Oleksii Novikov1, Mariia Shreider1, Iryna Stopochkina1 and Mykola Ilin1
                         1
                           National Technical University of Ukraine "Igor Sikorsky KPI", Beresteiskyi Ave, 37, Kyiv, 03056, Ukraine

                                           Abstract
                                           This work focuses on enhancing the toolkit for simulating cyber attacks on energy facilities. The paper
                                           examines models of typical attacks on energy systems, specifically accounting for an attacker’s ability to
                                           distort control system signals, manipulate control measurements, and alter measurement signals related to
                                           the state of the facility. A threats model for a critical infrastructure energy facility is proposed that refers to
                                           attack techniques. The approach considers integrity-breaking attacks expression as a function dependent on
                                           unknown parameters. Criteria are introduced to enable parametric identification of integrity compromising
                                           attack parameters, based on measurement data and constraints on process behavior. Stability conditions for
                                           a typical automatic gain control system under cyber attack are analyzed. An algorithm for identifying attack
                                           parameters is proposed. Computer simulations of facility processes under various attack types were
                                           conducted, appropriate software was developed, and conclusions were drawn regarding the impact of attacks
                                           on facility resilience.

                                           Keywords
                                           energy facilities, cybersecurity attacks, FDI attacks, models, resilience 1


                             1. Introduction
                         The AGC system is highly dependent on open communication infrastructure, such as the SCADA
                         system, which increases its operational efficiency and responsiveness, but at the same time makes it
                         more vulnerable to cyber attacks. Network technologies have many advantages, but all their defects
                         – insufficient security, outdated protocols and software, and weak authentication mechanisms –
                         create new opportunities for attackers. Therefore, the vulnerable points of the system are the inputs
                         and outputs of the control center, that is, the communication channels through which data is
                         transmitted [1].
                            Due to the need for rapid operation, the system does not employ complex algorithms for verifying
                         and evaluating measurement data. Attackers can exploit this to manipulate data without sophisticated
                         calculations. By knowing certain characteristics, an adversary can identify other unknown
                         parameters of the system. In this paper, we demonstrate how this can be done, based on principles
                         described in [2, 3].
                            Moreover, high coordination between interconnected control zones enhances productivity but
                         also means that a sufficiently powerful cyberattack on one zone can adversely impact the entire
                         power system.
                            Cyber attacks on energy supply facilities amplify and deepen the effects of physical attacks for
                         maximum destructive impact. Understanding the limits of resilience to cyber influences is crucial in
                         developing effective protective mechanisms and preventive measures. However, existing research [4-
                         7] provides insufficient attention to the assessment of attack features or parameters.
                            The cyber vulnerabilities of AGC systems stem from data transfer mechanisms and protocol
                         weaknesses. A taxonomy of these attacks was proposed in [8-10]. The paper [11] provides a detailed
                         description of existing attack types on the advanced measurement infrastructure of smart grids,
                         focusing on both IT (Information Technology) and OT (Operational Technology) systems. We


                         ITS-2023: Information Technologies and Security, November 30, 2023, Kyiv, Ukraine
                             o.novikov@kpi.ua (O. Novikov); marshr-ipt23@lll.kpi.ua (M.Shreider);
                         i.stopochkina@kpi.ua (I.Stopochkina); m.ilin@kpi.ua (M.Ilin)
                            0000-0001-5988-3352 (O. Novikov); 0009-0006-8621-5521 (M. Shreider);
                         0000-0002-0346-0390 (I. Stopochkina); 0000-0002-1065-6500 (M. Ilin)
                                      © 2023 Copyright for this paper by its authors.
                                      Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).




CEUR
                  ceur-ws.org
Workshop      ISSN 1613-0073
Proceedings
                                                                                                                                                          35
consider the entire AGC cyber-physical system, with particular emphasis on its OT features, and
consider these attacks in terms of the necessary knowledge about cyber-physical system parameters.
   The main classes of cyber threats for AGC system of energy facility are:

   1. DoS (Denial of Service), DDoS (Distributed Denial of Service), and time delay attacks
   (targeting availability) [4, 5].
   2. Replay attacks (targeting integrity) [6].
   3. FDI (false data injection) and covert attacks (targeting integrity) [6,7].
   In wartime, these cyber attacks are often combined with physical attacks on critical infrastructure
facilities [12]. Developing algorithms for calculating attack parameters remains a crucial task for
understanding the resilience limits of the facility and for investigating cyber incidents.
   The findings of this work will contribute to more accurately fulfilling the guidelines of document
[13] regarding the identification of adversary tactics, techniques, and procedures used to circumvent
controls, along with other cybersecurity objectives.

