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# Data-Driven Fast Frequency Control using Inverter-Based Resources

Etinosa Ekomwenrenren, John W. Simpson-Porco, Evangelos Farantatos, Mahendra
Patel, Aboutaleb Haddadi, and Lin Zhu E. Ekomwenrenren is with the Department
of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON
N3L 3G1 CA (e-mail: etinosa.ekomwenrenren@uwaterloo.ca).J. W. Simpson-Porco is
with the Department of Electrical and Computer Engineering, University of
Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, CA (e-mail:
jwsimpson@ece.utoronto.ca)E. Farantatos, M. Patel, A. Haddadi and L. Zhu are
with the Electric Power Research Institute, Palo Alto, CA 94304 USA (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>lzhu@epri.com).Funded under EPRI Project #10009168: Wide-Area Hierarchical
Frequency and Voltage Control for Next Generation Transmission Grids.

###### Abstract

To address the control challenges associated with the increasing share of
inverter-connected renewable energy resources, this paper proposes a direct
data-driven approach for fast frequency control in the bulk power system. The
proposed control scheme partitions the power system into control areas, and
leverages local dispatchable inverter-based resources to rapidly mitigate
local power imbalances upon events. The controller design is based directly on
historical measurement sequences, and does not require identification of a
parametric power system model. Theoretical results are provided to support the
approach. Simulation studies on a nonlinear three-area test system demonstrate
that the controller provides fast and localized frequency control under
several types of contingencies.

###### Index Terms:

frequency control, low inertia, distributed control, renewable energy, smart
grid, next generation control.

##  I Introduction

Akey objective in power system operations is the maintenance of a stable
system frequency and the quick restoration of power balance [1]. However, the
increasing penetration of inverter-connected renewable energy resources (RESs)
[2] is resulting in adverse effects on power system frequency regulation [3].
By replacing conventional synchronous generators, along with their
synchronized rotational mass, the increasing proliferation of RESs reduces the
system inertia, resulting in a faster rate of change of frequency (ROCOF) and
lower frequency nadir during contingencies (i.e., a deeper drop in frequency).
Furthermore, with some RESs not participating in primary frequency control[4]
while displacing synchronous generators which do, the aggregate effective
primary control response in the system is reduced [5, 6]. When combined with
the net load variability these intermittent and variable RESs introduce, the
upshot is larger and more frequency deviations, making it increasingly
difficult for system operators to maintain frequency within acceptable limits,
such as the extreme variations noted by the California Independent System
Operator in the so-called ’duck chart’ [7].

There has been extensive research into the negative dynamic effects of reduced
inertia in the power system due to increased renewables, with suggested
solutions such as virtual inertia emulation and services [8, 9]. Equally
important to consider is the problem of maintaining the average frequency
close to nominal during normal operation, with regulation performance being
quantified by regulatory authorities in Control Performance Standards (CPS)
[10, 11]. For this, it is essential to consider the local system inertia,
primary control response, and primary control deadband, as these have the
greatest effect on the average frequency deviations [6]. This fosters the need
for localized fast frequency control strategies, which take into cognizance
local system model and parameter information. Traditionally, the Automatic
Generation Control (AGC) system employs a centralized approach to maintain
average frequency deviations within desired limits for each balancing
authority area. This is achieved by generating control signals at a central
control center. However, due to the sheer size of the balancing authority
area, maintaining an accurate dynamic system model becomes an arduous task.
Consequently, the AGC system relies on classical frequency bias constant
methods [12], which, while effective to some extent, limit its speed and
utility for rapid frequency control.

If accurate parametric models are available, then modern model-based
controller design approaches can be successfully used to enable IBR
participation in local fast frequency control. In [13] the authors developed
and validated such a controller based on the principles of active disturbance
estimation and rejection; as our work here builds directly on this, further
details are deferred to Section II. This scheme and other model-based
frequency control approaches, such as model predictive control [14, 15] and
robust optimal control [16, 17] can provide good closed-loop control
performance and reduce average frequency deviations. However, obtaining
accurate parametric models may be prohibitively difficult in practice [18],
ultimately limiting the performance of model-based designs. For example,
simulations in [13] show deterioration in control performance (e.g., post-
disturbance settling time and overshoot) when there are parametric mismatches
between the true system model and the model used for design.

Data-driven or data-based controller design methods provide a promising
alternative in this regard. Proposals to address the issue of power system
frequency control using data-driven control can be broadly divided into two
categories: indirect and direct. In an indirect approach, historical data from
the power system is used to explicitly identify a system model, and a
controller is then designed based on that model (e.g., [19, 20]). The indirect
approach has the advantage of providing an explicit, interpretable model of
the system, which can aid in understanding the particular frequency response
dynamics. However, even selecting an appropriate parametric model to fit is a
difficult trial-and-error process, particularly in modern power systems with
diverse and quickly evolving components. Additionally, there is evidence that
the intermediate identification step may lead to poorer closed-loop
performance than recent direct approaches [21].

In a direct data-driven or model-free approach, a frequency controller is
designed directly based on recorded or online real-world data, without
explicit identification of a system model. One broad approach in this category
is adaptive dynamic programming or reinforcement learning [22, 23, 24]. Here,
control actions are taken to maximize some form of cumulative reward. However,
reinforcement learning approaches are limited by their sensitivity to hyper-
parameter selection, the complex training process required to determine the
weight coefficients of the trained agent, which in turn relies on a
significant amount of historical sampled data [25, 26].

In contrast with reinforcement learning, a suite of alternative direct data-
driven control approaches have recently appeared [27, 25, 28, 29], and derive
from a branch of control theory called behavioral systems [30]. These
techniques allow for direct control while being sample efficient, and often
come with rigorous performance guarantees. While the specific controllers
differ between approaches (e.g., model-predictive [25], linear-quadratic, [28]
etc.), these approaches all rely on the so-called _fundamental lemma_ of
behavioral systems, which states that a single recorded trajectory is
sufficient to capture the underlying dynamic model of the system if the input
signal is rich enough to excite all system modes [30]. Our proposed controller
is also based on this principle.

#### Contributions

This paper provides direct data-driven controller designs which enable IBRs to
participate in providing geographically localized fast frequency control. The
key components in our approach are novel designs for _data-driven disturbance
estimators_ : dynamic algorithms which provide online estimates of the net
real power imbalance within a specified control area. Local IBRs are then
quickly redispatched within their operating limits to eliminate the imbalance.

