Download Harmonic/State Model Order Reduction of Nonlinear Networks

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Transactions on Power Delivery
Harmonic/State Model Order Reduction of Nonlinear
Networks
Abner Ramirez, Senior Member, IEEE and J. Jesus Rico, Member, IEEE
Abstract—This paper presents a practical approach to obtain
reduced-order models for electric networks expressed in the
dynamic harmonic domain (DHD). The proposed method reduces
the DHD system of ordinary differential equations first in the
harmonic sense via a voltage-based frequency-scan procedure
and subsequently in the state sense via truncated balanced
realizations. The harmonic-selection and truncation stage
substantially reduces the original full-size system. The statereduction process is applied to the already harmonic-reduced
system, achieving further reduction. Two networks are used as
case studies to demonstrate the reduction properties of the
proposed model order reduction methodology.
Keywords: Electromagnetic transient analysis, frequencydomain analysis, harmonic analysis, reduced order systems.
M
I. INTRODUCTION
ODEL order reduction (MOR) methods provide a
reduced-order model from an original full-size model of
a system. The reduced-order model has to fulfill three major
characteristics: a) it should reproduce the principal dynamics
of the original full-order system with high fidelity, b) it should
have a reduced number of state variables compared to the fullsize system, and c) system simulations have to be
computationally more efficient than in the original system [1][2].
Formerly, MOR techniques were developed for linear
time-invariant (LTI) systems. These techniques can be broadly
classified as: i) proper orthogonal decomposition (POD)
methods [3], ii) truncated balanced realization (TBR) methods
[4], and iii) Krylov subspace methods [5].
Krylov-based methods have been the basis to reduce linear
time-varying (LTV) systems [6]-[7]. However, reducing a
nonlinear system represents a challenge and remains as open
topic.
Several of the above mentioned techniques have been
extended to reduce nonlinear systems via linearization, Taylor
polynomial expansion, bilinearization of the nonlinearity, and
representation of the nonlinearity as functional Volterra series
expansion [2]. These approximations, which are limited to
quadratic expansions in practical applications, enclose the
reduced-order systems to be valid around the assumed
operating point. Also, linearized reduced-order models are
only applicable to weakly nonlinear systems. Some POD- and
TBR-based methods have been proposed to reduce nonlinear
systems [8], [9], [10]. As drawback, these methods lack of an
efficient representation of nonlinear components in the
reduced-order model. Also, the reduced model is obtained at
high cost due to the necessity of computing snapshots which
A. Ramirez is with CINVESTAV campus Guadalajara, Mexico, 45019 (email: [email protected]).
J.J. Rico is with Universidad Michoacana de San Nicolas de Hidalgo, Mexico
(e-mail: [email protected])
involve time domain simulations of the original full-order
system [2]. To partially overcome the weakly nonlinearity
condition, nonlinear systems are approximated by a suitable
weighted combination of linearized models. The procedure to
generate a quasi-piecewise-linear model of the nonlinear
system has been named trajectory piecewise-linear (TPWL)
method [2].
MOR techniques, although initially conceived for control
systems, have been recently expanded to several areas of
engineering [11]-[12]. They also have found interesting
applications in the power systems area [13]-[18]. An
additional application proposed in this paper is the MOR of
nonlinear networks involving harmonic dynamics.
In power systems, nonlinear loads act as frequency
converters. The generated frequencies penetrate a network and
produce waveform distortion and potential resonance
conditions. With the aim of analyzing dynamic behavior of
harmonic frequencies in transient conditions, the network can
be modeled in the dynamic (extended) harmonic domain
(DHD) [19]. A major drawback of DHD models is that the
dimensions of the state-space system increases according to
the number of harmonics involved in the analysis.
