though poorly damped.

Figure 12.6 shows the time response of the IEEE 14-bus system without PSS,

with PSS and with retarded PSS with

τω

=71 ms. As already known from [46],

the trajectory of the system without PSS enters into a stable limit cycle after the

line outage while the system with PSS is asymptotically stable. The behavior of the

system with delayed PSS is similar to the case without PSS, i.e., presents a limit

cycle trajectory. This result was expected as can be justiﬁed as follows. For

τω

→∞,

the PSS control loop behaves like an open-loop system, which is unstable. Therefore,

there exists at least one critical delay value between zero and inﬁnity which causes

the zero-delay stable PSS to transition to instability. This critical delay value can

be computed via the small-signal stability analysis corresponding to HB. Moreover,

the added value of the time domain simulations is to show that the system trajectory

enters into a limit cycle rather than diverging.

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 21

The time domain simulations for the retarded system are obtained using an ITM,

adapted to include delays as discussed in Section 12.3.2. Same results can be ob-

tained using the Pad´

e approximants with p=q=6, which, as outlined in Section

12.3.1, is the standard order used in commercial software.

It has to be noted that, for ﬁnite-dimensional power system models (as in the case

of standard DAE), HBs, which are co-dimension one local bifurcations, are generic.

In other words, HBs are expected to occur given certain loading conditions and syn-

chronous machine controllers. However, the case of inﬁnite-dimensional dynamics

such as delay systems requires further analysis to conclude on the genericity of the

bifurcation points. This is currently an open ﬁeld of research.

Figure 12.6: Rotor speed

ω

of machine 5 for the IEEE 14-bus system with a 20% load increase

and for different control models following line 2-4 outage at t=1 s.

12.5.1.3 Small-signal Stability Map

Applying the approaches summarized above, the small-signal stability region for the

delayed IEEE 14-bus system is found as in Fig. 12.7. The shaded regions indicate

stable equilibria. The shape of the stable region is similar to the one of the simpliﬁed

system depicted in Fig. 12.3 only for small positive values of

τω

, but for larger values

of

τω

, there is noticeable difference. Moreover, inspecting Fig. 12.7, one ﬁnds out as

a general rule that in order to keep the IEEE 14-bus system in the stability region for

larger

τω

values, one should decrease Kw. Furthermore, there exists a region on the

stability map, corresponding to relatively large values of

τω

, for which the system

stability can still be maintained with the selection of negative values of the PSS gain

Kw. Finally, it is remarked that a properly damped response, assuming that 5% is an

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22 Chapter 12

adequate damping threshold, is attainable in this case, as indicated by the dark gray

region in Fig. 12.7.5

0.0 0.1 0.2 0.3 0.4 0.5

Delay τω[s]

−5.0

0.0

5.0

10.0

15.0

20.0

PSS controller gain Kw

Figure 12.7: Stability map of the Kw-

τω

plane for the IEEE 14-bus system. Shaded regions

are stable. Dark shaded regions indicate a damping greater than 5%.

As can be seen in the Figure 12.7, the stable region for

τω

∈(0.2,0.5)ms shows

a cusp for (

τω

,Kw)≈(0.3325,−5.067). This is a bifurcation point: the descending

branch remains stable for

τω

>0.3325, but the other branch cuts the stability region.

No stable points can be found for Kw<−5.067.

To solve the stability map shown in Fig. 12.7, the software Dome [48] has

been used, which implements the frequency domain approach discussed in Section

12.3.3.1. The number of points of the Chebyshev differentiation matrix is N=10,

which leads to a matrix Mof order 520 ×520. On a Dell Precision T1650 equipped

with 4-core Intel Xeon CPU 3.50GHz and 8 GB of RAM, the solution of the right-

most eigenvalue spectrum for each given point in the parameter space takes around

0.45 s. The small-signal stability boundaries has been calculated using a simple

bisection method with a tolerance of 10−3on the real part of the critical eigenvalues.

