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8/2/2019 Simple Modeling and Identification Procedures for Black-Box Behavioral Modeling of Power Converters Based on

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2776 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 12, DECEMBER 2009

Simple Modeling and Identification Procedures forBlack-Box Behavioral Modeling of Power

Converters Based on Transient Response AnalysisVirgilio Valdivia, Student Member, IEEE, Andres Barrado, Member, IEEE, Antonio Lazaro, Member, IEEE,Pablo Zumel, Member, IEEE, Carmen Raga, Student Member, IEEE, and Cristina Fernandez, Member, IEEE

AbstractToday, black-box behavioral models of power con-verters are becoming interesting for system level simulation ofpower electronics systems. These models can be used to evaluatethe response of systems that are composed of commercial convert-ers, since they can be fully parameterized by analyzing the actualconverter response. To optimize the required computational re-sources, these models should be as simple as possible. Furthermore,the identification of the parameters should be carried out easily,looking for simple experiments and straightforward adjusting al-

gorithms. Easy modeling and identification procedures, based on atransient response analysis, are proposed in this paper. Using thismethod, a large-signal black-box behavioral model of a powerconverter is composed of reduced-order transfer functions, whichare identified by analyzing the step response of the converter. Bothan actual commercial dcdc converter and a line-commutated rec-tifier have been modeled and identified by means of this approach,in order to validate the proposed procedures.

Index TermsBehavioral modeling, black-box modeling,reduced-order model, system identification.

I. INTRODUCTION

TODAY, power electronics systems with multiple converters

and loads are growing dramatically (e.g., onboard powerelectronics-based systems for aircrafts) [1][7]. The system

level analysis is complex [8][10], and simulation tools are

needed to carry it out.

Therefore, models of the power converters oriented to system

level simulation (also known as behavioral models) are needed.

These models should be as simple as possible in order to op-

timize the simulation time. This need was first discussed two

decades ago [1], and several works have been published about

modeling of converters for system level simulation and their ap-

Manuscript received March 6, 2009; revised June 12, 2009. Current versionpublished December 28, 2009. This work was supported in part by the Span-ish Ministry of Education and Science under the Research Project Dise no yModelado de Sistemas Electronicos Aeroespaciales. Nivel Subsistema (Code:DPI2006-14866-C02-02) and by the private contract with the European Aero-nautic Defense and Space CompanyConstrucciones Aeronauticas SociedadAnonima under the Research Project High-Voltage Direct-Current Load Distri-bution System (Code: 04-AEC0527-000050/2005) financed by The EuropeanRegional Development Fund via the Aerospace Sector Plan of the Communityof Madrid. This paper was presented in part at the IEEE Applied Power Elec-tronics Conference 2009. Recommended for publication by Associate Editor J.Jatskevich.

The authors are with the Power Electronics Systems Group, Electronics Tech-nology Department, University Carlos III of Madrid, Leganes 28911, Spain(e-mail: [email protected]; [email protected]; [email protected];[email protected]; [email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TPEL.2009.2030957

plication [1], [11][18]. However, the system designer does not

usually have access to all necessary data to parameterize these

models(e.g., a system composed of commercial converters), so a

black-box approach is required, and identification procedures

of the models are also needed, in order to parameterize them.

A black-box behavioral model of a power converter is a

mathematical model oriented to system level analysis. These

models only compute the needed signals to analyze the responseof the power system. They are defined by a set of elements that

could have no physical interpretation, and they are identified by

analyzing the response of the modeled converter.

First works on behavioral models and their identification,

focused on power electronics, have been recently proposed.

1) References [19][21] propose a model for dcdc convert-

ers composed of one nonlinear static network and two

linear dynamic networks (a WienerHammerstein struc-

ture). The static network models the converter steady-state

behavior in a nonlinear way (efficiency, static load and line

regulations, and constant power source behavior). The dy-

namic networks are linear time invariant (LTI), composed

of passive elements (R, L, and C), and model the inrushcurrent, high-frequency input impedance, and the output

impedance. In addition, models of the protections, start-

up, and other features are also proposed. The model is

entirely parameterized by means of the datasheet data, or

equivalent tests.

2) References [22][25] propose the model shown in Fig. 1.

It is an unterminated two-port-network model composed

of four dynamic systems: the output impedance Zo , theaudiosusceptibility Go , the back current gain Hi , and theinput admittance Yi . Vss is the steady-state output volt-age. Each dynamic system can be composed of either a

linear model or a nonlinear model, so the dynamic behav-ior of dcdc converters with nonlinear dynamics can be

accurately modeled by means of this approach. The identi-

fication of each dynamic system is based on the converter

frequency response, which is obtained using a network

analyzer, a linear amplifier, and an isolation transformer.

However, the required hardware equipment is relatively

expensive and may not always be available.

3) Other references propose behavioral models related to

power systems. In [26], behavioral models of protection

devices (solid-state power controllers) are described and

in [27], a model of the electromagnetic interference gen-

eration in power converters is proposed.