In Section III we present two data-driven disturbance estimator designs. Both
designs are based directly on recorded system data, and do not require a
parametric system model. The two designs trade off between simplicity and
robustness/performance. The first design, proposed in our preliminary work
[31], uses a simple linear update law to estimate the disturbance, and
requires tuning of only a single parameter. Extending [31], here we provide
theoretical guarantees supporting this design. The first design serves as a
stepping stone to our second approach, which is an optimization-based
estimation procedure. The second design has a higher computational burden,
requiring the solution of a convex optimization problem at each time-step, but
(i) is less sensitive to noise in the recorded data, (ii) is less sensitive to
strong nonlinearity in the system dynamics (e.g., governor deadbands), and
(iii) shows superior performance in simulation studies. As the formulations
are general, we outline specifically how these methods are applied to the
frequency control problem under consideration. Compared, for instance, to the
recent data-driven load-frequency controller proposed in [29], we do not make
the strong assumption that a measurement of net load demand is available; our
approach is based only on direct measurements of area frequency and net power
flow out of the control area.

In Section IV we extensively validate our designs via simulations on a
detailed nonlinear three-area power system. Several scenarios are examined,
included load increases, heavy renewable penetration, generation trips, and
three-phase faults. The tests demonstrate that our approach provides fast and
effective frequency control for the bulk grid, and outperforms our recent
model-based design [13].

##  II Review: Model-Based Fast Frequency Controller [13]

We briefly review the key architectural aspects of the model-based
hierarchical fast frequency controller developed and extensively tested in
[13]. The key criteria driving the design are a desire for (i) design
simplicity, (ii) fast localized response to achieve frequency regulation, and
(iii) localized use of system data and measurements to minimize latency and
maximize data privacy.

In the scheme of [13], the grid is divided into small local control areas
(LCAs). Within each LCA, a _disturbance estimator_ processes frequency and
area tie power flow measurements $\Delta\omega$ and $\Delta P_{\rm tie}$
(incorporating estimated delays) in order to detect frequency events. The
estimator generates a real-time estimate $\Delta\widehat{P}_{\rm u}$ of the
net unmeasured active power imbalance $\Delta P_{\rm u}$ within the LCA, and
an allocation mechanism optimally redispatches local IBRs to correct the
imbalance; see [13, Section III-A] for further details on the power allocator.
A block diagram of this LCA controller is shown in Figure 1.

Local Control Area Disturbance
---
Estimator
Power
---
Allocator
LCA
ControllerIBRs$z^{-\tau_{\mathrm{m}}}$$z^{-\tau_{\mathrm{m}}}$$z^{-\tau_{\mathrm{m}}}$$z^{-\tau_{\mathrm{c}}}$$\Delta\widehat{P}_{\rm
u}$$\Delta P_{\mathrm{tie}}$$\Delta P_{\mathrm{u}}$$\Delta\omega$$\Delta
P_{\mathrm{ibr}}^{\rm set}$$\Delta P_{\mathrm{ibr}}$ Figure 1: Block diagram
of area control structure for each LCA. Dashed lines denote sampled signals.

In situations where local resources are insufficient, a higher-level
coordinating controller facilitates the provision of additional power support
from neighboring LCAs; as this higher-level coordinating controller is not our
focus in this article, we refer to [13, Section III-B] for further details.

The disturbance estimator within the LCA controller is the core design
component of the approach. The estimator is a Luenberger-type state observer,
and its design requires a parametric model describing the LCA frequency
dynamics. For practical reasons, a simple model with few parameters is
strongly preferred, and the designs in [13, Section II], used the second-order
_frequency response model_

$\begin{aligned}
2H\Delta{\dot{\omega}}&=-\tfrac{1}{R_{\mathrm{I}}}\Delta\omega+\Delta
P_{\mathrm{m}}-\Delta P_{\mathrm{u}}-\Delta P_{\mathrm{tie}}+\Delta
P_{\mathrm{ibr}}\\\ T_{\rm R}\Delta{\dot{P}_{\mathrm{m}}}&=-\Delta
P_{\mathrm{m}}-R_{\rm
g}^{-1}(\Delta{\omega}+T_{\mathrm{R}}F_{\mathrm{H}}\Delta{\dot{\omega}})\end{aligned}$

(1)

which considers local aggregated parameters, such as total LCA inertia $H$,
total IBR and generator primary control gains $R_{\rm I}$ and $R_{\rm g}$,
aggregated turbine-governor time constant $T_{\rm R}$, and aggregated high-
pressure turbine fraction $F_{\rm H}$.

##  III A Data-Driven Control Approach for Area-Based Fast Frequency Control

The model-based design of Section II requires an explicit and accurate model
of the frequency dynamics of each LCA. In practice, this requirement poses at
least two major challenges. First, an appropriate class of parametric models
must be selected; this step balances simplicity vs. accuracy, and will become
increasingly difficult as RESs with black-box power electronic controls
proliferate. Second, the parameters of the model must be selected or fit; this
procedure itself is challenging, with associated bias-variance trade-offs
[32].

To address these issues, in this section we develop two direct data-driven
design approaches to supplant the model-based design approach described in
Section II. In essence, the idea is to replace the crude parametric LCA model
(1) with a non-parametric model based on time-series data collected from the
system. This time-series data is directly used to design a disturbance
estimation scheme, without passing through an explicit system identification
step.

Section III-B describes our first data-driven disturbance estimation approach,
which fuses ideas from linear estimator design and behavioral systems theory.
The resulting disturbance estimator is described by a linear update rule, and
requires tuning of only one scalar gain. To improve robustness to grid
nonlinearities and inexact data collection procedures, our second design
approach in Section III-C extends this linear estimation procedure with an
optimization-based estimation procedure. Finally, in Section III-D we describe
how these general estimation ideas are adapted for the particulars of power
system frequency control and integrated into the hierarchical control
framework outlined in Section II.

