A Two-Stage Mixed-Integer Nonlinear Framework for Assessing Load-Redistribution False Data Injection Effects in AC-OPF-Based Power System Operation

Abstract

Load-redistribution false-data-injection (LR-FDI) attacks can degrade power-system operation by reshaping the perceived nodal demand pattern, thereby inducing congestion-aware redispatch and economic inefficiency while preserving the net system load. Prior LR-FDI studies commonly adopt bilevel/Stackelberg formulations with a continuous attack vector and an embedded operator response; however, these formulations often (i) do not represent explicit compromised-load selection, (ii) become computationally restrictive when combinatorial target sets are considered, and (iii) offer limited transparency for structured, stage-wise attack planning. This paper proposes a sequential two-stage attacker–operator framework for LR-FDI vulnerability assessment that integrates sparse load compromise decisions with screening-regularized attack synthesis and post-attack operational evaluation. In Stage-1, a mixed-integer nonlinear program identifies economically influential load buses via binary selection and determines admissible perturbation magnitudes under total-load conservation and proportional shift bounds. To confine the attacker-side search region and avoid economically exaggerated solutions, a screening-derived conservative operating-cost ceiling is first estimated through a parametric load-sensitivity analysis and then used to regularize the attack-synthesis step. In Stage-2, the system operator’s corrective redispatch is evaluated by solving an active-power-oriented economic dispatch model with nonlinear network-consistent assessment of operational outcomes. Using the IEEE 24-bus RTS, results show that the hourly operating-cost deviation reaches ≈0.2% in the most adverse feasible cases, and the cumulative daily impact approaches ≈5% only under selectively realizable compromised-load patterns, accompanied by a nearly 80% increase in total active-power transmission losses relative to the base case. Overall, the framework yields a practically grounded quantification of conditionally severe economic and network stress under coordinated LR-FDI scenarios and provides actionable insight for prioritizing vulnerable load locations for protection and monitoring.

1. Introduction

Modern power systems have become high-value cyber targets because the impact-to-effort ratio for adversaries has risen sharply with grid digitalization, decentralization, and legacy–modern integration. First, pervasive digitalization—SCADA/EMS/DMS, AMI, PMUs, market automation, and vendor/cloud-hosted applications—has enlarged the cyber-physical attack surface well beyond control centers to the substation and grid edge. Second, structural decentralization via DERs, prosumers, microgrids, and EV charging multiplies telemetry/actuation points and third-party trust relationships, creating many opportunities for integrity manipulation. Third, decades-old OT protocols and devices (e.g., IEC 60870-5-104, DNP3) now coexist with enterprise IT stacks and APIs, producing heterogeneous security postures and supply-chain exposure. In this regard, false-data injection (FDI) stands out: by crafting measurement integrity perturbations that slip past residual-based bad-data detection, an adversary can steer dispatch and flows without overt availability outages [1,2,3].

Figure 1 presents the conceptual architecture of the cyber kill chain, which models the lifecycle of cyber attacks through three major phases and seven sequential stages, namely reconnaissance, weaponization, delivery, exploitation, installation, command and control, and actions on objectives. These stages represent the fundamental principles governing most cyber intrusions. In the context of smart grids, this framework provides a structured basis for analyzing how adversaries infiltrate communication infrastructures, manipulate system data, and influence operational decisions [4]. Accordingly, the proposed work adopts this standardized framework to systematically model LR-FDI attack progression and assess its impact on power system operation. The increasing integration of electric vehicles introduces additional operational and market-coupled complexity into modern power systems, further motivating the need for robust cyber-physical monitoring, attack analysis, and mitigation frameworks [5].

In recent years, the energy and power sector has been persistently targeted by nation-state and state-sponsored cyber adversaries across the globe, reflecting its growing strategic importance as critical national infrastructure. Accordingly,

Table 1. Chronology of significant cyber-physical attacks on energy infrastructure around the globe [6].

Year	Event/Malware	Country	Sector/Plant Type	Impact and Capability Loss	Refs.
2010	Stuxnet	Iran	Nuclear (Thermal)	Physical destruction of 1000+ centrifuges via SCADA.	[7]
2012	Shamoon	Saudi Arabia	Oil and Gas (Thermal)	Wiped 30k+ workstations; halted Aramco business ops.	[8]
2013	Havex	Global	Energy Grid/ICS	Espionage and mapping of energy infrastructure	[9]
2015	BlackEnergy 3	Ukraine	Power Grid	First cyber-blackout: ≈230,000 customers lost power	[10]
2016	Industroyer	Ukraine	Substation (Grid)	1 h blackout in Kyiv via automated protocols	[11]
2017	Triton/Trisis	Saudi Arabia	Petrochemical	Targeted Safety Instrumented Systems (SISs) to induce physical damage	[12]
2019	Guri Event	Venezuela	Hydroelectric	Massive national blackout; attributed to infrastructure decay/cyber factors	[13]
2021	DarkSide	USA	Fuel Pipeline	5500 m pipeline shut down; East Coast fuel disruption	[14]
2023	SektorCERT	Denmark	22 Energy Firms	Coordinated firewall compromise; firms shifted to island mode	[15]
2024	Cactus	France	Energy Management	Data exfiltration from Schneider Electric sustainability division	[16]
2025	Bremanger	Norway	Hydroelectric	State-sponsored intrusion targeting dam gate controls	[17]

Note: Entries are compiled from publicly available reports; attribution remains contested for some incidents where explicitly indicated.

presents a chronological summary of major cyber-physical attacks on energy infrastructure worldwide [6]. Consequently, these persistent and sophisticated attack campaigns have exposed critical research gaps, thereby necessitating deeper exploration of emerging and underexplored domains in cyber-physical security for modern power and energy systems.

Within the FDI family, load-redistribution (LR) attacks are particularly insidious: they conserve total demand while reshaping its spatial profile across buses, provoking costly redispatch, covert thermal stress, and LMP distortions—effects confirmed beyond DC linearization and into AC-consistent analyses [18,19]. Most attack-generation papers cast LR-FDI in a Stackelberg (bilevel) form where the attacker jointly optimizes a continuous attack vector anticipating the operator’s OPF/SCOPF response. While expressive, this single-shot design (i) rarely encodes explicit, discrete access to meters/substations, (ii) induces large, nonconvex MINLPs that scale poorly, and (iii) implicitly assumes the attacker “arrives” only at the moment of injection—limiting stealth pre-planning and realistic intrusion workflows [19,20].

In particular, market and dispatch-facing consequences of coordinated load-altering behavior have been examined, and have shown how ICT and IoT-exposed controllable loads can create new pathways for market manipulation and arbitrage incentives under coordinated attack patterns [21]. Complementary work models dynamic load-altering attacks that destabilize grid frequency and develops control-oriented mitigation strategies, highlighting that load attacks are not purely static “redistribution” constructs but can be strongly time-coupled and stability-critical [22]. On the detection side, data-driven and unsupervised approaches to identify stealthy FDIA behavior can evade traditional screening, including expectation–maximization learning and neighborhood-based anomaly ranking with interpretability mechanisms [23,24]. In parallel, resilience-focused studies analyze moving-target defense for improving PSSE robustness to FDIAs [25], and propose defense strategies for cyber-physical EV grids where high variability and cyber exposure increase attack surfaces [26]. Broader reviews consolidate these developments by emphasizing (i) the deep coupling in cyber-physical power systems and the need for vulnerability assessment [27], (ii) the expanding attack surface in distributed economic dispatch due to evolving communication topology and information security risks [28], and (iii) the growing role of AI/ML-based methods across smart-grid cybersecurity and state-estimation anomaly handling [29,30].