This paper presents a practical (part heuristic)
methodology to reduce the size of a nonlinear DHD system in
both harmonic and state senses. The harmonic reduction is
achieved via a voltage-based frequency-scan procedure in
which the response of the linear part of the system, due to
injection of unitary voltages of distinct frequencies to the
nonlinear load terminals, is computed and numerically
analyzed. Once the maximum harmonic is defined by the
frequency-scan procedure, the harmonic-reduced system is
grouped into linear and nonlinear parts. The linear subsystem
is then reduced in the state sense via TBR [4]. It is shown, via
numerical experiments, that the first reduction (in harmonics)
provides a sufficiently accurate model leaving the state
reduction as optional. The used frequency-scan procedure is
similar to the computation of equivalent admittance matrix,
viewed from a specific bus, used in filter design, iterative
harmonic analysis, and frequency-dependent equivalents [20].
In this paper, the main objective of the frequency-scan
procedure is to calculate the harmonic contribution of the
DHD state vectors aimed to MOR.
TBR relies on the analysis of principal components;
singular value decomposition (SVD) is the fundamental
computational machinery [4], [21]. The backbone of TBR is to
transform the original system to a new coordinate system
where it becomes internally balanced. This is achieved by
making the system’s controllability and observability
Gramians equal and corresponding to a diagonal matrix. The
diagonal matrix contains the Hankel singular values, which
represent its degree of controllability/observability. The
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Transactions on Power Delivery
balanced transformation permits to obtain a reduced-order
system via direct truncation of the balanced system [1], [4].
The proposed methodology permits to reduce nonlinear
systems valid for the complete operating range. Also, there is
no need to simulate the full-size system to obtain snapshots, as
in traditional techniques. The preliminary harmonic reduction,
via the sweeping-type procedure, is efficiently performed
through the evaluation of the HD transfer function via a
frequency-by-frequency solution scheme. The final
harmonic/state reduced system results in an efficient model for
dynamic harmonic analysis.
II. DYNAMIC HARMONIC DOMAIN FUNDAMENTALS
The DHD technique considers slowly (within a
fundamental period) time-varying Fourier coefficients
arranged in matrix/vector format. This consideration permits
to convert an instantaneous-variable state-space system into a
harmonic-variable state-space system.
Without loss of generality, consider the LTI scalar system
x  ax  bu ,
y  cx  du .
The DHD defines variable x in (1) as
x(t )  x h (t )e jho t    xo (t )    xh (t )e jho t ,
(1a)
(1b)
(2a)
and its corresponding derivative with respect to time as
x (t )  x h (t )e  jho t  jho x h (t )e  jho t    xo (t )  
 xh (t )e jhot  jho xh (t )e jhot
. (2b)
where h represents the highest harmonic under analysis and o
the fundamental frequency. Application of definition (2) into
(1) results, after equating coefficients of same exponential
terms, into (with some abuse of notation)
x  Sx  ax  bu ,
y  cx  du ,
(3a)
(3b)
where variables are now complex-valued vectors with timevarying coefficients, e.g.,
x  [ x h (t )  xo (t )  xh (t )]T ,
(4a)
where T denotes transpose. S is called the operational matrix
of differentiation defined by [19], [22]
S  diag{ jho , ,  jo , 0, jo , , jho },
(4b)
The evolution of harmonics with respect to time can be
obtained from (3) and the corresponding instantaneous values
are calculated by assembling a Fourier series as in (2a). The
steady-state of the dynamic system is readily computed by
setting to zero the derivative in (3).
III. HARMONIC MODEL ORDER REDUCTION
To describe the first part of the proposed method, consider
a general network, represented as in Fig. 1. The enclosed part
of the network can involve RLC elements with a given
topology. Alternatively, it can consist of a network
approximated by rational functions interpreted as a set of
ordinary differential equations (ODEs). As input, the network
has harmonic voltage and current sources, and nonlinear loads;
see Appendix A for basic time-domain and frequency-domain
relations of nonlinear loads considered in this paper.
Based on the network of Fig. 1 and considering the
nonlinear loads represented as current injections, the following
LTI system can be obtained (the D-term is for convenience
omitted):
(1a)
x  Ax  Bu ,
y  Cx ,
(1b)
where input u contains both known harmonic voltage/current
sources and unknown nonlinear injected currents.
