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Conference Paper

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February 2019

A Hybrid Framework Combining ModelBased and Data-Driven Methods for 

Hierarchical Decentralized Robust 

Dynamic State Estimation

Preprint

Marcos Netto,

1 Venkat Krishnan,2 Lamine Mili,1

Yoshihiko Susuki,

3 and Yingchen Zhang2

1 Virginia Polytechnic Institute and State University

2 National Renewable Energy Laboratory 

3 Osaka Prefecture University

To be presented at the 2019 IEEE Power and Energy Society 

General Meeting (IEEE PES GM) 

Atlanta, Georgia

August 4–8, 2019 
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Conference Paper

NREL/CP-5D00-72685

February 2019

A Hybrid Framework Combining ModelBased and Data-Driven Methods for 

Hierarchical Decentralized Robust 

Dynamic State Estimation 

Preprint

Marcos Netto,1 Venkat Krishnan,2 Lamine Mili,1

Yoshihiko Susuki,3 and Yingchen Zhang2

1 Virginia Polytechnic Institute and State University

2 National Renewable Energy Laboratory 

3 Osaka Prefecture University

Suggested Citation 

Netto, Marcos, Venkat Krishnan, Lamine Mili, Yoshihiko Susuki, and Yingchen Zhang. 

2019. A Hybrid Framework Combining Model-Based and Data-Driven Methods for 

Hierarchical Decentralized Robust Dynamic State Estimation: Preprint. Golden, CO: 

National Renewable Energy Laboratory. NREL/CP-5D00-72685. 

https://www.nrel.gov/docs/fy19osti/72685.pdf. 
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1

A Hybrid Framework Combining Model-Based and

Data-Driven Methods for Hierarchical Decentralized

Robust Dynamic State Estimation

Marcos Netto, Student Member, IEEE, Venkat Krishnan, Senior Member, IEEE, Lamine Mili, Life Fellow, IEEE,

Yoshihiko Susuki, Member, IEEE, and Yingchen Zhang, Senior Member, IEEE

Abstract—This paper combines model-based and data-driven

methods to develop a hierarchical, decentralized, robust dynamic

state estimator (DSE). A two-level hierarchy is proposed where

in the lower level comprises robust, model-based, decentralized

DSEs. The state estimates sent from the lower level are received

at the upper level, where they are filtered by a robust data-driven

DSE after a principled sparse selection. This selection allows us to

shrink the dimension of the problem at the upper level and hence

significantly speed up the computational time. The proposed hybrid framework does not depend on the centralized infrastructure

of the control centers; thus it can be completely embedded into

the wide-area measurement systems. This feature will ultimately

facilitate the placement of hierarchical decentralized control

schemes at the phasor data concentrator locations. Also, the

network model is not necessary, thus a topology processor is

not required. Finally, there is no assumption on the dynamics

of the electric loads. The proposed framework is tested on the

2,000-bus synthetic Texas system, and shown to be capable of

reconstructing the dynamic states of the generators with high

accuracy, and of forecasting in the advent of missing data.

Index Terms—Compressed sensing, data-driven dynamical systems, dynamic state estimation, Kalman filtering, Koopman mode

decomposition, sparse selection.

I. INTRODUCTION

The large-scale deployment of phasor measurement units

(PMUs) and other grid-edge metering devices, such as smart

meters [1] and micro-PMUs [2], is enabling the application of

data analytics in electric power systems. This is fueling interest

in data-driven methods that can enhance power system’s

reliability and resilience [3]. Simultaneously, and perhaps most

importantly, we are witnessing an unprecedented increase in

the share of renewable energy sources [4], [5], which are

inherently intermittent and uncertain. The consequent increase

in the stochastic dynamics of the net load challenges the

traditional deterministic model-based, real-time monitoring

and control methods applied in the legacy systems [6], [7].

This work was supported in part by CAPES Foundation, Ministry of

Education of Brazil, under grant BEX13594/13-3.