0885-8993/$26.00 2009 IEEE


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VALDIVIA et al.: SIMPLE M ODELING AND IDENTIFIC ATION PRO CEDU RES FO R BLAC K-BOX BEHAVIORAL MOD ELING 2777

Fig. 1. Black-box behavioral model of a dcdc converter.

These approaches are oriented to model both the input and

the output signals average behavior, like conventional average

models.

In this paper, easy modeling and identification procedures for

black-box models of power converters are proposed. These

models are composed of reduced-order dynamic systems, which

allow optimizing the required computational resources for sim-

ulation [1], [23]. On the other hand, the proposed identificationprocedure is based on analyzing the converter step response,

which is obtained from simple experiments, carried out with in-

expensive equipment and straightforward adjusting algorithms

(see Fig. 2).

The proposed procedures allow modeling and identifying

large-signal models of these dynamic systems, either with a

linear or nonlinear dynamic response.

The paper is organized as follows.

1) Section II explains the proposed modeling method. It is

explained for the output network of power converters with

controlled dc output voltage, and it can also be applied

similarly to model other networks.

2) Section III deals with the identification of the models.Existing methods are reviewed and discussed, and identi-

fication methods are proposed.

3) Section IV shows experimental results. The proposed pro-

cedures are used to identify a black-box behavioral

model of two converters output: a commercial step-down

dcdc converter and an actual line-commutated rectifier,

which provides a 270 Vdc bus for an experimental test

bench of an aircraft power system. Some step tests are

also shown and discussed.

II. MODELING METHOD

A. Structures of the Dynamic Systems that Compose

a Black-Box Model

As explained before, a black-box dynamic model of a power

converter can be composed of several dynamic systems (see

Fig. 1). However, these dynamic systems can exhibit either a

linear or a nonlinear behavior. The next structures can be used

to model them.

1) Linear Models: If the dynamic system can be consid-

ered linear, or nearly linear, it can be modeled as a single

LTI model. An LTI model can be defined by a single transfer

function.

For example, this approach is useful to model the output

impedance of current-mode controlled (CMC) converters since

it can be assumed linear in many practical cases (see Fig. 3).

2) Nonlinear Models: If the dynamic systems exhibit a non-

linear behavior in the converter range of operation, a nonlinear

approach should be considered.

1) Polytopic Structures: Reference [24] shows that polytopic

structures are a suitable choice in modeling the nonlinear

dynamics of a power converter. A polytopic structure is

composed of a set of local small-signal LTI models, and

each one is parameterized for a certain operating condi-

tion. Each local model is weighted as a function of the con-

verter operating point by means of weighting functions,

and the response of all of them is combined to compute the

model response. For example, this approach is useful to

model the output impedance of voltage-mode-controlled

buck-derived converters, since it depends on the input volt-

age in many cases (e.g., converters without feedforward

compensations). Fig. 4 shows an example, in which Zo

is composed of three small-signals local models obtainedon three different input voltages (ViQ 1 , ViQ 2 , and ViQ 3 ).A deeper explanation of polytopic structures applied on

power electronics modeling can be found in [24] and [28].

2) Structures Based on Nonlinear Static Networks Connected

with Linear Dynamic Networks: References [19][21],

[25] have shown that nonlinear static networks connected

with linear dynamic networks can be applied in model-

ing converters, which exhibit a nonlinear static behavior.

They can be used to model the static load and line regula-

tions, the constant power source behavior of the regulated

converter (if the input voltage increases, the input current

decreases), and the efficiency. For example, [22] proposesa Wiener structure to model the back current gain Hi (seeFig. 1) in order to consider the constant power source be-

havior and the efficiency. Fig. 5 shows the use of this kind

of structures to model Zo . It is composed of a static non-linear networkRc(io), whose output is a function of io ,and an LTI model LTIZo , defined by a transfer function.This structure allows modeling the static load regulation

in a nonlinear way.

Though these structures have been shown in modeling the

output impedance (see Figs. 35), they can also be applied in

modeling the other dynamic systems that compose a black-

box behavioral model.

Therefore, LTI models are suitable to compose both linear andnonlinear dynamic black-box models, using the explained ap-

proaches. This paper is focused on modeling and identification

of these LTI models.

B. Model Characterization Through the Converter Response

The converter response has to be analyzed in order to choose

the proper LTI parameters of the models.

The two-port network model shown in Fig. 1 is an unter-

minated model [22], so the dynamic systems that compose it

only should model the internal converter dynamics. The experi-

ments should be chosen in such a way that the influences of the


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Fig. 2. Proposed procedures applied over the identification of the output impedance of a black-box behavioral model.

Fig. 3. Output impedance of a power converter defined by a single LTI model.

external equipments (source and load) are minimized [22], [23].

Furthermore, the experiment has to be informative enough,

meaning that the obtained data allow an accurate fitting of the

LTI model in the frequency bandwidth of interest. As a conse-

quence, the choice of a proper input signal is a critical task [29].For example, reference [22] proposes a set of experiments to

measure the frequency response of the dynamic systems that

compose the unterminated model. A sinusoidal input signal is

applied by means of a network analyzer and a linear amplifier.

