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Small-signal stability and time-domain analysis of delayed power systems

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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., Hcontrol) 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(xRn) the state variables, y(yRm) the
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4Chapter 12
algebraic variables, and u(uRp) 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=fxx+fxdxd+fyy+fydyd(12.5)
0=gxx+gxdxd+gyy(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=A0x+A1x(t
τ
) + A2x(t2
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=fxfyg1
ygx(12.8)
A1=fxdfyg1
ygxdfydg1
ygx(12.9)
A2=fydg1
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
InA0A1e
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=A0x(t) +
ν
i=1
Aix(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
InA0
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 L1allows 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)L1
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, pqis imposed in (12.17). If p=q, the coefficients aiand biare
obtained by the following iterative formula:
a0=1,ai=ai1
pi+1
i·(2pi+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
=103s. Then, one obtains a9=b9=5.6679 ·1011 and
τ
9=
1027, which leads to a9·
τ
9=5.6679 ·1038. 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., 2126 1.18 ·1038 . 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+···+bp1
τ
p1˜xp+bp
τ
p˙
˜xp(12.19)
where:
˙
˜xi=˜xi+1,i=1,2,...,p1 (12.20)
and
ap
τ
p˙
˜xp=u(a0˜x1+a1
τ
˜x2+···+ap1
τ
p1˜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
=1
γ
x(t(1)h)(12.25)
and without a substantial loss of generality, we assume a constant step length hfor
x(t(1)h)values. The Jacobian matrix of (12.24) is given by:
A=In
β
h f x
β
h f y
gxgy(12.26)
The coefficients
γ
and
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 4/31/3 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
CIn
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
N1 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= (Nk)
τ
, with
k=1,2,...,Nand
τ
=
τν
/(N1). 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,...,N1, 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=
ii
Ak,i,(12.39)
where iis 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) = A0x(t) + A1x(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 zRn·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
rIn),
B1=hR ·(AA1),(12.43)
and
R= (Inr hAA0)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)(AA
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
=
ρ
P
(
ε
)xp+,
ε
[0,1)(12.48)
where
P
=
σ
k=
ρ
,k̸=
ε
k
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., Kn. 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
=pmpe(
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
vv,(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
qs
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
,ej
τω
) =
ω
2+jA
ω
ej
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= ((1T1
T2
)(Kw
ω
(t
τω
) + v1)v2)/T2
˙v3= ((1T3
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
01.76765 ±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 103on 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((z)/(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
56
10
11
45 76
360
37
225 1696
1800 2
225 +6
75
2
5+6
10
37
225 +1696
1800
11
45 +76
360 2
225 6
75
14
96
36
4
9+6
36
1
9
4
96
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=q8.
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 es
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=g1
ygxxg1
ygxdxd(12.61)
Substituting (12.61) into (12.5) one has:
˙x= ( fxfyg1
ygx)x+(12.62)
(fxdfyg1
ygxd)xd+
fydyd
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(t2
τ
), differ-
entiating (12.63) leads to:
0=gxxd+gxdx(t2
τ
) + gyyd(12.64)
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30 Chapter 12
In steady-state, for any instant t0,x(t0) = x(t0
τ
) = x(t02
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=g1
ygxxdg1
ygxdx(t2
τ
)(12.65)
and, substituting (12.65) into (12.62), one obtains:
˙x= ( fxfyg1
ygx)x+(12.66)
(fxdfyg1
ygxdfydg1
ygx)xd+
(fydg1
ygxd)x(t2
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(xixj),i̸=j
1
2
xi
1x2
i
,i=j̸=1,N1
2N2+1
6,i=j=0
2N2+1
6,i=j=N
(12.68)
where c0=cN=2 and c2=··· =cN1=1. For example, D1and D2are:
D1=0.50.5
0.50.5,with x0=1,x1=1.
and
D2=
1.52 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 ABis an
mp ×nq block matrix as follows:
AB=
a11B··· a1nB
.
.
.....
.
.
am1B··· amn B
(12.69)
For example, let A=1 2 3
3 2 1 and B=2 1
2 3 . Then:
AB=B2B3B
3B2B B =
214263
234669
634221
694623
Note that AB̸=BA.
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32 Chapter 12
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