This work was authored in part by the National Renewable Energy

Laboratory (NREL), operated by Alliance for Sustainable Energy, LLC, for the

U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308.

Funding provided by the U.S. Department of Energy Office of Electricity

(FIA181867). The views expressed in the article do not necessarily represent

the views of the DOE or the U.S. Government. The U.S. Government retains

and the publisher, by accepting the article for publication, acknowledges that

the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide

license to publish or reproduce the published form of this work, or allow

others to do so, for U.S. Government purposes.

M. Netto and L. Mili are with the Bradley Department of Electrical and

Computer Engineering, Virginia Polytechnic Institute and State University,

VA 22043 USA (e-mail: mnetto@vt.edu).

V. Krishnan and Y. Zhang are with the National Renewable Energy

Laboratory (NREL), Golden, CO 80401 USA.

Y. Susuki is with the Department of Electrical and Information Systems,

Osaka Prefecture University, Sakai, Osaka, Japan.

Therefore, power engineers are looking to exploit the fast

sampled measurement data to advance grid modeling, state

estimation, forecasting, and controls through data analytics

[8], [9], including by proposing data-driven dynamic state

estimators (DSEs) [10], [11], which can track system dynamic

states [12].

In this paper, we develop a two-level, hierarchical, decentralized, robust DSE by combining model-based and datadriven methods. The model-based decentralized DSEs at the

lower level provide the necessary data for performing datadriven model identification at the upper level by using the

Koopman mode decomposition (KMD) [13], which further

allows for dynamic stability assessment and modal analysis of

nonlinear dynamical systems [14]. The paper also illustrates

the application of the proposed framework for large-scale

systems by using compressed sensing [15] to find sparse state

estimate selection. Because the use of all state estimates at the

upper level could be prohibitive for high-dimensional systems,

we rely on compressed sensing to find a sparse selection of

state estimates, following the work of Brunton et al. [16], [17].

The proposed hybrid framework provides the opportunity to

devise powerful tools by combining concepts from dynamical

systems, estimation, and control theory. Firstly, the use of

decentralized DSEs and KMD makes it independent of the

network model; hence, it does not require a topology processor. Secondly, by virtue of the data-driven KMD, the method

does not need to make assumptions about the underlying

load model dynamics. Thirdly, and most notably, it can be

completely embedded into the wide-area measurement systems

instead of being an add-on to the energy management systems

installed at the control centers. This attribute will ultimately

facilitate the hierarchical control design of electric power

systems [18] with the placement of control schemes at the

phasor data concentrator (PDC) location, thereby exploiting

the synergies between agile, low-latency, decentralized and

holistic centralized monitoring and control architectures.

The paper proceeds as follows. Section II summarizes

relevant literature and motivation for this work. Section III

briefly introduces the data-driven robust DSE, and Section IV

presents the sparse selection approach. Numerical results are

discussed in Section V. The conclusions and directions for

future research are given in Section VI.

II. BACKGROUND AND MOTIVATION

Following the work of Modir and Schlueter [19], several

DSEs [20]–[23] have been proposed based on the conceptual

idea of transmitting all the PMU measurements to a centralized

location, such as the control center, where the DSE is supposedly installed. We refer to this configuration as the centralized

1

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2

PMU

Gen. Bus 1

PMU

Load Bus 1

PMU

Gen. Bus k

Control Center

Gen. Owner 1

Composite

load 1 Gen. Owner k

PDC

c-DSE

Time-tagged

Network measurements

model

(a) Centralized DSE (c-DSE)

PMU

Gen. Bus 1

PMU

Gen. Bus 2

PMU

Gen. Bus k

Gen. Owner 1 Gen. Owner 2 Gen. Owner k

d-DSE 1 d-DSE 2 d-DSE k

Control center Time-tagged

measurements

Time-tagged

state estimates

(b) Decentralized DSE (d-DSE)

PMU

Gen. Bus 1

PMU

Gen. Bus 2

PMU

Gen. Bus k

Control Center

Gen. Owner 1 Gen. Owner 2 Gen. Owner k

PDC

DSE 1

Time-tagged

measurements DSE 2 DSE k

(c) Centralized distributed DSE

Fig. 1. Comparison of the conceptual idea of different dynamic state estimators proposed in the literature. Note, in all the cases, system-level stability

assessments such as inter-area oscillations [27], [28] will be done in a centralized manner, either supported by a centralized or decentralized DSE.