2. State of the Art and Limitations in Load Redistribution FDI Attack Frameworks

2.1. Foundations of FDI and Load Redistribution Attacks

The foundations of false data injection (FDI) attacks were established through the characterization of unobservable attack conditions in power system state estimation. Liu et al. demonstrated that an attacker capable of constructing an injection vector aligned with the measurement Jacobian can remain undetected by conventional bad-data detection mechanisms [1]. Subsequent work by Kosut et al. quantified the operational implications of integrity manipulation under realistic detection thresholds [2].

Load-redistribution (LR) attacks refined this paradigm by enforcing total-load conservation while inducing economically or physically adverse outcomes, such as transmission overloads, congestion rent shifts, or dispatch cost escalation [18]. By preserving net demand balance, LR-FDI attacks maintain frequency stability and avoid immediate suspicion, thereby formalizing the stealth–impact trade-off that underpins much of the subsequent literature [1,2,3]. On this basis, the proposed framework implements literature-grounded stealth-oriented LR-FDI scenarios, formulated under the adopted unobservability assumptions, and examines their impact on system operation and economic performance.

2.2. Bilevel Attacker–Operator Formulations

A dominant stream of research models FDI and LR-FDI attacks as bilevel optimization problems in which the attacker maximizes economic cost, overload severity, or market distortion at the upper level, while the system operator responds through an OPF or SCOPF lower-level problem [20]. Early studies relied primarily on DC approximations for computational tractability. Later extensions incorporated AC-consistent OPF models and nonlinear state estimation, revealing that voltage magnitude limits, reactive power balance, and network nonlinearities significantly reshape feasible attack regions and detection signatures [19,20].

Recent studies have also explored convexification strategies to alleviate the computational burden of nonconvex power-system optimization models. Liang et al. developed steady-state convex bi-directional converter models for hybrid AC/DC networked microgrids, where least-squares-based approximation was used to convexify converter-efficiency behavior and additional convex conditions were derived to prevent simultaneous rectification and inversion. Their results show that such reformulation can greatly improve computational efficiency while retaining operational relevance. Although this work is not focused on cyber attacks, it highlights the broader value of relaxation-based modeling for improving scalability in complex dispatch and decision-making problems [31]. While these bilevel formulations are mathematically rigorous, they commonly reduce to single-shot mixed-integer nonlinear programs (MINLPs) through Karush–Kuhn–Tucker (KKT) reformulations. Such approaches implicitly assume that the attacker possesses complete system knowledge including topology, parameters, and measurement access—prior to solving the optimization problem. Moreover, attacker decision variables are typically modeled as continuous injection magnitudes rather than discrete, meter-level access decisions. Consequently, these models emphasize optimal impact under perfect knowledge rather than explicitly modeling which loads or meters are realistically compromisable.

2.3. TriLevel and Defender-Aware Extensions

Trilevel attacker–defender–operator frameworks extend bilevel models by embedding protective resource allocation or hardening strategies within a hierarchical structure. These formulations evaluate worst-case impacts under rational adversaries and limited defensive budgets, often incorporating robust optimization or moving-target defense concepts [3,32].

Although insightful for infrastructure planning, trilevel formulations substantially increase computational complexity and continue to treat attack vectors as continuous decision variables. The abstraction remains largely equilibrium-based, with the entire cyber-physical interaction compressed into a unified optimization problem. As a result, feasibility within the optimization framework implicitly equates to guaranteed attack success.

Recent studies have extended LR-FDI analysis beyond the original attack formulations by considering local attacks under incomplete network information, game-theoretic attack–defense interaction, and reliability-oriented consequence assessment [33,34,35]. These works advance vulnerability analysis and impact quantification; however, they still largely assess attack construction and system consequences within unified or tightly coupled modeling abstractions. From a cybersecurity perspective, such formulations do not explicitly separate reconnaissance-driven scenario construction from subsequent operator-side post-attack assessment, thereby overlooking stage-wise feasibility constraints.

2.4. Cyber Kill Chain Perspective and Sequential Modeling Gaps

The cyber kill chain framework conceptualizes attacks as sequential processes spanning reconnaissance, weaponization, delivery, exploitation, installation, command-and-control, and actions on objectives [36]. Most bilevel and trilevel LR-FDI models deviate from this paradigm by collapsing reconnaissance and execution into a single equilibrium problem. In doing so, they assume that the attacker instantaneously acquires the measurement Jacobian H and computes an optimal injection vector $a=Hc$ without accounting for stochastic learning, partial system observability, or reconnaissance failure.

This modeling abstraction neglects epistemic uncertainty in topology discovery, parameter estimation errors, limited meter compromise, and time-varying system configurations. Consequently, the literature predominantly quantifies the impact of an already-constructed attack rather than modeling the structured process through which attack capability is developed.

2.5. Discrete Access, Sparsity, and Staged Formulations

Mixed-integer techniques have appeared in the context of protected-meter placement, cardinality constraints, and sparse attack construction. However, explicit per-load binary compromise decisions integrated with load-redistribution constraints remain comparatively underexplored. Furthermore, a clear methodological separation between (i) reconnaissance-driven target selection and (ii) magnitude sizing with AC-consistent recourse has not been systematically developed in LR-FDI research.

Recent advances in AC feasibility preservation and convex relaxation theory [32,37] provide the computational foundation for stage-wise formulations that remain tractable while respecting network physics. A staged framework that first identifies economically influential load buses under load-redistribution constraints and subsequently evaluates AC-OPF-consistent operator response aligns more closely with observed intrusion processes. Such separation enables explicit modeling of feasibility bounds, partial attack failure, and bounded economic impact—dimensions largely absent in the classical single-shot bilevel or trilevel LR-FDI optimization literature.

Motivated by these gaps, this paper proposes a sequential two-stage LR-FDI framework with an active-power-oriented economic dispatch model. In Stage-1, feasible and economically influential load-bus targets are identified under load-redistribution and stealth constraints, while a load-sensitivity-based conservative operating-cost cutoff is used to restrict the attack-generation space. In Stage-2, the attacked load vector is transferred to the operator model to evaluate the resulting operating-cost escalation and associated network and market impacts. This staged formulation better reflects cyber kill chain progression, improves computational tractability, and preserves AC-operational realism.

The contributions of this paper are summarized as follows:

• A sequential two-stage LR-FDI assessment framework is proposed in alignment with the cyber kill chain paradigm, wherein reconnaissance-driven attack planning and post-attack operational assessment are modeled as distinct but sequentially linked optimization stages. This formulation differs from conventional single-shot bilevel and trilevel LR-FDI models by explicitly separating attack construction from operator-side impact evaluation.
• The Stage-1 attacker model integrates explicit binary load-selection variables with LR-FDI and stealth constraints, thereby enabling realistic representation of selective load compromise. In addition, a load-sensitivity-based conservative operating-cost cutoff is introduced to restrict the attack-generation space and improve the physical credibility of the cost-maximization process.
• The proposed framework enables quantitative evaluation of economically influential and feasible LR-FDI scenarios in an active-power-oriented economic dispatch model with nonlinear network constraints. By transferring the attacked load profile from Stage-1 to Stage-2, the framework captures the resulting operating-cost escalation and associated network and market impacts, including variations in line flows, transmission losses, bus voltages, and LMPs.

