Noise Resilient Learning for Attack Detection in
Smart Grid PMU Infrastructure
Prithwiraj Roy∗ § , Shameek Bhattacharjee† § , Sahar Abedzadeh† and Sajal K. Das∗
∗ Missouri University of Science and Technology, Rolla, MO, USA,† Western Michigan University, Kalamazoo, MI, USA
E-mail: przhr@mst.edu; shameek.bhattacharjee@wmich.edu; sahar.abedzadeh@wmich.edu; sdas@mst.edu

Abstract—Falsified data from compromised Phasor Measurement Units (PMUs) in a smart grid induce Energy Management
Systems (EMS) to have an inaccurate estimation of the stateof the grid, disrupting various operations of the power grid.
Moreover, the PMUs deployed at the distribution layer of asmart grid show dynamic fluctuations in their data streams,
which make it extremely challenging to design effective learning
frameworks for anomaly based attack detection. In this paper,
we propose a noise resilient learning framework for anomaly
based attack detection specifically for distribution layer PMU
infrastructure, that show real time indicators of data falsifications
attacks while offsetting the effect of false alarms caused by thenoise. Specifically, we propose a feature extraction frameworkthat uses some Pythagorean Means of the active power froma cluster of PMUs, reducing multi-dimensional nature of the
PMU data streams via quick big data summarization. We also
propose a robust and noise resilient methodology for learning
thresholds based on generalized robust estimation theory of
our invariant feature. We experimentally validate our approachand demonstrate improved reliability performance using twocompletely different datasets collected from real distribution level
PMU infrastructures.
Index Terms—Anomaly Detection, Big Data Management,
Learning based PMU Security, Smart Grid Security.

I. I NTRODUCTION
Modern smart electrical grid is an example of a large scaleInternet of Things (IoT) domain, in which advanced IoT
telemetry devices like Phasor Measurement Units (PMUs) are
deployed for collecting high resolution time series measurements. Such measurements offer highly improved real timesituational awareness of the state of the grid compared to theolder Supervisory Control And Data Acquisition (SCADA) infrastructure measurements. Another critical difference between
the PMU and SCADA technologies is how they gauge theangle measurements. While in PMU, the angle measurements
are made directly, the SCADA system measure the voltage
angles by using voltage measurements, active and reactive
power, network parameters, and a reference angle. Thus thequality of the results depend heavily on network parameters
for SCADA system which are not always precise [2]. Thus
the PMUs have become the critical cornerstones in the smart
grid architecture. Additionally, several internet based communication technologies for sending such data and receiving thecorresponding actions are being developed for a vision of an
adaptive, reliable, and efficient modern electrical grid.
§ Equal contribution

The PMUs are IoT endpoint devices positioned at criticallocations at both the transmission and distribution layers of asmart grid infrastructure. The PMUs record time-synchronizedmeasurements (voltage, current, phase angles - collectively
called synchrophasor data) and send them to an aggregatorcalled Phasor Data Concentrator (PDC), which in turn relay
such data to a Local Control Center (LCC). Multiple LCCs
allow smart grid operators to localize and infer the type, time,and location of a fault or disturbance as well as support criticaloperations such as state estimation.
Data Integrity Threat in PMU: Data integrity in PMUs is
extremely critical since the control centers base their decisions(control or actuation) directly on these measurements; theoutput of various applications also uses these measurements
indirectly, such as economic dispatch and contingency analysis. Now with the process of digitalization, these systems havebecome increasingly more complex and have been exposed topotential security threats which could lead to a fatal effect on
the power grid network. These threats are posed by numerous
parties such as hackers, ex-employees, competitors, and evenmaintenance personnel [3]. Various reports on cyber-attacks
in power distribution systems are threatening the security andreliability of these operations. For example, a report of the
US National Research Council highlights potential multi-state
blackouts as a result of coordinated False Data Injection (FDI)
attacks on the power systems [6]. An attack on the Ukrainian
power grid resulted in the loss of power for 225,000 customers
in three different territories which lasted for several hours[7]. The Stuxnet worm that attacks particular programmable
logic controller (PLC) on the vulnerable windows computersaffected more than 100,000 industrial components [8].
The widely used IEEE C37.118-2 protocol for synchrophasor data communication between the PMUs and PDC isreported to be vulnerable to cyber-attacks [11], [13]. In fact,
most synchrophasor data transmission happens on insecure IP
networks. Strong encryption is not feasible due to the latency
criticality of PMU applications and high data resolution thattypically generate 50-120 samples of 12 physical quantities
per second. All of the above reasons increase the chances ofdata falsification attacks once the adversaries laterally intrude
into the PMUs. Furthermore, the possibility of transduction
attacks causing data falsification on such telemetry data makes
cryptography and network traffic based attack detection ineffective [25]. Therefore, more data-driven anomaly based data
falsification attack detection is a viable cybersecurity approach

that depends on the unraveling of suspicious patterns.
A. Motivation and Challenges
The primary motivation of this paper is fueled by theneed for an anomaly based FDI attack detection scheme thatis resistant to outliers caused by either measurement and/or
process noise that may bias the accurate learning of the underlying structure of distribution level PMU data under benign
operating conditions. Additionally, the resultant frameworkneeds to be light-weight and automated, and it should be able
to detect attacks quickly that would work under settings of
distribution level power grid systems.
Challenges of Distribution Level PMUs: The specificchallenge of distribution level PMUs is that the data underbenign conditions change readily under legitimate benign
conditions, as opposed to transmission level PMUs, where
the dynamics of PMU data streams show inherently fewer
variations. The distribution level PMU data streams are
particularly prone to unpredictable variations since they are
directly connected to the customer layer. The customer layer
contains an eclectic mix of loads from campuses, residences,and businesses that create higher variations in current data
streams, which are affected by not only physics but also
human behavior. Additionally, distributed energy sources (e.g.,
solar panels) increase uncertainty in the measurement patterns
collected from PMUs. These issues make the approaches usedin transmission level PMUs for anomaly detection unreliablein the distribution level PMUs setting.
Challenges of Outliers and Noise: After careful synthesisof the existing literature we found that most works in FDI
attacks on PMU, have small duration of study that ranged
from a few minutes to a few days [15], [21], [22], [24]). In
the real world, however, this attack detection needs to run overmuch longer time horizons and be effective.
Keeping that in mind, we tried to apply our previous
approach from our preliminary conference version [1] on the
EPFL dataset that has duration in the order of months insteadof a few minutes to few days. In our preliminary work, we hadfound success with both false alarms and attack detection on an
LBNL dataset that was 11 days long. However, we found that
it failed to achieve any acceptable performance on the much
longer duration PMU dataset from the Ecole Polytechnique
Federale De Lausanne (EPFL), Switzerland.
We believe that the root causes of such failure can be
attributed to outliers and process noise. This motivated us toextend our previous method with robust learning. Specifically,[26] conducted studies on the EPFL dataset and found andquantified noise due to various environmental drivers andevents that create changes in the measured data. The paper
also proposed a method to quantify noise in the PMU data.
Following that method, we tried to quantify noise for bothdatasets: LBNL and EPFL. We found that Signal-to-Noise(SNR) is higher for EPFL data compared to LBNL data. For
example, the SNR for EPFL data is 7.6 while the same for
LBNL data is 5.4. Moreover, we have also used a MedianAbsolute Filter [26] to demonstrate the presence of significant
noise in both datasets as shown in Fig. 1. It is evident that thefiltered measurements (orange line) closely follow the actual
signal for the LBNL data. However, for the EPFL data, thefiltered signal is quite far from the actual signal envelope
confirming that the later dataset contains more variations thatdeviate from a central tendency, and it also confirms thatfiltration causes the filtered data to be far away from theactual data and this caused an increase in false alarms. We
tried to understand the underlying cause of this and found thefollowing:
There are many routine benign events like series capacitor
switching, load transferring, etc. [44] in the power grids thatcause sudden short lived perturbations in the PMU measurements. To properly model the benign system behavior, we
ought to take into account the measurement changes causedby routine events. If we use a noise filter such as the popularly
used Median Absolute Filter [26], [42], [43], many such events
get eliminated from the filtered signal along with occasional
outliers. This produces an overcompensated model of benign
behavior that does not reflect the actual routine behavior
during benign operation. This increases the occurrences offalse alarms when our preliminary method is applied in thetesting/generalization phase with filtered data. This makes timeseries anomaly based attack detection frameworks unusable inpractice. Also, the noise filtering technique in [26] requires
the Filter Order, M , which is manually extracted from thedata. This implies the filter order is dependent on the data
under consideration and it adds computational overhead andmanual intervention to the process. Therefore, adding existing
filtration approaches is not viable in our setting.