DSE; see Fig. 1a for more details. The assumptions are that

the system is fully observable by PMUs and that the Kronreduced network model (KNM) [24] is accessible. Although

this may be true for independent system operators (ISOs)

and regional transmission organizations (RTOs), access to the

KNM is a hurdle for local utilities and transmission operators

because of limited data sharing across neighboring systems.

One alternative is to build dynamic reduced-order models [25];

unfortunately, this approach will increase the existing model

uncertainties. Further, the KNM assumes that the electric loads

are modeled as constant admittances, thereby not allowing for

capturing the rich dynamics of composite electric loads [26].

Another important point that has been overlooked is that a

topology processor is required to determine the KNM in real

time; the existing topology processors embedded on the static

state estimators cannot supply this demand. Nevertheless,

although of reduced dimension, the graph of the KNM is

full [24] rather than sparse, hence sparsity techniques cannot

be exploited. The computational burden is another important

barrier to the adoption of model-based centralized DSEs.

As an attempt to overcome the aforementioned problems,

model-based decentralized DSEs have been investigated; see,

for instance, [29], [30]. The idea is to individually estimate the

states of each generator at their own location, as in Fig. 1b.

Given that each decentralized DSE is designed for a particular

generation location, only a PMU at the generator terminal is

assumed. The decentralized DSE requires neither the model

of the network nor the model of the electric loads, and it

is computationally inexpensive because of a reduced number

of involved state variables. Further, it offers the additional

benefit of performing model calibration [31] without taking the

generator offline, which is an appealing attribute from the system’s reliability and economic standpoints. However, although

a decentralized DSE allows for local stability assessment,

control, and protection, for wide-area stability assessment and

control, the DSE states will still need to be communicated

to centralized processing, like any other architecture shown

in Fig. 1; and, although the data transmissions in this case

are not raw measurements but actual system states, it is

still necessary to ensure data integrity, cybersecurity, and

robustness against bad or missing information at the receiving

end, i.e., at the control center. Additionally, in the event of loss

of communication with a decentralized DSE, the states of the

associated generator must be forecast. To circumvent some

of these issues, Paul et al. [32] proposed a computationally

distributed but physically centralized DSE, wherein the conceptual idea is to send all the measurements to a centralized

location instead of performing the estimation separately for

each generator; see Fig. 1c. Although ingenious, the approach

in [32] relies on the reference signals of voltage and power

that are accessible only to ISOs and RTOs, thereby precluding

its adoption by transmission companies and alike utilities. The

state forecasting capability is also not available.

Given the promise that decentralized DSEs have in terms of

reducing the computational burden on increasing proliferations

of sensors and data, and given its favorable trade-off between

local and wide-area assessments, we delve deeper into the

decentralized DSE. Presuming that model-based decentralized

DSEs are available, we pose the following questions:

• Suppose that the state estimates are sent to a centralized

location. How could one verify the data integrity of the

received state estimates without relying on any model?

• How could one forecast the dynamic states in case of a

communication problem with a given decentralized DSE?

To address these questions, we suggest the use of the robust,

data-driven DSE, which is presented next.

III. ROBUST KOOPMAN KALMAN FILTER

This section introduces the data-driven method, namely, the

Koopman operator-based method that enables studying nonlinear dynamical systems without relying on any model. Further,

the developed robust Koopman Kalman filter (KKF) [10],

[11] lays the foundation for ensuring that the state estimates

received from the decentralized DSEs will be robustly filtered.