The LTI system (1) can be readily transformed to the
harmonic domain (HD) [22] resulting in (without change of
notation)
(2a)
Sx  Ax  Bu ,
y  Cx ,
(2b)
where S corresponds to a set of n (where n represents the
number of HD states) differentiation matrices arranged in
block-diagonal form. The kth differentiation diagonal matrix
has the form as in (4b) [22]. From (2), the HD input/output
relation results in:
(3)
y  C ( S  A) 1 Bu ,
The sweeping-type procedure relies on calculating the
output y based on the currents injected by nonlinear loads.
This is achieved by applying voltages of the type
v (t )  sin( h o t ) , for h = 1, 2, …hmax, at all nonlinear loads
terminals. This procedure is described with more detail in
Appendix A. The sweeping procedure is computationally
implemented in a frequency-by-frequency fashion, resulting in
an efficient computation. It is mentioned that distinct
magnitudes of applied voltage to nonlinear load terminals
provide different harmonic current magnitudes. The
application of the unitary voltages is justified by the fact that
the examples in this paper utilize unitary sources and per unit
values. For different applied voltage amplitudes, the
magnitude of the truncation plane in the sweeping-type
procedure has to be modified accordingly to obtain proper
results.
Assuming C an identity matrix, the outlined frequencydomain sweeping procedure provides the magnitude of all HD
state variables. Fig. 2 shows illustrative numerical results for
case study 1, Section IV. In Fig. 2a, the x-axis presents the
harmonic states, each with 2hmax+1 harmonic frequencies
(considering positive, negative, and DC frequencies), the yaxis indicates the injected harmonics, and the z-axis shows the
corresponding obtained output, or, equivalently, the
magnitudes of all states. Fig. 2b presents the obtained output
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Transactions on Power Delivery
of all harmonic states when v (t )  sin(5 o t ) is applied to
nonlinear loads terminals.
IV. MODEL ORDER REDUCTION VIA BALANCED
REALIZATIONS
This second part permits to further reduce the nonlinear
system, in the state sense, by grouping linear and nonlinear
parts and by applying the TBR technique [2] to the linear part
of the system. Details follow.
The base system corresponds to the already harmonicreduced set of ODEs expressed in the DHD and separated as
follows:
Fig. 1. Illustrative general network.
 x L    AL
 x NL   A21
A12   x L   BL  ,
u

ANL   x NL   BNL 
x
y  C L C NL   L  ,
 x NL 
(4a)
(4b)
harmonic state 1
2
harmonic state 12
y (pu)
1.5
1
25
0.5
20
15
0
600
10
500
400
300
states
5
200
100
0
0
injected harmonics
(a)
1.8
0.2
harmonic state 1
y (pu)
1.6
where subscript b denotes balanced variables. According to
the TBR technique, and based on the magnitudes of the
Hankel singular values provided by the transformation in (15),
system (5) is truncated to an order r, becoming in
0.1
1.4
0
0
100
y (pu)
1.2
200
300
states
400
500
1
0.8
x Lr  ALr x Lr  A12 r x NL  BLr u .
0.6
0.4
(6)
harmonic state 12
truncation criterion line
0.2
0
0
where AL, A12, and A21 are constant matrices; xL and xNL
contain harmonic vector states corresponding to linear and
nonlinear state variables, respectively; u contains now known
harmonic voltage/current sources only. The linear subsystem
of (4), i.e., the triplet (AL, BL, CL), is balanced via the
similarity transformation xb = Tbx, as proposed by Moore [4],
see Appendix B. Applying Tb to the first subsystem of (4)
results in
(5)
x Lb  ALb x Lb  A12 b x NL  BLb u ,
100
200
300
states
400
500
600
(b)
Fig. 2. Frequency-domain sweeping results for case study 1 (a) injecting
harmonics 1st to 23rd and (b) injecting 5th harmonic only.