(a)

(b)

Fig. 1: Noise in Active Power: (a) LBNL (b) EPFL.
Then we tried the EPFL dataset with small duration, butsince this small duration is not representative of all benign
conditions, it produced high rate of false alarms when tested
over a duration of months. In contrast, when we increased
the size of the training set in the EPFL, the detection rate
dropped because the thresholds learning was getting biasedby more outliers. To explain the link between longer durationand this development, we believe that if a longer durationof training data is taken into consideration, there is a higher
chance events/environmental drivers that create outliers manifest themselves. It is a simple result of unbiased sampling.
If one considers a small training and testing dataset, one may
get lucky because the small portions being considered does
not have any event/drivers that lead to outliers in the training
and testing. This favors good performance but is not robust.
Since outliers in the benign data prevent accurate learning thegeneral representation of the underlying structure of benign
operating conditions, it affects the performance of anomaly
detection techniques which depend on characterizing the latentstructure of benign data. The root cause of this paradox seems
to be sampling bias although there may be other unknown
things like any topology change that might have happened in
the test bed which we may be unaware of.
Therefore, we concluded that we need to develop
noise/outlier resilient learning of the anomaly detection
thresholds that can represent the most general bounds in theunderlying structure of benign operations in the anomaly detection frameworks for distribution level PMUs; which enables
more accurate anomaly based attack detection.
B. Contributions of this WorkIn this paper, we first discuss multiple attack strategies fordata falsification in smart grid PMU infrastructure. Then we
propose a semi-supervised learning framework for anomaly
based data falsification attack detection in the distribution level
PMU network that contains a set of PMUs under the control
of a PDC. The proposed framework is divided into two parts:(1) the feature engineering generates a latent space of featuresin a low dimension, which models the multivariate streams of
raw PMU data into one learnable feature across time; and (2)robust learning of anomaly detection threshold and criterion
from the generated feature space that contains outliers.
Specifically, for feature engineering, we propose the useof active power as the process variable and use the ratio of
harmonic mean to arithmetic mean of the active power ona strategically calculated spatial and temporal granularity as
an invariant. Such spatial and temporal granularities lead totime series invariance under no attack, but show changes undervarious kinds of data falsification attacks. Next, we convert theinvariant into stateful residuals which is a feature that balances
false alarm versus detection trade-off.
For robust learning of attack detection criterion from thefeature representation, we propose to use robust estimation
theory to derive the optimal thresholds that characterize thebenign operating condition in the presence of noise and heavy
tailed nature of the feature distribution. Finally, we validate
our work by using two real PMU datasets. The first dataset
was collected by the Power Standards Lab (PSL) at Lawrence
Barkley National Lab (LBNL) campus in Berkeley, CA, whichdeveloped high-precision µ-PMUs deployed at multiple utilities and LBNL campus locations on a 12 kV distribution
grid [16]. The second dataset was collected by the smart grid
infrastructure monitoring project on a 20 kV distribution gridin the EPFL campus, Switzerland across several months [20].
These two large datasets corroborate the generalizabilityof our novel approach. The main benefits of our approach
are to provide a practical framework to identify the presenceof FDI attacks in the distribution level PMU architecture.
The framework (i) is noise resilient, real time, light-weight,
and semi-supervised; (ii) enables quick identification; (iii)simultaneously works for a variety of data falsification attack
types; and (iv) works for multiple data sources with different
distribution and frequency ranges.
Paper Organization: The rest of this paper is organized asfollows. Section II introduces the system model and describes
the datasets, while Section III discusses the threat model.
Section IV and Section V respectively present the proposed

framework and experimental results. Section VI reviews therelated works and Section VII presents further discussion on
the scope and applicability of the proposed appraoch. Finally,
Section VIII concludes the paper.
II. S YSTEM M ODELIn this section, we discuss the PMU system architecture anddifferent datasets used for modeling and validation.
A. PMU System Architecture
Now we layout the architecture of a PMU network on a 3phase AC system. For any phase j, the PMU devices which areessentially IoT devices that measure time-stamped (t) voltage
and current magnitudes denoted by Vt (j), It (j) and the phaseangles θtV (j), θtI (j) respectively. In our paper, we use the term‘load’ as synonymous to the current magnitude per phase.
The PMUs are deployed at strategic locations of the distribution layers of a smart electrical grid which send theirdata to PDC which in turn send sychrophasor data to LCC.
The estimated state from LCC informs power generation,transmission, and distribution strategies and applications suchas early fault detection and load balancing. The architectureof a typical PMU-PDC infrastructure is shown in Fig. 2.

Fig. 2: Architecture of a PMU Infrastructure.
B. Dataset Description
To validate our contribution, we used two datasets viz. EPFL
and LBNL with contrasting characteristics, to ensure that our
method generalizes across datasets. In each of the 3 phases
4 physical quantities are being measured by a PMU device:
voltage magnitude, current magnitude, phase angle of voltage,
and phase angle of current. Therefore, in both datasets each
PMU is producing 12 streams of data.
EPFL Dataset: This data was collected for a smart gridproject by the Ecole Polytechnique (EPFL), Switzerland [20].
It contains data from five PMUs deployed on a 20 kV distribution layer. The PMU devices are named as: PMU2, PMU3,
PMU4, PMU5, PMU6 at a data resolution of 50 samples per
second. We have taken 6 months of data (Sep-14 to Feb-15)
for our model for training and testing, which the longest in theliterature of PMU data integrity attack detection research. The
work [26], showed that this data-set contains a lot of noise.
LBNL Data: We used another dataset collected by thePower Standards Lab (PSL) located at Lawrence Berkeley National Lab (LBNL), USA, which was used in our preliminary
work [1]. We have included the LBNL dataset for performancecomparison. This dataset is collected from three high precisionµ-PMUs deployed at multiple LBNL campus locations on

12 kV distribution grid [16]. Each µ-PMU device is givena name for unique identification: A6, Grizzly and Bank514.
The measurements are taken a rate of 120 Hz with time stamp
accuracy of 100 ns, for 11 days (from 1-Oct-15 to 11-Oct-15).
In this paper, we are specifically interested in a decentralized anomaly detection that runs on a PDC or a LCC andfacilitates early attack detection from a bunch of PMUs thatare geographically proximate in terms of the PMU network.
III. T HREAT M ODELS
Telemetry attack data on PMU are not available sincethese are protected systems. Threat models for data integrity
attacks in PMU have been most simulated and emulated in
the literature in two competing ways. Some works emulate
specific strategies but the perturbation amount is random,
while others use a strategic perturbation amount to not violate
physical bounds of operation (typically applies on voltage
and phase data). There is some criticism of this approachin that the attacks are random in nature without quantifiable
ways to emulate a threat landscape that would allow security
performance evaluation. The other types of PMU data integrity
attacks consider optimal attacks or a specific instance of attack
hiding [30]. While papers on hiding attacks from bad datadetectors exist, the operational impact is not emphasized.
We believe that threat models should ideally parameterize
if possible all instances of data falsification attacks and show
the performance of a defense framework as a function of
those parameters to understand the performance limits. This isbecause a defender can never know what kind of attack will
happen from what adversary. Therefore, assuming a specificattack instance or strategy and then proposing solutions validated against those attacks are realistic. Many adversaries act
in a bounded rational and sometimes what defenders would
perceive as irrational, which creates the paradox between
expectations and reality.
Therefore, to avoid this pitfall we have used a generalized attack emulation technique proposed in several earlier
papers [4], [5] that parameterizes the data integrity threat
landscape into the following features: (i) Attack types (ii)
Attack Strength (iii) Attack Scale (iv) Attack Strategy. All
possible instances/values of these features are implemented in
the attack simulation to create a super set of attack combinations that do not have optimistic or pessimistic assumptions
about adversaries’ actions or capabilities. Our method is tested
against all the whole super set and performance is reported
accordingly. Below we provide details of how we created thedata integrity attack landscape for this work.
Attack Types: Attack types indicate how data is changed
from each PMU and the impact objective behind data falsification. Organized adversaries may inject false data from one or
multiple compromised PMUs concurrently by changing either
data at rest, or in transit from the PMUs to the LCC. Any of
the four streams (Voltage Magnitude, Voltage Angle, Current
Magnitude, Current Angle) of any phase may be falsified.
However, in this paper’s validation, we show adversary
falsifies the ‘current magnitude’ because the current data
stream at the distribution level changes constantly over awide range even under benign conditions. This is not thecase with the voltage and phase angle values. For example,
in power grids, the electrical devices and equipment that areconnected to the network are often very sensitive to voltage.
Hence, sudden increase or drops in voltage damage such
expensive devices. To counter that, voltage stabilizers are usedin electric grid network to keep the voltage stable. Thus, if thecurrent is compromised, that does not necessarily impact thevoltage. Instead power measurements get impacted because of
the current modifications. Small margin attacks hide behindthese randomnesses. While our method may work for any
falsification (as reasoned later with active power), all our
results and validation assume current data falsification.
Let Iti (act) be the actual current magnitude measured byi-th PMU at time slot t and let Iti be its advertised value thatreached the PDC. Under no attacks, Iti = Iti (act), while in thepresence of attacks, the reported value Iti can be perturbed in
the following ways:
Deductive Attack: The current magnitude from the i-thcompromised PMU at time t is reduced from its actual
value such that Iti (act)−Iδt , where Iδmin ≤ Iδt ≤ Iδmax ,where the Iδt is a perturbation value that is sampledaccording to a strategic statistical distribution (which is
explained under attack strategies), bounded within theinterval [Iδmin , Iδmax ] with Iδmin > 0.
If the current magnitude drops in a certain phase, it
disrupts the efficiency of the grid by reducing the power
utilization factor (or efficiency, η) which indicates theratio of mechanical input power to the generated electrical
output power. This creates a motivation to falsify current
measurements with a deductive attack type.
• Additive Attack: Similarly, an additive data falsification
may be injected by modifying the current magnitude
Iti = Iti (act) + Iδt from a compromised PMU, whereIδt is chosen with a similar strategic principle as mentioned in the deductive attack type. An increase in thecurrent on a given phase will trigger a drop in the phasevoltage. If the current magnitude increases beyond asafe limit, the concerned phase is shed, or the load is
switched/distributed to other phases. To keep the voltage
from dropping more, power needs to be injected in agiven phase. The above reasons create a motivation for
additive perturbation of current measurements collectedfrom PMUs. The attack type can be launched by a rival
utility to make the control center believe in a sudden
increase in load which might lead to load shedding inthat particular phase.
• Alternating Switching Attack: The adversary alternates
between additive and deductive falsification for equalamounts of time, ensuring the same average of perturbation amounts Iδt over the time domain. In such a case,the total average additive and the total average deductive
perturbation amounts over a particular attack time period
will balance each other. This statistical balancing makes
it hard for many cumulative change point based statisticalanomaly detectors to detect such attacks. Fig. 3a demonstrates the impact on the current magnitude if a PMU iscompromised with such an attack. The grid is affected in•the same way as additive and deductive types would forthe respective durations, but will escape some statisticaldetectors.
i• Mirroring Attack:Here the attacker monitors the It for agiven time period and then he replaces the then actual current measurements with a mirror image of the previously
recorded Iti over a previous period [9]. The mirroring
attack makes sense when the adversary wants to hide
the detection of a legitimate event or change point by
the PDC/LCC. By mirroring the old perfectly believablevalues, the PDC/LCC never receives the updated realmeasurements that would have enabled them to detecta significant change in the state of the network. Fig.
3b shows the impact of mirroring attack on the current
magnitude, by hiding a grid state with a sudden increasein current magnitude, by replaying a mirror image of thejust previously recorded readings.