A. The Koopman Operator

Consider a discrete-time autonomous dynamical system:

xk = f(xk1), (1)

where the state x is an element of the state space S ⇢ Rn,

f : S ! S is the discrete map, and k 2 Z is the time index.

Define g : S ! R, a vector-valued observable in (1). The

Koopman operator, K, is a linear transformation on this vector

space:

Kg(xk) = g  f(xk) = g(f(xk)). (2)

The Koopman eigenvalues, μi, and the Koopman eigenfunctions (KEFs), 'i, of K are defined as follows:

K'i(xk) = μi'i(xk), i = 1, 2, ... (3)

2

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3

If all the elements of g lie within the span of the KEFs,

then we have:

g(xk) = X1

i=1

'i(xk)i = X1

i=1

'i(x0)iμk

i , (4)

where i are the Koopman modes [13], and {μi, 'i, i}, i =

1, 2, ..., are referred to as the Koopman tuples. The interested

reader is referred to [13] for more details on the derivation of

(4). For power systems, because of the existence of multiple

attractors, we estimate a subset of the Koopman tuples, such

that g(xk) ⇡ Pq

i=1 'i(xk)i = Pqi=1 'i(x0)iμk

i .

B. Koopman Canonical Coordinates [33]

Consider the discrete-time autonomous dynamical system:

xk = f(xk1), yk = h(xk), (5)

where x and f are defined as in (1), y 2 Rm is the system

observation vector, and h : S ! Rm. Let Fq = span{'i}q

i=1

be a subset of the KEFs, such that x, y 2 Fq. Then, we have:

xk ⇡ Xq

i=1

'i(xk1)(x)

i μi, yk ⇡ Xq

i=1

'i(xk1)(y)

i μi.

(6)

Now, suppose that the Koopman tuples are ordered such

that complex conjugate pairs appear adjacent to each other. It

can be shown [11] that a nonlinear change of coordinates is

given by TK : Rn ! Rq, expressed as:

TK(x) = x = ⇥

x1, ..., xq

⇤> , (7)

where ·

> denotes the transpose of a vector,

8

<

:

xi = 'i, if 'i is real-valued,

xi = 2<{'i} and xi+1 = 2={'i}, if ('i, 'i+1)

form a complex conjugate pair,

<{'i} and ={'i} are the real and imaginary parts of 'i,

respectively. Using (7), and after some algebraic manipulation

[11], we have:

xk = ⌦ xk1, xk = (x)

xk, yk = (y)xk, (8)

and, remarkably, the nonlinear dynamical system in (5) can

be mapped to the linear dynamical system (8). The KEFs

define the Koopman canonical coordinates (KCC). If 'i is

real-valued, ⌦i,i = μi. If ('i, 'i+1) form a complex conjugate

pair, then we have:



⌦i,i ⌦i,i+1

⌦i+1,i ⌦i+1,i+1 

=

 <{μi} ={μi}

={μi} <{μi}



,

and, thus, ⌦ is a block diagonal matrix.

The matrix (x) 2 Rn⇥q is a mapping between the states in

the KCC, x, and the original state space coordinates, x. If 'i

is real-valued, (x)

:,i = (x)i . Conversely, if ('i, 'i+1) form a

complex conjugate pair, (x)

:,i = <{(x)i }, (x):,i+1 = ={(x)i }.

The matrix (y) 2 Rm⇥q is a mapping between the states

in the KCC, x, and the observation vector in the original state

space coordinates, y. If 'i is real-valued, (y)

:,i = (y)i . If

('i, 'i+1) form a complex conjugate pair, (y)

:,i = <{(y)i },

(y)

:,i+1 = ={(y)i }.