The remainder of this paper is organized as follows.

Section 3 discusses the structural characteristics of the proposed two-stage optimization framework, with emphasis on its mathematical behavior, computational challenges, and corresponding remedial considerations. Section 4 presents an analytical comparison between conventional bilevel formulations and the proposed bi-stage framework, thereby clarifying their conceptual distinctions in the context of the existing LR-FDI literature. Section 5 develops the complete mathematical formulation of the proposed methodology, including both optimization stages and their one-way coupling mechanism. Section 6 presents the numerical results and discussion, highlighting the economic and operational impacts observed under the analyzed attack scenarios. Finally, Section 7 concludes the paper by summarizing the key findings and major inferences.

3. Structural Characteristics of the Two-Stage Optimization Framework

In the proposed LR-FDI framework, Stage-1 is formulated as an attacker-side optimization problem that seeks economically severe yet feasible load-manipulation patterns under load-conservation, proportional shift-bound, and selective compromise constraints. Due to the simultaneous presence of binary selection variables and nonlinear network-coupled relations, this stage is treated as a mixed-integer nonlinear programming (MINLP) problem. Stage-2 represents the operator-side post-attack corrective redispatch problem and is formulated as a nonlinear programming (NLP) model subject to network and operational constraints. The framework is therefore interpreted as a sequential two-stage attacker–operator assessment model, and the main mathematical and computational issues associated with both stages, together with the adopted solution strategies, are discussed in the following subsections.

3.1. Computational Challenges in Convex Function Maximization

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be convex. While minimizing f over a convex set $\mathcal{X}$ is a convex program, the maximization

$$\underset{x\in \mathcal{X}}{max}f\left(x\right)$$

is generally nonconvex: the hypograph of a convex function is nonconvex, so local maxima need not be global, leading to multiple stationary points and poor algorithmic guarantees [38,39]. In practice this entails:

• Loss of global optimality guarantees: first-order and interior-point methods no longer certify global optimality; ascent can stall at saddles or non-global local maxima [38].
• Computational hardness: convex-function maximization over polyhedral/convex regions is NP-hard in general [40].
• Unboundedness risks: if $\mathcal{X}$ is not compact or regularized, f may diverge to $+\infty$ along feasible directions [39].

3.2. Possible Remedies

• Reformulation and global optimization: solve $min(-f)$ via deterministic global methods (spatial branch-and-bound, convex relaxations) to obtain certified solutions [41,42,43,44].
• Difference-of-convex (DC) programming: represent $f=g-h$ with $g,h$ convex and apply CCCP/DC algorithms to seek high-quality stationary points [45].
• Convexification and relaxations: build bounding problems via convex over-/under-estimators (e.g., $\alpha$BB) that yield global bounds and enable pruning [41,42].
• Mixed-integer structure: if binaries/selectors are present, strengthen relaxations using perspective/extended formulations to tighten bounds [46].
• Heuristics/stochastic search: evolutionary methods, SA and PSO can be used when scale or modeling precludes exact global methods; often paired with bounding for certification [44].

For convex-objective maximization, Luo and Mehrotra develop a geometric branch-and-bound method with piecewise-linear approximations, proving $\epsilon$-convergence and computing upper/lower bounds at nodes [43]. Ben-Tal and Roos analyze optimality conditions and algorithms tailored to convex-function maximization [40]. Broader 2024–2025 surveys review state-of-the-art bounding and matheuristics integrating learning/strong reformulations [44,47].

3.3. Using a Relaxation to Get an Upper Bound

Suppose ${\mathcal{X}}_{\mathrm{tight}}\subseteq {\mathcal{X}}_{\mathrm{loose}}$ and consider the same convex f. Then

$$\underset{\mathbf{x}\in {\mathcal{X}}_{\mathrm{tight}}}{max}f\left(\mathbf{x}\right)\le \underset{\mathbf{x}\in {\mathcal{X}}_{\mathrm{loose}}}{max}f\left(\mathbf{x}\right)$$

(1)

Thus, solving the less constrained (relaxed) problem yields a valid upper bound on the objective of the original (more constrained) problem. This monotonicity directly powers branch-and-bound, cutting planes, and outer-approximation methods: at each node/relaxation, solve an easier problem to get an upper bound, keep the best feasible value as a lower bound, and prune when bounds close [38,43,44].

Based on the issues identified in Section 3.1 and the remedies discussed in Section 3.3, this work employs a relaxed upper-bound approach to address the non-concave cost maximization in Stage-1. The operating cost is evaluated through load sensitivity analysis by varying the aggregate demand within the feasible range defined by generator capacity limits, thereby estimating the maximum attainable cost without directly solving the non-concave formulation. This approach alleviates local optimality and numerical instability while preserving physical feasibility.

4. Bilevel vs. Bi-Stage (Proposed)

In optimization and game theory, a Stackelberg-style framework refers to a hierarchical leader–follower model in which one decision maker (leader) acts first, anticipating the optimal reaction of another decision maker (follower). This concept originates from Heinrich von Stackelberg’s work in 1934 [48].

Table 2. Comparison between classical Stackelberg (bilevel) LR-FDI models and proposed two-stage LR-FDI formulation.

Aspect	Classical Stackelberg (Bilevel) LR-FDI Models	Proposed Two-Stage LR-FDI Formulation
Mathematical Structure	${\displaystyle \underset{a}{max}F(a,x^{*}\left(a\right))\mathrm{s}.\mathrm{t}.x^{*}\left(a\right)=arg\underset{x}{min}f(x,a)}$ $$\left.\begin{array}{cc}\hfill \mathrm{Attacker}& \to \mathrm{Leader}\hfill \\ \hfill \mathrm{Operator}& \to \mathrm{Follower}\hfill \end{array}\right\}$$	${\displaystyle \underset{z,\delta }{max}F(z,\delta ,P^{*}(z,\delta ))}$ Where: Stage-1 selects attackable load set z (binary) and perturbations $\delta$. Stage-2 embeds AC-OPF response $P^{*}$ under those constraints.
Load Controllability	Load buses implicitly continuous; no explicit selection of which loads to manipulate (excludes load manipulation/availability uncertainty).	Binary selection vector $z_i\in \{0,1\}$; attacker explicitly chooses subset of loads to compromise before optimizing perturbation size (includes load manipulation/availability uncertainty).
Coupling of Stages	Leader solves using lower-level KKT of AC-OPF but assumes continuous feasible set; attack and system response loosely coupled.	Stage coupling explicit: Stage-1 pre-screens feasible attack set; Stage-2 simultaneously enforces AC-OPF and stealth constraints within that chosen set.
Complexity and Tractability	Bilevel reformulated as single-level MILP/MINLP using KKT and big-M; typically large, nonconvex, and solver-hard.	Two-stage decomposition reduces search: binary load selection first, then continuous AC-OPF re-dispatch; empirically more scalable for medium/large grids (e.g., IEEE-24 RTS).
Detection Awareness	Stealth constraints often generic: residual threshold, $\mathbf{H}a=0$, or $\parallel r\parallel \le ϵ$.	Allows scenario-specific stealth rules at Stage-1 (e.g., false load redistribution balance, zero column-sum) and economic stealth tuning at Stage-2.
Attacker Knowledge	Requires global Jacobian $\mathbf{H}$ matrix to design a; assumes attacker can fully foresee OPF.	Relaxes full knowledge: only selective bus/load access pattern and feasible shift bounds needed at Stage-1; still leverages AC-OPF but can use partial network knowledge.
Economic Impact	Focus: maximize generation cost/congestion rent increase.	Same objective but with targeted controllability; higher impact with smaller compromised set—beneficial for stealthy, resource-limited adversary.

illustrates the comparison between the classical Stackelberg (bilevel) LR-FDI Models and the proposed two-stage LR-FDI models. Table 2 illustrates the comparison between the classical Stackelberg (bilevel) LR-FDI models and the proposed two-stage LR-FDI models.