(a)

(b)

Fig. 4: Attack on Current Magnitude of PMU2 (a) Additive
Ramp; (b) Deductive Ramp.
(c) Random Strategy: This attack involves the addition/subtraction of positive values generated by a uniform
random function to the actual measurements [5]. The lower(Iδmin ) and upper (Iδmax ) bounds for selection are provided
to the function as an input. While this is simple, that does notchange the resultant shape of the load distribution drastically,making it a less obvious attack.
 i
if t ∈/ ∆a
It (act),
Iti =
Iti (act) ± rand(Iδmin , Iδmax ), if t ∈ ∆a .
Attack Scale: This includes how many PMUs in a decentralized system are involved in data falsification at the sametime which can affect the accuracy of outcomes.

(a)

(b)

Fig. 3: Attack on Current Magnitude of PMU2: (a) Alternating
Attack; (b) Mirroring Attack.
Attack Strength: We denote Iδavg as the average marginof false data per compromised PMU device. The Iδavg is a
strategic parameter that is selected by an adversary depending
on how much damage it wants to inflict on the electricalgrid. We keep this as an uncontrolled variable to test detection sensitivity since there could be various applications of
PMU data. We consider that the attack falsification sampleIδt ∈ [Iδmin , Iδmax ] being sampled from a distribution with acertain strategy (see later) whose long term average is Iδavg
The Iδmin , Iδmax , Iδavg extremes are informed by known limitsof each quantity physically possible and bad data detector [30].
Attack Strategies: We consider three types of attack strategies that are employed by the attacker for an attack period ∆a :(a) Step Strategy: In this case the adversary modifies all
samples to higher (additive) or lower (deductive) values byIδc in the
period, ∆a [34].
 attack
Iti (act),
if t ∈/ ∆ai
It =Iti (act) + Iδc , if t ∈ ∆a .
(b) Ramp Strategy: A ramp attack involves gradual falsification of the actual measurements. Here adversary gradually
increases the Iδt in each time slots to reach Iδmax and thenagain gradually decreases Iδt [12]. Here λr is the gradient of
the introduced ramp attack. Based on the adversary’s intent,
this attack can also be both additive and deductive in nature.
Fig. 4a and 4b shows the impact on current magnitude for
additiveand deductive ramp attacks respectively.
if t ∈/ ∆a Iti (act),
Iti (act) + λr .t,
if t ∈ ∆a /2
Iti =
 i
It (act) − λr .(∆a − t), if ∆a /2 ∈ t ∈ ∆a .

IV. P ROPOSED F RAMEWORKWe divide the proposed framework into the following
stages: (1) PMU-specific process variable (rather than streamspecific); (2) Invariant based anomaly detection metric building by optimizing spatial and temporal granularities of theprocess variables; (3) Building stateless and stateful residuals;(4) Learning thresholds of the stateful residual in a noise
resilient manner to establish the stateful residual’s metric
under benign conditions; 5) Determine the detection criteria
parameters, based on learning from the training and crossvalidation steps, and apply it on the testing set. An overall flow
diagram of the proposed framework is shown in the following
Fig. 5.

Fig. 5: Flow Diagram of the Proposed Framework.
A. Active Power as Process Variable
As indicated earlier, there are 12 streams of data per PMU,
and each stream is generated at rates as high as 50 to 120
sample data points per second. This shows that the variety
of data and velocity of the data are extremely large. If aframework requires monitoring each of these multiple data
streams per PMU individually, the anomaly monitoring of allthese streams separately increases the computational cost andlatency in anomaly detection analytics. Given the high velocityof the data, quick and lightweight big data summarization isrequired.
To address the above challenge, we propose the phase wise
active power calculated from synchrophasor data streams per
PMU, as the process variable over which the data driven
invariant is calculated. The active power (p(j)) per phase from
PMU measurements is calculated using the following standard
power equations:p(j) = V (j)I(j) cos θ(j)(1)

where j ∈ {1, 2, 3} denote the phases and V (j), I(j), θ(j) arevoltage magnitude, current magnitude, and θ(j) = θtV (j) −θtI (j) is the angle difference between voltage and current
phases respectively, for the j-th phase. This reduces the complexity of monitoring each stream separately unlike existing
works. Another advantage is that any deliberate falsification of
the voltage or current (both in terms of magnitude and phase)will impact the active power, and hence we can potentiallydetect an attack on any of the data streams from PMUs.
Therefore, for our anomaly detection, we propose to use
the phase wise monitoring of the active power p(j) as astarting point. Given that our analysis is phase specific we have
dropped the phase j from subsequent equations for simplicity.
Winsorization and Normalization: We apply winsorizationof α% on the derived active power dataset during the training
to remove instantaneous disturbances typically caused due toelectromagnetic transients which ensures that we learn themost general benign patterns. Please note, this is done only
for the training set.
As mentioned earlier, PMUs are deployed at different locations of the grid network that attach themselves to multiple
kinds of customer loads. Due to this reason, the active powerfrom each PMU is not in the same range and varies across
PMUs.Variables that are measured on different scales do
not contribute equally to the feature set. Therefore, we donormalization via MinMax Scaling. We also add a constantvalue (= 1) as per Eqn. 2 to keep all the values >= 1
which is required for Harmonic Means to work in the intended
manner. For notation simplicity, we used P instead of Pscaled
to indicate active power for the rest of the paper.
Pscaled =p − min(p)+ 1.
max(p) − min(p)However, as shown in the Fig. 6a, the time series of mean
active power of PMUs vary greatly over time windows, making
it difficult to distinguish legitimate changes from a malicious
one. Any moving average or smoothing technique either loses
sensitivity for a small margin of attacks (since the moving
average does not reflect the changes beyond already existing
deviations) or has a high number of false alarms. To avoid thiswe need a new approach for these smart living CPS.

(2)

B. Invariant for Anomaly Detection Metric
For real time anomaly detection in CPS, it has been established that a metric which is invariant under normal operating
conditions (without any attack) is ideal for attack detection [4],[31]. However, unlike tightly controlled industrial CPS applications, the distribution level synchrophasor data is affected byrandomness in renewable power outputs and heterogeneous
consumer types. This causes traditional statistical invariants
to have high randomness. As shown in Fig. 6a the arithmetic
mean of active powers of the time series is not stationary.
Prior works such as [17] propose the use of derived smoothingstatistics of the arithmetic mean (such as ARMA, EWMA,
CUSUM control charts) for time series anomaly detection.

(a)

(b)

Fig. 6: Illustrations of AM and HM/AM: Weekdays: Day
1,4,5,6,7; weekends: 2,3 (a) AM; (b) HM/AM.
Recently, in [4], for a different problem relating to smart
metering, we had shown that the ratio of harmonic mean
and arithmetic mean of a positively correlated set of randomvariables exhibit invariance in their time series even as the individual means show non-stationarity. Furthermore, [4] provedthat any data perturbation in any variable of this correlated set
will cause the ratio to lose its invariance and show deviations.
However, this stability is guaranteed for appropriately correlated variables only. Hence, our domain specific innovation
is to understand how to make it work on active power
derived from the PMUs. To achieve this, we aim to find aclustering with an appropriate spatial and temporal granularitythat maximizes the correlation between active powers on agiven phase across different PMUs in a distribution grid, which
ensures invariance in the following metric.
The following framework applies for every phase individually because specific utilities are connected to individual
phases which are predefined and they do not switch between
phases. Since our anomaly detection depends on unravelingan inconsistency in the system’s behavior, the inconsistency
check applies only within a given phase. Since the two phases
are connected to different consumer loads, it is not possible
to do anomaly detection since there is no way to establish thecausal link to distinguish a legitimate change from a malicious
one. However, within the same phase, since they are attachedto loads belonging to similar consumer loads, the method is
feasible per phase separately.