However, this paper deals with modeling and identification of

the two-port network based on the step response of the converter.

It focuses on the output network of converters with dc output

voltage, and it is worth noticing that the discussed methods can

also be applied to model other networks in a similar way.

In this proposal, the step input signal has been selected, since

step tests can be easily carried out in power electronics labo-

ratories with low-cost equipment, and the measured transient

Fig. 4. Output impedance of a power converter defined by a polytopicstructure.

Fig. 5. Output impedance of a power converter defined by a nonlinear staticnetwork connected with a linear dynamic network.

waveforms can be used to fit accurately reduced-order models

in the frequency bandwidth of interest.

The necessary tests to parameterize the output network are

the following. Notice that the step slew rate limits the frequency

bandwidth to be identified.


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1) An output current step is applied while the input volt-

age is kept constant in order to parameterize the output

impedance Zo .2) An input voltage step is applied while the output current

is kept constant in order to parameterize the audiosuscep-

tibility Go .In practice, it is not possible to keep these quantities abso-

lutely constant. The input voltage will be perturbed under a load

step test because of the output impedance of the voltage source.

On the other hand, the output current will be perturbed under

an input voltage step because of the load. These effects could

affect the measurements, but their influence can be assumed

negligible if both the voltage source and the load are properly

selected, and the perturbation of the input voltage and the output

current, respectively, is low enough. If these perturbations are

found to be significant, methods proposed in [23] can be applied

to remove this influence. Some actual implementations of these

tests are discussed in Section IV.

C. Proposed Reduced-Order Models

Once the converter response has been measured, the number

of parameters that compose each LTI model can be found. As

explained earlier, an LTI model can be defined by means of a

transfer function (1), where m n.

LTI(s) =bm s

m + bm1sm1 + + bo

ansn + an1sn1 + + ao. (1)

Power converters usually have a high number of poles and

zeros in their closed-loop dynamic transfer functions. This factis a handicap: selecting the optimal number of parameters of

each dynamic system would be difficult and high-order models

could be derived, requiring a high computational cost. However,

this procedure can be simplified if the following fact is taken

into account: the response of a dynamic LTI system can be

analyzed as the sum of first-order and second-order subsystems

response [30]. The first-order subsystems allow modeling the

real poles of the system, and the second-order subsystems allow

modeling the complex poles. Therefore, using partial fraction

expansion (1) can be rewritten as (2).

LTI(s) = K0 +i

K1is + 1i

+j

K21js + K22js2 + 22j2js + 22j

.

(2)

This approach is simple and suitable when reduced-order

models of the dynamic systems are required. The main first-

order and second-order evolutions can be identified in the step

response, and the LTI model can be composed of the sum of the

corresponding first-order and second-order transfer functions

(K0 = 0, when m < n).Regarding regulated converters with dc output, a zero at the

origin appears in the transfer functions ofZo and Go , becausethese converters have a pole at the origin in their control stages

to minimize steady-state error. Therefore, (1) can be rewritten

Fig. 6. Overdamped transient response of a CMC buck converter under a loadcurrent step (switching model).

as (3), (b0 = 0).

Zo(s), Go(s) =i

K1is

s + 1i+j

K21js2 + K22js

s2 + 22j2js + 22j

.

(3)

This zero means that measuring thestep response is equivalent

to measure the impulse response of a system with the same

parameters but without that zero. As a consequence, the step

response can be analyzed as the sum of the impulse response of

first-order and second-order subsystems, and the model can be

fully characterized (an LTI model is completely defined by its

impulse response).

Two kinds of step responses can generally be found: over-

damped (real poles) and underdamped (both real and complexpoles).

1) Overdamped: A typical overdamped response of a power

converter is shown in Fig. 6. For example, this kind of response

can be found at the output voltage of CMC dcdc converters,

when load steps are applied.

In most cases, overdamped responses can be properly approx-

imated by the sum of the responses of two first-order subsystems

(4).

LTI1 (s) =K11s

s + 11+

K12s

s + 12. (4)

As a consequence, the converter output impedance of thisexample can be approximated by means of (4). Fig. 7 shows

how this response is approximated by the sum of two first-

order subsystems responses: the slowest exponential evolution

is mainly modeled by the slowest subsystem fos1, and the

initial time interval of the response is mainly modeled by the

fastest subsystem fos2.

The approximated model and the true system are compared

in Fig. 8. It can be concluded that the response of the reduced-

order model is quite close to the converter response. Moreover,

the order of the adjusted model is significantly lower: the true

system is a sixth-order system, and the adjusted model is a

second-order system.


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Fig. 7. Overdamped response approximated as the sum of two first-ordersubsystems responses.

Fig. 8. Output impedance of a CMC buck converter: true system (averagesmall signal model, sixth-order, blackline) and reduced-order model (second-

order, gray line).

Fig. 9. Underdamped transient response of a VMC buck converter under aload current step (switching model).

If both real poles are very close, the system can be modeled by

means of (5), which corresponds to a critically damped second-

order system.