C. Robust Koopman Kalman Filter

In the KKF form, (8) becomes:

xk = ⌦xk1 + wk1,

yk = (y)

xk + vk, (9)

where w stands for the system process error, and v denotes

the measurement noise. In what follows, we rely on [34] to

solve (9). The interested reader is referred to [11] for details

on the derivation of the robust KKF.

Now, the practical experience suggests that the synchronous

generators oscillate coherently; see Fig. 2. This is equivalent of

saying that the dynamics of electrical power systems evolve on

a low-dimensional attractor. In the next section, this property

is leveraged to perform a sparse selection of the state estimates

used to identify the KKF, which is applicable to larger power

systems.

IV. SPARSE SELECTION OF STATE ESTIMATES

Following Manohar et al. [17], suppose that a state x evolving according to the nonlinear dynamics (1) has a compact

representation in a transform basis . In a universal basis

 2 Rn⇥n, x might have a sparse representation:

x = s, (10)

s 2 Rn is a sparse vector. In a tailored basis r 2 Rn⇥r,

such as a basis defined by a proper orthogonal decomposition,

x might have a low-rank representation:

x = r`, (11)

` 2 Rr. We seek to find a matrix C 2 Rp⇥n consisting of a

small number (p ⌧ n) of optimized measurements:

b = Cx, (12)

b 2 Rp, which facilitates the accurate reconstruction of s or `,

and thus x. Note that C = [e1 e2 ... ep ]

>, where ei is the

unit vector with a unit entry at index i and zeros elsewhere.

Combining (10) and (12) yields:

b = (C ) s = ✓s. (13)

Eq. (13) is referred to as the compressed sensing problem.

Conversely, combining (11) and (12) yields:

b = (C r) ` = ✓`. (14)

If C is properly structured such that ✓ is well conditioned,

it is possible to solve for the low-rank coefficients ` given the

measurements b in (14) as follows:

`

ˆ=

(

✓1b = 

C r

1

b, p = r,

✓†b = 

C r

†

b, p > r,

(15)

✓† denotes the Moore-Penrose pseudoinverse of ✓. Thus, x can

be estimated as xˆ = r`

ˆ. From (15), one seeks columns of

 r corresponding to point sensor locations in the state space,

ei , that optimally condition the inversion of the matrix ✓.

The structure of the elements ei affect the condition number

of C and consequently of M = ✓>✓. The condition number

of the system might be indirectly bounded by optimizing the

3

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4

spectral content of M using its determinant, trace, or spectral

radius. For example, we have:

⇤ = arg max

,||=p

|det M| = arg max



Y

i

|i(M)|

= arg max



Y

i

i(M), (16)

where i and i are, respectively, the i-th eigenvalue and singular value of M. The QR factorization with column pivoting

decomposes a matrix M 2 Rm⇥n into a unitary matrix Q, an

upper triangular matrix R, and a column permutation matrix

C, that is:

MC> = QR. (17)

The key idea from [17] is, when applied to an appropriate

basis, the QR pivoting procedure provides an approximate

greedy solution method for the optimization in (16), also

known as a submatrix volume maximization because the

matrix volume is the absolute value of the determinant.

A. Sparse Selection of State Estimates

In [17], the QR pivoting procedure is proposed as a tool to

optimize sensor placement, in particular for the reconstruction

of high-dimensional states from point measurements given

tailored bases. Instead of measurements, here, we have access

to the state estimates received from the robust, model-based,

decentralized DSEs. Our objective is to shrink the number of

state estimates used by the robust KKF so as to speed up its

processing.