(a)

(b)

Fig. 7: (a) PMU clustering using correlation among the PMUs;(b) MAD over time windows.

1) Optimizing Spatial Granularity: Intuitively, a group of
PMUs connected to the same feeder or serving proximate
geographical areas should exhibit some interdependence in the
PMU data streams. We opt to utilize the pairwise Pearson
correlation coefficient to identify clusters that show somelevel of positive correlation. Please note, higher the desired
level of invariance, the higher is the required level of positive
correlation. We calculate Pearson’s correlation among all possible pairs of PMU devices in the data to find groups having
a maximum correlation. In the EPFL dataset, the mean ofhourly correlations between PMU 2, 3, 4, 5 are shown in
Fig. 7a. It is evident from the mean correlations that PMU2,
PMU3, and PMU5 show a strong correlation and can beconsidered in a single cluster. The correlation identifies PMUs
to be clustered under one instance of the anomaly detection
technique. A similar clustering for the LBNL data is available
in our previous work [1]. However if the PMU infrastructureis larger than the kind of real deployments considered in ourwork, then to address the scalability challenge in grouping
clusters of PMUs with maximum positive correlation, detailed
plan on it, is provided in Sec. VII-B.
2) Optimizing Temporal Granularity: Now we investigate
on selecting the appropriate time granularity over which theratio metric is to be calculated. The time granularity should beselected in such a way that the invariance in the ratio metric ismaximized. In other words, we want to minimize the measureof dispersion in the ratio statistic. Therefore, our approach is
to solve the following search problem:
T = argminM AD(Qr (T ∗ ))

(3)

T∗

where M AD(Qr (T ∗ )) is the median absolute deviation of theresulting ratio time series with different time granularity andWe choose T ∗ that minimizes the MAD of the ratio time series(shown in Fig. 7b).
After the above process, we get a clusters of PMUs of size
N. Since the same principle applies on each cluser the restof the paper’s methodology discusses from a cluster specificperspective. Let Pt = [Pt1 , ..., PtN ] denote the active power
for a phase from N PMUs selected to be in the same cluster
at time slot t. We have taken second wise average of active
power for our analysis, thus t = 1 second.
3) Harmonic to Arithmetic Mean Ratio: Let the harmonic mean (HMt ) and arithmetic mean (AMt ) of Pt at timeslot t be defined as:
N
N
X
1X i
i −1
HMt = N (Pt )and AMt =P .
(4)
N i=1 ti=1
We calculate HMt and AMt for slot t over a time window
T of length n slots. Then we calculate the average HMt to
AMt ratio, Qr (T ), at the end of each window as follows:Pn
HMtr
Q (T ) = Pt=1(5)n
t=1 AMt
where 0 ≤ Qr (T ) ≤ 1, as HMt ≤ AMt . Compared tonormal arithmetic mean proposed Qr (T ) shows high stability
as depicted in Fig. 6b.

(a)

(b)

Fig. 8: Illustration: (a) Safe Margins (b) RUC(T) samples.
C. Stateless and Stateful Residual based Threshold Design
Now, the anomaly detection needs to identify a proximate
spatial region around the ratio time series that specifies the
behavior of the invariant under no attacks. Usually, a threshold based approach is considered by tracking the difference
between the actual time series value and its smoothed value
over time. However, a simple threshold based approach, cannot decrease both false alarms and misdetections simultaneously [17]. Hence, we propose a two-tier approach withstateless and stateful residuals.
1) Stateless Residuals: The stateless residual denotes theinstantaneous difference between the ratio metric and a timevarying parameter per time window T . We compute the meanµr and Median Absolute Deviation (MAD), mQ , from theprobability distribution of ratio values Qr (T ) for each PMU
cluster (shown in Fig. 8a).
We use MAD as a scale parameter for designing the statelessresidual instead of the standard deviation (SD), because MAD
is more robust to outliers in the training ratio samples. The
MAD is more robust than SD since it is based on a squared
error from the mean, so a finite number of outliers can increase
SD easily compared to MAD, thus reducing sensitivity to small
attack strengths.
Stateless residual is parameterized as κ = ϵmQ whereϵ ∈ (0, 5], such that κ ∈ (0, 5mQ ] and mQ is the MAD.
Intuitively, larger κ values produce wider safe margins, thus
reducing false alarms but increasing misdetection and viceversa. Hence, a trade-off is necessary for choosing a thresholdthat will automatically generalize into lowering false alarms
while not sacrificing the detection sensitivity.
Our framework calculates a parameterized ’stateless residual’ ∇(T ) using two values; Γl (T ), and Γh (T ) around theobserved instantaneous ratio values Qr (T ) (called safe margin), on each time window on the training dataset, such that:Γh (T ) = Qr (T ) + ϵmQ .

(6)

Γl (T ) = Qr (T ) − ϵmQ .

(7)

Now the an instantaneous stateless residual ∇(T ) which isthe ’signed residual distance’ between the observed ratio andthe stateless residuals is calculated by: r Q (T ) − Γh (T ), if Qr (T ) > Γh (T );Qr (T ) − Γl (T ), if Qr (T ) < Γl (T );∇(T ) =
(8)0,otherwise.
The value of ∇(T ) could be positive (or negative) dependingon whether the ratio sample observed is above (or below) theupper (or lower) safe margin Γh (T ) (or Γl (T )). Thus, ∇(T )is zero when the ratio observed is within [Γh (T ) , Γl (T )].
2) Stateful Residuals: Now the framework maintains thesum of residuals between the ratio value and the Γh (T ) andΓl (T ) over a sliding frame of past K time windows. We denote
this sum as RU C(T ). As depicted in Fig. 8b, this RU C(T )shows more stability compared to stateless residuals. So, theframework calculates the sum of residuals, RU C(T ) over asliding frame of past K time windows as:TX
RU C(T ) =∇(f ).
(9)acts as a latent space), that contains outliers that do not
represent the most general underlying structure of benign data,
but also are not related to attacks. The robust learning needs tobalance these two seemingly opposing perspectives. Hence, theconclusion is that neither we can exclude outliers completely,
nor include outliers completely. From the above, we need an
approach of robust learning of the detection thresholds that is
a compromise between these extremes.

f =T −K

RU C(T ) samples form the feature set which serves an the
input to the learning of anomaly detection threshold andthereby attack detection criterion. Note that the RU C(T ) may
have both negative and positive values. The aim of the learning
is to find an upper and a lower thresholds that identify the safe
operating region. Any RUC(T) in the test set beyond the safe
operating region is an indication of an attack. However, theprocess of identifying the safe operating region is not easily
since the RU C(T ) features characterize the underlying benign
structure of a noisy system.
D. Theory of Resilient Learning of Detection Criterion
This subsection is not part of the solution framework but
contains theoretical underpinnings of the resilient learning of
anomaly detection threshold from the latent space of statelessresiduals that are derived from the raw data. Although, not partof framework implementation, it is important to discuss here
to completely appreciate the learning of the anomaly detection
thresholds. For framework implementation, this subsection is
not relevant and the reader should refer to the next subsection.
The probability distribution of RUC values from the twodatasets is shown in Fig. 9. We observed that RUC values
follow a heavy tailed distribution. This is particularly pronounced for the EPFL dataset that has a much longer duration
and reveals the true nature of real world PMU data with ahigher number of outliers compared to the measure of central
tendency (Check Fig. 1a and Fig. 1b). This was an indicationthat our previous approach [1] is unlikely to offer reliable
and robust attack detection performance. This is because large
outliers on the tails of the RUC learning feature bias the best
fit threshold line towards itself thus widening the limits. We
tested this experimentally to verify and it showed severe true
positive detection performance degradation under the EPFL
dataset when compared to the LBNL dataset. What happened
in our preliminary conference work for the LBNL dataset, we
just got lucky with a smaller dataset, like other works using
the same or similar duration datasets. This is because external
drivers of process noise may not manifest them during that
small duration being considered. However, active power which
is our derived process variable that is influenced by the current
data/load drawn, is directly dependent on occasional unpredictable seasonal/weather events that are not representative of
the most general pattern but also correspond to the presenceof attacks. These occasional unpredictable events create aprocess noise in our active power variable. This indirectly gets
captured into the RU C(T ) stateless residual metric (which

(a)

(b)

Fig. 9: Dataset Distribution: (a) EPFL Data; (b) LBNL Data.
Theoretical Intuition of Robust Estimator Let us assume
that there are m training input RUC(T) values each denotedby rk , such that k ∈ 1, · · · , m. Our job is to learn best fit
threshold say τopt based on the known rk . The search space ofτopt is the parameter estimation search space denoted by τ . For
simple linear regression the typical parametric assumption isa Gaussian distribution under maximum likelihood estimation.
The estimate of the parameter is the one that maximizes thelikelihood function. More generally, the likelihood function
for parameter estimation is specified under an parametricassumption on a distribution of choice.
To understand the best parametric distribution of choicefor our purpose, we tried fitting the RUC sample distribution
against known distribution models in parametric statistics(such as normal, beta, gamma, t-distribution) using a Kolmogorov–Smirnov test [29] to identify the best fitted distribution model. We calculated the sum of squared error foreach distribution model for model selection and found that thestudent t-distribution, gives the lowest sum of squared errors
compared to all other distributions. Hence, the parametric
distribution of choice for us was the student t-distribution.
Following the theory of t-distributions, we know that the pdf
of a t-distribution is not expressed in terms of the raw input(ri in our case) but a standardized input. The standardizing
involves the error between the raw input and the parameterthat is being estimated, divided by the standard deviation of
the raw input. In our problem, this takes the recognizable formrk −τ
σ . However, [28] observes that this does not take into
account varying scale ranges of the input. Therefore, a moregeneral form of the t distribution is more accurate. Therefore,
letting rk − τ = sk , the pdf of the student’s t-distribution with
auxiliary scale correction is given by:
ν+1Γ( ν+1
sk 2 − 22 )fτ (sk ) = p1+(10)ν(cσ)2νπ(cσ)2 Γ( ν2 )where Γ is the gamma function, ν is degree of freedom, sk
is error difference between a candidate parameter choice τand the training point rk , σ is the standard deviation of the rk
values, and c is a tuning constant for auxiliary scale correction.