2. Cyber attack models in AGC systems
Paper [1] examines a two-area power system and its dynamic model equations, demonstrating system
behavior under abnormal conditions and analyzing the types of attacks that can disrupt the power
system.
    In paper [4], a dynamic model of a single-area load-frequency control (LFC) system is presented,
focusing on the principles of sustainable operation. The study addresses time-delay attacks and DoS
(Denial of Service) attacks, providing equations for the main system components under DoS attack
conditions.
    Paper [5] expands on DoS attacks by exploring data integrity attacks as well. It proposes a multi-
area scheme with a control center, presenting detailed LFC equations and describing the main types
of attacks.
    Paper [6] discusses power grid control strategies, with particular emphasis on time-delay threats
and replay attacks. The authors derive stability bounds for systems subjected to these attacks.
    In paper [7], a different class of cyber attacks is explored: robust stealth covert attacks. The study
includes a simulation example and uses a mathematical approach to calculate attack parameters for
adversaries.
    Paper [8] addresses cyber-physical reliability using game theory, incorporating probability factors
into the calculations.
    Paper [9] focuses on technical aspects of cyber attacks, reviewing examples, countermeasures, and
a taxonomy of attack types. A section is dedicated to the use of machine learning algorithms for attack
detection.
    In paper [11], a detailed taxonomy of IT (Information Technology), OT (Operational Technology),
and AMI (Advanced Metering Infrastructure) attacks is provided, along with an overview of papers
that propose approaches to counter these attacks.
    Paper [12] examines DoS and DDoS models, emphasizing that these attacks may have different
impacts when combined with physical attacks by adversaries during wartime.
    Simulation models of cascading effects in power grids under cyber attack are discussed in paper
[14].
    Paper [15] investigates various attack strategies, mathematical models, and methods for assessing
system vulnerabilities.
    The authors of paper [16] delve into the interconnected AGC systems and existing frequency
deviations, advancing the study in this area.
    Existing research reveals a gap in deterministic mathematical approaches, based on control theory
methods, for not only identifying stability bounds but also uncovering unknown attack parameters.
The current work aims to address this gap by developing relevant algorithm.



                                                                                                       36
   Paper [13] provides guidelines and compliance directions for reporting cyber incidents in critical
infrastructure. This document offers guidance that could be reinforced by mathematical analyses and
studies, particularly in the field of restoring attack parameters. The findings of the current study could
provide the necessary numerical data for addressing these challenges.

    3. Cyber threats to the AGC system
Let us examine the structural features of the AGC (Automatic Gain Control) system that make it
susceptible to attacks. The AGC system operates within a communications infrastructure, facilitating
data transmission between control centers and control zones. Sensor measurement data is sent to the
control center, where an error signal is generated and then transmitted back to the control area. The
local controller subsequently calculates the power control signal.
   Real-time data collection can be achieved through remote terminal units (RTUs) or intelligent
electronic devices (IEDs) positioned at critical locations (such as power stations and substations)
within the control zone.
   The SCADA (Supervisory Control and Data Acquisition) system collects and aggregates this data
and relays it to the control center via communication channels using various protocols, such as DNP3
(Distributed Network Protocol), IEC 61850, and IEC 60870-5-104. Similarly, signals from the control
center are transmitted back to the control zone. A general diagram of a single-area power zone under
DDoS attack conditions is presented in [15], with specific points highlighted where other types of
attacks (particularly FDI attacks) could be applied (Fig. 1).




Figure 1: AGC System with External Communications. Arrows (1), (2), (3), (4), and (5) indicate points
where a cyberattack can be applied. Potential targets include the communication network (1) and
(3), internal communication lines (5), the AGC control center (2), and the programmable logic
controller (4). An adversary could impact measurements (5), control signals 𝑢(𝑡), and the system state
𝑥(𝑡).

   Let us compile a list of common attacks on the AGC system, linking specific attack types to
technique classes from the MITRE ATT&CK® Matrix for ICS, as shown in Table 1. In Table 1, CIA
refers to confidentiality, integrity, and availability, respectively.