###  III-A Background on Data-Driven System Representation

Consider the controllable finite-dimensional discrete-time linear time-
invariant (LTI) model

$\displaystyle x(t+1)$ $\displaystyle=Ax(t)+Bu(t)+B_{d}d(t)$ (2)
$\displaystyle y(t)$ $\displaystyle=Cx(t)+Du(t)$

with time $t\in\mathbb{Z}_{\geq 1}$, state $x(t)\in^{n}$, control input
$u(t)\in^{m}$, disturbance input $d(t)\in^{q}$, and measured output
$y(t)\in^{p}$. We assume the matrices $(A,B,B_{d},C,D)$ of (2) are _unknown_ ,
and hence the model (2) cannot be directly used for simulation, analysis, and
feedback design purposes. _Behavioral systems theory_ provides a set of tools
for constructing a data-based representation of the dynamic system (2) using
input and output measurements. We refer to [33] for a recent survey, and
mention only the essential concepts here.

As notation, if $(z(1),z(2),z(3),\ldots)$ is a m-valued signal defined for
positive time, we write $z\in(^{m})^{\mathbb{Z}_{\geq 1}}$. The starting point
is to diminish the role of the state, and consider all possible input-output
sequences $(u(t),d(t),y(t))$ which are compatible with (2), called the
_behaviour_ :

$\displaystyle\mathscr{B}$
$\displaystyle=\left\\{(u,d,y)\in(^{m+q+p})^{\mathbb{Z}_{\geq
1}}\,:\,\exists\,x\in(^{n})^{\mathbb{Z}_{\geq 1}}\,\,\text{s.t.}\right.$ (3)
$\displaystyle\qquad\left.\sigma x=Ax+Bu+B_{d}d,\,\,y=Cx+Du\right\\},$

where $(\sigma x)(t)=x(t+1)$ is the shift operation. The behaviour (3)
describes the system (2) as a subspace of the vector space of all possible
input-output signals, and (2) is a _state-space representation_ of
$\mathscr{B}$. The _order_ of the system, denoted by $n(\mathscr{B})$, is the
smallest possible state dimension of the representation (2). Given a
representation of minimal order, the _lag_ of $\mathscr{B}$, denoted by
$\ell(\mathscr{B})$ is the smallest integer $\ell$ such that the matrix
$\mathcal{O}_{\ell}=\mathrm{col}(C,CA,\ldots,CA^{\ell-1})$ has rank
$n(\mathscr{B})$.

Let $\mathscr{B}_{T}$ denote the restriction of the behaviour to trajectories
of finite length $T\in\mathbb{Z}_{\geq 1}$, i.e., input-output sequences of
length $T$. Suppose that we have collected $T$-samples of _input-output data_
$w^{\rm d}=(u^{\rm d},d^{\rm d},y^{\rm d})\in\mathscr{B}_{T}$ from the system.
This data may be directly used to create a non-parametric representation of
the model (3). To do this, let $L\leq T$ be a positive integer, and organize
the data into the _Hankel matrix of depth $L$_, given as

$\mathscr{H}_{L}(u^{\rm d})=\begin{bmatrix}u^{\rm d}(1)&\cdots&u^{\rm
d}(T-L+1)\\\ \vdots&\ddots&\vdots\\\ u^{\rm d}(L)&\cdots&u^{\rm
d}(T)\end{bmatrix}\in^{mL\times(T-L+1)},$

with analogous definitions for $\mathscr{H}_{L}(d^{\rm d})$ and
$\mathscr{H}_{L}(y^{\rm d})$. The input data $(u^{\rm d},d^{\rm d})$ is said
to be _persistently exciting of order $L$_ if
$\mathrm{col}(\mathscr{H}_{L}(u^{\rm d}),\mathscr{H}_{L}(d^{\rm d}))$ has full
row rank; this captures the idea that the inputs are sufficiently rich and
sufficiently long. The _Fundamental Lemma_ [30] states that if the input data
is persistently exciting of order $L+n(\mathscr{B})$, then _any possible_
length $L$ input-output sequence $(u,d,y)\in(^{(m+q+p)})^{L}$ can be expressed
as

$\begin{bmatrix}\mathscr{H}_{L}(u^{\rm d})\\\ \mathscr{H}_{L}(d^{\rm d})\\\
\mathscr{H}_{L}(y^{\rm d})\end{bmatrix}g=\begin{bmatrix}u\\\ d\\\
y\end{bmatrix}$ (4)

for some vector $g\in^{T-L+1}$. The linear equation (4) is a _data-based
representation_ of the system (2), and can be leveraged for prediction and
control [33].

###  III-B Design #1: Linear Data-Driven Disturbance Estimator

We now consider (2) as a model for each LCA. We assume that $d(t)$ is a
constant unknown disturbance signal, which for us will model mismatch between
generation and load. In our context, $x(t)$ would consist of states of
generators, converters, loads, and associated control systems, $u(t)$ would be
commands to IBRs, and $y(t)$ would be available measurements such as frequency
deviation. Since (2) would represent the system _including_ the action of
primary controllers, the model (2) will be assumed to be internally
exponentially stable, i.e., $A$ will have eigenvalues within the unit circle.

The design goal is to produce a real-time estimate $\hat{d}(t)$ of the unknown
disturbance $d(t)$. Our proposed estimator design consists of two steps:

  1. (i)

a _data-driven forward prediction_ $\hat{y}(t)$ of the output $y(t)$;

  2. (ii)

a _linear update rule_ for $\hat{d}(t)$ using $\hat{y}(t)$ and the true system
measurement $y(t)$.

To generate a prediction of the output for time $t$, we will leverage (4), and
assume that historical data $(u^{\rm d},d^{\rm d},y^{\rm d})$ is available.
Model-based prediction using, e.g., (2) would require the specification of an
initial condition. In the data-driven setting, the initial condition is
implicitly defined by using recent online samples of input and output data
[27]. Let $T_{\rm ini}\geq\ell(\mathscr{B})$ be the length of the
initialization data, and define the vectors

$\displaystyle u_{\rm ini}$ $\displaystyle=\mathrm{col}(u(t-T_{\rm
ini}),\ldots,u(t-1))$ (5) $\displaystyle\hat{d}_{\rm ini}$
$\displaystyle=\mathrm{col}(\hat{d}(t-T_{\rm ini}),\ldots,\hat{d}(t-1))$
$\displaystyle\hat{y}_{\rm ini}$ $\displaystyle=\mathrm{col}(\hat{y}(t-T_{\rm
ini}),\ldots,\hat{y}(t-1)).$