In parametric estimation, it is well known that optimizing
the likelihood function is the same as optimizing the loglikelihood function. Therefore, we can write the log-likelihood
function for estimation as ρ = −log(f (sk )) . In general, theparameter estimate in MLE is found by empirical risk minimization problem which can be expressed by the following
optimization problem.
argminτmX

ρ(sk ) where ρ(sk ) = −logf (sk )

(11)

k=1

With some algebra and derivation it can shown that negative
natural logarithm of pdf of t- distribution is given:
!
s2k
ν+1
ln 1 ++ constant(12)
ρ(sk ) =2
ν(cσ)2Given a degrees of freedom ν, we can derive a generalized
MLE which is an M-estimator of estimated parameter τ byfinding the solution to the above empirical risk minimizer
which considers ρ for all training data points i ∈ {1, N } suchthat the M-estimator is defined as a solution of following msummation equation for τ under a t-distribution
Theory of M-estimators in robust estimation states thatfinding the parameter estimate via minimizingP
log-likelihoodmfunction
is
the
same
as
findingthesolutiontok=1 ψ(sk ) =
Pm
ψ(r
,
τ
)
=
0
k
k=1
where ψ(sk ) =∂ρ(sk )
∂sk

(13)

With some algerbra and derivation, it can be shown that the∂ρderivative ∂s(of Eqn. 12) is Influence Function(IF) of ankM-estimator under a t-distribution [28] as a parametric choice
for learning the estimate.
ν+3∂ρ(sk )=
sk 2 (sk )∂skν + ( cσ)

(14)

In theoretical robust statistical estimation, derivative of a costfunction with respect to the error residual (that is directly
dependent on the input) is called the influence function. So
we can say the following is an influence function.
=⇒ IF =

∂ρ
ν+3=
−τ 2 (rk − τ )∂rk)ν + ( rkcσ

(15)

Now in our problem, we have just parameter that is free tochange statistically and we only sample one RUC(T) value in awindow. Therefore, according degrees of freedom definition int-distribution small values of degree of freedom give heavier
tail to the t-distribution and for bigger values it resembles a
Gaussian distribution. Therefore, from both perspectives, ourpick of the degrees of freedom ν = 1 is sound. Different
large values of RUC will affect the value of IF of the location
estimator τ which will be influenced highly. However, theselarge values of ruc will have less affect the learnt best fit ofτ under this the t-distribution as the parametric choice for the
MLE and this IF. Therefore, putting ν = 1, in Eqn. 10, we
get the influence function appropriate for our purpose as:4(rk − τ )
IF =
(16)
−τ 2
1 + ( rkcσ)

From the above equation, let cσ = β, be a scaling hyperparameter. This modeling relaxation is useful since in at-distribution the standard deviation is biased due to small
number of large outliers and its true value is often unknown.
Given, we know already that rk −τ = sk , the relaxed influencefunction can be written as:4(sk )
IF = sk 2(17)(β ) +1
To figure out the most appropriate estimator for our problem, we need to take the integral of the influence function withrespect to sk . This follows from theoretical robust estimation
theory, since the derivative of the estimator with respect tothe input is the influence function, we can do the integral of
the influence function to obtain the appropriate estimator. Bytaking integral over Eqn. 17 and with considering parametercorrection for auxiliary scale we find
Z

4sk
dsk
1 + ( sβk )2

(18)
2

β
du.
Let u = 1 + ( sβk )2 , then it can be shown dsk = 2skPlugging it back into the integral and substituting the value ofu, it can be shown that we getsk
β 2 log(1 + ( )2 ) = L(19)
βWe found that the above functional form has been reported
in computer vision and robust statistics literature as a CauchyLorentzian function. Given, the above analytical reasoning, wewere convinced that among the several possible approaches toresilient learning, that this is the most appropriate choice of theestimator that is robust to the heavy tails in our derived feature
space RU C(T ). Therefore, we reason the use of Eqn 19 as themost appropriate choice of loss function to learn and predict
the standard limit thresholds.
Analyzing Performance Overhead with Robust Estimator
We need to analyze whether introducing the Cauchy
Lorentz loss in place of the previously used L1 normin our previous work, comes at a sacrifice of increased
overhead?. Interestingly, the Cauchy Lorentz loss turns out
to be differentiable at all points, unlike the L1 norm (inthe previous method), which not differentiable at all points
due to gradient shattering. Therefore, to get exact optimal
result, the previous method required a brute force approachto learn the two thresholds. In contrast, our new choice ofCauchy Lorentz is convex and differentiable at all points,
and therefore gradient descent directly applies. Therefore, our
new modification does not require a brute force approach tofind the minima. Now, it is well known that gradient descent
has equal to (in the best case) or less overhead to converge
to a minima compared to a brute force approach. While wetheoretically optimized why the L1 should be changed toCauchy Lorentz to improve robustness against outliers andnoise, the side benefit of this design change is that it does notcome at the cost of increased overhead to learn the thresholds
that dictate the attack detection criterion.

E. Robust Threshold Learning Approach
We need to learn an upper threshold τmax and a lower
threshold τmin from the RUC values, which act as unknown
model parameters to be estimated. The RUC values which go
beyond these thresholds indicate attacks as they do not confirm
the underlying structure learned during benign operations, In
our previous work [1], we used a simple regression to calculate
the best fit line for the upper and lower thresholds. However,given the larger time horizon dataset of EPFL with more PMUs
and more outliers due to process noise, that approach cannot
compensate for outlying RU C values, which form the training
data. Given the heavy tailed nature of the RUC latent feature,
we use the derived loss function under an MLE with a tdistributed fitting.
Therefore, in our training, we replace the regular regression
ordinary least squares with 19. Additionally, the negative andpositive RU C values have a different range. Therefore, we
introduce the scaling hyper-parameter β for the 19 should becross-validated separately as β+ in RU C + and β− in RU C −data training. Therefore, we calculate both upper standard limit(τmax ) and lower standard limit (τmin ), for different values ofthe optimization parameter β+ for τmax (and β− for τmin ).
Another key observation we had is that the number ofpoints was not symmetrically distributed around the candidate
parameter τ . Some outlying points are still representing somebenign and legitimate behavior of the PMU system. Ignoringthem completely in the learning process, would yield anoverly restrictive system. Therefore, we need to treat the error
between the data point and the candidate fit τ .
Algorithm 1 Robust Learning of τmax
Input: RU C(T )=Training Input, [τ ]= All Candidate Parameter Choices, β+ = scaling hyper-parameter , w1 , w2 =Quantile Regression Weights
Result: τmax
for τ ∈ [τ ] docost+ = 0
for r ∈ RU C(T ) do
if r > 0 thens=r−τ
if s >= 0 then
2 2
cost+ : log(1 + ( s∗w
β+ ) )
end
if s < 0 then
1 2
cost+ : log(1 + ( s∗w
β+ ) )
end
end
end
2
T otalCost : β+
∗ cost+end
τmax = argminτ (T otalCost)
Hence, for learning τmax , since we do not want the higher
residual r values to be completely ignored by the robust loss
function, we add a higher weight w2 to the error corresponding
to s > 0, which happens when r > τ . As we have calculated
τmax and τmin separately, in our algorithm we simply used r
and s instead of rk and sk respectively. In contrast, we assign

the errors s a weight w1 < w2 , if s < 0, which corresponds
to points r < τ or the points with smaller r.
Description of Algorithm 1: First, for each potential τ outof [τ ], we calculate the total loss over positive RU C(T )(which is essentially RU C + (T )) by assigning different
weights (w1 and w2 ) depending on whether the RU C(T )is higher or lower than the τ . The regression errors are
represented by (r−τ = s), which is the difference between thecandidate parameter fit. If the data point r is outward and thecandidate fit τ is inward, it corresponds to the situation wheres > 0 while learning the upper threshold τmax . However,the current choice of τ will be equivalent to a false alarm,since we are learning over benign data. Due to the base rate
fallacy, we know that false alarm reduction is more criticalfor anomaly based intrusion detection systems. Therefore, thescenario s > 0 needs to have a different weight w2 , comparedto the opposite scenario (where s <= 0 that corresponds to a
missed detection). Therefore, in our learning framework, firstly
w2 ̸= w1 and w2 > w1 and balances the false alarm trade-off
in a way that handles base rate fallacy. Since loss function
quantifies goodness of fit per training data point, the positionof the data point w.r.t to a candidate parameter fit plays a role.
Across all training data points, it acts as a weighted sum of
Cauchy loss where the weights w1 and w2 are controlled by
the position of the point. The Cauchy loss helps reduce theimpact of outliers while the quantile weights help reduce thechances of false alarms while doing so.
Cross Validation: Finally we use a cross validation set withminimum δavg that we target to detect and calculate false
alarms (F A) and Mis-Detection (M D). Specifically, we use
the following optimization to select optimal τmax and thecorresponding β+ . Algorithm 1 provides the optimization of
the upper standard limit. A similar approach is used for lower
standard limit (τmin ) as well (when r < 0), with minorchanges to keep the same logic.
For learning the hyperparameters β+ , w1 , w2 , we use across validation set which uses the following criterion:
argmin (d1 F Acv + d2 M Dcv )(20)