4.1. Mathematical Definition

A Stackelberg game is formally expressed as a bilevel optimization problem:

$$\begin{array}{cc}\hfill \underset{x}{min}& F(x,y)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& y\in arg\underset{y}{min}f(x,y)\hfill \\ \hfill & g(x,y)\le 0,h(x,y)=0,\hfill \end{array}$$

(2)

where

• x represents the leader’s decision variables;
• y represents the follower’s decision variables;
• $F(x,y)$ is the leader’s objective function;
• $f(x,y)$ is the follower’s objective function.

The leader internalizes the follower’s best response mapping $y^{*}\left(x\right)$ while optimizing $F(x,y)$.

4.2. Application to Power System Cybersecurity

In cyber-physical power systems, Stackelberg games have been widely used to model attacker–defender interactions. The attacker plays the role of the leader, deciding the false data injection (FDI) attack vector while anticipating how the system operator (follower) will redispatch generation or redirect the flow using AC-OPF/PF. The operator solves OPF to minimize operating cost based on corrupted measurements.

4.3. Importance of Stackelberg-Style

• Realistic modeling of adversaries: Attacker has system knowledge and anticipates defense strategies.
• Detection-aware attacks: Stealth constraints can be embedded in the leader’s problem.
• Economic impact quantification: Allows us to study how attacker maximizes system cost deviation while the operator minimizes cost.

Stackelberg-style models are widely used in security-constrained power flow, unit commitment under cyber threats, LR-FDI attack analysis, and infrastructure defense planning.

4.4. Overview of the Proposed Sequential Framework

The proposed methodology is organized as a sequential two-stage framework for LR-FDI attack synthesis and post-attack operational assessment, as illustrated in Figure 2. In contrast to conventional bilevel or trilevel formulations, the proposed structure does not embed the operator response within the attacker problem. Instead, it follows a forward stage-wise progression in which the attacker first constructs a feasible attack scenario and the resulting attacked operating point is subsequently evaluated from the system-operator perspective. This one-way progression is consistent with the staged attack evolution represented through the cyber kill chain paradigm adopted in this work.

Stage-1 represents the attacker-side model and is executed in two phases. In Phase 1, an upper bound on the achievable operating cost, denoted by $O{F}_{max}$, is estimated through aggregate-demand sensitivity analysis over the physically admissible operating range. This bound is not itself the attack solution; rather, it serves as a physics-consistent cutoff that restricts the search region for the subsequent attack-synthesis phase. In Phase 2, the actual LR-FDI scenario is synthesized by selecting the compromised load buses, determining the admissible perturbation vector, and constructing the attacked load profile $L_i^{\mathrm{new}}$ subject to the load-redistribution, sparsity, and stealth-related constraints imposed in the formulation.

The outputs of Stage-1 form the interface with Stage-2. In particular, the selected compromised load set, the corresponding perturbation pattern, and the attacked load vector $L_i^{\mathrm{new}}$ are transferred to Stage-2 as fixed inputs. Stage-2 then represents the operator-side post-attack assessment problem, wherein the system response is evaluated under the attacked load condition using the active-power-oriented economic dispatch model with nonlinear network constraints. This stage quantifies the resulting operating cost and the associated network- and market-level consequences, including variations in generator dispatch, line flows, transmission losses, bus voltages, and locational marginal prices.

From an optimization viewpoint, the overall framework is therefore not a single equilibrium-based formulation, but a sequentially linked optimization architecture: Stage-1 is posed as an MINLP because it combines binary load-selection decisions with nonlinear operational constraints, whereas Stage-2 is formulated as an NLP-based post-attack operational assessment problem. No iterative feedback loop from Stage-2 to Stage-1 is considered in the present work; hence, the framework should be interpreted as a sequential attacker–operator interaction model rather than a game-theoretic co-optimization structure.

Accordingly, Figure 2 summarizes the complete workflow of the proposed methodology, while the equation references shown in the flowchart correspond to the detailed mathematical formulations presented in the subsequent section. Based on this sequential architecture, the detailed formulations of Stage-1 and Stage 2 are presented next.

Stage-1 to Stage-2 Forwarding Protocol

To make the sequential mapping explicit, the forward transfer of attacker-side outputs to the operator-side model is summarized below. Here, cardinality refers to the number of compromised loads, whereas a scenario refers to the complete LR-FDI realization obtained for a given cardinality.

• Input: Base-case system data, admissible LR-FDI limits, network and operating constraints, and the total number of load buses $N_L$.
• Stage-1, Phase-1: A preliminary load-sensitivity analysis is carried out to estimate a screening-derived operating-cost reference ceiling, which is subsequently used to regularize the attacker-side search.
• Stage-1, Phase-2: The attacker-side MINLP is solved repeatedly for each admissible compromise cardinality $k=2,3,\dots ,N_L$. For each fixed k, the formulation determines the feasible LR-FDI scenario corresponding to that cardinality, namely:

  (a)

  The compromised-load subset;

  (b)

  The binary selection pattern;

  (c)

  The associated manipulation magnitudes;

  (d)

  The attacked load vector.

  The obtained realization is stored as the representative attacker-side scenario for that cardinality.
• Stage-2, Phase-3: Each stored attacked-load realization is forwarded individually to the operator-side model as fixed exogenous input, and Stage-2 is solved independently to quantify the corresponding post-attack economic and operational impact.
• Result interpretation: This process is repeated over the admissible cardinality range so that multiple scenarios are evaluated, but only one scenario is injected into Stage-2 at a time. The final outcomes are therefore reported scenario-wise/cardinality-wise, and the most severe overall case is identified by comparing the corresponding Stage-2 results.

5. Problem Formulation

The proposed framework is formulated within the active-power-oriented economic dispatch model with nonlinear network constraints of power system operations. The baseline operational model follows the active-power-oriented economic dispatch model, which provides the standard cost structure for generation scheduling under network and operating constraints [49]. Therefore, the operating cost function, common to all formulations considered in this work, is defined in generalized form as

$$OF=\left\{\sum _{g\in {\Omega }_G}\left(a_g{P}_g^2+b_g{P}_g+c_g\right)\right\}$$

(3)

where $OF$ denotes the total generation operating cost; $P_g$ represents the active power output of generator g; $a_g$, $b_g$, and $c_g$ are the quadratic cost coefficients of generator g; ${\Omega }_G$ denotes the set of generating units.