Note that $\hat{d}_{\rm ini}$ and $\hat{y}_{\rm ini}$ are formed based on our
past _estimates_ of the disturbance and output. In (4), we consider
trajectories of length $L=T_{\rm ini}+1$. We partition $u,d,y$ in (4) as

$u=\begin{bmatrix}u_{\rm ini}\\\
u(t)\end{bmatrix},\quad\hat{d}=\begin{bmatrix}d_{\rm ini}\\\
\hat{d}(t)\end{bmatrix},\quad y=\begin{bmatrix}y_{\rm ini}\\\
\hat{y}(t)\end{bmatrix},$

and correspondingly partitioning the rows of the Hankel matrices in the same
fashion as

$\mathscr{H}_{L}(u^{\rm d})=\begin{bmatrix}U_{\rm ini}\\\
U_{f}\end{bmatrix},\,\,\mathscr{H}_{L}(d^{\rm d})=\begin{bmatrix}D_{\rm
ini}\\\ D_{f}\end{bmatrix},\,\,\mathscr{H}_{L}(y^{\rm
d})=\begin{bmatrix}Y_{\rm ini}\\\ Y_{f}\end{bmatrix}.$

With these choices, (4) can be re-expressed as

$\mathscr{H}_{\rm red}g:=\begin{bmatrix}U_{\rm p}\\\ D_{\rm p}\\\ Y_{\rm p}\\\
U_{f}\\\ D_{f}\end{bmatrix}g=\begin{bmatrix}u_{\rm ini}\\\ \hat{d}_{\rm
ini}\\\ \hat{y}_{\rm ini}\\\ u(t)\\\
\hat{d}(t)\end{bmatrix},\qquad\hat{y}(t)=Y_{f}g.$ (6)

The first set of equations is solved for the unknown $g$, and the prediction
$\hat{y}(t)=Y_{f}g$ is immediately obtained. If the underlying data-generating
system is LTI and the collected data are exact, the Fundamental Lemma
guarantees that (6) is consistent and the computed response matches the
system’s response exactly, provided $T_{\rm ini}\geq\ell(\mathscr{B})$ [30].

With the output estimate generated, the disturbance estimate is now updated
according to the feedback rule

$\hat{d}(t+1)=\hat{d}(t)-\varepsilon L(\hat{y}(t)-y(t)),$

where $L\in^{q\times p}$ is the estimation gain and $\varepsilon\in(0,1)$ is a
tunable parameter which controls the rate of adjustment. Putting everything
together, we can compactly express the overall disturbance estimator as

$\displaystyle\hat{y}(t)$ $\displaystyle=\mathcal{P}\cdot\mathrm{col}(u_{\rm
ini},\hat{d}_{\rm ini},\hat{y}_{\rm ini},u(t),\hat{d}(t))$ (7a)
$\displaystyle\hat{d}(t+1)$ $\displaystyle=\hat{d}(t)-\varepsilon
L(\hat{y}(t)-y(t))$ (7b)

where $\mathcal{P}=Y_{f}\mathscr{H}_{\rm red}^{\dagger}$ is the _prediction
matrix_ and $\mathscr{H}_{\rm red}^{\dagger}$ denotes the pseudoinverse of
$\mathscr{H}_{\rm red}$. As $\mathcal{P}$ depends only on historical data, it
can be computed once and stored, and thus implementing (7) simply amounts to
matrix-vector multiplication.

The final issue to address concerns the tuning of the estimator gain $L$ and
parameter $\varepsilon$ in (7). Our tuning recommendation is
$L=G(1)^{\dagger}$, where $G(1)=C(zI_{n}-A)^{-1}B_{d}|_{z=1}$ is the _DC gain_
of the system (2) from input $d$ to output $y$. This selection will be
justified in our theory to follow, and the matrix $G(1)$ can be obtained
directly from the same historical data used to construct $\mathcal{P}$ in (7);
see [34, Thm. 4.1] for details on that construction. We can now give a
theoretical result concerning convergence of the disturbance estimator (7).

######  Theorem III.1 (Data-Driven Disturbance Estimator).

Consider the disturbance estimator (7) for the system (2) under all previous
assumptions. Assume further that $G(1)=C(I_{n}-A)^{-1}B_{d}$ has full column
rank, and set the estimator gain as $L=G(1)^{\dagger}$. Then there exists
$\varepsilon^{\star}>0$ such that for all
$\varepsilon\in(0,\varepsilon^{\star})$, $\hat{d}(t)\to d(t)$ exponentially as
$t\to\infty$.

The disturbance estimator (7) provides a _completely model-free_ solution to
disturbance estimation problem; the only required tuning is the single scalar
parameter $\varepsilon\in(0,1)$. An implication of Theorem III.1 is that one
may tune the estimator (7) by starting $\varepsilon$ small and slowly
increasing it; the proof can be found in Appendix A.

######  Remark III.2 (Singular Value Thresholding).

In practice, the system generating the data which is used to build
$\mathscr{H}_{\rm red}$ in (7) may contain nonlinearity, and the measurements
will be corrupted by measurement noise; this will be the case in our
subsequent case studies. Both of these effects will compromise performance of
the design (7). It has however been observed that low-rank approximations of
Hankel matrices reduce the effects of noise in data-driven control, and
enhance generalization [25]. In implementation, we compute the singular value
decomposition of $\mathscr{H}_{\rm red}$ and retain only the dominant singular
values and vectors, to obtain a low-rank approximation
$\tilde{\mathscr{H}}_{\rm red}$ [35]. We then use
$\mathcal{P}=Y_{f}\tilde{\mathscr{H}}_{\rm red}^{\dagger}$ in (7), which,
empirically, greatly increases the robustness of the approach. $\square$

###  III-C Design #2: Optimization-Based Data-Driven Disturbance Estimator

The advantage of (7) is simplicity, as it involves only linear update rules at
each time step. We now outline a more flexible optimization-based disturbance
estimation procedure which can achieve improved performance at the cost of
higher implementation complexity. The key idea is to formulate the disturbance
estimation problem as a regularized optimization problem. In particular, the
use of regularization affords us more flexibility to select a better model
class in terms of behaviour and complexity to better capture the dynamics of
the true underlying system [33].