Based on the obtained output y, we can conclude that not
all harmonics penetrating into the nework present a substantial
magnitude and we can establish a criterion for choosing the
maximum harmonic for the harmonic-reduced system. This
criterion consists on assuming a truncation plane in Fig. 2a
and selecting all harmonics that contribute to state magnitudes
above that plane. This criterion is shown in Fig. 2b as a line.
A frequency-domain sweeping-type method is proposed in
[23]. The method in [23] is based on representing the linear
network by a Thévenin equivalent, injecting harmonic
voltages, and monitoring the interaction of those injections
with the Thévenin equivalent. The first reduction step of the
proposed method can be applied to the following cases: a)
when the linear part of the network is represented as a
Thévenin equivalent seen from the nonlinear loads terminals
and b) when a rational approximation, expressed as in (1),
represents the linear part of the network. This paper’s proposal
takes advantage of b) to incorporate a second reduction
process via TBR.
Similarly, for the second subsystem of (4) we obtain the
reduced system
(7)
x NL  A21r x Lr  ANL x NL  BNL u .
Combining (6) and (7), the final harmonic/state reducedorder system is
 x Lr    ALr
 x NL   A21r
A12 r   x Lr   BLr  ,

u
ANL   x NL   B NL 
x
y  C Lr C NL   Lr  .
 x NL 
(8a)
(8b)
The fact that the nonlinear subsystem matrices (ANL, BNL,
CNL) remain untouched by the balanced transformation, as can
be observed from (4) and (8), originates a source of error in
addition to the one from harmonic truncation, Section II. To
date, there is no criterion for error bound in MOR of nonlinear
systems; this issue is left for future research and addressed in
this paper in a numerical way only.
As for this second reduction process, electronic devices
are, as nonlinear loads, isolated from the linear system. Their
DHD transient simulation can be readily handled by including
in (8) their proper switching function representation.
As numerically demonstrated in the case studies in
Sections IV and V, the first reduction process can be deemed
as sufficient to obtain an accurate reduced-order system. In
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Transactions on Power Delivery
V. CASE STUDY 1
An arbitrary lumped-parameters network, Fig. 3, is
presented in this Section to illustrate the proposed MOR
procedure. Note that a larger network, or a larger set of ODEs
given by rational approximation, does not modify the
mathematical structure of the proposed methodology.
A. System Configuration
Consider the test network of Fig. 3. A current source,
representing current injection harmonics from an electronic
device, is connected to node 2. The nonlinear reactors at nodes
4 and 5 follow the polynomial flux/current relations:
inl1  0.11  1.5113 ,
inl 2  0.1 2  1.5213 .
The rest of parameters are given in Table I. Twenty three
harmonics (positive and negative) are used to model the fullsize system in the DHD. The linear part of the network
consists of 12 state variables which are assigned to inductor
currents and capacitor voltages. Thus, accounting for the two
nonlinear loads variables, the dimensions of the full-size DHD
system is of 14 × (23 × 2 + 1) = 658 states.
TABLE I
NETWORK PARAMETERS DATA
Parameter, value (pu)
Ro = 0.001, Lo = 0.001
R1 = 0.01, L1 = 0.05, C1 = 0.02
C2 = 0.04
R2 = 0.02, L2 = 0.1,
L3 = 0.2,
C3 = 0.2
R3 = 0.1,
R4 = 0.01, L4 = 0.05, C4 = 0.02
L5 = 0.2,
C5 = 0.2
R5 = 0.1,
R6 = 10
C123 = C1 + C2 + C3
R7 = 1
C245 = C2 + C4 + C5
R8 = 10
R9 = 1
R10 = 1
B. Harmonic Reduction
The original time-domain system (1) is transformed to the
HD, as in (2), and the input/output relation of the network, as
given by (3), is evaluated up to harmonic 23th via the sweeping
process described in Section II. The magnitudes of y, given in
per unit (pu) values, are presented in Fig. 2, Section II. As
mentioned in Section II, a truncation criterion of 0.2 pu is
applied to the magnitudes of y, Fig. 2. This yields a maximum
harmonic of 5. Hence, the DHD harmonic-reduced system
becomes of dimensions 14 × (5 × 2 + 1) = 154 states, i.e.,
about 23% of the full-size DHD system. A second reduction
process, based on TBR, follows.