To properly determine the whole stable region, an eigenvalue analysis for a grid

of points in the rectangle deﬁned by

τω

∈[0,0.5]ms and Kw∈[−5,20]has been

carried out. Note that we have not detected other stable regions beside the one shown

in Fig. 12.7. However, even if such regions would exist under different parametric

settings, they would be “islanded” regions, and thus would be likely unreachable as

system trajectories could not get to such disjointed stable regions without passing

5Damping ratio

ζ

is approximated here as the negative cosine of the angle formed by the com-

plex vector deﬁned by the imaginary and real part of the stable rightmost complex eigenvalues z;

ζ

=−cos(arctan(ℑ(z)/ℜ(z))).

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 23

through an unstable path. Hence the stability region depicted in Fig. 12.7 is the only

one that has practical interest.

There are a number of practical implications of the delayed PSS which should be

noted.

•The system studied in Fig. 12.3 remains stable for 0 ≤

τ

≤1.11 s, provided

that the gain Ais properly adjusted. However, although stable, the response of

the system in terms of damping can be unacceptable for high values of

τ

. In

fact, as

τ

increases, Ahas to decrease to keep the system stable. In the limit

case

τ

→∞, PSS control loop behaves like an open loop system, and hence the

system transient behavior is driven by the sole synchronous machine, which is

generally poorly damped.

•In case of remote PSS input signals (see, for example, [45]), estimating the

value of time delay would allow properly tuning the gain Aso that the effect

of the delay on the system dynamic response could be minimized.

•The effect of delays depends on ˜

Kand thus on relevant parameters of the

synchronous machines such as pm,vhand e′

q. This fact can be taken into

account to deﬁne a proper tuning of Ain case of changes in the operating point

of the synchronous machine.

•To intentionally add delay to a control loop is generally not acceptable. How-

ever, the stability map shown in Fig. 12.7 suggests that, in case the mea-

sured PSS input signal is affected inevitably by a relatively large delay (e.g.,

τ

∈(0.1,0.3)s), then it could be convenient to introduce an additional delay,

and to accordingly change the control gain Kw, in order to improve the over-

all system small-signal stability. This adaptive control requires an estimation

of the delay and, apart from that, it can be easily implemented by means of

a look-up table based on the results obtained from the small-signal stability

analysis. Some results along the lines of “delay-scheduling” can be found

in [49].

12.5.2 All-island 1479-bus Irish System

The numerical techniques presented in Section 12.3 work satisfactorily for small

size systems, e.g., few tens of state variables and few tens of delays. For example,

basically identical results are obtained for the IEEE 14-bus system discussed in the

previous subsection regardless of the adopted technique for both time domain in-

tegration and small-signal stability analysis. Based on our experience, Chebyshev

discretization and Pad´

e approximants always provide good results for small systems.

TIO is also accurate provided that Nis increased with respect to the Chebyshev

method. For example, N=7 is acceptable for small power systems. On the other

hand, LMS provides good results if his relatively small. For example, h=0.1 s

appears acceptable for small power systems. However, standard benchmarks are too

small, not allowing us to draw sensible conclusions on the robustness and the accu-

racy of the techniques discussed in Section 12.2 for large scale eigenvalue problems.

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24 Chapter 12

Eigenvalue stiffness and the numerical rounding errors play a crucial role as the size

of the problem scales up, as shown in this section.

In this case study, the techniques in Section 12.3 are compared through a dy-

namic model of the all-island Irish transmission system set up at the UCD Electric-

ity Research Centre. The model includes 1,479 buses, 1,851 transmission lines and

transformers, 245 loads, 22 conventional synchronous power plants with AVRs and

turbine governors, 6 PSSs and 176 wind power plants. The topology and the data of

the transmission system are based on the actual real-world system provided by the

Irish TSO, EirGrid, but dynamic data are guessed and based on the knowledge of the

technology of power plants.

12.5.2.1 Small-Signal Sensitivity Analysis

The objective of this subsection is to compare the robustness of different methods

for the small-signal stability analysis of a large DDAE. With this aim, constant

time-delays are artiﬁcially included in most regulators of the all-island Irish sys-

tem, as follows. All bus terminal voltage measurements of the Automatic Volt-

age Regulators (AVRs) of the synchronous machines include delays in the range

τ

AVR ∈(5,15)ms [27]. The input frequency signal of PSS devices is delayed in the

range

τ

PSS ∈(50,250)ms [2]. The reheater of the turbine governors of thermal power

plants is modeled as a pure delay in the range

τ

RH ∈(3,11)s.