β+ ,w1 ,w2

where d1 , d2 are trade-off weights. The d1 represents theimportance of reducing false alarms and d2 represents theimportance of reducing missed detection. We have d1 +d2 = 1
and d1 > d2 such that we give more importance to the false
alarm rate compared to misdetection rate. This is because theprobability of an actual attack is very low, and seemingly low
false alarm rates, do not necessarily indicate a good usableattack detector. In the end, we choose a threshold τmax (andτmin ) which minimizes the total loss for the corresponding
RUC inputs.
F. Detection Criterion in Test Set
The main idea behind attack detection is that RUC in thetest set (RU C(T C )) should not deviate from the standard limit
obtained from the training set. We first calculate the statelessresiduals for each time window of the testing set T C suchthat Γh (T C ) = Qr (T h ) + κopt and Γl (T C ) = Qr (T h ) − κopt ,where κopt is the margin that resulted in the optimal standard

limit. The historical value of the ratio on that time window
Qr (T h ), where T c is the current time window and T h is thecorresponding time window in the training set, Γhigh (T c ) andΓlow (T c ) are the safe margins at T c of the test set.
From Γh (T C ) and Γl (T C ), we calculate the RU C(T C )using Eqn. 21. Then we check whether RU C(T C ) violates
the standard limit range identified during the training set.
∈ [τmin , τmax ], No Anomaly;c
RU C(T ) :(21)∈/ [τmin , τmax ], Anomaly.
TABLE I: Parameter Descriptions and Values
Parameter
Spatial Granularity
Temporal Granularity
Sliding Frame
Scaling
Weights
Trade-off Weights

SymbolNT
Kβ+ , β−w1 , w2
d1 , d2

EPFL
PMU 2,3,5600
10.0008,.0008.5, 2
1/3, 2/3

LBNL
PMU 1,360
5.006,.002.5, 2
1/3, 2/3

A. Snapshot Results
For all snapshot results, we have the attack strength of
δavg = 0.5 amps on the current magnitude of a single
PMU (PMU2), phase 1, from the EPFL dataset, and theattacks continue for a period of 2 hours from the test set.
We select a small time period to prove the visual intuitionand explainability. Rigorous experimental evaluation with thewhole duration of the testing set is reported in the performance
evaluation subsection. We do not show the snapshot results on
the LBNL dataset since most of the corresponding results are
already available in our previous work [1].
We first show the impact of the deductive type of attacks
with a step strategy. Fig. 10a confirms the deviation of the ratio
metric beyond the set safe margin, and the Fig. 10b, shows that
the RU C(T ) feature during the test set, successfully violates
the learnt standard limit thresholds τmax and τmin from thetraining and cross-validation.

V. E XPERIMENTAL E VALUATION
We have used both the LBNL [16] and EPFL [20] dataset
for our experimental results. For the LBNL dataset, the total
available data is of 11 days from 1-Oct-2015 to 11-Oct-2015.
We have used the first 7 days of data as training and next 2
days as cross validation and the final 2 days as our testing set.
For the EPFL dataset, we have considered a total of 6months of data (Sep-14 to Feb-15) in our experiment. The
first 17 weeks are selected as training and next 4 weeks as
cross validation and the final 4 weeks of data as testing set.
We conducted extensive experiments for different falsification margins and attack strategies. We divide this section
into two parts: (1) Snapshot Results These results are illustrative results to aid in the explain-ability of how and why
the framework is successful in detecting the attacks undervarious attack types and strategies and what kind of signatures
are observed under attacks; (2) Performance Evaluation thatshows the true positive rate versus the false alarm acrossvarying attack margins. The basic parameters selected for the
EPFL data are shown in Table I and the same information for
the LBNL data is available in our previous work [1].
Details of Testing Set and Attacks: For false alarm performance, we calculate the base rate false alarm, which is thefalse alarm rate in the presence of no attacks throughout
the whole testing set. For attack detection performance, we
introduced 100 (20 for LBNL data) short term attacks of length
2 hours distributed throughout the test period. We continue
to do these attacks on different PMUs. Then we check theaverage detection accuracy by measuring the fraction of total
attacked samples which violated the standard limits to prevent
bias in the time periods selected for attacks. The specifics
of different attack parameters are discussed while explaining
snapshots and performance evaluation under each attack type,strength, scale, and strategy. This is done because we are
exploring the full strategy space of possible FDI attacks,
instead of a specific instance. Such a parameterized approachto threat modeling leads to a more unbiased evaluation.

(a)

(b)

Fig. 10: Deductive Attack with δavg = 0.5 amps (a) Ratio
Snapshot; (b) RU C(T ) Snapshot.
Similarly, for additive type of attack with step strategy,
the behavior of the ratio and the stateful residual under adeductive attack is shown in Fig. 11a and Fig. 11b respectively.
The conclusions on successful deviation and violation of theanomaly detection criterion via the learned thresholds can bevisually confirmed.

(a)

(b)

Fig. 11: Additive Attack with δavg = 0.5 amps (a) Ratio
Snapshot; (b) RU C(T ) Snapshot.

(a)

(b)

Fig. 12: Alternating Switching Attack with δavg = 0.5 amps(a) Ratio Snapshot; (b) RU C(T ) Snapshot.

Fig. 12a and Fig. 12b. validate our success in creating
deviations in both the ratio and thereby the RU C(T ) metricto violate the learned attack detection criterion under analternating switching attack type that as discussed in our threat
model.
(a)(b)

Fig. 15: Performance under Additive Attack for varying
Strengths: (a) EPFL Data (b) LBNL Data

(a)

(b)

Fig. 13: Anomaly Detection for Mirroring Attack for ∆a = 1
hour (a) Ratio Snapshot; (b) RU C(T ) Snapshot.
Finally, the detection of the mirroring attacks are shown in
Fig. 13a and Fig. 13b. To implement such attacks we haveselected a random period of one hour of a PMU and used themirror image of the captured time series for the next hour.
B. Performance Evaluation
For performance evaluation, we first generate the ROC curvethat characterizes the trade-off between the probability of
attack detection vs. the probability of false alarm.
To remove attack period selection bias we introduced continuous attacks for the whole testing period with different
attack margins and measured the % of sample points outside
the standard limit boundaries for different scalar factors (ϵ).
We introduced attacks on each PMU and have taken the
average detection and false alarm rate to remove the bias ofa selected PMU for an attack scale of 1 PMU and 2 PMU
compromised out the cluster of 3 PMUs.
1) Under Various Attack Types and Strengths: While we
do not limit attack margins within any arbitrary range to keep
an unbiased evaluation, we stopped reporting performance forattack strengths where our missed detection rate becomes 50%
or worse, since that would mean a random coin toss will
be a better detector. Hence, the lowest attack strength shown
corresponds to this performance limit.

the margin is 0.25 amps the method performs poorly with amisdetection rate of >= 0.5, which is the evasion point of our
method. However, any attack strength above 1.5 amps reaches
a 0.98 detection rate even as the false alarm rate is 0.025. Suchlow false alarm rates are a significant improvement given thatis the main challenge of anomaly based approaches.
2) Under Various Attack Strategies: For any anomaly
detection mechanism, it is essential to investigate the false
alarms raised and ROCs cannot always be the primary evaluation metric for that. So to compare this, we first introduced 100(20 for LBNL data because of its shorter duration) short term
attacks (of length 2 hours) on different PMUs and checked theaverage detection accuracy by measuring the fraction of total
attacked samples which violated the standard limits. Then we
calculated the false alarm raised in the presence of no attacks
in the selected attack periods.

(a)Fig. 16: Performance under Alternating Attack for varying
Strengths: (a) For EPFL Data; (b) For LBNL Data.