C. Balanced Reduction
The obtained harmonic-reduced network is now
represented in the DHD, as in (4), considering only the first 5
harmonics, as dictated by the sweeping process. Note that (4)
includes now the two nonlinear components within the
network and not as current sources. The state reduction
procedure described in Section III is applied then to the linear
subsystem of the harmonic-reduced system; the Hankel
singular values given by the TBR procedure are presented in
Fig. 4. Based on the obtained singular values, the linear
subsystem of the harmonic-reduced system is further reduced
to order 88 in states by neglecting balanced states
corresponding to singular values below 104. Hence, the final
harmonic/state reduced-order system, as in (8), results in
dimensions of 88 + 4 × (5 × 2 + 1) = 132 states (note that the
“4” applies to two flux and two voltage variables related to the
nonlinear loads). This results in about 20% of the full-size
DHD system.
0
10
magnitude
our experience, TBR helps to wring the remaining order
reduction opportunities from the linear part of the system.
−5
10
−10
10
0
20
40
60
80
100
120
Hankel singular values
Fig. 4. Hankel singular values corresponding to the linear subsystem of the
harmonic-reduced system, case study 1.
D. Transient Simulation
This Subsection presents a transient simulation of the
network in Fig. 3 when sources:
v(t )  sin(o t )  0.2sin(3o t )
and
i (t )  0.4[sin(o t )  0.015sin(3o t )  0.16sin(5o t )
 0.12sin(7o t )  0.007 sin(9o t )  0.09sin(11o t )]
Fig. 3. Lumped-parameters network used for case study 1.
are simultaneously applied at t = 0 s. The resultant transient
voltages at nodes 4 and 5 are shown in Fig. 5, calculated with
both the original time-domain system (continuous line), as in
(1), and with the harmonic/state reduced system (dashed line),
as in (8). Both waveforms in Figs. 5(a) and 5(b) show an
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Transactions on Power Delivery
0.8
reduced−order DHD
original TD
0.6
voltage (pu)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2
4
6
8
10
12
14
time in electrical angle (rad)
16
18
20
(a)
1
1
0.9
0.8
0.7
magnitude (pu)
acceptable agreement; the rms errors, taking as basis the time
domain results, are of 0.038 and 0.052 for the transient
voltages at nodes 4 and 5, respectively. Also, the results by the
full-order DHD model have been verified with the timedomain model. However, the cpu-time by the DHD full-order,
for which an arbitrary 23th order could have been chosen by a
power engineer, and by the DHD harmonic/state reducedorder simulations are of 85.1 s and 4.5 s, respectively. The
cpu-time by the harmonic reduction process is 0.84 s which is
about 100 smaller than the cpu-time by the simulation of the
DHD full-order system.
Fig. 6 presents the corresponding harmonic dynamics (1st,
rd
3 , and 5th harmonics) obtained with both the full-order DHD
system (continuous line) and the harmonic/state reduced order
DHD system (dashed line). Harmonic dynamics cannot be
obtained by the time-domain model (1) in a direct way,
requiring a post-processing routine. On the other hand,
harmonic dynamics are directly available from the harmonic
state vectors in DHD models.
first harmonic
0.6
0.5
0.4
0.3
third harmonic
0.2
fifth harmonic
0.1
0
0
2
4
6
8
10
12
14
time in electrical angle (rad)
16
18
20
(b)
Fig. 6. Harmonic content of voltages (a) at node 4 and (b) at node 5.
______
full-order DHD system,    reduced-order DHD system
E. Harmonic Reduction Including TCRs
As further experiment, the network of Fig. 3 has been
modified by replacing the current source at node 2 and the
nonlinear load at node 5 by thyristor controlled reactors
(TCRs) with reactor values of 0.25 pu and 0.2 pu and firing
angles of 170o and 140o, respectively. We have adopted the
TCR representation in the HD as proposed in [22]. The
nonlinear load at node 4 remains the same as in the
unmodified network.