Table 1
Energetic facility cyber attacks
   Attack,     Affect     Description      Attack pre-       Information          Target         Sub-
 technique       ed                        conditions         gathering                         system
     ID        (CIA)
    DoS           A      Data flood of        Partial             —            Channels for     IT, OT,
  (T0814)                 the internal      knowledge                          measurement       AMI
                                              about                               s and


                                                                                                       37
  Attack,   Affect    Description      Attack pre-     Information        Target         Sub-
technique     ed                       conditions       gathering                       system
    ID      (CIA)
                      network and        software,                      commands,
                        services         hardware                         system
                                          versions,                       services
                                            open
                                         interfaces
  DDoS        A        DoS from            Partial          —          Channels for     IT,
 (T0814)               multiple         knowledge                      measurement      AMI,
                        sources             about                          s and        OT
                                         software,                      commands,
                                         hardware                         system
                                          versions,                       services
                                            open
                                         interfaces
   FDI       IA        False data          Normal        System        Measurement      OT, IT,
 (T0836,               injection            mode      reactions and    transmission     AMI
  T0868,                               features and   measurements       channels
 T0830)                                   anomaly
                                           ranges
                                        knowledge
 Replay      CIA     Replaying real        Partial     Sensors and     Measurement      OT,
 (T0856,                  data          knowledge      signals data     and control     AMI
 T0830)                                     about                         signals
                                          protocol                     transmission
                                         timelines                       channels
 Covert      IA      Hidden attack      System full    Sensors and     Channels for     OT
 (T0836,                                knowledge     actuators data     measure-
  T0868,                                                                ments and
 T0830)                                                                 commands

  Time        A       Introducing          Partial          —          Channels for     OT,
 Delays               time delays       knowledge                      measurement      AMI
 (T0814,                                   about                         s, control
 T0830)                                   protocol                     signals, and
                                         timelines                      commands

Physical     CIA        Destroying        Partial      Gathering all   Physical parts   OT, IT,
attacks              infrastructure,    knowledge     the data using     of critical    AMI
(T0879)                intercepting    about system       social       infrastructure
                      control under                    engineering,        facility
                        biometrical                    geolocation
                          features,                     detection
                     controlling the
                         locks and
                     other physical
                           objects
Spoofing     CI           Identity       Network            —           IoT devices,    IT, OT,
 (T0856,              spoofing due      protocols                      PLCs, control    AMI
 T0830)                  to lack of    knowledge,                          center,
                     authentication      access to                        network
                                       transmitted                         objects
                                           data



                                                                                               38
   Attack,    Affect     Description      Attack pre-      Information         Target         Sub-
 technique      ed                        conditions        gathering                        system
      ID      (CIA)
   Sniffing     C      Access to data    Access to the   Obtaining any        Network       IT, AMI
   (T0842,             transfer nodes      network       usable data for      channels
    T0887,                 to sniff       channels          further
    T0801,                                                 intrusion
    T0830)
 TSA (time      IA     Synchronizing     Knowledge       Obtaining local     Channels of    OT
 synchroni              signal delay        about        time on target        signals
    zation               (replaying        protocol          object         transmission
   attack)                 signals)      peculiarities
   (T0868)
  Malware      CIA     Taking control         Full         Keylogging      Software and     IT, OT,
  (TA0108,                  under         knowledge       and gathering     hardware of     AMI
   TA0104,               controllers       of object      all accessible       critical
   TA0110,                and other      architecture,         data        infrastructure
   TA0111,             cyber-physical     and partial                          facility
   TA0103,              elements, or      knowledge
   TA0102,               software of       of system
   TA0109,                  critical     vulnerabili-
   TA0100,             infrastructure         ties
   TA0101,              facility. Can
   TA0107,                realize all
   TA0106,                 types of
  TA0105)                  possible
                         techniques




    4. AGC mathematical models
   In this section, we present generalized mathematical models in state space, building on previous
works [5,6]. The primary vectors under consideration include malicious intrusion into the system
state via control parameters and measurement parameters (see Fig. 1). We then focus on the FDI (False
Data Injection) class of attacks and develop an algorithm to identify attack parameters under certain
assumptions. Additionally, we discuss the adversary's potential extended knowledge of the system.

1.1.    Initial undisturbed system model
    We consider an initial undisturbed system with control, which is described by equations system
in state space:

                                 𝑥 (𝑡) = 𝐴𝑥(𝑡) + 𝑘𝐵𝑢(𝑡) + 𝐹,                                     (1)
where 𝑥 is system state; 𝑢 is control; F is source function (energy supply from/to neighboring zones);
k is a parameter of control influence intensity.
    We have to notice, that in the general description, state vector 𝑥(𝑡) can contain the components
of frequency deviation Δ𝑓 , regulator, turbine, and tie-line power deviations as it was proposed in
[6]. But we consider the scalar values.
    If the control depends on 𝑦 measurements:
                                        𝑢(𝑡) = −𝐶 𝑦(𝑡),
where measurements depend on the state:
                                         𝑦(𝑡) = 𝐶 𝑥(𝑡).