LTI1 (s) =K21s

(s + 21 )2 . (5)

2) Underdamped: A typical underdamped response of a

power converter is shown in Fig. 9. For example, this kind

of responses can be commonly found at the output voltage of

voltage-mode controlled (VMC) power converters, when load

steps are applied.

Most underdamped responses can be properly approximated

by the sum of the responses of first-order subsystems and a sec-

ond order subsystem. In this example, the sum of two first-order

subsystems, fos1, and fos2, and a second-order subsystem

sos allows obtaining a good fit (6).

LTI2 (s) =K11 s

s + 11+

K12s

s + 12+

K21s2 + K22s

s2 + 2221s + 221. (6)

The added second-order subsystem corresponds to a damped

sinusoidal response, which could present a certain phase delay,

which is related to term K21 . Regarding the example shownin Fig. 9, no delay is found, so K21 = 0, and the response isapproximated, as shown in Fig. 10.

The output impedance of the power converter and the ap-

proximated model are compared in Fig. 11. The model matches

properly the true system, though the order reduction is less than

in the previous example: the true system is a fifth-order system,

and the adjusted model is a fourth-order system.Fig. 12 shows an example of an underdamped response in

which a phase delay is found. It corresponds to the output volt-

age of a VMC boost converter, when an input voltage vi step isapplied (vi vi). The reduced-order model and the true sys-tem match properly in the bandwidth of interest (approximately

up to 5 kHz in this particular example, Fig. 13), and the time

waveforms are very close (see Fig. 12).

In this case, only one first-order subsystem was necessary to

obtain a proper adjustment, so a third-order model is used (7),

and a significant model order reduction is achieved.

LTI2 (s) =K11s

s + 11+

K21s2 + K22s

s2 + 22121s + 221. (7)


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Fig. 10. Underdamped response approximated as the sum of two first-orderand one second-order subsystems responses.

Fig. 11. Output impedance of a VMC buck converter: true system (averagesmall-signal model, fifth-order, black line) and reduced-order model (fourth-order, gray line).

Fig. 12. Underdamped transient response of a VMC boost converter, underinput voltage step. Responses of both the switching model (black line) and theadjusted model (gray line).

Fig. 13. Audiosusceptibility of a VMC boost converter: true system (averagesmall-signal model, fifth-order, black line) and reduced-order model (third-order, gray line).

In some cases, a transient response close enough to a pure

second-order system response can be found. Then, both first-

order subsystems can be avoided in order to adjust the reduced-

order behavioral model (8).

LTI2 (s) =K21s

2 + K22s

s2 + 22121s + 221. (8)

Table I summarizes the proposed reduced-order models. They

can be applied to model the dynamic systems, which define a

power converter dc output in most practical cases. Only in some

cases, if the proposed reduced-order models were not suitable

to obtain an accurate enough fitting, higher order models would

be necessary.


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TABLE IPROPOSED REDUCED-ORDER MODELS FOR Go AND Zo

III. IDENTIFICATION

Reduced-order models have been proposed in Section II.

However, an identification method is needed to parameterize

them. In this section, the main LTI systems identification meth-

ods are reviewed and discussed. Finally, an alternative one is

proposed.

A. Brief Review of LTI Systems Identification Methods

A high number of LTI models identification methods have

been reported in literature. There are two main methods: non-

parametric and parametric [29], [31]:

1) Nonparametric Methods: Nonparametric methods are

used to identify models that are not defined by a finite numberof parameters. Nonparametric models are curves or functions,

such as the frequency response (Bode, Nyquist or Nichols plots)

or the impulse response.

These methods can be classified in two groups.

a) Impulse response estimation methods: These methods

are based on the transient response. The identification

is performed by analyzing the impulse response, the

step response, or by means of correlation techniques.

b) Frequency response estimation methods: These meth-

ods are based on the frequency response analysis and

the spectral analysis.

Nonparametric methods are easy to use, and any assump-

tion about the identified system structure is not required. Theyare used in some power electronics applications, e.g., autotun-

ing in digitally controlled converters [33], [34]. However, the

resulting models are not suitable for simulation.

2) Parametric Methods: The parametric methods are applied

to identify models, which are defined by a set of parameters.

They are generally based on the least-squares method and the

extensions of it, and they can be applied to fit models using both

transient response and frequency response data [29], [31], [32].

The parametric models can be defined by means of several

structures.

1) Transfer Function Models: Common structures of transfer

function models are autoregressive with external input,

autoregressive moving average with external input, output

error (OE), and Box-Jenkins.

2) State Space Models: These methods allow adjusting LTImodels accurately, which can be applied for simulation.

The selection of a proper input signal is a critical task,

in order to reach a fine adjustment of the model in the

frequency bandwidth of interest. Moreover, proper model

structure and number of parameters have to be selected.

Parametric methods are used to identify the power converters

black-box models in [22][25] from frequency response data.

B. Proposed Identification Method

Regarding system level simulation, parametric identificationmethods seem to be the proper ones, since the identified models

can be directly applied for simulation.