(a)
(a)

(b)

Fig. 14: Performance under Deductive Attack for varying
Strengths: (a) EPFL Data (b) LBNL data
Fig. 14a and 14b depicts the ROC of deductive type ofattacks with step strategy for EPFL and LBNL dataset respectively for different values of δavg . Similarly, Fig. 15a and 15b
depicts the same ROC for additive type of attacks with step
strategy for both the datasets. For low margin of attacks inthe EPFL setup, as shown in Fig. 14a and Fig. 15a, when

(b)

(b)

Fig. 17: Performance under Mirroring Attack for varying
Strengths: (a) For EPFL Data; (b) For LBNL Data.
Fig. 16a and Fig. 16b shows the performance of our model
under Alternating types of attacks for EPFL and LBNL data
respectively. For Alternating Type 1 the adversary uses additive
attack for the first half and then uses deductive attack for thesecond half of the attack period. Alternating Type 2 indicatesthe opposite of Type 1 where the adversary uses deductive first
and then uses the additive attack. Now for mirroring type ofattacks, we have selected different lengths of attack periods.

We capture the data from a PMU of the selected time period
and used the mirror image of the captured data for the sameamount of time and we implemented 100 such attacks. The
detection rate with various attack period length is shown in
Fig. 17a and Fig. 17b.
(a)

(b)

Fig. 21: Performance when 2 PMUs attack simultaneously(EPFL): (a) Additive Attack; (b) Deductive Attack.

(a)

(b)

Fig. 18: Performance under Step Strategy: (a) For EPFL Data;(b) For LBNL Data.
Fig. 18a and Fig. 18b shows the true detection averaged
over all the PMUs for EPFL and LBNL dataset respectively forstep attack strategy. It is evident from the plots that our method
achieves detection accuracy ≥ 0.95 for both the deductive andadditive attack types on both datasets. Similarly, Fig. 19a andFig. 19b shows the detection rate under Ramp attack strategy
averaged over all the PMUs for EPFL and LBNL dataset
respectively. Now, Fig. 20a and 20b depicts the performanceagainst random attacks for both the dataset.

(a)

(b)

Fig. 19: Performance under Ramp Strategy: (a) For EPFL
Data; (b) For LBNL Data.

(a)

(b)

Fig. 20: Performance under Random Strategy: (a) For EPFL
Data; (b) For LBNL Data.
3) Scale Sensitivity: Figs. 21a. and 21b prove that the attack
detection still works with very low false alarms even when 2
out of the 3 PMUs in the cluster are compromised with false
data injection.
4) Improvement with Resilient Learning: We have compared our previous [1] and the proposed noise resilient learning
found improved performance. We have simulated the sameattack strategy and attack margin on the EPFL dataset andcompared the ROC curves.

(a)

(b)

Fig. 22: Comparison Between Old and New Method on EPFL
dataset: (a) For Additive Attack; (b) For Deductive Attack.
Fig. 22a and 22b shows the comparison of the previous andthe new method for additive and deductive attacks respectively.
It is evident that although our detection rate (or true positive)
drops a little compared to the old method, we can still reduce
false alarms (false positive) significantly. As we mentioned in
Section IV.D, since the actual probability of a PMU system being under attack is less (base rate fallacy), therefore, we should
not ignore the importance of the false alarms improvement.
Our new method produces much better false alarm performance compared to our old model. As depicted in Fig.
22a and Fig. 22b, the new model reduces the average false
alarm rate by more than 80% (from .105 to .019). However,
we have observed minuscule degradation in missed detection
performance of the new method. For any attack margin >= 1
amps, we have seen only 4.9% (for additive type of attacks)and 4.2% (for deductive type of attacks) drop in the average
missed detection performance which is much lesser than the
average improvement of the false alarm performance.
5) Real Timeliness of Detection: To demonstrate thereal-time nature of the detection method, we have estimatedthe Average Attack Detection Time for different attack types
and strategies. For the EPFL data, as we have shown in
our parameter description table, the temporal granularity is
estimated to be 600 seconds. The average detection time forthis EPFL data is shown in Fig. 23. It is evident from Fig. 23athat even with our relatively higher temporal granularity, theaverage delay is less than 12 minutes even for low margin (.25amps) of additive, deductive, or alternating attacks with step
strategy. However, for a low margin of random attacks, ourmodel produces an average delay of less than 17 minutes asshown in Fig. 23b.
For the LBNL data, the average detection time is even less.
First of all, as the LBNL data is relatively stable, the temporal
granularity is estimated to be 60 seconds. Based on that, theaverage delay in attack detection is shown in Fig. 24a for

(a)

(b)

Fig. 23: Average Attack Detection Time for EPFL Data: (a)Step Strategy (b) Ramp and Random Strategy.

(a)

(b)

Fig. 24: Average Attack Detection Time for LBNL Data: (a)Step Strategy (b) Ramp and Random Strategy.
additive, deductive, or alternating attacks with step strategy.
For a low margin of ramp attacks, our model produces anaverage delay of less than 7 minutes as shown in Fig. 24b.
This illustration corroborates the near real-time applicability
and quick FDI attack identification of our proposed model.
6) Run Time Complexity of Our Method: For thedemonstration of how light-weight our framework is we have
explored the average running time (measured in seconds) onan Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz 16 GB RAM
computer for training and testing of the proposed model. The
running time of the crucial steps of training and testing isshown in Fig. 25. The 95% of the total training time is spenton the optimal τmax , τmin calculation for each β+ and β−
respectively. However, once the training is done and all theparameters are learned, the complete testing phase with 4
weeks of data takes only 88.22 seconds. We can therefore
conclude that the method does not much time or resource

(a)

(b)

Fig. 25: Average Running Time: (a) Training (b) Testing.
VI. R ELATED W ORK
Most of the works related to PMU data falsification canbe broadly divided into two categories - state estimation
based detection, and machine learning based detection. Stateestimation based approaches are typically model driven. For

example, in [12], a mechanism based on continuous monitoring of phase-wise voltage in an equivalent transmission
line was proposed, for detecting data falsification in PMUs.
However, they require two PMUs deployed at both ends ofthe transmission line. More importantly, we found that the
PMU data streams at transmission level were inherently stablemaking anomaly detection a less challenging problem.
In [38], the authors have proposed a multisensor track-level
fusion-based model prediction (TFMP) to estimate electromechanical power oscillations modes during data-injection attacks. However, it did not discuss the detection of FDI attacks,
instead, the algorithm is utilized to remove bias and noise
while accurately extracting the system parameters. Also the
proposed method only monitors phase angles and is tested on
a IEEE 39-Bus New England System. Similarly, The [39] proposed a Bayesian-based approximated filter (BAF) to improve
the immunity of the monitoring of power oscillation against
data-injection attacks. The predictive distribution property ofthe this algorithm supports monitoring power oscillation in thepresence of FDI with high probabilities of information loss.
As both of these papers only monitors the phase angles, they
are not suitable to detect FDI attacks in other streams such as
current or voltage magnitude.
In [40], the authors presented an algorithm to detect line
outage by measuring the phase angles and then comparing it
to a threshold value to identify the event. However, they use
actual topology of the network as one of the input parameterswhich is very critical information and very hard to obtain.
Also, their approach is said to be applicable in transmission
line network whose measurement data is inherently stablecompared to distribution line network, which we have considered. In [41], the authors proposed a convex optimization
method for detecting line failures and exploit Bayesian regression to develop an algorithm for probabilistically detecting line
failures after an attack using partial noisy measurements. Their
model is numerically validated against DC power flow whichmainly focuses on the event on line tripping which is a special
case of FDI attack.
In [32], first it is demonstrated that the conventional bad datadetectors are less effective against sophisticated FDI attacks
mainly because they are based on the classical weighted leastsquare estimation where redundancy is a must in detecting thebad data [36]. Then a decentralized homomorphic computation
paradigm is proposed along with a hierarchical knowledge
sharing algorithm to facilitate the secure ciphertext calculationof state estimation. However, the method is computationallyheavy and no real data is used to demonstrate the efficacy ofthe proposed approach. In [33], an FDI detection method basedon the Kullback–Leibler distance (KLD) which calculates the
distance between two distributions, p and q. p is derived
from the measurement variation between the current and theprevious time step and q is derived from the historical data.
However, the minimum attack margin that is demonstrated iscomparatively higher than our approach and it is not explained
how the proposed method can be implemented in a PMUPDC setting. Similarly, in [21] the authors designed statistical
testing with sliding windows that detect anomalies in a single
stream by generating an equivalent circuit to identify a specificevent. Please note, each PMU generates 12 streams of datawhen deployed in a 3-phase grid network, thus creating ahuge computational overhead for stream specific model driven
approaches.In [37], a novel approach is presented to enhance
the resilience of wide-area control systems against the FDI
attacks. The temporal prediction attribute of the proposedmodel is able to parry the FDI attacks while estimating andcontrolling the voltage magnitude. However, the proposedmodel only validated against the voltage magnitude, so there
is a need to implement models/methods for the other data
streams as well.
On the other hand, machine learning based approaches
such as [22] propose to use clustering of the sets of events
distinguish different events from one another. However, thedesigns are built on a single data stream, thus, can result incomputational overhead. The survey article, [35], discussed
different ML models, both supervised and semi-supervised,and unsupervised ML methods. However, the high computational overhead and the lack of validation on real dataraises the question of reliability and the possibility of using
these methods in real time. The [15] proposed a decision
tree based anomaly detection scheme to differentiate between
normal tripping and malicious tripping by training on specificattack samples. However, it is not feasible to generate 100%of all the possible legitimate line tripping cases for training
in [15]. In [18] a density-based local outlier factor (LOF)
analysis was used to detect the anomalies among the data, todescribe spatio-temporal outliers among all the synchrophasor
measurements from the grid. However, this method might not
be able to detect attacks in real time, and in their proposed
method the authors have only considered an attack on voltage
magnitude. The [27] tries to combine both the approaches
together by proposing the extraction of events signatures suchas line faults and trips, generation and load fluctuations toevery cycle of state estimation and then use a Convolutional
Neural Network (CNN) data filter to validate the PMU data.
Another work, [23], uses a rule-based mechanism to detect
perturbations in each data stream independently by identifying
optimal placements of PMUs. Most of these works use a single
dataset for a few minutes or days or use simulated datasetsfrom controlled testbeds that have lesser noise.
A critical analysis of all previous works on the detectionof PMU data falsification revealed that current data falsification for PMU streams was not investigated. Furthermore, we
found that, unlike transmission level PMUs, the distribution
level PMU’s current synchrophasor data shows high dynamic
variations in benign conditions, making anomaly detection
challenging. Finally, all previous defenses are stream specificin the sense that they only work for either voltage or current
falsification. Since each PMU contains 4 streams and has 3phases, a stream specific defense will require 12 different
defense models that need complex cross-coordination.