• Constraints (Generalised)

• A. Power balance constraint

$$\sum _{g\in {\Omega }_G^i}{P}_g=\mathrm{Load}.$$

(4)

• B. Generating unit constraint

$$P_g^{min}\le P_g\le P_g^{max}\forall g\in {\Omega }_G^i.$$

(5)

• C. AC line real-power flow

$$P_{ij}={\displaystyle \frac{V_i^2}{Z_{ij}}}cos\left({\theta }_{ij}\right)-{\displaystyle \frac{V_i{V}_j}{Z_{ij}}}cos\left({\delta }_i-{\delta }_j+{\theta }_{ij}\right)\forall (i,j)$$

(6)

• D. Nodal power balance

$$\sum _{g\in {\Omega }_G^i}{P}_g-L_i=\sum _{j\in {\Omega }_i}{P}_{ij},\left({\lambda }_i\right),\forall i$$

(7)

• E. Node voltage constraint

$$V_i^{min}\le V_i\le V_i^{max}\forall i\in {\Omega }_N.$$

(8)

• F. Line power flow constraint

$$-P_{ij}^{max}\le P_{ij}\le P_{ij}^{max}\forall (i,j)\in {\Omega }_L.$$

(9)

where $\mathit{Load}$ represents the total system active power demand. Further, $P_{ij}$ denotes the active power flow from bus i to bus j; $V_i$ and $V_j$ are the voltage magnitudes at buses i and j, respectively; $L_i$ denotes the active power demand at bus i, and ${\lambda }_i$ denotes the locational marginal price (LMP) at bus i, obtained as the dual variable corresponding to the nodal active power balance constraint. A complete list of symbols, variables, parameters, and associated quantities is given in Appendix A.

5.1. Stage-1: Attacker Model

Stage-1 is formulated as the attacker-side optimization block of the proposed sequential framework. Its purpose is to construct a feasible and economically influential LR-FDI scenario by maximizing the operating-cost function over the admissible attack space. Stage-1 is implemented in two sequential phases; In Phase-1, a load-sensitivity-based analysis is performed to estimate a conservative operating cost ceiling that is used to regularize the subsequent attacker-side search. In Phase-2, this estimated bound is embedded into the attacker model to restrict the search space and to synthesize the final LR-FDI attack vector under the prescribed operational and stealth constraints. Thus, Phase-1 serves as a preparatory bounding step, whereas Phase-2 performs the actual attack generation.

The attacker objective in Stage-1 is expressed as:

$${\mathrm{OF}}_1=max\left\{OF\right\}$$

(10)

However, directly solving the above maximization problem over the full nonlinear feasible region is computationally difficult. Therefore, prior to the attack-synthesis step, an upper-bound estimate of the attainable operating cost is obtained through load sensitivity analysis. Such upper-bound-based treatment has also been discussed in the literature as one of the possible practical remedies for handling computationally challenging attack-oriented optimization problems. In the present work, this bound is estimated by parametrically varying the aggregate system demand within its physically feasible range and evaluating the associated operating cost under the corresponding feasibility constraints.

5.1.1. Phase-1: Load Sensitivity Analysis for Upper Bound Estimation

Let the aggregate system demand be varied over the feasible interval bounded by the aggregate lower and upper generation capabilities.

Then, the parametric load level is defined as

$$\mathrm{Load}=\sum _{g\in {\Omega }_G^i}{P}_g^{min}+{\displaystyle \frac{\alpha -1}{\beta -1}}\sum _{g\in {\Omega }_G^i}\left(P_g^{max}-P_g^{min}\right)$$

(11)

with iteration index $\alpha \in \{1,\dots ,\beta \}$.

Where $\alpha$ denotes the load-sensitivity iteration index and $\beta$ denotes the total number of parametric load levels considered. This parametrization linearly interpolates the total demand from the minimum aggregate generation limit to the maximum aggregate generation limit and is used only for sensitivity analysis.

Accordingly, the lower and upper operating-cost cutoffs are obtained as

$$O{F}_{min}=OF\left(Loa{d}_{min},P_g^{min}\right)$$

(12)

and

$$O{F}_{max}=OF\left(Loa{d}_{max},P_g^{max}\right)$$

(13)

where $OF(·)$ denotes the operating-cost value evaluated under the specified aggregate loading and generation-limit setting. The quantity $O{F}_{max}$ is subsequently used as a screening-derived operating-cost cutoff to confine the attacker-side search region.

5.1.2. Phase-2: LR-FDI Attack Synthesis

Using the upper-bound estimate obtained in Phase-1, the attacker-side optimization in Phase-2 is restricted as

$$O{F}_1\le O{F}_{max}$$

(14)

which reduces the feasible search space and improves the computational tractability of the attack-generation problem. Under this restriction, the attacker synthesizes the LR-FDI scenario by selecting the compromised load buses, determining the admissible perturbation pattern, and constructing the attacked load vector.

The nodal power balance under the attacked load condition is written as

$$\sum _{g\in {\Omega }_G^i}{P}_g-L_i^{\mathrm{new}}=\sum _{j\in {\Omega }_i}{P}_{ij},\left({\lambda }_i\right),\forall i$$

(15)

where $L_i^{\mathrm{new}}$ denotes the attacked demand at bus i. The attacked load is expressed as

$$L_i^{\mathrm{new}}=L_i^{\mathrm{old}}+b\left(i\right),\forall i$$

(16)

with the adjustment factor defined as

$$b\left(i\right)=a\left(i\right)y\left(i\right),\forall i$$

(17)

where $a\left(i\right)$ denotes the perturbation magnitude and $y\left(i\right)$ is the binary load-selection variable indicating whether bus i is compromised. In this way, Phase-2 captures selective load compromise under the LR-FDI setting while preserving the total-load redistribution logic and the corresponding feasibility constraints.

The total compromised load selectivity is then quantified as

$$loadsAlter=\sum _{i}y\left(i\right)$$

(18)

The load selectivity relation in Equation (18) determines the number of compromised load buses participating in a particular attack realization. Accordingly, each feasible combination of selected buses obtained through the binary vector $y\left(i\right)$, together with its associated proportional adjustment $a\left(i\right)$, net load modification $b\left(i\right)$, and attacked load vector $L_i^{\mathrm{new}}$, is treated as one synthesized LR-FDI attack scenario. Let such scenarios be indexed by $s\in {\Omega }_S$, where ${\Omega }_S$ denotes the set of feasible attack scenarios generated in Stage-1. Thus, it labels a feasible attack realization corresponding to a particular compromised-load pattern and its associated attacked load vector.

The load-redistribution nature of the attack is preserved through the net-load conservation condition

$$\sum _{i}b\left(i\right)=0$$

(19)

thereby ensuring that the total system demand remains unchanged after redistribution. In addition, the proportional adjustment at each compromised load is bounded as

$$a\left(i\right)\in \left[-\chi L_i^{\mathrm{old}},\chi L_i^{\mathrm{old}}\right],\forall i$$

(20)

where $\chi$ denotes the admissible fractional perturbation limit. These bounds enforce realistic flexibility limits on the load modifications and prevent physically implausible adjustments.

In addition to the attack-specific constraints introduced in this phase, the Phase-2 formulation is also governed by the generalized operational constraints given in Equations (5), (6), (8) and (9), which define the generator limit, AC real-power flow, node voltage, and line-flow constraints of the underlying network operating environment.

Hence, Stage-1 should be interpreted as a sequential attacker-side construction process: Phase-1 estimates a practical conservative operating-cost cutoff through aggregate-demand sensitivity analysis, and Phase-2 uses that bound together with nodal balance, selective load compromise, net-load conservation, and proportional adjustment constraints to generate feasible and economically influential LR-FDI attack scenarios.