1) Transfer function models can be adjusted using these

methods, from step transient response.

2) Section II proposes the parameters that should compose

them.

In this section, two identification methods are proposed: the

first one is based on split the step response as a sum of subsystem

responses, and a sequential identification of them. The second

one is based on the least-squares method and the extensions of

it.

1) Method-Based on Sequential Identification of Subsystems:

This method is suitable to identify those structures proposedin Section II. If an input signal close to an ideal step is ap-

plied to the converter, the measured response can be split as

a sum of first-order and second-order responses, and the cor-

responding subsystems can be easily identified following a se-

quential procedure. The main advantage of this approach is the

ease of implementation, since the identification can be carried

out, following a simple procedure and no data preprocessing is

required.

The proposed procedure is defined by a set of steps described

shortly. Fig. 14 shows its application to identify the buck sim-

ulated response, as shown in Fig. 9, and can be applied in a

similar way to model other kind of responses.


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Fig. 14. Identification iterative procedure applied on the response shown in Fig. 9.

In addition, the detailed application of the method to identify

a commercial converter is shown in Section IV.1) Step 1: Subtract the steady-state value ss from the actual

response ar.

2) Step 2: Find a time interval at the end of the transient

response in which only the response of a single subsystem

is dominant and identify this subsystem. In this case, an

exponential evolution is found, so a first-order subsystem

fos1 is fitted.

3) Step 3: Subtract from ar-ss the effect of the identified

subsystem fos1. Identify a new dominant subsystem at

the end of the new response. An underdamped response

is clearly found in this case, so a second-order subsystem

sos is fitted.

4) Step 4: Subtract from ar-ss-fos1 the effect of the

identified subsystem sos and identify the remainingsubsystems (in this case, another first-order subsystem

fos2).

Once the subsystems have been identified, the LTI model can

be composed of the sum of the identified subsystems.

The identification of first-order and second-order subsystems

can be carried out in many ways (e.g., by means of normalized

plots and analytical expressions) [30]. In this paper, a set of

simple analytical expressions is given.

1) Identification of First-Order Subsystems: First-order sub-

systems are defined by two parameters: a gain K1 anda time constant 1 = 1/1 . Hence, both parameters can

be identified by selecting two points P11 (y11 , t11 ) and


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P12 (y12 , t12 ) into the transient response of the subsys-tem, and applying (9) (u is the height of the step).It is recommended to select these points separately

enough, in the interval in which this subsystem is dom-

inant, in order to achieve a proper fit. Both Step 2 and

Step 4 (Fig. 14) illustrate the identification of first-order

subsystems.

y(s) = u(s)LTI1 (s) =u

s

K1s

s + 1y(t) = uK1e

t 1

1 =ln(y11/y12 )

t12 t11K1 =

y12u

et1 2 1 . (9)

2) Identification of Second-Order Critically Damped Subsys-

tem: Critically damped systems are defined by a constant

K2 and an angular frequency 2 . This subsystem shouldbe identified when an overdamped response, in which both

real poles are very close, is found. The identification can

be carried out by selecting the peak point of the transient

response P2

(y2, t

2), and applying (10).


s

K2s

(s + 2 )2

y(t) = uK2tet 2

2 =1

t2K2 =

y2u

et2 2

t2. (10)

3) Identification of Second-Order Underdamped Subsys-

tems: The second-order subsystems transfer functions are

parameterized by identifying their gains K21 and K22 ,oscillation 2 , and damping factor 2 . The identificationcan be carried out by measuring the damped oscillation

frequency d , selecting two maximum and/or minimumpoints (either local or absolute), into the transient response

of the subsystem, P21 (y21 , t21 ), P22 (y22 , t22 ), and apply-ing (11). Step 3 (see Fig. 14) illustrates the identification

of second-order underdamped subsystems.


s

K21s2 + K22s

s2 + 222s + 22

y(t) = uGet2 2 sin2

1 22 t +

2 =ln

y2 1y2 2

1(t2 2t2 1 )d

1 +

ln y2 1

y2 2 1

(t2 2t2 1 )d2

= atan

1 222

dt21 2 =

d1 22

G =y21u

et2 1 2 2

sin(dt21 + )K21 =

tan()1 + tan2 ()

G

K22 = K21 22 +G

1 + tan2 ()d . (11)

Therefore, the main advantage of this method is the ease of

use. It can be used when an input signal close to an ideal step

can be applied.

Fig. 15. OE model.

If the response of each subsystem is separate enough from the

response of the others, an accurate fit can be performed using

the proposed method. However, if the subsystems responses are

not separate enough, finding the best fit could be more difficult

and the following methods may be used instead of it.

2) Least-Squares Method and the Extensions of It: Paramet-

ric identification based on the least-squares method and the ex-

tensions of it can be generally applied to fit the transfer function

models, which compose the black-box model, from measured

step responses. There are several structures of transfer func-

tion models, and they differ on how they model the dynamicrelation between input, output, and disturbance. A disturbance

model is not considered in this paper, so the OE model is sug-

gested (see Fig. 15), in which B(Z)/F(z) is the LTI system to be

identified [29].