                                                                                                   39
     Then:

                                 𝑥 (𝑡) = (𝐴 − 𝑘𝐵 )𝑥(𝑡) + 𝐹,                                    (2)
where 𝐵 = 𝐵𝐶 𝐶 .
                                           𝐵 = 𝐵𝐶 𝐶 .
    For stability, the matrix 𝐴 − 𝑘𝐵 has to be negatively defined or at least, non-positively defined.
This depends on eigenvalues 𝜆 of this matrix that can be defined from equation det(𝐴 − 𝑘𝐵 − 𝜆𝐼) =
0. Suppose that 𝐴 − 𝑘𝐵 is negatively defined for a sufficiently large 𝑘. Then the necessary condition
that this property becomes invalid at some 𝑘 , i. e. , the largest eigenvalue changes its sign 𝜆(𝑘 ) =
0 is
                                       det(𝐴 − 𝑘𝐵 ) = 0.                                         (3)
     That can be used to find a critical value 𝑘 .

1.2. Attack on system measurements and instability conditions
determining
   Let 𝜉(𝑡) be the distortion introduced to the measurements by an attacker. The measurements are
given by
                                    𝑦(𝑡) = 𝐶 𝑥(𝑡) + 𝜉(𝑡),
then
              𝑢(𝑡) = −𝐶 𝑦(𝑡) = −𝐶 𝐶 𝑥(𝑡) + 𝜉(𝑡) = −𝐶 𝐶 𝑥(𝑡) − 𝐶 𝜉(𝑡).
   Thus, equation (1) takes the form

                            𝑥 (𝑡) = 𝐴 − 𝑘𝐵 − 𝑘𝐵 𝜉(𝑡) 𝑥(𝑡) + 𝐹,                                 (4)
where
                                          𝐵 = 𝐵𝐶.
  If 𝑥(𝑡) is known, identifying the attacker’s intervention 𝜉(𝑡) becomes a standard fitting problem.
Otherwise, it is necessary to determine 𝑥(𝑡) simultaneously with 𝜉(𝑡) when 𝑦(𝑡) is known.
  The problem can be simplified if we know etalon values 𝑥 ∗ , 𝑦 ∗ , which allow us to eliminate 𝐹:
                                     𝑧(𝑡) ≡ 𝑥(𝑡) − 𝑥 ∗ (𝑡);
                                 𝑧 (𝑡) = 𝐷𝑧(𝑡) + 𝐷 𝜉(𝑡)𝑥(𝑡);
                               𝜉(𝑡) = 𝑦(𝑡) − 𝐶 𝑥 ∗ (𝑡) + 𝑧(𝑡) .                                  (5)
  From here:

                 𝑧 (𝑡) = 𝐷𝑧(𝑡) + 𝐷 {𝑦(𝑡) − 𝐶 [𝑥 ∗ (𝑡) + 𝑧(𝑡)]}[𝑥 ∗ (𝑡) + 𝑧(𝑡)],                 (6)
or
                   𝑧 (𝑡) = 𝐷𝑧(𝑡) + 𝑓(𝑡) + 𝐷 {𝑦(𝑡) − 𝐶 [𝑥 ∗ (𝑡) + 𝑧(𝑡)]}𝑧(𝑡),                    (7)
where
                              𝑓(𝑡) = 𝐷 {𝑦(𝑡) − 𝐶 [𝑥 ∗ (𝑡)]}𝑥 ∗ (𝑡).
   Assuming the effect of disturbances is small, successive approximations can be considered for
equation (6). For the zero approximation, we set

                                            𝜉(𝑡) = 0;
                                            𝑧(𝑡) = 0.
     In the first approximation, we neglect the quadratic term by 𝑧 :
                          𝑧 (𝑡) = 𝐷𝑧(𝑡) + 𝑓(𝑡) + 𝐷 {𝑦(𝑡) − 𝐶 𝑥 ∗ (𝑡)}𝑧(𝑡),

                                    𝑧 (𝑡) = 𝐷 𝑧(𝑡) + 𝑓(𝑡),                                        (8)

where
                                 𝐷 = 𝐷 + 𝐷 {𝑦(𝑡) − 𝐶 𝑥 ∗ (𝑡)}.
Assuming



                                                                                                      40
                                         𝑧(0) = 0,
we can find 𝑧 (𝑡) by numerically solving the linear equation.
  Given a set of measurements:

                                     𝑦 ∗ (𝑡) = 𝐶 𝑥 ∗ (𝑡),
which characterizes normal process flow (solution of equation (2) or (4) when 𝜉(𝑡) ≡ 0), we assume
the adversary aims to maximize damage, causing 𝑥 ∗ (𝑡) becomes unstable. The control problem for
critical infrastructure systems is to prevent such scenarios through control measures and by
comparing 𝑦(𝑡) and 𝑦 ∗ (𝑡).
    To detect intrusions caused by additional adversarial distortions, an additional criterion can be
added to the measurement system to identify deviations from the normal process flow (e.g., electricity
supply):


                            𝐽(𝑦) =     (𝑦(𝑡) − 𝑦 ∗ (𝑡)) 𝑑𝑡 → 𝑚𝑖𝑛.
   Let ℰ   be threshold such that
                                        𝐽(𝑦) ≥ ℰ ,
signals abnormal system behavior. For discrete measurements:
                                                                                                (9)
                              𝐽=     (𝑦(𝑡 ) − 𝑦 ∗ (𝑡 )) ⟶ 𝑚𝑖𝑛.

   Next, let us determine 𝜉(𝑡) that leads to system instability. Such a problem can arise in cyber
incident investigation, especially when trying to uncover adversarial actions aimed at destabilizing
the system. We can use
                                    det(𝐷 + 𝐷 𝜉(𝑡)) = 0                                       (10)
where
                                       𝐷 ≡ 𝐴 − 𝑘𝐵 ,
                                        𝐷 ≡ −𝑘𝐵𝐶 .
This allows us to define 𝜉(𝑡).
   In equation (8), the addition of

                                       𝑦(𝑡) − 𝐶 𝑥 ∗ (𝑡)
is small because the distortions introduced by the adversary are minor and can be neglected in the
first approximation.
    Thus, from (8) we can write:


                               𝑧 ( ) (𝑡) ≈ 𝑒    𝑒           𝑑𝑡 ;

                      𝜉(𝑡) ≈ 𝑦(𝑡) − 𝐶 𝑥 ∗ (𝑡) + 𝑒       𝑒      𝑓(𝑡 )𝑑𝑡 .
In the next approximation, we substitute 𝑧 ( ) (𝑡) in the last term of (7).
    As Table 1 shows, some attacks require knowledge of system functioning and parameters. Using
the principles of parametric identification outlined above, and having access to measurement data, an
adversary can infer unknown parameters (e.g., 𝐴, 𝑘, or 𝐵 from (1)). Thus, intercepting measurement
information may enable more dangerous attacks, such as covert attacks.

1.3.    Attack on system state and parameter identification
  Let us consider a typical attack on the system state that involves false data injection (FDI) by
manipulating system control parameters.




                                                                                                      41
   In FDI attacks, a scaling parameter 𝜉 is used to alter control [5], allowing the adversary to influence
system requlation.
   Under attack, system (1) takes the form:

                         𝑥 (𝑡) = 𝐴𝑥(𝑡) + 𝑘𝐵𝑢(𝑡) + 𝜉𝑢(𝑡) + 𝐹,                                      (11)
                                    𝑢(𝑡) = −𝐶 𝑦(𝑡),
                                     𝑦(𝑡) = 𝐶 𝑥(𝑡).
Rewriting the state equation, we have:
                           𝑥 (𝑡) = 𝐴 − 𝑘𝐴 𝑥(𝑡) + 𝜉𝑢(𝑡) + 𝐹,
where
                                       𝐴 ≡ 𝐵𝐶 𝐶 .
  Let us rewrite the state equation in the form

                                    𝑥 (𝑡) = 𝑀𝑥(𝑡) + 𝐹,
where
                                   𝑀 ≡ 𝐴 − 𝑘𝐴 − 𝜉𝐶 𝐶 .
From the condition
                                         det 𝑀 = 0 ,                                           (12)
we can obtain the critical value of 𝜉 that leads to system instability.
   To illustrate the process of restoring attack parameters of the cyber incident, let us consider the
generalized case of the system (11):