Algorithm 1 Sparse selection of state estimates

1: procedure

2: r svd(xˆ)

3: if p == r then

4:  pivot( r) . [Q, R, pivot] = qr( r)

5: else if p>r then

6:  pivot( r >

r ) . [Q, R, pivot] = qr( r >r )

7: b xˆ . xˆ = xˆ1:p

After b is determined as in Algorithm 1, we make use of

(7) such that:

TK(b) = x = ⇥

x1, ..., xq

⇤> . (18)

V. NUMERICAL RESULTS

We carry out simulations on the 2,000-bus synthetic

Texas system [35] comprising 544 synchronous and 87 nonsynchronous generators; the latter are modeled as variablespeed wind generators with full converters and do not contribute to the electromechanical modes. A three-phase shortcircuit is applied to Bus 1017 and cleared after 10 milliseconds. Fig. 2 shows the rotor angle of the synchronous

generators relative to the system average angle, with removed

mean. We observe that 456 of 544 synchronous generators

present coherent dynamics, whereas 88 of 544 synchronous

generators do not contribute at all to the dynamics; see

constant line at zero degrees in Fig. 2.

Upon application of the KMD explained in Section III, the

10 most important Koopman eigenvalues–i.e., the ones with

the smaller damping coefficient–are shown in Fig. 3. They

were estimated using 10 seconds of rotor speed estimates. The

black crosses indicate results obtained with all 544 estimates

0 5 10 15

Time (seconds)

-2

-1.5

-1

-0.5

0

0.5

1

 (degrees)

Fig. 2. Rotor angle, , of the 544 synchronous generators relative to system

average angle, after removing the mean.

obtained from the decentralized DSE (assuming we have

local PMU measurements), whereas the blue circles indicate

results obtained with 240 principled selected estimates from

the sparse state measurement selection technique explained

in Section IV. The remaining 5 seconds of the time domain

simulation are used for testing. We consider three different

scenarios, as presented next.

-2 -1.5 -1 -0.5 0

{ }

-15

-10

-5

0

5

10

15

{ }

all estimates

selected estimates

5% damping

10% damping

20% damping

30% damping

Fig. 3. The Koopman modes calculated using estimates of the rotor speed of

the synchronous generators.

1) No outliers: The results are shown in Fig. 4a. True state,

received state estimate, and filtered state estimate, respectively,

refer to the outcome of the time-domain simulation, the data

received from the robust model-based decentralized DSEs, and

the outcome of the robust KKF.

2) Impulsive noise: The results are shown in Fig. 4b. We

observe that the robust KKF is able to suppress the impulsive

noise.

3) Loss of the communication link with a decentralized

DSE: The results are shown in Fig. 4c. This case demonstrates

the forecasting capability of the robust KKF.

VI. CONCLUSIONS AND FUTURE RESEARCH

The proposed hybrid framework offers a balance between

model-based and data-driven methods, and it has several

important advantages compared to other methodologies. It

is completely independent of the network model, does not

attempt to model the dynamics of the loads, and, most importantly, does not depend on the infrastructure that is exclusively

available at the control centers.

4

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5

10 11 12 13 14 15 16

Time (seconds)

-1

0

1

 (radians/second)

10-4

True state

Received state estimate

Filtered state estimate

(a)

10 11 12 13 14 15 16

Time (seconds)

-1

0

1

 (radians/second)

10-4

True state

Received state estimate

Filtered state estimate

(b)

10 11 12 13 14 15 16

Time (seconds)

-1

0

1

 (radians/second)

10-4

True state

Received state estimate

Filtered state estimate

(c)

Fig. 4. Rotor speed deviation, !, of the synchronous generator 4. (a) No outlier. (b) Impulsive noise at t = 12 seconds. (c) Loss of communication with

the decentralized DSE between t = 12 and t = 13 seconds.

Although promising, the proposed approach requires further

investigations. One important aspect is the system observability at the upper level, i.e., the sparse selection must be

conditioned to a certain degree of redundancy. We will rely

on the correspondence between the Koopman operator and the

Lie derivatives to pursue this effort. In addition, the estimation

of the KEFs [36] is an important open problem and must be

addressed. Future work will also look into the applicability of

using other observables from sensors in the upper level system

identification by either partially or completely bypassing the

lower level DSE in certain scenarios to gain increased agility.

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5

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