5.2. Stage-2: System Operator Model

After completion of Stage-1, the synthesized LR-FDI attack scenario(s) are transferred to the system-operator layer for post-attack operational assessment. Accordingly, Stage-2 is formulated as the operator-side optimization problem in which the attacked load vector obtained from Stage-1 is treated as a fixed exogenous input. In contrast to Stage-1, no attack-synthesis decision is performed here; rather, the purpose of Stage-2 is to evaluate the economically optimal system response under the attacked operating condition.

As defined in Stage-1, each feasible attack realization is indexed by $s\in {\Omega }_S$, where scenario s corresponds to a specific compromised-load pattern identified through Equation (18) and the associated LR-FDI adjustment variables. Hence, a scenario s represents one feasible selected attack vector characterized by its own attacked load profile $L_{i,s}^{\mathrm{new}}$. For a chosen scenario s generated in Stage-1, the Stage-2 objective is expressed as

$$O{F}_2=min\left\{OF\right\}$$

(21)

The corresponding nodal active-power balance under the attacked load condition is written as

$$\sum _{g\in {\Omega }_G^i}{P}_g-L_{i,s}^{\mathrm{new}}=\sum _{j\in {\Omega }_i}{P}_{ij},\left({\lambda }_i\right),\forall i$$

(22)

where $L_{i,s}^{\mathrm{new}}$ denotes the attacked load at bus i for scenario s.

Thus, the term selected scenario refers to one feasible LR-FDI attack realization synthesized in Stage-1 and then passed to Stage-2 for operator-side evaluation. Since multiple feasible attack scenarios may be generated in Stage-1, corresponding to different combinations and numbers of compromised load buses, Stage-2 is solved one scenario at a time by treating the associated attacked load vector as fixed input data. In this way, the Stage-1 to Stage-2 coupling remains one-way: Stage-1 constructs the scenario, while Stage-2 evaluates its operational consequences without any iterative feedback to the attacker model.

In addition to the attacked-load nodal balance in (22), the Stage-2 optimization remains subject to the underlying network operational constraints introduced earlier in the generalized formulation. In particular, the generator-limit, AC real-power flow, node-voltage, and line-flow constraints given in Equations (5), (6), (8) and (9) are retained in Stage-2, while the power-balance relation is enforced through the scenario-specific attacked-load formulation in (22). Under these constraints, Stage-2 computes the operator’s post-attack redispatch and quantifies the resulting economic and network impacts, including changes in operating cost, line active-power flows, transmission line losses, bus-voltage magnitudes, and LMPs.

5.3. Solution Strategy and Solver Selection

The proposed framework is solved sequentially across two optimization stages. In Stage-1, Phase-1, a parametric load-sensitivity analysis is performed to estimate the conservative operating-cost cutoff used to guide the subsequent attacker-side search. In Stage-1, Phase-2, both relaxed and non-relaxed MINLP formulations are investigated to compare attack realizability and solution behavior. The relaxed formulation is solved using BARON in the GAMS environment. Since BARON is a deterministic global optimizer for MINLPs, any global optimality statement in this stage is limited to the relaxed formulation and only to the extent certified by the solver within the reported optimality tolerance, primal bound, and dual bound [50]. The non-relaxed formulation is solved using DICOPT, which is adopted to handle the mixed-integer nonlinear structure of the selective load-compromise model [51]; accordingly, these solutions are interpreted as solver-obtained computational solutions for comparative vulnerability assessment rather than as globally optimal solutions. In Stage-2, the operator-side post-attack assessment is formulated as an NLP and solved using CONOPT, a local nonlinear programming solver suitable for large-scale nonlinear dispatch problems without integer variables [52]. Since no convex reformulation or relaxation is introduced for Stage-2, no convexity or global optimality claim is made for this stage, and the obtained solution is interpreted as a locally optimal NLP solution under the adopted solver settings.

All simulations were implemented in GAMS Distribution 38.3.0 on a 64-bit MS Windows platform. The computations were carried out on an HP Elite Tower 600 G9 Desktop PC equipped with a 12th Gen Intel Core i7-12700 processor (2.10 GHz) and 16 GB RAM, running Windows 11 Pro (64-bit, Version 25H2, OS build 26200.8037).

5.4. Scope and Modeling Considerations

To clarify the scope of the proposed framework, the main modeling assumptions are summarized as follows:

• Active-power-focused attack modeling: The proposed framework focuses on LR-FDI attacks in active-power dispatch and load redistribution, with the objective function defined in terms of generation operating cost. Although AC network constraints are retained, reactive power and voltage regulation are not modeled as independent attack decision variables.
• Offline attacker-side screening in Stage-1: Stage-1 is treated as an offline attacker-side screening and scenario construction step, and therefore assumes access to the system model required for constrained attack synthesis, including network topology, generation cost characteristics, and operating limits. However, this knowledge is used only to identify feasible and economically influential attack scenarios; only the synthesized attacked-load realization is passed to the subsequent operator-side assessment stage.
• Sequential one-way stage coupling: The proposed methodology follows a sequential two-stage structure with one-way information transfer from Stage-1 to Stage-2. Stage-1 constructs feasible LR-FDI attack scenarios, while Stage-2 evaluates the corresponding post-attack system response. No iterative feedback loop from Stage-2 to Stage-1 is considered.
• Interpretation of stealthiness: In this work, the term stealth refers only to load-conserving and bounded LR-FDI redistributions over selected compromised loads under the adopted attack-accessibility assumptions. Accordingly, the proposed formulation does not explicitly enforce classical state-estimation unobservability, residual-based bad-data-detection (BDD) evasion, or residual-threshold constraints.

5.5. Benchmarks and Evaluation

The updated IEEE RTS-24 as shown in the Figure 3 Bus System provides transparent generator cost curves, limits, and topology variants for OPF/market/security studies, making it well-suited for comparing continuous LR, cardinality-limited LR, and binary-selective two-stage LR-FDI on cost uplift, overload severalties, LMP shifts, and stealth statistics [53]. The IEEE 24-bus RTS is adopted in this work as a proof-of-concept benchmark to evaluate the proposed framework under AC-consistent recourse and explicit sparse attack selection; large-scale validation is deferred to future work.

6. Results and Discussion

This section presents and analyzes the numerical results obtained from the proposed sequential two-stage framework. For clarity, the discussion is organized according to the three computational components of the methodology: (i) Stage-1 Phase-1, which evaluates the load-sensitivity-based conservative operating-cost cutoff; (ii) Stage-1 Phase-2, which analyzes the attacker-side synthesis of feasible and economically influential LR-FDI scenarios; and (iii) Stage-2, which examines the system-operator response to the selected attacked-load realization in terms of post-attack operational and economic behavior.

6.1. Stage-1 (Phase-1): Load Sensitivity Analysis

Substage-1 evaluates the sensitivity of system operating cost with respect to variations in aggregate system demand. The total load is parametrically varied within the physically admissible range defined by the sum of generator minimum and maximum capacities. For each admissible load level, the system is operated under feasibility and network constraints, and the corresponding generation cost is computed.

The results reveal a clear nonlinear trend between aggregate demand and operating cost, as shown in Figure 4. In particular, as the system load approaches the upper feasible operating region, the operating cost increases more sharply. This behavior is associated with greater utilization of higher-cost generation units and the reduced flexibility of lower-cost units near their operating limits.

Furthermore, the load-sensitivity curve indicates two broad operating regions. Over the low-to-moderate loading range, cost variation remains relatively gradual and predictable. However, near the upper loading region, incremental increases in demand produce disproportionately larger increases in operating cost, indicating heightened economic stress. These observations support the use of load sensitivity analysis as a practical means to estimate a conservative operating-cost cutoff without directly solving the Stage-1 maximization problem.