To begin, consider the previous development leading up to equation (6). Even
if the system of equations (6) is consistent, it will generally have
infinitely many solutions [27, 21]. The prediction matrix $\mathcal{P}$ in (7)
is given by $\mathcal{P}=Y_{f}\mathscr{H}_{\rm red}^{\dagger}$, and
corresponds precisely to taking the _least squares_ solution of the first
equation in (6) as

$\displaystyle g^{\star}=\operatorname*{argmin}_{g}$
$\displaystyle\quad\|g\|_{2}^{2}$ $\displaystyle\operatorname{subject~{}to}$
$\displaystyle\quad\mathscr{H}_{\rm red}g=\mathrm{col}(u_{\rm
ini},\hat{d}_{\rm ini},\hat{y}_{\rm ini},u(t),\hat{d}(t))$

and then substituting to obtain $\hat{y}(t)=Y_{f}g^{\star}$. When using noisy
data from a non-LTI data-generating system, it is advantageous to robustify
this least-squares problem by adding regularization [21]. To this end, for the
equation $\mathcal{H}_{\rm red}g=\xi$, we have

$g=\mathscr{H}_{\rm
red}^{\dagger}\xi\qquad\Longleftrightarrow\qquad(I-\mathscr{H}_{\rm
red}^{\dagger}\mathscr{H}_{\rm red})g=0.$

Thus, with $\mathcal{Q}=\mathscr{H}_{\rm red}^{\dagger}\mathscr{H}_{\rm red}$,
a least squares solution for $g$ also arises from minimizing the objective
function $\|(I-\mathcal{Q})g\|_{2}^{2}$ subject to the linear constraint
$\mathscr{H}_{\rm red}g=\xi$.

Our disturbance estimation approach is now to _intentionally bias_ this least
squares solution, by introducing additional objective functions quantifying
the prediction error along with regularization on $g$. This intentional
biasing exploits the bias-variance trade-off from system identification [32],
leading to reduced overfitting in the estimation procedure. With the same
notation and set-up as in Section III-B, at time $t$ we solve the convex
optimization problem

$\displaystyle{\operatorname*{min}_{\hat{d}(t),\hat{y}(t),g}}$
$\displaystyle\quad\|y(t)-\hat{y}(t)\|_{2}^{2}+\lambda_{1}\|(I-\mathcal{Q})g\|_{2}^{2}+\lambda_{2}\|g\|_{2}$
(8) s.t. $\displaystyle\qquad\begin{bmatrix}U_{\rm p}\\\ D_{\rm p}\\\ Y_{\rm
p}\\\ U_{f}\\\ D_{f}\\\ Y_{f}\end{bmatrix}g=\begin{bmatrix}u_{\rm ini}\\\
\hat{d}_{\rm ini}\\\ y_{\rm ini}\\\ u(t)\\\ \hat{d}(t)\\\
\hat{y}(t)\end{bmatrix},$

where $\lambda_{1},\lambda_{2}\geq 0$ are tuning parameters. The problem (8)
combines the prediction and estimation steps from (7) into one formulation,
jointly generating the output prediction $\hat{y}(t)$ and the disturbance
estimate $\hat{d}(t)$. The first objective function term attempts to match the
prediction $\hat{y}(t)$ to the measured output $y(t)$. Increasing
$\lambda_{1}$ encourages a least-squares solution for $g$, similar to that
used in (7), while increasing $\lambda_{2}$ regularizes the solution; this
reduces overfitting [32] and improves estimation robustness for noisy
measurements and non-LTI dynamics. While a theoretical estimation error
analysis for (8) is outside the scope of this article, the approach is
strongly justified by recent advances in regularized data-driven control [33],
and performance will be extensively tested in Section IV.

###  III-D Specialization to Area-Based Fast Frequency Control using Inverter-
Based Resources

We now describe the adaptation of our general data-driven disturbance
estimation methods to the fast frequency control architecture described in
Section II. Consider a large interconnected power system which is divided into
several small LCAs. Each LCA has local IBR’s that can be re-dispatched by the
operator, subject to their real-time capacity limits. Since each LCA is
geographically small, the effect of a power imbalance within the LCA on the
frequency is approximately independent of the specific nodal location of the
imbalance within the LCA. Therefore, it is assumed that power disturbances and
generation are aggregate, and effectively lumped at a single bus. Put
differently, disturbance and control signals enter through the same channel,
and thus $B=B_{d}\in^{n\times 1}$ in (2).

The following selections are made for inputs and outputs: the measurement
$y(t)=\Delta\omega(t)\in$ is a single local measurement of frequency
deviation, and the disturbance $d(t)=\Delta P_{\rm u}\in$ models aggregate
unmeasured generation-load imbalance within the LCA. The input $u(t)$ to the
system consists of the measured tie-line flow $\Delta P_{\mathrm{tie}}(t)$ out
of the LCA, as well as the sum of all IBR power set-points $\Delta
P_{\mathrm{ibr}}(t)$.

Historical data must be used to build the Hankel matrices used in both
estimators. As the control and disturbance channels are lumped, during the
collection of historical data, the sum of IBR set-point changes, exogenous
load/generation changes, and inter-LCA tie-line flow changes must be recorded.
Further discussion of options for data collection is deferred to Section IV-A.

As a result of the above, the estimator (7) simplifies to

$\displaystyle\Delta\hat{f}(t)$
$\displaystyle=\mathcal{P}\cdot\mathrm{col}(\Delta v_{\rm
ini},\Delta\hat{f}_{\rm ini},\Delta v(t))$ (9a)
$\displaystyle\Delta\hat{P}_{\rm u}(t+1)$ $\displaystyle=\Delta\hat{P}_{\rm
u}(t)-\varepsilon L(\Delta\hat{f}(t)-\Delta f(t)),$ (9b)

where $\Delta v=\Delta P_{\rm ibr}-\Delta P_{\rm tie}-\Delta P_{\rm u}$, is
the aggregated input,
$\mathcal{P}=Y_{f}\left[\begin{smallmatrix}U_{p};\,Y_{p};\,U_{f}\end{smallmatrix}\right]^{\dagger}$,
is the prediction matrix, and $L\in$ is now a scalar. Analogously, the
optimization-based estimator (8) becomes