Identification algorithms of OE models have been widely dis-

cussed in literature [29], [31], [32], and there are commercial

tools that allow identifying them (e.g., MATLAB System Identi-

fication Toolbox [35]). The OE model identification algorithms

are useful even when the step signal is not too close to an ideal

step (the identified frequency bandwidth will be limited by the

step slew rate). However, the required fitting algorithms may be

complex and specific software tools may be required, as well as

data preprocessing.

Second-order to fourth-order models are suitable to fit

most step responses accurately, according to that exposed in

Section II.

C. Linear and Nonlinear Dynamic Systems Identification

The LTI models can be used to model both linear and

nonlinear dynamics of power converters, as explained in

Section II.

1) Linear Dynamic Systems Identification: if the identified

dynamic system can be supposed linear, the correspond-

ing LTI model can be identified by means of conven-tional parametric methods, or the method proposed in this

section.

2) Nonlinear Dynamic Systems Identification: If the dynamic

system is nonlinear, the nonlinear structures can be iden-

tified as follows.

a) Polytopic structure: Small step tests applied on dif-

ferent operating conditions should be performed.

For example, the identification of a VMC buck con-

verter audiosusceptibility is shown in Fig. 16. It de-

pends on the input voltage operating point, so small

input voltage steps are applied on three operating

input voltage values ViQ 1 , ViQ 2 , and ViQ 3 .


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Fig. 16. Polytopic structure of VMC buck audiosusceptibility.

b) Structures based on a nonlinear static model con-

nected with a linear dynamic model: The static non-

linear model can be identified by means of the anal-ysis of the steady-state response of the converter at

different operating points. After that, the effect of

the static network is subtracted from the measured

response, and finally the LTI model is identified.

IV. EXPERIMENTAL RESULTS

The proposed procedures have been applied to obtain a

black-box behavioral model of the output of two kinds of

converters: A step-down commercial dcdc converter and a line-

commutated rectifier.

A. DCDC Switching Converter

The Texas Instruments PTN78020WAZ switching converter

[36] has been modeled. The converter has been set to provide

a regulated output voltage Vo = 2.5 V, and the considered op-erating ranges during the tests have been Vi = [815 V] andPo = [413W], respectively. Both an outputcapacitor of 470Fand an input capacitor of 2.2 F have been externally placed.

In this case, both the audiosusceptibility Go and the outputimpedance Zo have been considered, and their dependence onthe operating point has been evaluated. The applied tests are

shown in Fig. 17.

1) The load step test [see Fig. 17(a)] allows identifying the

output impedance of the model. A MOSFET IRF740 hasbeen used to apply the step. A certain oscillation appears

at the end of the step because of using resistors. However,

if the output current waveform is close to an ideal step,

the proposed identification method III.B.1 can be used.

A step-down test is preferred because the oscillation is

smaller, so the output current waveform is closer to an

ideal step. Several loads and input voltage values have

been set in order to evaluate the Zo dependence on bothVi and Io operating points. The inputvoltage source shouldhave low enough output impedance to ensure that the input

voltage perturbation is low enough, and its influence on

the output voltage transient response is negligible. In this

Fig. 17. Step tests applied to identify the black-box model of the dcdcconverter: (a) load step and (b) input voltage step.

Fig. 18. Behavioral black-box model of the actual dcdc converter: (a)model structure, (b) static load regulation, (c) weighting functions of the poly-topic structure, and (d) LTI models parameters.

case, a voltage source HP 6012B has been used. On the

other hand, output current tests can be also carried out

by means of an electronic load with a high slew rate, if

available.


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Fig. 19. LTI model identification using the proposed procedure.

Fig. 20. Comparison between the actual measured response and the identified model response. Output current perturbations are applied while the input voltageis kept constant. Dark lines: actual measured response. Color lines: Model response: vo yellow. io blue. (a) Load step-down response used to identify Zo .

Vi = 8 V. (b) Load step-up response used for validation. Vi = 15 V. (c) Exponential load current response used for validation Vi = 10 V.

2) The input voltage step test [see Fig. 17(b)] allows identify-

ing the audiosusceptibility of the model. An input voltage

step with high enough slew rate is applied by closing the

switch connected in parallel with the diode. An electronic

load has been connected at the output, in a way that the

output current can be assumed constant during each test.

The HP 6050 A has been used in this case and the ioperturbation has been found to be negligible. The Go de-pendence onVi and Io operating points has been evaluatedby setting different values of them.

Following results were obtained.1) Output ImpedanceZo : It hasbeen verified that the response

dependence on Io and Vi operating points is not significant.However, static load regulation has been measured.