The frequency scan procedure described in Section III is
applied to the modified circuit obtaining the results presented
in Fig. 7. For this case, the maximum harmonic, using a
truncation criterion of 0.2 pu, is 7. Note that the maximum
harmonic for the unmodified network is 5. The BR reduction
and the transient simulation for the modified network follow
the same procedure as the original network and, because space
limitations, will not be repeated here.
original TD
reduced−order DHD
0.8
0.6
harmonic state 1
0.4
harmonic state 12
voltage (pu)
2
0.2
y (pu)
1.5
0
−0.2
1
25
0.5
−0.4
20
15
0
600
500
−0.6
−0.8
0
10
400
300
states
2
4
6
8
10
12
14
time in electrical angle (rad)
16
18
injected harmonics
5
200
100
20
0
0
(a)
(b)
Fig. 5. Transient voltages (a) at node 4 and (b) at node 5.
1.6
harmonic state 1
1.4
1
1.2
0.9
1
y (pu)
0.8
magnitude (pu)
0.7
first harmonic
0.6
0.8
0.6
harmonic state 12
0.5
0.4
0.4
truncation criterion line
0.2
0.3
third harmonic
0.2
0
0
0
0
fifth harmonic
0.1
2
4
6
8
10
12
14
time in electrical angle (rad)
(a)
16
18
20
100
200
300
states
400
500
600
(b)
Fig. 7. Frequency-domain sweeping results including TCRs (a) injecting
harmonics 1st to 23rd and (b) injecting 5th harmonic only.
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Transactions on Power Delivery
46% of the full-size DHD system.
VI. CASE STUDY 2
A. Circuit Description
The industrial system presented in [25] is adopted for this
case study, Fig. 8. The voltage source corresponds to
v(t )  cos(o t ) . The current sources at buses 101, 201, and
301 of the industrial system can be assumed, as in [25], to
correspond to AC/DC converters [25]. The total current drawn
by the three current sources is 8 kA and it is distributed as
75%, 20%, and 5% for buses 101, 201, and 301, respectively
[25]. All three current sources have been assumed with the
same harmonic content, indicating in Table II the percentage
values of their fundamental current component.
TABLE II
HARMONIC COMPONENTS OF CURRENT SOURCES
h
3
5
7
9
11
13
15
Ih(%)
1.5
16
12
0.7
9
6
0.5
In this paper, two nonlinear loads are added to the system at
buses 24 and 34 with corresponding current/flux relations:
inl1  0.11  0.513 ,
inl 2  0.1 2  0.5 23 .
The original full-size DHD system considers 15 harmonics
and the dimensions of the corresponding system of ODEs are:
Matrix A: nh  nh =1860  1860
Matrix B: nh  hi = 1860  124
Matrix C: ho  nh = 93  1860
where n = 60 corresponds to the number of states (including
two for the nonlinear loads), h = 31 represents the length of
the HD vector for a single variable (considering positive and
negative frequencies, and DC), i = 4 and o = 3 are the number
of inputs and outputs, respectively. The outputs correspond to
the current delivered by the generator and the voltages at
buses 24 and 34, all in per unit; the resultant direct
transmission matrix is zero.
B. Harmonic Reduction
Similarly to case study 1, the original time-domain system
(1) is transformed to the HD, as in (2), and the input/output
relation of the network, as given by (3), is evaluated up to
harmonic 15th via the sweeping process described in Section
II. The truncation criterion is set to 0.2 pu yielding a
maximum harmonic of 7. Hence, the DHD harmonic-reduced
system becomes of dimensions 60 × 15 = 900 states, i.e.,
about 50% of the full-size DHD system. A second reduction
process, based on TBR, follows.
C. Balanced Reduction
The harmonic-reduced network, involving only the first 7
harmonics, is represented as in (4) and used for the TBR-based
MOR stage.