The model of some variable-speed wind turbines includes a frequency regulation

that receives as input the frequency of the center of inertia of the system. The model

of the frequency regulator is based on the transient frequency control described in

[50]. The frequency signal is assumed to be similar to those of PSS devices, hence

τ

TFC ∈(50,250)ms.

Finally, 20% of the loads are assumed to provide a frequency regulation. In

other words, 20% of loads are assumed to be equivalent thermostatically controlled

heating systems. The dynamic model of these loads and their control is based on [51]

and [52], respectively. Again, the input frequency signal is delayed and, in analogy

with PSS devices, delays are chosen in the interval

τ

TCL ∈(50,250)ms.

The delay ranges considered in this case study are summarized in Table 12.3. In

total, the system contains 296 delays ranging in the interval (0.005,11)s. This wide

range is chosen with the purpose of determining the accuracy and the performance of

the methods presented in Section 12.2. The resulting DDAE are stiff in terms of both

device and regulator time constants, which span a range from tens of milliseconds to

tens of seconds, and pure time-delays.

The order of the system, i.e., the number of state and algebraic variables, depends

on the model. Table 12.4 shows system statistics for four different models, namely,

no delay; constant delays; Pad´

e approximant with p=q=6; and Pad´

e approximant

with p=q=10. The only DDAE is the model where delays are implemented as

in (12.3), as Pad´

e approximants transform the delays into a set of linear differential

equations.

It is noteworthy that the DDAE is also the model with the lower number of vari-

ables. This is due to the fact that, in the standard model with no delays, delays are

actually modeled as a simple lag transfer function, each of which introduces a state

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 25

Table 12.3: Ranges of time-delays included in the all-island Irish system

Device Delayed Signal Delay Range [s]

Primary voltage regulator bus voltage

τ

AVR (0.005,0.015)

Power system stabilizer frequency

τ

PSS (0.05,0.25)

Reheater of steam turbines steam ﬂow

τ

TG (3,11)

Wind turbine freq. reg. frequency

τ

TFR (0.05,0.25)

Therm. controlled load frequency

τ

TCL (0.05,0.25)

Table 12.4: Number of variables for the all-island Irish system

Model Type State vars Algeb. vars

No delays DAE 2,239 7,478

Constant delays DDAE 1,935 7,338

Pad´

e approx. (p=q=6) DAE 3,415 7,929

Pad´

e approx. (p=q=10) DAE 4,399 7,929

variable. Note also that, the lag transfer function is, in turn, the Pad´

e approximant

with p=1 and q=0. Higher order Pad´

e approximants lead to a substantial increase

of the order of the system, and hence of the computational burden of the initialization

of system variables and time domain simulations.6Transient analysis is out of the

scope of this chapter but the latter remark has to be kept in mind when choosing the

power system models.

All simulations are obtained using Dome, a Python-based power system analysis

toolbox [48]. The Dome version used in this case study is based on Python 3.4.1,

NVidia Cuda 7.0, Numpy 1.8.2, CVX OP T 1.1.7, MAGMA 1.6.1, and has been exe-

cuted on a 64-bit Linux Fedora 21 operating system running on a two Intel Xeon 10

Core 2.2 GHz CPUs, 64 GB of RAM, and a 64-bit NVidia Tesla K20X GPU.

Table 12.5 shows the 20 rightmost eigenvalues for the all-island Irish transmis-

sion system using different system models and techniques. For reference, the ﬁrst

column also shows the 20 rightmost eigenvalues of the non-delayed model. This

system does not show any poorly damped mode, i.e., a mode whose damping is be-

low 5%. Columns 2-5 of Table 12.5 show the results obtained using the Chebyshev

discretization, the discretization of the Time Integrator Operator (TIO), the Linear

Multi-Step (LMS) approximation, and the Pad´

e approximants. Two cases are shown

for the latter, namely, p=q=6 and p=q=10. Both Chebyshev and TIO dis-

cretizations use a grid of order N=7. This number is considered a good trade-off

between accuracy and computational burden. The interested reader can ﬁnd further

details on the accuracy of the Chebyshev and TIO discretizations in [27] and [39],

respectively. For the discretization of the TIO, a ﬁfth order Radau IIA method is

6Pad´

e approximants also lead to increase the number of algebraic variables because the output udof

the approximated transfer function (12.17) is algebraic, as shown by (12.19).