$\displaystyle{\operatorname*{min}}$ $\displaystyle\,\,\|\Delta
f(t)-\Delta\hat{f}(t)\|_{2}^{2}+\lambda_{1}\|(I-\mathcal{Q})g\|_{2}^{2}+\lambda_{2}\|g\|_{2}$
(10) s.t. $\displaystyle\,\,\begin{bmatrix}U_{\rm p}\\\ Y_{\rm p}\\\ U_{f}\\\
Y_{f}\end{bmatrix}g=\begin{bmatrix}\Delta P_{\rm ibr,ini}-\Delta P_{\rm
tie,ini}-\Delta\hat{P}_{\rm u,ini}\\\ \Delta f_{\rm ini}\\\ \Delta P_{\rm
ibr}(t)-\Delta P_{\rm tie}(t)-\Delta\hat{P}_{\rm u}(t)\\\
\Delta\hat{f}(t),\end{bmatrix},$

The imbalance estimate $\Delta\hat{P}_{\rm u}(t)$ from either method is then
used to redispatch the local IBRs in the LCA via the optimal power allocation
algorithm presented in [13].

##  IV Simulation Studies

We validate our designs by applying them to the three-area nonlinear test
system illustrated in Figure 2. Each LCA of the test system is based on the
the IEEE 3-machine 9-bus system given in [1], with the interconnection
parameters and active power dispatch info similar to [13]. In the modified
test model, two synchronous generators (SGs) in area one have been replaced
with a photovoltaic (PV) array and a Wind (WT) plant. Similarly, one SG each
in areas two and three are replaced with a PV farm. The PV array and wind
turbine plant are simplified models represented by non-dispatchable converter-
based units, which are parameterized using wind power and solar irradiance
data from [36, 37]. To facilitate frequency control, two dispatchable IBRs
have been added in each LCA. In addition, static var compensators (SVCs) and
synchronous condensers have been added to areas 1 and 2/3 to support the
voltage. All SGs and dispatchable IBRs in the system are set to have a 5%
speed droop curve on their respective bases, with a 36 mHz primary control
deadband. The pre-disturbance generation/demand in the system is approximately
800 MW.

Figure 2: Three-LCA test system.

###  IV-A Offline Data Collection and Controller Tuning

As described in Section III, our estimators require a library of historical
data generated from a _persistently exciting input_ that must be collected
_before_ the online implementation of the control. Examples of common
persistently exciting inputs from the literature include, pseudo random binary
sequence, autoregressive moving average sequence, sum of sinusoids, and white
noise [38, 39]. Among these, white-noise derivatives are most commonly used in
power system identification studies, such as measured ambient power
fluctuations [40], and low-power injected probing signals such as the low-
level pseudo-random noise (LLPRN) in [41] and the band-limited white noise in
[42]. In terms of what sources should be actuated for this data collection,
there are several theoretically-equivalent options for the purposes of this
work, including

  1. (i)

apply low-power probing modulations to IBRs during calm system conditions
(i.e., during times of minimal unmeasured generation/load changes), and meter
the resulting frequency and tie-line power changes,

  2. (ii)

hold IBR set-points constant, record ambient load power consumption changes
(or injected pseudo random white noise that mimics such changes), and meter
the resulting frequency and tie-line power changes,

or obvious variations/combinations of these. Still other possibilities, such
as using historical load estimates as proxy data, are of interest, but are
outside of our scope in this study. In our testing to follow, we pursue option
(i); we refer the reader to Remark III.1 in [13] for a discussion on the
feasibility and market incentives that makes this choice viable.

We now turn to the design of the low-power probing signal for IBR set-point
changes. In this work, we modeled our probing injection signal after the LLPRN
in [41], where we have combined a sum of sinusoids and band-limited white
noise. Each IBR within each LCA is commanded with the set-point changes shown
in Figure 3, given by

$\Delta P_{\rm ibr}(t)=\sin(12\pi t)+w(t)\qquad(\text{in MW}).$ (11)

The signal consists of a sinusoidal perturbation of 1 MW ($1.76\times 10^{-3}$
p.u.), with band-limited white noise $w(t)$ with noise power of $\approx$
$0.2\times 10^{-3}$ p.u.111Further investigation into excitation signal design
is outside our scope, but see [43, 44] for recent theoretical results.

While we stress that the choice of probing signal is not unique, with our
choice of signal, we are able to utilize the aggregated power input $u=\Delta
P_{\mathrm{ibr}}-\Delta P_{\mathrm{tie}}$ and output $y=\Delta\omega$ data for
each LCA recorded for only $10$ seconds at a sampling rate of $0.1$ seconds,
which is significantly shorter than the duration of 1200 seconds for ambient
data and 600 seconds for LLPRN reported in the literature [41].

Figure 3: Persistently exciting IBR set-point change for data collection
phase.

Regarding the tuning parameters, we used $T=101$ historical data points for
each LCA, collected sequentially with only one LCA being excited at a time.
The length of recent past data used in (9) and (10) was $T_{\rm ini}=7$;
larger values were found to produce no benefit. The controller gain
$\varepsilon$ in (9) was set via tuning to $\varepsilon=0.2$ by starting from
a small value and increasing until satisfactory performance was reached. For
the penalty parameters in (10), we set $\lambda_{1}$ to a large value of
$1\times 10^{8}$, according to the insights from Section III-C, and
$\lambda_{2}=1\times 10^{2}$ was set via tuning by gradually increasing its
value until no noticeable improvement in performance was observed.

###  IV-B Simulation Scenarios

We consider four different testing scenarios, which aim to highlight the
diverse challenges that can arise in a power system, including renewable
resource variability, sudden changes in load demand, and equipment failures.
The scenarios are

  1. (1)

response to sudden load changes of different sizes,

  2. (2)

response to solar and wind farm variability,

  3. (3)

response to a three-phase-to-ground fault, and

  4. (4)

response after loss of a conventional generation unit.