Applying the TBR-based MOR procedure described in
Section III to the linear subsystem of the harmonic-reduced
system results in a final harmonic/state reduced-order system,
as in (8), with dimensions of 870 states. This results in about
Fig. 8. Diagram of industrial network used in case study 2, taken from [25]
D. Transient Simulation
A transient analysis is carried out aimed to compare results
by both the original time-domain system, as in (1), and by the
harmonic/state reduced system, as in (8). The transient
analysis assumes that all four sources, Fig. 8, are
simultaneously applied at t = 0 s, considering zero initial
conditions; a time-step for the simulation of both full- and
reduced-order systems is set to 0.01.
The transient waveforms corresponding to voltages at
buses 24 and 34, for both the time-domain and the DHD
reduced-order systems, are shown in Fig. 9 where a good
agreement of both models can be observed. The rms errors,
taking as basis the time-domain results, are of 0.044 and 0.047
for the transient voltages at buses 24 and 34, respectively.
Similarly to case study 1, the full-order DHD model has
been verified with the time-domain model yielding the same
results. The cpu-times by the DHD full-order and by the DHD
harmonic/state reduced-order simulations are of 1080 s and
234 s, respectively; thus the former system resulting in more
than four times slower than the reduced-order system.
Fig. 10 presents the corresponding harmonic dynamics
(showing only 1st, 3rd, and 5th harmonics) of the voltage at bus
24 obtained with both the full-order DHD system (continuous
line) and the harmonic/state reduced order DHD system
(dashed line). The harmonics waveforms in Fig. 10 are
obtained directly from the corresponding HD voltage vector
elements in a step-by-step fashion.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRD.2015.2496899, IEEE
Transactions on Power Delivery
VIII. CONCLUSIONS
1
original TD
reduced−order DHD
0.8
0.6
voltage (pu)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8
10
12
14
time in electrical angle (rad)
16
18
20
(a)
1
reduced−order DHD
original TD
0.8
0.6
voltage (pu)
0.4
0.2
This paper presents a methodology to reduce a nonlinear
system in both harmonics and states. The reduced-order model
achieves computational savings while preserving accuracy. In
the development of the methodology, the authors have taken
into account experience and state of the art methods in linear
systems theory to yield leaner frequency equivalents than
those reported in the power systems literature.
The proposed method has been successfully validated with
time-domain simulation of full-order systems that in theory
contains all possible harmonics. Also, the proposed technique
has shown to be effective, it avoids cumbersome computations
of snapshots, projections, etcetera, and provides an attractive
platform for harmonic dynamics analysis of actual distributed
networks. With respect to this scenario, interaction between
multiple power electronic devices in the proposed method has
been left as future topic.
0
REFERENCES
−0.2
−0.4
[1]
−0.6
−0.8
−1
0
[2]
2
4
6
8
10
12
14
time in electrical angle (rad)
16
18
20
(b)
Fig. 9. Transient voltages (a) at node 24 and (b) at node 34.
[3]
1.2
[4]
magnitude (pu)
1
[5]
0.8
first harmonic
0.6
[6]
0.4
[7]
0.2
third harmonic
0
0
______
2
4
6
fifth harmonic
8
10
12
14
time in electrical angle (rad)
16
18
20
Fig. 10. Harmonic content of voltage at bus 24.
full-order DHD system,    reduced-order DHD system
VII. DISCUSSION
The MOR methodology proposed in this paper has to be
applied first on harmonics and then on states. Otherwise, if we
apply first MOR to states via TBR, the sense of a “complete”
HD vector is lost [22] since TBR focuses on states. In other
words, HD vectors will not have a complete set of positive and
negative harmonic coefficients due to truncation of harmonic
states; this will result also in loss of complex conjugacy.
Networks larger than the ones presented in this paper
follow the same MOR methodology, expanding only
dimensions of the equations. This can be a computational
limitation for very large systems [24]. Nevertheless, diakoptic
techniques can be adopted in the harmonic reduction process.
Also, Krylov-based methods can be used as preliminary MOR
before applying TBR [1].