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26 Chapter 12

Table 12.5: 20 rightmost eigenvalues for the all-island Irish system

No delay Chebyshev Discr. Discr. of TIO LMS Approx. Pad´

e Approx. Pad´

e Approx.

N=7N·r=21 h=0.2 s p=q=6p=q=10

−0.00010 −0.00010 −3.16992 0.91568 −0.00010 14370.508

−0.02500 −0.02500 −3.46994 0.82393 −0.02500 2166.5568

−0.02646 −0.02650 −3.54846 0.58361 −0.02848 1545.1549

−0.03780 ±0.32935i −0.03780 ±0.32935i −3.79015 0.36998 −0.03780 ±0.32935i 1540.3456

−0.05475 −0.05475 −3.79481 0.29701 −0.05475 1445.2436

−0.06615 −0.06100 ±0.32755i −3.85081 0.10980 −0.06615 1434.9052

−0.08759 ±0.10409i −0.06615 −3.86392 −0.00327 −0.08759 ±0.10409i 1019.4456

−0.11681 −0.08759 ±0.10409i −4.25558 −0.05199 −0.10759 ±0.33539i 891.50938

−0.12665 ±0.34150i −0.11445 ±0.78025i −4.33068 −0.09677 −0.11681 795.91920

−0.13055 ±0.17132i −0.11681 −4.52052 −0.13551 −0.12906 ±0.34552i 724.39851

−0.13922 −0.12818 ±0.34639i −4.68635 −0.15511 −0.13380 ±0.17103i 648.25856

−0.13950 −0.13455 ±0.17176i −4.80909 −0.23989 −0.13417 625.18431

−0.13978 −0.17139 −4.84030 −0.32102 −0.17474 ±0.27121i 593.37327

−0.14008 −0.17358 ±0.27051i −5.24457 ±0.35652i −0.34557 −0.17504 587.83144

−0.14027 −0.17504 −5.26514 −0.45854 −0.18411 ±0.78161i 533.95381

−0.14048 −0.18208 ±0.81259i −5.67946 ±0.81568i −0.55539 −0.18562 528.11686

−0.14062 −0.18316 ±0.81807i −5.74580 −0.67482 −0.18892 519.93536

−0.14081 −0.18562 −5.80760 −0.73128 −0.20000 497.91600

−0.14104 −0.18877 ±0.81637i −5.98648 −0.95327 −0.20483 ±0.87988i 456.93850

−0.14119 −0.18892 −6.10122 −0.97517 −0.20944 ±0.36519i 420.89130

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 27

Table 12.6: Computational burden of different methods to compute eigenvalues using GPU-

based Magma library

Model Settings Matrix setup Matrix order LEP sol.

No delays 1.18 s 2,239 11.91 s

Cheb. discr. N=7 29.4 s 13,545 12.69 m

Discr. of TIO N·r=21 7.07 h 40,635 50.73 s

LMS approx. h=0.2 s 7.48 m 32,895 20.83 s

Pad´

e approx. p=q=6 2.01 s 3,415 35.21 s

Pad´

e approx. p=q=10 2.78 s 4,399 76.75 s

used, with r=3 along with the following Butcher’s tableau:

2

5−√6

10

11

45 −7√6

360

37

225 −169√6

1800 −2

225 +√6

75

2

5+√6

10

37

225 +169√6

1800

11

45 +7√6

360 −2

225 −√6

75

14

9−√6

36

4

9+√6

36

1

9

4

9−√6

36

4

9+√6

36

1

9

Finally, an Adams-Bashforth 6th order method is used for the LMS approximation,

with the following coefﬁcients:

α

= [1,−1,0,0,0,0]

β

= [0,1901/720,−1387/360,109/30,−637/360,251/720]

A time step h=0.2 s is used for the LMS approximation.