For all scenarios, we integrate our disturbance estimators into the
hierarchical fast frequency control architecture proposed in [13] and compare
the model-based disturbance estimator of that work against the data-driven
disturbance estimators presented in this paper. We term the controller
described in (9) as the Linear Data-Driven Disturbance Estimator (LDDE) and
that in (10) as the Optimization-Based Data-Driven Disturbance Estimator
(ODDE); the ODDE is the default data-driven controller presented in the
figures when no other context is given. As a baseline, we compare to the
response without any supplementary control scheme, where frequency support is
provided only through the primary droop control action of both generators and
IBRs. Additionally, we compare against the response obtained by implementing
standard automatic generation control (AGC) on the three-area system in Figure
2.222See [13, Remark II.4] for extensive discussion on the distinctions
between the proposed approach and traditional AGC. In Scenario #1, we have
compared the ODDE against all the alternatives listed above, and demonstrate
its performance premium relative to the LDDE. In the remainder of the
scenarios, we focus the plots on comparing the better estimator (ODDE) against
the model-based approach presented in [13]. Finally, the data collection and
real-time simulation steps include measurement noise, modelled as zero-mean
white noise of standard deviation $10^{-6}$ p.u. for frequency deviation and
$2\times 10^{-2}$ p.u. for inter-area power flow measurements; these represent
realistic noise on the variables scaled for their typical values (e.g., [42]).

###  IV-C Scenario #1: Step Load Changes

This scenario evaluates the performance of our controller in response to step
load changes of varying magnitudes – 14 MW and 60 MW – applied in Area 2.

At $t=10$s, a small load change of 14 MW is applied at bus 8 in area two. The
size of the disturbance is chosen such that the resulting frequency deviation
is below the 36 mHz deadband of the generator primary control systems. The
frequency response and disturbance estimate of the system are plotted in
Figure 4, while the net-tie line deviations and IBR power outputs are
displayed in Figure 6. For clarity in differentiating the alternative
approaches, a zoomed-in frequency response plot is shown in Figure 5.

Figure 4: Frequency and disturbance estimate during a 14 MW load change at bus
8 in Area 2. Figure 5: Zoomed-in frequency plot of the contingent area showing
the control alternatives during a 14 MW load change at bus 8 in Area 2. Figure
6: Tie-line deviation and active power profiles during a 14 MW load change;
dashed lines in the lower plots indicate the responses under model-based
estimation.

Using both the model-based and data-driven disturbance estimators, the
disturbance was quickly identified to originate in Area 2 and promptly
corrected by adjusting the setpoints of the local IBRs, with minimal impact on
other areas. Overall, the frequency was restored quickly, and the variables in
the non-contingent areas returned to their pre-disturbance state due to the
decentralized nature of the control scheme. The plots also demonstrate that
the proposed optimization-based data-driven estimator outperforms the linear
data-driven and model-based estimators in terms of a higher nadir and faster
settling time for the post-contingency frequency. We believe the improved
performance of the optimization-based estimator relative to the linear
estimator is due to its ability to better capture the dynamics of the true
underlying system in terms of behaviour and complexity.

The plots in Figures 7, 8, and 9, which display the frequency response,
disturbance estimate, net tie-line deviations, and IBR outputs in response to
a step load change of 60 MW applied at bus 8 in Area 2 at $t=10s$, lead to the
same conclusion about the controller’s performance as the previous scenario
with a 14 MW load change. This demonstrates that the proposed controller
exhibits superior performance for step load changes both inside and outside
the governor deadband range.

Figure 7: Frequency and disturbance estimate during a 60 MW load change at bus
8 in Area 2. Figure 8: Zoomed-in frequency plot of the contingent area showing
the control alternatives during a 60 MW load change at bus 8 in Area 2. Figure
9: Tie-line deviation and active power profiles during a 60 MW load change;
dashed lines in the lower plots indicate the responses under model-based
estimation.

In general, the results show that the robust data-driven approach presented in
this study outperforms the model-based approach and other alternatives,
quickly localizing and compensating for large and small disturbances.

###  IV-D Scenario #2: High Renewable Resource Fluctations

In this scenario, we aim to demonstrate the effectiveness of our data-driven
approach in the presence of renewable variability using realistic wind and
solar irradiance data. To simulate the solar irradiance component, we use data
from the Oahu solar measurement grid 1-year archive [37], containing 1-second
measurements of solar irradiance. We select a slice of data from July 31, 2010
(see Figure 10). These values are used to simulate a converter-interfaced PV
farm in Area 2.

Figure 10: Solar irradiance and wind power data representing suitably scaled
slices of data on 31 July 2010 and 15 Aug 2019 from [37] and [36]
repositories.

For the wind farm component, we use 4-second resolution wind power
measurements from the wind power dataset accessible at [36], originally
obtained from the Australian Energy Market Operator (AEMO). We select a slice
of data from August 15, 2019 (see Figure 10). These values are used to
simulate a converter-interfaced wind farm in Area 1.

At $t=10s$, we introduce a step load change of $40$ MW in Area 2 and
simultaneously introduce the new the solar irradiance levels and wind power in
accordance with the real-world data. The frequency response, disturbance
estimate, net tie-line deviations, and IBR outputs of the power system for
this scenario are displayed in Figures 11, 12, and 13.

Figure 11: Frequency and disturbance estimate during high renewable resource
fluctuations in multiple areas. Figure 12: Zoomed-in frequency plot during
high renewable resource fluctuations in multiple areas. Figure 13: Tie-line
deviation and active power profiles during high renewable resource
fluctuations in multiple areas.; dashed lines in the lower plots indicate the
responses under model-based estimation.

The simulation results show that the optimization-based data-driven estimator
is able to quickly estimate the highly variable power imbalance in real-time,
allowing the frequency to be kept close to the nominal value throughout the
simulation period. The performance of our designed data-driven estimator is
quantified in Table I, which shows the root mean squared frequency deviation
for each generator in the system. The results show that the data-driven
disturbance estimator with regularization performs the best.

Control type | RMSE (Hz) | Total (Hz)
---|---|---
| G1 | G2 | G3 | G4 | G5 |
Data-based w/ reg. | 0.0065 | 0.0065 | 0.0064 | 0.0066 | 0.0065 | 0.0325
Data-based w/o reg. | 0.0071 | 0.0073 | 0.0072 | 0.0074 | 0.0073 | 0.0365
Model-based | 0.0081 | 0.0081 | 0.0080 | 0.0080 | 0.0081 | 0.0403
AGC | 0.0236 | 0.0225 | 0.0233 | 0.0225 | 0.0233 | 0.1151
Droop only | 0.1885 | 0.1883 | 0.1884 | 0.1884 | 0.1884 | 0.9420

TABLE I: Root Mean Square Error for Control Alternatives.