                              𝑥 (𝑡) = A𝑥(𝑡) + B(𝜉)𝑢(𝑡) + 𝐹;                                 (13)
                                         𝑥(0) = 𝑥 ,                                         (14)
where 𝑥 represents the system state, 𝑢 is the control function, 𝐹 is the source function, and 𝜉
describes the intensity of adversary’s intrusion. The dependency 𝐵 on 𝜉 is assumed to be known.
   Suppose the adversary’s goal is defined by the criterion under conditions (13), where
𝑃(𝑡), 𝑄(𝑡) are given functions, 𝑥 (𝑡) represents the process state boundaries, and 𝑢 (𝑡) is the
desired control target of the adversary. We assume that 𝑥 (𝑡) and 𝑢 (𝑡) are known:
                                                                                            (15)
          𝐽(𝑢) =      [𝑃(𝑡) 𝑥(𝑡) − 𝑥 (𝑡) + 𝑄(𝑡)(𝑢(𝑡) − 𝑢 (𝑡)) ]𝑑𝑡 → 𝑚𝑖𝑛.
  Setting 𝑧(𝑡) = 𝑥(𝑡) − 𝑥 (𝑡); 𝑣(𝑡) = 𝑢(𝑡) − 𝑢 (𝑡), equation (1) can be reformulated as:
                                𝑧 (𝑡) = 𝐴𝑧(𝑡) + 𝐵(𝜉)𝑣(𝑡),                                (16)
                                        𝑧(0) = 𝑧 ,                                       (17)
where 𝑧 = 𝑥 − 𝑥 (0) and ideally
                       𝐹(𝑡) + 𝐴𝑥 (𝑡) + 𝐵(𝜉)𝑢 (𝑡) − 𝑥 (𝑡) = 0.                            (18)
Then, expression (15) is transformed to
                                                                                         (19)
                          𝐽=     [𝑃(𝑡) 𝑧(𝑡) + 𝑄(𝑡)(𝑣(𝑡)) ]𝑑𝑡.
   The objective is to determine the feedback between 𝑧(𝑡) and 𝑣(𝑡) that the attacker introduces
into the system to achieve the goal (15). This enables: 1) predicting the magnitude of adversary actions
to train anomaly detection systems, and 2) recovering details of adversary actions from known
incident characteristics (𝑥 (𝑡), 𝑢 (𝑡)).
   Introducing Lagrange multiplier, we have:

                                                                                                  (20)
                       𝛿𝐽 =      {2𝑃(𝑡)𝑧(𝑡)𝛿𝑧(𝑡) + 2𝑄(𝑡)𝑣(𝑡)𝛿𝑣(𝑡)
                  + 𝜆(𝑡)[𝛿𝑧 (𝑡) − 𝐴𝛿𝑧(𝑡) − 𝐵(𝜉)𝛿𝑣(𝑡)]}𝑑𝑡 =
     =     {[2𝑃(𝑡)𝑧(𝑡) − 𝜆 (𝑡) − 𝜆(𝑡)𝐴]𝛿𝑧(𝑡) + [2𝑄(𝑡)𝑣(𝑡) − 𝜆(𝑡)𝐵(𝜉)]𝛿𝑣(𝑡)}𝑑𝑡 +
                                     +𝜆(𝑇)𝛿𝑧(𝑡).
   From the condition 𝛿𝐽 = 0, we select 𝜆(𝑡) so that:



                                                                                                         42
                                𝜆 (𝑡) = 2𝑃(𝑡)𝑧(𝑡) − 𝜆(𝑡)𝐴,                                        (21)
                                         𝜆(𝑇) = 0                                                 (22)
and
                                        2𝑄𝑣 = 𝜆𝐵,                                                 (23)
                                       𝑣 = 𝑄 𝜆𝐵.                                                  (24)

   Equation (16) then becomes:

                             𝑧 = 𝐴𝑧 + 𝐵𝑄        𝜆𝐵, 𝑧(0) = 𝑧 .

   This problem is reduced to equations (21) and (25). However, this system is inconvenient because
the conditions apply for 𝑡 = 𝑇 and 𝑡 = 0, respectively. To simplify it, substitute 𝜆 = 𝐿𝑧, where
𝐿(𝑇) = 0:
                           𝑧 = 𝐴𝑧 + 𝐵𝑄 𝐿𝐵𝑧, 𝑧(0) = 𝑧 .                                       (25)

                         𝐿 𝑧 = 2𝑃𝑧 − 𝐴𝐿𝑧 − 𝐿 𝐴𝑧 + 𝐵𝑄 𝐿𝐵𝑧 .
                                                1                                              (26)
                       𝐿 = 2𝑃 − 𝐴𝐿 − 𝐿 𝐴 + 𝐵𝑄 𝐿𝐵 , 𝐿(𝑇) = 0
                                                2
    Thus, we can solve (26) for 𝐿 numerically with the condition at 𝑡 = 𝑇 and then, with the known
𝐿, solve equation (25) to find 𝑣 = 𝑄 𝐿𝐵𝑧. This solution minimizes the expression (19).
    Given 𝑣 and 𝑧, we can investigate the minimal values of J with respect to the attack parameter ξ
using equations (15) and (16).
    Applying the gradient method allows for a more efficient parameter identification process
compared to a “brute force” calculation approach. The convergence ratio for the gradient procedure
is estimated in [17]. Additionally, the conjugate gradient method [18] can be used as an alternative in
step 6 of the algorithm.
    The algorithm steps are as follows:

   1.   Set an initial arbitrary value 𝜉 .
   2.   Find 𝐿 from equation (26).
   3.   Determine z using equation (25).
   4.   Calculate 𝑣 from equation (24), with 𝜆 = 𝐿𝑧.
   5.   Calculate 𝐽(𝜉 ) using equation (19), with 𝑣 = 𝑄        𝐿𝐵(𝜉 )𝑧.
   6.   Update 𝜉    :𝜉     =𝜉 +𝜏          .

   7.   If         ≤ ℰ , proceed to step 8. Otherwise, return to step 2 for the next iteration.
   8. The parameter value 𝜉 will then satisfy (15) with precision ℰ.
   A similar algorithm can also be used by a malicious actor to identify unknown parameters of the
system. For this, only system state measurements are needed.

      5. Computer simulation results
  Using the presented models, we generated dynamics graphs of FDI attacks. For the simulations,
we developed a Python software package.

1.4.    Stability violation features
   In Fig. 2, we illustrate the normal situation for the AGC. Here, we consider a one-component state
x, representing frequency deviation Δf, and constant values of ξ, which could generally time-
dependent. For a two-component state, see the example in Fig. 3.




                                                                                                         43
Figure 2: Undisturbed system, 𝜉 = 0




Figure 3: Undisturbed system, two-component state, 𝜉 = 0

   To identify the parameter ξ that meets a certain criterion J (see Fig.4), the proposed algorithm can
be applied. In certain cases, some J samples may not contribute to the rapid convergence of the
algorithm. However, in a significant number of cases, the proposed algorithm proves to be
numerically efficient.




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Figure 4: Criterion J sample with a minimum at 𝜉 = 1

   Fig. 5 shows that with small values of attack parameter, malicious influence may be subtle, making
these attacks difficult for anomaly detection systems to detect. Such attacks typically target the
software components of cyber-physical systems, aiming to insert false data into monitoring systems.




Figure 5: Malicious influence with 𝜉 = 1




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Figure 6: Malicious influence with 𝜉 = 5

   Attacks with larger values of scaling attack parameter can be detected effectively by monitoring
systems due to noticeable changes in state pattern. For such attacks, cyber defenders should not only
detect but also react quickly to mitigate potential damage. High values of the scaling parameter can
pose risks to hardware components by threatening system stability. As shown in Figs. 7-9, with
certain values of 𝜉, system state becomes unstable. The threshold value 𝜉 = 9.4 (corresponding to the
Fig. 8) can be calculated with necessary accuracy from (12). In the case of measurement intrusion, the
stability boundary is determined by (10).




Figure 7: System state remains stable




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Figure 8: Attack with threshold value of the parameter




Figure 9: System state loses stability


1.5.    Illustration of malicious activity at a specific time
    Figures 10 and 11 illustrate scenarios where malicious influence "activates" at a specific time rather
than initially. We observe a minor spike with low scaling parameter values (Fig. 10) and a clear change
in the pattern with more significant influence intensity (Fig. 11).
    Depending on the attacker's goal, small impacts can also lead to serious consequences as a result
of tampering, affecting intrusion detection systems.




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Figure 10: Frequency deviation pattern under attack parameter 𝜉 = 1




Figure 11: Frequency deviation pattern under attack parameter 𝜉 = 7

    6. Conclusions
   Computer simulation results indicated that attacks with low values of the scaling parameter are
not a threat to system stability but are challenging for anomaly detection systems to detect. Such
attacks could be used by malicious actors to incrementally falsify historical data or poison machine-
learning-based modules.
   We derived the conditions for stable system operation based on the values of the attack parameter.
Additionally, an algorithm was proposed for estimating the control intensity of FDI attacks, enabling
the collection of quantitative data on malicious strategies to support system resilience.
   An analysis of typical attack patterns in modern energy facilities showed that certain classes of
attacks require full knowledge of the system. This information (e.g., system parameters) can be
indirectly recovered using control theory principles, similar to the algorithm proposed in this paper
for identifying unknown attack parameters. This highlights the risks posed by "sniffing" as a method
for gathering measurement data, underscoring the need for preventive measures to prevent sniffing.
Most data transfer protocols in AGC systems lack confidentiality by default, making them vulnerable.
   The proposed approach and algorithm can be used for numerical incident investigations, providing
solid foundations for response strategies. Future research could focus on studying combined attack
types and enhancing detection methods.