A. Scope of the Framework
First, the presence of a positive correlation among the PMUssending data to the same PDC/LCC is a required property forthis framework to work. Intuitively, if a group of PMUs areconnected to the same feeder or serving proximate geographical areas with similar types of customers, then the current
drawn exhibit some level of positive correlation. Since voltage
and phase angle needs to be regulated within a specified
range, the positive correlation in the current implies a positive
correlation for the active power on which our invariant isbuilt. The framework will work on distribution level PMU
infrastructure, where and when there is the presence of groupsof PMU that are positively correlated.
Second, our framework is useful for distribution level PMUs
rather than transmission level PMUs, where the data varies
more readily due to direct integration with loads and positive
correlation may be observed. This does not mean our proposed
method would not work on transmission levels PMU clusters.
However, as the PMU data is more stable in the transmission
line network and the PMUs connected to the same feeder are
strongly correlated, our previous method, which we proposedin our conference version [1] would work. Other simple
conventional methods based on statistical learning, moving
average can also be applied to the FDI attack detection ontransmission line PMU-PDC architecture.
Third, in our datasets we observed weak cyclostationarity
in the PMU data streams. Since the active power at the
distribution level is directly affected by cyclical patterns ofhuman behavior in terms of using electricity on a day-to-day
basis. For example, on the weekdays the power measurements
peak during mid-day but on the weekends they stay relatively
stable throughout the day (See Fig. 6a).
However, if the PMUs connected to the same PDC/LCC
are not positively correlated, then either the clusters can not
be created or clusters will be formed based on very weakor no positive correlation among the PMUs. This will cause
the invariants to have a high variation even under benign
conditions, which will prevent us from distinguishing between
benign behavior and attacks. Similarly, the positive correlationis also somewhat tied to the cyclical nature of the use ofelectricity by customer loads. This allows us to use the time
context successfully, by using a historical mean value of theratio invariant on a certain time window of the day that is notarbitrarily different. If for some reason, this is not present, theperformance may degrade. Finally, if all PMUs within a cluster
are compromised and the attackers have the same attack type,strength, and strategy, then our method cannot detect because
if every node in the cluster is compromised, the misalignment
in the space time covariance structure [4] required to create
deviation is not possible.
B. Applicability for Tackling Large Scale PMU infrastructureVII. D ISCUSSION
In this section, we discuss the broader scope, limits ofapplicability, and how to scale up our framework for large
scale PMU infrastructure as the electric grids become more
and more PMU enabled.

For our framework to scale to larger system than the small
testbed deployments, we need to group the large scale system
into positively correlated clusters with a causal link. The
causal link could be come from topology knowledge, i.e. if
two PMUs are connected to completely different feeders with

completely different loads (e.g. residential versus industries or
DER microgrids versus non-DER integrated customers). Thisis because a positive correlation may not have a causal link.
So first group the large scale distribution PMU system, suchthat PMUs within each group has similar customer types/load
types. Within each such group say C, further divide it into
clusters ck , such that each cluster maximizes the positive
correlation between the active power of the PMUs that arepart of that cluster. Such positive correlation maximization isrelated to achieving high invariance under benign conditions
but deviations under attacks as shown in our previous work [4],[14]. Let Cor(pi , pj ) represent the correlation between theactive power data from any two PMUs i and j and let p(min)be a threshold that represents a minimum lower bound on
positive correlation such that PMUs can only be eligible tobecome part of the cluster if their correlation is higher than
this threshold.
X
X
Cor(pi , pj )
max
ck ∈C

s.t.

{pi ,pj }∈ck

Cor(pi , pj ) > p(min)(22)

where the ck is any candidate cluster. The above problem
can be solved by converting the PMU system into a graphwhere the edges represent the correlation level and the vertices
represent PMUs. We showed in our previous work in a different context of city scale transportation systems [14], that thisclustering could be done via a region growing approximation
algorithm, although the optimization problem in Eqn. 22 is
NP-hard. Once done, our framework proposed in this paper,
can be applied to each of these clusters formed separately(i.e. train and test on each cluster separately). In this way, ourmethod can apply to large scale PMU systems.
VIII. C ONCLUSIONSIn this work, we presented a noise resilient real time learning
framework for attack detection that can detect current magnitude falsification in distribution level PMU data streams. We
showed that winsorized harmonic to arithmetic mean ratios
can allow a feature space whose benign operating region may
be reliably learned even in the presence of noise via robust
estimation theoretic approaches. We proved that given thecharacteristics of the reduced feature space in the form ofstateless residuals, we obtain the Cauchy-Lorentz loss function
as the best learning estimator for learning the attack classification criterion. We found that our method shows detection
accuracy of more than 95% for false alarm rates of less than
5% for various attack types with two different PMU datasets.
The performance reported is with a dataset of 4 weeks, which
is a longer time horizon compared to other works in this area.
Furthermore, this work validates with two different datasets
giving credence to the generality of the approach in the context
of distribution level PMUs.
Acknowledgments: This work was supported by National
Science Foundation grants: SATC-2030611, SATC-2030624,
OAC-2017289, DGE-1433659, CNS-1818942.

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Prithwiraj Roy received his PhD in Computer Science from Missouri University of Science and Technology, Rolla, USA in 2021, and BS in Electronics
and Communication from Bengal Engineering andScience University in 2010. His current researchinterest includes smart grid security and information
propagation in complex networks and he works as
Data Scientist in Global Action Alliance Inc.

Shameek Bhattacharjee received his PhD and MS
degrees from the University of Central Florida, Orlando, in 2015 and 2011, respectively and BS fromthe West Bengal Univ. of Tech, India, in 2009. He
is currently an assistant professor with the Dept.
of Computer Sc. at Western Michigan University.
Between 2015-2018, he worked as a post-doctoral
researcher with Missouri Univ. Sc & Tech. His
current research interests include cybersecurity incyber-physical systems, internet of things, and artificial intelligence, particularly in topics such as
anomaly detection, trust models, dependable machine learning. He is arecipient of the Provost Fellowship and IEEE PIMRC Best Paper Award.
He serves as a TPC member in various leading conferences such as IEEEICDCS, IEEE SECON, IEEE/ACM Middleware, and a regular reviewer inleading journals such as IEEE Trans. on Mobile Computing, IEEE Trans. of
Dependable and Secure Computing, IEEE J. of Sel. Areas in Commm.

Sahar Abedzadeh is currently a PhD student in
the Department of Computer Science at WesternMichigan University, Kalamazoo. Prior to this, she
received her MS (in 2017)and BS degrees (in 2014)
in Mathematics from K.N. Toosi University of Technology and Ferdowsi University of Mashhad in Iran.
Here research interests include AI for cybersecurity, Cybersecurity Data Science, Applied Machine
Learning.

Sajal K. Das is a professor of Computer Scienceand the Daniel St. Clair Endowed Chair at theMissouri Univ. of Sc. and Tech., where he was
the Chair of Computer Science Dept. during 20132017. His research interests include cyber-physicalsystems, IoT, cybersecurity, pervasive and mobile
computing, wireless sensor networks, and parallel
computing, among others. He has made fundamental
contributions to these areas and published extensively in high quality journals and peer-reviewed
conference proceedings. He holds 5 US patents andcoauthored 4 books, such as Handbook on Securing Cyber-Physical Critical
Infrastructure: Foundations and Challenges, and Principles of Cyber-Physical
Systems: An Interdisciplinary Approach. His h-index is 96 with more than
37,000 citations. He is a recipient of 12 Best Paper Awards at conferences
like ACM MobiCom and IEEE PerCom, and numerous awards for teaching,mentoring and research including the IEEE Computer Society’s TechnicalAchievement Award for pioneering contributions to sensor networks andmobile computing, and University of Missouri System President’s Award
for Sustained Career Excellence. Dr. Das serves as the founding Editor-inChief of Elsevier’s Pervasive and Mobile Computing Journal, and as Associate
Editor of the IEEE Transactions on Dependable and Secure Computing, IEEE
Transactions on Mobile Computing, IEEE/ACM Transactions on Networking,and ACM Transactions on Sensor Networks. He is an IEEE Fellow.