Therefore, Zo has been modeled by means of an LTI modelconnected with a nonlinear static network [see Fig. 18(a)], which

have been parameterized as follows.

a) First, the static load regulation has been characterized, by

measuring the steady-state output voltage along the output

power range, in order to parameterize Rc(io). Detaileddata are shown in Fig. 18(b).

b) After that the LTI model is parameterized. A load step-

down test has been applied, in which the output current

waveform is very close to an ideal step. First, the effect of

Rc(io) has been subtracted from the measured waveform.Second, a fourth-order model corresponding to (6) was ad-

justed using the proposed identification method 3.2.1 (see

Fig. 19). The points in the transient response have been

selected according to the criteria exposed in Section III. It

is worth noticing that the difference between the gains of

both first-order subsystems allow modeling accurately the

initial sudden rise due to the output capacitor equivalent

series resistance. Models data are given in Fig. 18(d).

Fig. 20 shows a comparison between the adjusted model re-sponse and the actual converter response when the output cur-

rent is perturbed while input voltage is kept constant. Both load

step-down and load step-up responses have been compared [see

Fig. 20(a) test has been used for identification and Fig. 20(b)

has been used for validation]. In addition, an exponential out-

put current has been set up by means of the electronic load, in

order to validate that the model works properly not only when

load step tests are tested, but also when other current transient

waveforms are applied [see Fig. 20(c)].

2) Audiosusceptibility Go: It has been verified that the Godependence on Io operating point can be assumed negligible.

However, a significant dependence on the input voltage was


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Fig. 21. Comparison between the actual measured response and the identi-fied model response. Input voltage perturbations are applied while the outputcurrent is kept constant. Dark lines: actual measured response. Color lines:model response: vo yellow.vi green. (a)Vi step-up response usedto iden-tify GoVi = 8 . Io = 2 A. (b) Vi step-up response used to identify GoVi = 11 .5 .Io = 2 A.

found (the higher the input voltage, the lower Go ). On the otherhand, no significant static line regulation has been measured.

Therefore, a polytopic structure has been selected to model

it [see Fig. 18(a)]. The range partitioning and the weighting

functions have to be properly selected [24], [28]. Generally,

more partitions should mean higher model accuracy. On the

other hand, the sum of the weighting functions has to be always

equal to one.

As a simple example, an equally spaced partition of the input

voltage range in two regions and triangular weighting functions

were chosen [see Fig. 18(c)]. The input voltage steps have been

applied on each weighting function maximum point. Since the

Fig. 22. Comparison between the actual measured response and the identified

model response.Both input voltage and output current perturbations are applied.Dark lines: actual measured response. Color lines: model response:vo yellow.vi green. io blue(note thatboth measured and simulatedcurrentsare totallyoverlapped).

Fig. 23. Load step tests applied on the actual line-commutated rectifier.

magnitude of the steps is low enough, the identification of eachsmall signal LTI model is accurate.

A second-order model is suitable to get a good fit in this case

(8). The absolute maximum and minimum points have been

chosen as P21 and P22 , respectively, in order to apply (11) inall cases. Fig. 21(a) and (b) shows the transient responses used

to identify two of the three LTI local models. Detailed data of

them are given in Fig. 18(d).

Finally, both vi and io are simultaneously perturbed in orderto validate the behavior of the whole model (see Fig. 22). It can

be seen that both responses are very close, so it is concluded that

the output of the dcdc converter has been properly modeled.

B. Line-Commutated Rectifier

This converter generates the dc bus of a distributed power

system for an aircraft test bench. The rated power is 40 kW. The

input ac voltage of the converter is obtained from the ac grid, so

its magnitude is assumed to be constant, and Go is neglected.The proposed procedures are significantly useful in this case.

1) A reduced-order model of the converter is suitable in order

to optimize the required computational cost, because the

power distribution system is composed of a high number

of loads and converters.

2) Several resistive loads are implemented in the test bench,

so load steps can be easily carried out (see Fig. 23). The


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Fig. 24. Behavioral black-box model of the actual rectifier: (a) model struc-ture, (b) static load regulation, and (c) LTI model parameters.

Fig. 25. Comparisonbetweenthe responses of theactual rectifier andthe iden-tified model when load steps are applied. Blue lines: actual converter response;red and brown lines: adjusted model response. (a) 20%40% of rated load step.

(b) 30%70% of rated load step.

io waveform is close to an ideal step, so the proposedidentification method III.B.1 can be used.

3) The output voltage waveform has a significant ripple,

in which the harmonic content is close to the main fre-

quency oscillation of the transient response [interval of

time (0.050.15 s)]. Using the proposed identification

method, a signal preprocessing, which could be complex,

is avoided.

The load steps have been applied using different resistive

loads, in order to evaluate the Zo dependence on Io operating

point. It has been verified that it is negligible (the converter oper-

ates in continuous conduction mode in the considered operating

range). However, static load regulation has been measured, so

Zo was modeled by means of a nonlinear static model connectedwith an LTI model structure [see Fig. 24(a)].

A fourth-order LTI model has been chosen again (6), although

a second-order model (8) would also have been a proper choice

because the computational cost is optimized at the expense of a

low accuracy loss. The model data are given in Fig. 24(c). The

second-order model is adjusted assuming that it is dominant at

time0.1 s, and both first-order subsystems are added to obtain

a finer fit. A comparison between the transient response of the

actual rectifier and the identified model is shown in Fig. 25,

showing that both of them are close enough.