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References
[1] M. Vrakopoulou, P.M. Esfahani, K. Margellos, J. Lygeros, G. Andersson. Cyber-Attacks in the
     Automatic Generation Control. In: Khaitan, S., McCalley, J., Liu, C. (eds) Cyber Physical Systems
     Approach to Smart Electric Power Grid. Power Systems (2015). Springer, Berlin, Heidelberg. doi:
     10.1007/978-3-662-45928-7_11.
[2] P. S. Kundur, O. P. Malik, Power system stability and control, 2-nd ed., McGraw Hill Education,
     New York, 2022. URL: https://lccn.loc.gov/2021062400.
[3] P. L. Goethals, N, M. Scala, D. T. Bennett, Mathematics in Cyber Research, 2022. doi:
     10.1201/9780429354649
[4] Y. Shen, M. Fei, D. Du, Cyber security study for power systems under denial of service attacks,
     2017, SageJournals, Volume 41, Issue 6. doi: 10.1177/0142331217709528.
[5] A.M Mohan, N. Meskin, H. Mehrjerdi, A Comprehensive Review of the Cyber-Attacks and
     Cyber-Security on Load Frequency Control of Power Systems, 2020, Energies, 13(15), 3860. doi:
     10.3390/en13153860.
[6] C. Moya, On Cyber-Attacks against Modern Power Grids, Ph.D. Thesis, The Ohio State
     University, 2020. URL: https://dl.acm.org/doi/10.5555/AAI28890224.
[7] Х.Li, P.Zhang, H.Dong, Robust Stealthy Covert Attacks on Cyber-Physical Systems, 2022, IFAC-
     PapersOnLine, Volume 55, Issue 6, P. 520-525. doi: 10.1016/j.ifacol.2022.07.181.
[8] A. Ashok, A. Hahn, M. Govindarasu, Cyber-Physical Security of Wide-Area Monitoring,
     Protection and Control in a Smart Grid Environment, Journal of Advanced Research (2013), doi:
     http://dx.doi.org/10.1016/j.jare.2013.12.005.
[9] J. Ding, A. Qammar, Z. Zhang, A. Karim, H. Ning, CyberThreats to Smart Grids: Review,
     Taxonomy, Potential Solutions, and Future Directions. Energies, 2022, 15, 6799. doi:
     10.3390/en15186799.
[10] P. S. Kundur, O. P. Malik, Power system stability and control, 2-nd ed., McGraw Hill Education,
     New York, 2022. URL: https://lccn.loc.gov/2021062400.
[11] Kim, Yoonjib & Hakak, Saqib & Ghorbani, Ali. (2022). Smart grid security: Attacks and defense
     techniques, IET Smart Grid, 6. doi: 10.1049/stg2.12090.
[12] M. Ovcharuk, M. Ilin, Models of Denial of Service Attacks on Cyber-Physical Systems, 2023,
     Theoretical and Applied Cybersecurity, Vol. 5, No2 (2023). doi: 10.20535/tacs.2664-
     29132023.2.289459.
[13] Cyber Incident Reporting For Critical Infrastructure Act of 2022, Subtitle D – Cyber Incident
     reporting. URL: https://www.cisa.gov/sites/default/files/2023-01/Cyber-Incident-Reporting-
     ForCriticalInfrastructure-Act-o-f2022_508.pdf .
[14] G. Vedmedenko, I. Stopochkina, O. Novikov, M. Ilin, Cascading effects simulation for cyber
     attacks on the power supply network // XXI International Scientific and Practical Conference
     "Information Technologies and Security" (ITS-2021), 09.12.2021. URL: http://ceur-ws.org/Vol-
     3241/.
[15] S. Saxena, S. Bhatia, R. Gupta, Cybersecurity analysis of load frequency control in power systems:
     A survey. Designs, 2021, 5(3), 52. doi: 10.3390/designs5030052.
[16] G. S. Kamboj, R. Dhiman, R. Choudhary, 2015, Automatic Generation Control of Two Areas in
     Interconnected Power System, International Journal of Engineering Research & Technology,
     (IJERT) NCETEMS – 2015, Vol.3, Issue 10. URL: https://www.ijert.org/automatic-generation-
     control-of-two-areas-in-interconnected-power-system.
[17] Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge: Cambridge University
     Press. URL: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf .
[18] M.R. Hestenes, E. Stiefel, Methods of Conjugate Gradients for Solving Linear Systems. Journal of
     Research of the National Bureau of Standards, 1952. 49 (6): 409. doi:10.6028/jres.049.044.




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