To complete the comparison of the four techniques whose results are provided

in Table 12.5, Table 12.6 shows the computational burden of these techniques using

the GPU-based MAGM A library. The information given in Table 12.6 is the time

required to setup the full matrix for the eigenvalue analysis, the order of this matrix,

and the time required to solve the full Linear Eigenvalue Problem (LEP).

While all methods are necessarily approximated, the most accurate method to

estimate the spectrum of the DDAE can be expected to be the one based on Cheby-

shev discretization scheme. As indicated in [39], in fact, this approach shows a fast

convergence. Moreover, simulation results on large scale systems indicate that the

Chebyshev discretization does not require Nto be high [27]. The accuracy of other

methods can be thus deﬁned based on a comparison with the results obtained through

the Chebyshev discretization method. As shown in Table 12.6, the lightest compu-

tational burden is provided by Pad´

e approximants. However, the solution obtained

with p=q=6 shows some differences with respect to the Chebyshev discretization,

e.g., two poorly damped modes, namely −0.26201±6.3415iand −0.2746 ±5.9609i

do not appear in the solution based on the Chebyshev discretization. Both modes

show a damping lower than 5% and, through the analysis of participation factors,

both are strongly associated with ﬁctitious state variables introduced by the Pad´

e ap-

proximant, e.g., (12.20) and (12.21). This effect has to be expected as extraneous

oscillations are a well-known drawback of Pad´

e approximants. Finally, observe that

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28 Chapter 12

for the Pad´

e approximant with p=q=10, results are fully unsatisfactory due to

numerical issues. These are due to the extremely small values taken by coefﬁcients

in (12.19) and (12.21). For the considered case study, numerical problems show up

for p=q≥8.

Tables 12.5-12.6 also show that the techniques based on the TIO discretization

and LMS approximation are both highly inaccurate and time consuming. In particu-

lar, the TIO discretization requires a huge time to setup up the matrix SNof (12.42). It

is likely that the implementation of the algorithm that builds SNcan be improved us-

ing parallelization, which was not exploited in this case. However, the inaccuracy of

the results makes improving the implementation of this technique unnecessary. Note

also that the size of the computational burden of the LMS approximation strongly

depends on the time step hused in (12.47). The smaller the time step h, the more

precise the approximation, but the higher the computational burden. However, for

h<0.2, the size of the eigenvalue problem becomes too big and the MAGMA solver

fails returning a memory error. Unfortunately, in this case, h=0.2 s is too large to

obtain precise results.

A rationale behind the poor results shown by the TIO and LMS methods com-

pared to the Chebyshev discretization and Pad´

e method, is as follows. The Cheby-

shev discretization method works directly with (12.14) and is concerned solely with

approximating the e−

λτ

iterms. As such, the quality of the results depends only on

the quality of this approximation. In the same vein, Pad´

e approximants are con-

cerned with the approximation of e−s

τ

iterms in (12.15). On the other hand, the TIO

and LMS methods work with (12.12) and require approximations of both the ˙xterm

and the delay terms. Hence, the TIO and LMS approximations require a step size

small enough that the resulting difference equation is stable.

12.5.2.2 Time Domain Simulation

This section compares the numerical robustness of the time integration schemes for

DDAEs discussed in Section 12.3.2 as well as the DAE model based on Pad´

e ap-

proximants. With this aim, Fig. 12.8 shows the speed of the Center of Inertia (CoI)

for the all-island Irish system with stochastic wind speeds and a short-circuit occur-

ring at bus 1238 at t=1 s and cleared after 50 ms by removing line 1633. ITM and

BDF schemes provide essentially the same results using a time step of 0.01 s. BEM

results are consistent with those of ITM and BDF but slightly more damped, as was

expected due to the latent hyperstability of the BEM. On the other hand, the results

obtained with a DAE based on Pad´

e approximants with p=q=6 and ITM are not

satisfactory. The trajectory appears unstable and starts diverging even before the oc-

currence of the fault. Note that this divergence is caused by numerical instabilities.

In fact, the coefﬁcients of the Pad´

e approximants given by (12.18) make the Jacobian

matrix of the resulting DAE ill-conditioned. Similar to the case of small-signal sta-

bility analysis discussed in the previous section, such ill-conditioning does not occur

for relatively small systems with a reduced number of state variables, e.g., the IEEE

14-bus system and most IEEE dynamic benchmarks.