Accordingly, the maximum operating cost observed over the admissible load range is treated as a conservative reference cutoff for the subsequent attacker-side optimization.

6.2. Stage-1 (Phase-1): Attacker

The Stage-1 objective function, defined in Equation (10), is solved using both relaxed MINLP (R-MINLP) and non-relaxed MINLP (NR-MINLP) formulations.

The results presented in the

Table 3. Comparison of possible and actual load alterations for R-MINLP of Stage-1.

Possible Loads Alter	Actual Loads Alter	Load Nodes Altered	Operating Cost ($/h)	Total Line Losses (MW)
Base Case	–	–	85,418.61	44.13
2	2	(1, 6)	85,431.97	46.14
3	3	(1, 2, 3)	85,447.02	49.33
4	4	(1, 3, 6, 9)	85,451.07	50.33
5	4	(1, 3, 6, 9)	85,446.28	49.20
6	6	(1, 2, 3, 4, 6, 9)	85,458.70	52.16
7	6	(1, 2, 3, 4, 6, 9)	85,459.31	52.24
8	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
9	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
10	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
11	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
12	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
13	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
14	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
15	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
16	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96
17	8	(1, 2, 3, 4, 6, 8, 9, 10)	85,467.13	53.96

show that, under the relaxed MINLP formulation, the number of actual load alterations saturates at eight, even when the number of possible load alterations exceeds this value. Specifically, for all cases beyond eight loads, the optimization consistently selects the same eight load nodes. This saturation indicates a key limitation of the relaxed formulation. By weakening the binary decision structure, the relaxation restricts the solution space and prevents further expansion of coordinated load manipulation. As a result, the model fails to represent realistic attack scenarios, where an adversary with greater capability would be able to compromise more load nodes.

Therefore, to overcome this limitation and to better capture practical system vulnerabilities, the non-relaxed MINLP formulation has been adopted in the subsequent analysis.

Figure 5 presents the results of Stage-1 obtained using the non-relaxed MINLP formulation. It is evident that, in this case, the exact number of load alterations is preserved, reflecting the true combinatorial nature of the problem. Unlike the relaxed model, the non-relaxed formulation accurately identifies the precise number of loads that can be manipulated under given constraints.

Furthermore, the results clearly indicate which specific load nodes are selected to maximize the system operating cost.

The load-node combinations identified in the Stage-1 results correspond to those combinations that produced the highest operating-cost impact during the Stage-1 screening process. Their importance is not based on load size alone; rather, it arises from the joint effect of nodal demand level, electrical position in the network, interaction with marginal generators, and the extent to which load perturbation triggers congestion-aware redispatch under system constraints. Therefore, the identified subsets in Table 3 and

Table 4. Stage-2: Operational and economic impact under different LR-FDI attack cases with varying numbers of altered loads.

Possible	Load	Operating	Total
Load	Nodes	Cost	Line
Alter	Altered	($/h)	Losses (MW)
Base Case	–	85,418.61	44.13
2	(6, 13)	85,446.62	49.01
3	(2, 6, 13)	85,452.88	50.45
4	(2, 6, 13, 18)	85,455.32	50.99
5	(2, 4, 5, 6, 18)	85,468.41	54.04
6	(1, 2, 4, 6, 18, 20)	85,476.82	55.91
7	(1, 2, 4, 5, 6, 18, 20)	85,484.88	57.77
8	(1, 2, 3, 4, 5, 6, 15, 18)	85,500.51	61.70
9	(1, 2, 3, 4, 5, 6, 18, 19, 20)	85,507.00	62.70
10	(1, 2, 3, 4, 5, 6, 16, 18, 19, 20)	85,510.19	62.97
11	(1, 2, 3, 4, 5, 6, 9, 13, 18, 19, 20)	85,518.00	65.23
12	(1, 2, 3, 4, 5, 6, 9, 13, 15, 18, 19, 20)	85,518.88	65.43
13	(1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 18, 19, 20)	85,531.26	70.28
14	(1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 18, 19, 20)	85,550.37	77.11
15	(1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 16, 18, 19, 20)	85,565.01	77.94
16	(1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 15, 16, 18, 19, 20)	85,570.14	79.34
17	(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 18, 19, 20)	85,582.14	80.45

represent economically sensitive nodes under the given network and operating conditions. Accordingly, the identified subsets should be interpreted as scenario-dependent, optimization-derived high-impact load combinations under the studied operating condition and adopted solution environment, rather than as a universal or globally exhaustive ranking of nodal criticality.

The explicit integration of binary load-selection variables enables realistic load selectivity, providing greater flexibility in modeling adversarial behavior. This feature is consistent with practical scenarios, where simultaneous manipulation of all loads is rarely feasible. Instead, only a limited subset of strategically critical loads can be targeted. Hence, the non-relaxed MINLP formulation effectively captures the presence of such critical loads and offers a more realistic representation of coordinated load redistribution attacks.

6.3. Stage-2: System Operators Response

In Stage-2, the problem is formulated from the system operator perspective, with the objective of minimizing the overall operating cost subject to the underlying network and operating constraints. Under normal conditions, this formulation yields the economically optimal dispatch for the given load profile. In the proposed framework, however, the operator-side optimization is performed using the attacked load realization generated in Stage-1.

In Stage-1, the attacker strategically modifies load data through targeted LR-FDI attacks by breaching the communication channel and injecting manipulated profiles into the control system. Consequently, although the Stage-2 solution appears optimal computationally, it does not reflect the true physical operating condition of the system.

Table 4 illustrates the impact of these selective load alterations by reporting the variation in operating cost and transmission losses with respect to the number of altered load buses. The results indicate that targeted manipulation of economically influential load nodes can produce noticeable cost escalation and adverse operational effects, thereby demonstrating the sensitivity of optimization-based system operation to LR-FDI cyber attacks.

The operating cost values reported in

Table 5. Economic loss assessment of the system operator under LR-FDI Attacks.

Scenarios	Operating Cost ($/h)	Economic Loss (%)	Total Line Losses (%)
Base Case	85,418.61	–	–
2	85,446.62	0.79	11.06
3	85,452.88	0.96	14.32
4	85,455.32	1.03	15.54
5	85,468.41	1.40	22.46
6	85,476.82	1.64	26.69
7	85,484.88	1.86	30.91
8	85,500.51	2.30	39.81
9	85,507.00	2.48	42.08
10	85,508.19	2.57	42.69
11	85,518.00	2.79	47.81
12	85,518.88	2.82	48.27
13	85,531.26	3.17	59.26
14	85,550.37	3.70	74.73
15	85,565.01	4.11	76.61
16	85,570.14	4.26	79.79
17	85,582.14	≈5	82.30

are hourly operating costs ($/h). The corresponding daily economic impact is estimated by projecting the hourly cost difference over a 24 h horizon as

$$\Delta C_{\mathrm{day},\mathrm{s}}=\left(C_{\mathrm{attack},\mathrm{s}}-C_{\mathrm{base}}\right)\times 24$$

and the corresponding percentage economic impact is computed as

$$\%{\mathrm{Loss}}_{\mathrm{s}}=\left({\displaystyle \frac{\Delta C_{\mathrm{day},\mathrm{s}}}{C_{\mathrm{base}}}}\right)\times 100$$

Here, $C_{\mathrm{base}}$ and $C_{\mathrm{attack},\mathrm{s}}$ denote the base-case and attacked scenario hourly operating costs, respectively. Thus, the reported daily value represents a 24 h equivalent projection based on the computed hourly operating-cost deviation.