V. CONCLUSION

Simple modeling and identification methods for black-box

behavioral models of power converters, based on the transient

response analysis, have been proposed.1) Some concepts about black-box behavioral modeling

have been discussed, showing that LTI models can be used

to compose large-signal black-box models of power

converters. When the behavior of the converter is linear,

a single LTI model is suitable. When the behavior of the

converter is nonlinear, nonlinear structures that combine

LTI models and nonlinear functions can be used.

2) A set of reduced-order LTI models has been proposed.

Reduced-order models are appropriate when large power

systems are analyzed, since the simulation time can be

optimized. Those models can be selected by simply an-

alyzing the step response of the converter, which can beobtained in an easy way.

3) The main identification procedures of LTI models have

been reviewed, and their applicability to the identification

of black-box models of power converters, based on the

step response, has been discussed. An alternative and sim-

ple method has been proposed. This method allows iden-

tifying the proposed reduced-order models, without need

of data preprocessing and complex-fitting algorithms.

4) Finally, black-box behavioral models of the output net-

work of both an actual commercial dcdc converter and

an actual line-commutated rectifier, which is applied in a

test bench of an aircraft power system, have been mod-

eled and identified by means of the proposed approaches.The experimental results show a proper response of the

identified models.

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Virgilio Valdivia (S09) was born in Barcelona in1983. He received the M.Sc. degree in electrical en-gineering in 2006 from the University Carlos III ofMadrid, Leganes, Spain, where he is currently work-ing toward the Ph.D. degree in electrical engineering.

Since 2005, he has been a Member of the PowerElectronics Systems Group, University Carlos III ofMadrid. His research interests include modeling andsimulation of power electronics systems and model-ing and design of integrated magnetic components

for power electronics converters.

Andres Barrado (M04) wasborn in Badajoz,Spain,in1968. Hereceived theM.Sc. degree inelectrical en-gineeringfrom the Polytechnic University of Madrid,Madrid, Spain, in 1994, and the Ph.D. degree fromthe University Carlos III of Madrid, Leganes, Spain,in 2000.

Since 1994, he has been an Associate Professor atthe University Carlos III of Madrid, and the Head ofthe Power Electronics Systems Group since 2001. Hiscurrent research interests include switching-modepower supply, dcdc and acdc converters, modeling

and control of systems, low-voltage fast transient response dcdc converters,and energy harvesting.

Antonio Lazaro (M03) was born in Madrid, Spain,in 1968. He received the M.Sc. degree in electri-cal engineering from the Universidad Politecnica deMadrid, Madrid, Spain,in 1995,and thePh.D.degreein electronic engineering from the University CarlosIII of Madrid, Leganes, Spain, in 2003.

Since 1995, he has been an Assistant Professor atthe University Carlos III of Madrid. He has been in-volved in power electronics since 1994, participatingin more than 30 research and development projectsforindustry. He is theauthor or coauthor of more than

100papers published in IEEE journals andconferences. He is theholder of threepatents. His current research interests include switched-mode power supplies,power factor correction circuits, inverters (uninterruptible power system andgrid-connected applications), and modeling and control of switching convertersand digital control techniques.


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Pablo Zumel (M06) received the B.S. degree inelectrical engineering from the University of Burgos,Burgos, Spain, in 1995, the first M.S. degree in elec-trical engineering from the Universidad Politecnicade Madrid, Madrid, Spain, in 1999, the second M.S.degree from the Ecole Centrale Paris, Paris, France,in 2000, and the Ph.D. degree from the UniversidadPolitecnica de Madrid, in 2005.

From 1999 to 2003, he was a Researcher in

the Division de Ingeniera Electronica, UniversidadPolitecnica de Madrid. Since 2003, he has been with

the Department of Electronics Technology, University Carlos III of Madrid,Leganes, Spain, where he is currently an Assistant Professor. His research in-terests include digital control in power electronics, power electronics systemmodeling, and educational issues on power electronics.

Carmen Raga (S06) was born in Madrid, Spain, in1976. She received the M.Sc. degree in industrial en-gineering from the University Carlos III of Madrid,

Leganes, Spain, in 2005.Since 2005, she has been an Assistant Professor atthe University Carlos III of Madrid. Her current re-search interests include switching-mode power sup-plies, modeling of dcdc and acdc converters, andpower distribution systems.

Cristina Fernandez (M05) wasbornin Leon,Spain,in 1972. She received the M.S. and Ph.D. degrees inelectrical engineering from the Universidad Politec-nica de Madrid (UPM), Madrid, Spain, in 1998 and2004, respectively.

From 1997 to 2003, she was a Researcher at theUPM. Since 2003, she has been an Assistant Pro-fessor in the Department of Electronic Technology,University Carlos III of Madrid, Leganes, Spain. Her

current research interests include contact-less powersupplies, modeling of power electronics systems, and

educational issues on power electronics.

simple modeling and identification procedures for “ black-box ” behavioral modeling of power...

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