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 29

0.0 1.0 2.0 3.0 4.0 5.0

Time [s]

0.994

0.995

0.996

0.997

0.998

0.999

1.0

1.001

ωCOI [pu]

ITM and BDF

BEM

Pad´e (p=q= 6)

Figure 12.8: Speed of the Center of Inertia (CoI) for the all-island Irish system with stochastic

wind speeds and a short-circuit occurring at bus 1238 at t=1 s and cleared after 50 ms by

removing line 1633.

12.6 Appendices

12.6.1 Determination of A0, A1and A2

This Appendix describes how (12.8)-(12.10) are determined based on (12.5)-(12.6).

From (12.6), one obtains:

∆y=−g−1

ygx∆x−g−1

ygxd∆xd(12.61)

Substituting (12.61) into (12.5) one has:

∆˙x= ( fx−fyg−1

ygx)∆x+(12.62)

(fxd−fyg−1

ygxd)∆xd+

fyd∆yd

In (12.62), one has still to substitute ∆ydfor a linear expression of the actual and/or

of the retarded state variable. To do so, consider the algebraic equations gcomputed

at (t−

τ

). Since algebraic constraints have to be always satisﬁed, the following

steady-state condition must hold:

0=g(x(t−

τ

),xd(t−

τ

),y(t−

τ

)) (12.63)

Then, observing that xd=x(t−

τ

),yd=y(t−

τ

), and xd(t−

τ

) = x(t−2

τ

), differ-

entiating (12.63) leads to:

0=gx∆xd+gxd∆x(t−2

τ

) + gy∆yd(12.64)

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30 Chapter 12

In steady-state, for any instant t0,x(t0) = x(t0−

τ

) = x(t0−2

τ

) = x0and y(t0) =

yd(t0) = y0. Hence, the Jacobian matrices in (12.64) are the same as in (12.6). Equa-

tion (12.64) can be rewritten as:

∆yd=−g−1

ygx∆xd−g−1

ygxd∆x(t−2

τ

)(12.65)

and, substituting (12.65) into (12.62), one obtains:

∆˙x= ( fx−fyg−1

ygx)∆x+(12.66)

(fxd−fyg−1

ygxd−fydg−1

ygx)∆xd+

(−fydg−1

ygxd)∆x(t−2

τ

)

which leads to the deﬁnitions of A0,A1and A2given in (12.8), (12.9) and (12.10),

respectively.

12.6.2 Chebyshev’s Differentiation Matrix

Chebyshev’s differentiation matrix DNof dimensions (N+1)×(N+1)is deﬁned

as follows. Firstly, one has to deﬁne N+1 Chebyshev’s nodes, i.e., the interpolation

points on the normalized interval [−1,1]:

xk=cosk

π

N,k=0,...,N.(12.67)

Then, the element (i,j)differentiation matrix DNindexed from 0 to Nis deﬁned as

D(i,j)=

ci(−1)i+j

cj(xi−xj),i̸=j

−1

2

xi

1−x2

i

,i=j̸=1,N−1

2N2+1

6,i=j=0

−2N2+1

6,i=j=N

(12.68)

where c0=cN=2 and c2=··· =cN−1=1. For example, D1and D2are:

D1=0.5−0.5

0.5−0.5,with x0=1,x1=−1.

and

D2=

1.5−2 0.5

0.5 0 −0.5

−0.5 2 −1.5

,with x0=1,x1=0,x2=−1.

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Small-signal Stability and Time Domain Analysis of Delayed Power Systems 31

12.6.3 Kronecker’s Product

If Ais a m×nmatrix and Bis a p×qmatrix, then Kronecker’s product A⊗Bis an

mp ×nq block matrix as follows:

A⊗B=

a11B··· a1nB

.

.

.....

.

.

am1B··· amn B

(12.69)

For example, let A=1 2 3

3 2 1 and B=2 1

2 3 . Then:

A⊗B=B2B3B

3B2B B =

214263

234669

634221

694623

Note that A⊗B̸=B⊗A.

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32 Chapter 12

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