Similarly, the percentage increase in total line losses reported in Table 5 is computed as

$$\%\Delta P_{\mathrm{loss},\mathrm{s}}=\left({\displaystyle \frac{P_{\mathrm{attack},\mathrm{s}}^{\mathrm{total}\mathrm{loss}}-P_{\mathrm{base}}^{\mathrm{total}\mathrm{loss}}}{P_{\mathrm{base}}^{\mathrm{total}\mathrm{loss}}}}\right)\times 100$$

where $P_{\mathrm{loss}}^{\mathrm{tot}}$ is the total system real-power loss, and $P_{\mathrm{base}}^{\mathrm{total}\mathrm{loss}}$ and $P_{\mathrm{attack},\mathrm{s}}^{\mathrm{total}\mathrm{loss}}$ denote the total system line losses under the base-case and attacked scenario operating conditions, respectively. Thus, $\%\Delta P_{\mathrm{loss},\mathrm{s}}$ represents the percentage increase in total line losses with respect to the base case.

Table 5 presents the operator-side economic impact assessment in Stage-2 under the LR-FDI scenarios generated in Stage-1. The results show that, as the number of altered load buses increases, the operating-cost burden on the system operator also rises due to the associated redispatch and congestion-management adjustments. In the studied cases, the hourly economic impact remains relatively small (≈0.2%), but its cumulative effect over a 24 h horizon reaches approximately $4000/day, corresponding to about ≈5% relative to the base-case daily operating cost. The normalized base-case operating cost is $25.09/MWh, computed as the ratio of the base case operating cost to the corresponding total energy served during the analysis horizon. These results indicate that the estimated economic impact remains bounded under the adopted formulation while still being operationally meaningful.

This represents a conditional and selectively achievable economic loss under the assumed attack-accessibility conditions.

Figure 6 illustrates the magnitude of load manipulations under different LR-FDI scenarios in Stage-2. The plot highlights how the synthesized attack realizations distribute load perturbations across buses relative to the baseline case, where no attack is present. It can be observed that larger perturbation magnitudes are associated with buses having greater operating-cost influence, while the overall load-redistribution constraint is maintained. This behavior is consistent with the coordinated and selective nature of the proposed attack-generation framework and helps interpret the attacker-side load-selection pattern identified in Stage-1.

6.4. Impact of the Proposed Sequential LR-FDI Attack Framework on System Operational Characteristics

Figure 7, Figure 8, Figure 9 and Figure 10 summarize the post-attack variations in line active-power flows, transmission losses, nodal voltage magnitudes, and locational marginal prices (LMPs) obtained from the Stage-2 operator-side assessment. The comparison spans the base case and representative LR-FDI scenarios involving two to seventeen compromised load buses. Collectively, these figures illustrate how the synthesized attack realizations affect network loading, operating losses, voltage profiles, and market signals under the modeled operating conditions.

Figure 7 illustrates the variation in line active-power flows under the analyzed LR-FDI scenarios. Relative to the base case, the attack realizations redistribute power across several transmission corridors, with the most severe deviations observed under the higher-order compromised-load cases. The min–max envelope further indicates that certain lines are more sensitive to LR-FDI-induced demand redistribution than others. These results suggest that the synthesized attack scenarios can introduce non-uniform network stress while preserving overall load conservation.

The corresponding effect on transmission losses is illustrated in Figure 8. A clear monotonic escalation in total active power line losses is observed as the number of compromised load buses increases, with losses rising to nearly 80%, (81 MW) above the base-case level under coordinated attack scenarios. Compared with the base case, the attacked scenarios exhibit noticeably higher line-loss levels on several transmission corridors, with the most severe deviations observed in the coordinated higher-order attack cases. This increase in losses is consistent with altered power-routing patterns and congestion-aware redispatch under compromised load inputs. Under the modeled cases, the post-attack operating point remains feasible, but with reduced transmission efficiency and increased network stress.

Figure 9 illustrates the variations in nodal voltage magnitudes across analyzed scenarios. Although the voltage profiles remain within the admissible operating range, noticeable deviations from the base case are observed at several buses. This indicates that the attack affects the post-dispatch operating point without immediately causing explicit voltage-limit violations. Such behavior is consistent with an economically disruptive but operationally bounded LR-FDI scenario.

The economic consequences of the attack scenarios are further reflected in the locational marginal price variations shown in Figure 10. As the number of manipulated load buses increases, larger spatial deviations in LMPs are observed across the network, particularly at buses associated with congestion-sensitive operating conditions. These price variations indicate that LR-FDI-induced load redistribution can influence market signals through its interaction with physical power flows and redispatch decisions.

Overall, the collective evidence from Figure 7, Figure 8, Figure 9 and Figure 10 indicates that the proposed sequential LR-FDI framework can increase network stress, operating losses, and LMP dispersion as the scale of load manipulation grows. At the same time, the analyzed post-attack operating points remain feasible under the imposed load-conservation and network-operating constraints. These results demonstrate that coordinated and selective LR-FDI scenarios can produce noticeable operational and economic consequences without requiring grossly infeasible system conditions.

7. Conclusions

This paper presented a sequential two-stage framework for assessing LR-FDI attacks in an active-power-oriented economic dispatch model-based power system operation. The proposed methodology separates attacker-side scenario synthesis from operator-side post-attack assessment, enabling the analysis of coordinated and selective load-redistribution attacks under a one-way stage-wise workflow. In Stage-1, feasible and economically influential compromised-load patterns are synthesized using a non-relaxed MINLP formulation together with a load-sensitivity-based conservative operating-cost cutoff used to regularize the attacker-side search. In Stage-2, the selected attacked-load realization is evaluated through the system-operator model to quantify its economic and operational consequences.

Results on the IEEE 24-bus reliability test system show that the non-relaxed Stage-1 formulation preserves the intended load-selective structure more effectively than the relaxed variant and provides a more suitable basis for screening-regularized vulnerability assessment. The Stage-2 analysis further indicates that coordinated LR-FDI scenarios can increase operating cost, transmission losses, network stress, and LMP dispersion while maintaining feasible post-attack operating points under the imposed model constraints. For the analyzed realizable cases, the hourly operating-cost deviation is about 0.2%, which accumulates to nearly 5% on a 24 h basis, while transmission losses increase by up to approximately 80% relative to the base-case level under the more severe coordinated scenarios.

Overall, the proposed framework provides a technically consistent and practically interpretable basis for analyzing LR-FDI-induced economic degradation and operational stress in an active-power-oriented economic dispatch model-based system operation.

8. Limitations and Future Scope

The present study is intended as a proof-of-concept assessment on the IEEE 24-bus RTS under the adopted LR-FDI formulation and active-power-oriented economic dispatch with nonlinear network-constraints model. Although the proposed decomposition improves interpretability and tractability relative to a monolithic formulation, the Stage-1 binary load-selection variables may still lead to combinatorial growth as system size and attack-space dimensionality increase. Moreover, validation on larger test systems and under broader attack settings remains necessary. Future work will therefore focus on scalability to larger IEEE test systems, computational acceleration of the Stage-1 search, and extensions toward stage-aware detection, mitigation, and more generalized LR-FDI scenarios.