For the presented case studies, the TBR-based MOR
yielded a further reduction of about 4%. However, the results
of these case studies may not be typical of general cases.
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Transactions on Power Delivery
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APPENDIX A
This appendix presents the basic time-domain and HD
models of the considered nonlinear loads. Also, it describes
the frequency-domain sweeping-type process used in the
harmonic reduction part of the proposed MOR methodology.
A. Nonlinear Reactor Model
Nonlinear loads are considered in this paper as nonlinear
reactors with the polynomial flux/current relation:
inl (t )   (t )   (t ) ,
p
(9)
where α and β are constant values. The HD counterpart of (9)
is given by [22]
I nl     p .
(10)
Additionally, the HD relation between flux and voltage is:
  D 1V ,
(11)
where D represents the differentiation matrix [22].
B. Frequency Scan Procedure
Based on the structure of HD vectors, [22], a typical
applied HD voltage (of frequency h) at a given nonlinear load
terminals is given by
Vh   0  0 1/ (2 j ) 0  0  0 1/ (2 j ) 0  0 . (12)
The voltage from (12) is used to calculate the flux, using
(11) which in its turn provides the HD nonlinear current, as in
(10). The voltage in (12) has to be applied to all nonlinear
loads present in the network. Then, all nonlinear current
sources are introduced in the HD vector u to compute y, using
(3).
The process described above is repeated for frequencies
related to harmonics h = 1, 2, …, hmax.
It is noted that the term Φp in (10) is calculated via HD
convolution [22]. It is also noted that y is calculated for all
states, i.e., considering C as an identity matrix, or,
equivalently y = x.
APPENDIX B
For completeness of the paper, the TBR basic relations are
presented in this appendix, details can be found in an
extensive number of references.
The controllability, Wc, and observability, Wo, Gramians of
a full-order model denoted by the triplet (A, B, and C), are
respectively given in the time-domain by

Wc   e At BBT e A t dt
T
0
,
(13a)

Wo   e A t C T Ce At dt
T
0
.
Both Gramians in (13) satisfy the Lyapunov relations:
AWc  Wc AT   BBT ,
AT Wo  Wo A  C T C .
(13b)
(14a)
(14b)
A similarity transformation Tb can be found from the
eigenvalue problem:
WcWo  Tb  bTb1 ,
(15)
such that the controllability and observability Gramians
become equal and diagonal [4], i.e.,
Wcb  Wob   .
(16)
The new coordinate system, given by (15), permits to
obtain an internally balanced system (Ab, Bb, Cb). Based on the
Hankel singular values provided by , truncation can be
applied to (Ab, Bb, Cb) by neglecting the smallest singular
values, resulting in the reduced-order system (Ar, Br, Cr).
Some algorithms for the computation of (15) can be found in
[24].
BIOGRAPHIES
Abner Ramirez (SM’07) received his B.Sc., M.A.Sc. and Ph.D. from
University of Guanajuato, Mexico, in 1996, University of Guadalajara,
Mexico, in 1998 and from the Center for Research and Advanced Studies of
Mexico (CINVESTAV) Campus Guadalajara, in 2001, respectively. He was a
postdoctoral fellow in the Department of Electrical and Computer Engineering
of the University of Toronto, Canada, from November 2001 to January 2005.
Currently, he is a Professor at CINVESTAV-Guadalajara. He is a member of
the Mexican Association of Professionals and Students A.C. (PLAPTSAC).
His interests are electromagnetic transient analysis in power systems and
numerical analysis of electromagnetic fields.
J. Jesus Rico (M’00) was born in Purepero, Michoacan, Mexico. He
received the M.Sc. degree from the Universidad Autonoma de Nuevo Leon in
1993, and the Ph.D. from the University of Glasgow in 1997. Since 1989, he
is with Universidad Michoacana de San Nicolas de Hidalgo, where he is
currently a professor of the Electrical Engineering graduate program. His
research interests center on electrical power systems, harmonic analysis, and
modelling and simulation of hybrid systems.
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