###  IV-E Scenario #3: Symmetric Three-Phase Fault

The essence of this scenario is to assess the performance of our control
approach under a severe contingency like a symmetrical three-phase line-to-
ground fault. The response of the system during the fault introduced at bus 8
in Area 2 at $t=2$s and cleared after $0.1$s is shown in Figures 14 and 15.
Note that despite the transients, the controller is able to discern that there
is no net load imbalance within the area. The results indicate that the
controller is able to effectively detect and respond to frequency events, and
the data-driven estimator’s performance is satisfactory and similar to the
model-based estimator.

Figure 14: Frequency and disturbance estimate during a three-phase fault in
Area 2. Figure 15: Tie-line deviation and active power profiles during a
three- phrase fault in Area 2; dashed lines in the lower plots indicate the
responses under model-based estimation.

###  IV-F Scenario #4: Loss of Generator

This scenario assesses the performance of our control approach under the loss
of generator G2 in area 2 at $t=10s$. When the generator G2 is lost, the
system experiences a disturbance, and the controllers respond to correct the
resulting power imbalance. The response of the system under both the model-
based and data-driven control is plotted in Figure 16. According to the
findings, the data-driven controller outperforms the model-based controller,
as demonstrated by its faster frequency settling time and lower overshoot.
Importantly, despite the data used to design the LCA controller having been
collected while G2 was online, the response indicates that the data-driven
controller is effective even in the face of significant changes in power
system composition and frequency response dynamics. This illustrates the
robustness of the design, and provides flexibility for system operators in
deciding how frequently they wish to collect new data to update the
controller.

Figure 16: Frequency and disturbance estimate during a generator G4 loss in
Area 3.

##  V Conclusions

We have proposed and validated through detailed simulations a robust data-
driven disturbance estimator that allows us to reliably compute the real-time
power imbalance in a highly nonlinear power system, and in the presence of
measurement noise. This data-driven estimate has been integrated in the
hierarchical frequency control architecture initially proposed in [13], to
provide a completely model-free approach to provide fast, localized frequency
regulation in the power system. An important direction for future research is
further investigation into the design of improved excitation input signals for
data collection, and integration of this approach with transmission and
distribution coordination schemes.

##  Appendix A Proofs

Proof of Theorem III.1: Under the stated assumptions of controllability, input
data persistency of excitation of order $T_{\rm ini}+1+n(\mathscr{B})$, and
sufficient initialization length $T_{\rm ini}\geq\ell(\mathscr{B})$, it
follows from [27, Prop. 6] that the output predictor (7a) produces precisely
the same values $\hat{y}(t)$ as the LTI system

$\displaystyle\hat{x}(t+1)$ $\displaystyle=A\hat{x}(t)+Bu(t)+B_{d}\hat{d}(t)$
(12) $\displaystyle\hat{y}(t)$ $\displaystyle=C\hat{x}(t)+Du(t)$

where the matrices may be taken to be the same as those in (2). The
disturbance esimator (7) may therefore be expressed as (12) with (7b), which
we rewrite together as

$\displaystyle\begin{bmatrix}\hat{x}(t+1)\\\ \hat{d}(t+1)\end{bmatrix}$
$\displaystyle=\underbrace{\begin{bmatrix}A&B_{d}\\\
0&I_{q}\end{bmatrix}}_{:=\mathcal{A}}\begin{bmatrix}\hat{x}(t)\\\
\hat{d}(t)\end{bmatrix}+\begin{bmatrix}B\\\
0\end{bmatrix}u(t)-\underbrace{\begin{bmatrix}0\\\ \varepsilon
L\end{bmatrix}}_{:=\varepsilon\mathcal{L}}(\hat{y}(t)-y(t))$
$\displaystyle\hat{y}(t)$
$\displaystyle=\underbrace{\begin{bmatrix}C&0\end{bmatrix}}_{:=\mathcal{C}}\begin{bmatrix}\hat{x}(t)\\\
\hat{d}(t)\end{bmatrix}+Du(t)$

where $y(t)$ is the measured output of (2). The above has the form of a
Luenberger observer, and standard estimation error analysis (e.g., [45])
implies that we will have $\hat{d}(t)\to d(t)$ exponentially, and irrespective
of the initial conditions, if

$\mathcal{A}-\varepsilon\mathcal{L}C=\begin{bmatrix}A&B_{d}\\\ -\varepsilon
LC&I_{q}\end{bmatrix}$

is Schur stable. Recall that, by assumption, $A$ is Schur stable, so $I_{n}-A$
is invertible; based on this define the invertible matrix

$T=\begin{bmatrix}I_{n}&(I_{n}-A)^{-1}B_{d}\\\ 0&I_{q}\end{bmatrix}.$

By similarity, $\mathcal{A}-\varepsilon\mathcal{L}C$ is Schur stable if and
only if $\mathcal{M}(\epsilon):=T(\mathcal{A}-\varepsilon\mathcal{L}C)T^{-1}$
is as well. Simple calculations show that $\mathcal{M}(\epsilon)$ evaluates to

$\mathcal{M}(\epsilon)=\begin{bmatrix}A+\varepsilon M_{1}&\varepsilon M_{2}\\\
\varepsilon M_{3}&I_{q}-\varepsilon LG(1)\end{bmatrix},$

where $M_{1},M_{2},M_{3}$ are constant matrices and
$G(1)=C(I_{n}-A)^{-1}B_{d}$. By assumption, $G(1)$ has full column rank and
$L=G(1)^{\dagger}$; thus, we have that $LG(1)=I_{q}$, and the $(2,2)$ block of
the above simplifies to $(1-\varepsilon)I_{q}$. Since $A$ is Schur stable, by
standard linear Lyapunov theory there exists a matrix $P\succ 0$ such that
$A^{\sf T}PA-P\prec 0$. Defining $\mathcal{P}=\mathrm{diag}(P,I_{q})$,
straightforward calculations and a use of the Schur complement lemma show that
$\mathcal{M}(\epsilon)^{\sf
T}\mathcal{P}\mathcal{M}(\epsilon)-\mathcal{P}\prec 0$ for all sufficiently
small $\varepsilon>0$, which establishes that $\mathcal{M}(\epsilon)$ is Schur
stable and completes the proof. $\square$

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