Power Flow Methods for Efficient Analysis of Modern Distribution Networks—Review

Abstract

Power Flow (PF) analysis is a fundamental tool in distribution systems since it determines the steady-state operating point for specified input conditions. Modern distribution networks face high distributed energy resource (DER) penetration and variable operating conditions, requiring repeated PF evaluations in time-series and scenario-based studies. Their unbalanced operation and high $R/X$ ratios can challenge conventional PF solvers, thereby requiring accurate, robust, and scalable methods. Prior studies have examined nonlinear distribution PF solvers and uncertainty-based formulations, but the review literature remains limited to specific categories and lacks a unified discussion of linearized models, numerical robustness, and acceleration techniques. Therefore, this paper presents a state-of-the-art review of PF methods for modern distribution networks, covering 205 studies published between 2000 and 2026. It summarizes conventional nonlinear PF formulations, reviews linearized models with their assumptions and applicability, and surveys numerical robustness strategies for improved convergence. The methods are compared according to their applicability to radial, weakly meshed, and unbalanced networks, while practical selection criteria are provided based on accuracy, convergence reliability, and computational requirements. Probabilistic, interval, and fuzzy approaches are also reviewed under renewable and load uncertainty. Finally, acceleration strategies for repeated PF evaluation are discussed, emphasizing sparse numerical implementations, topology-based schemes, and physics-informed surrogate models.

1. Introduction

Power flow analysis is a foundational tool in power system planning and operation as it determines the steady-state operating conditions of a network under specified injections and control actions. It supports evaluation of voltage profiles, line and transformer loading, network losses, and operating limit compliance and remains a core computational step in distribution system studies where radial or meshed topologies, high $R/X$ ratios, and unbalanced operation are common [1,2]. As distribution networks transition toward active operation with increasing penetration of distributed energy resources, storage, and flexible demand, power flow calculations must remain accurate and reliable over a wider range of operating conditions than those typically encountered in passive feeders [3]. Consequently, distribution-oriented power flow methods play a decisive role in modern distribution system planning studies and operational analysis.

Numerous past works have focused on deterministic power flow formulations that treat loads and generations as fixed inputs and compute the network state for a single operating condition [4]. Classical iterative solvers such as Newton–Raphson (NR) and Gauss–Seidel (GS) are widely used in transmission studies [5,6], but their convergence can be less reliable in distribution feeders, where high $R/X$ ratios, weaker voltage support, and stronger coupling between active power/reactive power and voltage magnitudes can lead to ill-conditioned behavior, particularly under heavy loading or when the initial point is not close to a feasible solution [7,8]. In addition, unbalanced three-phase modeling and inverter control mode switching can further challenge convergence under stressed operating conditions, and therefore backward–forward sweep techniques remain widely used for radial and weakly meshed distribution feeders due to their efficiency and reliable performance [9]. Practical distribution studies, therefore, require not only computational efficiency but also robust convergence behavior, particularly when the system operates near limits or when the model includes detailed three-phase representations and control functions.

With increasing penetration of variable renewable generation and evolving demand behavior, uncertainty in net injections has become a prominent aspect of modern distribution studies [10]. Probabilistic and uncertainty-based load flow formulations capture variability in photovoltaic generation, distributed resources, and emerging demand patterns, providing statistical descriptions of voltages, flows, and constraint violations rather than a single deterministic outcome [11,12]. Since these approaches typically require repeated evaluations over many samples, scenarios, or time steps, computational burden can become a primary limitation in large-scale studies, particularly when three-phase distribution models are employed [10]. Consequently, recent work [13,14] has emphasized scalable methods that maintain acceptable numerical performance and accuracy for large scenario sets, motivating acceleration strategies for time-series and scenario-driven applications.

Steady-state operating points in distribution networks are obtained through power flow computations for specific network conditions, whereas optimal power flow is based on the same AC equations but introduces an objective and operational constraints and optimizes controllable variables to obtain an optimal operating point [15,16]. This added optimization layer increases modeling and computational complexity and typically involves repeated power flow computations within iterative solution procedures. Although recent optimal power flow (OPF) review papers [15,16,17,18,19] devote extensive attention to optimization strategies, relaxations, and learning-aided formulations, their coverage of distribution-oriented power flow solution methods and practical convergence behavior is typically limited. Accordingly, this review focuses on power flow formulations and solution methods for modern distribution networks, including conventional nonlinear and linearized methods, numerical robustness, uncertainty-based formulations, and acceleration strategies. Higher-level applications such as optimal power flow [15,18,20], market-oriented operation [21], network reconfiguration [22], and real-time control [23] are outside the primary scope of this review and are discussed only where necessary to illustrate the role and application of PF models.

1.1. Contribution

Several papers have reviewed conventional distribution power flow methods and modeling assumptions for radial and meshed feeders [1,2,24]. Other works [11,25,26] have studied probabilistic and uncertainty-based load flow formulations under renewable and demand variability. However, existing reviews have generally focused on individual method categories, while dedicated discussion of linearized power flow families within the broader context of modern distribution-network analysis remains limited. As summarized in Table 1, this review addresses this gap by examining conventional and linearized power flow methods together with numerical robustness, uncertainty representation, and acceleration strategies. In addition to reviewing these categories, the paper relates their assumptions and numerical characteristics to practical method-selection criteria, including network topology, convergence behavior, approximation accuracy, and computational requirements. This broader scope facilitates the assessment of their applicability to different distribution-network studies. Motivated by this need, the objective of this paper is to provide a comprehensive review of power flow methods for modern distribution networks with high penetration of distributed energy resources.

The major contributions of this review are summarized as follows.

• A unified classification of distribution-oriented power flow methods is presented, covering conventional nonlinear methods and linearized approximations and comparing their main assumptions, applicability, advantages, and limitations.
• Robust power flow methods are examined with emphasis on improved numerical performance and consistent convergence under stressed and ill-conditioned operating conditions.
• Uncertainty-based power flow formulations are reviewed, including probabilistic power flow methods with numerical, analytical, and approximate techniques, as well as interval and fuzzy power flow models for bounded and imprecise uncertainty representations.
• Acceleration strategies for repeated power flow evaluation are surveyed, including sparse numerical techniques, topology-based updates, and surrogate-based approaches, with emphasis on physical consistency and scalability for large time series and scenario sets.
• The review also provides practical method-selection guidance by relating network characteristics and operating conditions to the accuracy, convergence, and computational requirements of different PF formulations, and it identifies research gaps related to model validity, uncertainty handling, and computational scalability in modern distribution networks.

1.2. Review Methodology and PRISMA-Based Study Selection

The review methodology followed the PRISMA guidelines for identifying, screening, and selecting relevant publications [27]. The adopted methodology comprises a systematic search strategy, predefined inclusion and exclusion criteria, and a structured literature-selection process. The review provides a qualitative technical synthesis of distribution-oriented power flow methods. Table 1 compares the scope of the present work with that of existing review papers.

1.2.1. Systematic Search Strategy

A systematic literature search was conducted in IEEE Xplore, Scopus, Web of Science, and ScienceDirect. The principal search period covered publications from 2000 to 2026, while earlier foundational studies were retained to describe established formulations and the historical development of power flow methods. The search used combinations of the terms “power flow” and “load flow” with “distribution network” and “distribution system”. Additional keyword combinations, including “linearized power flow”, “probabilistic power flow”, “convergence improvement”, and “computational acceleration”, were used to identify studies covering the main methodological areas of the review. Database filters were applied to restrict the results to the selected publication period and relevant power-system subject areas, with greater emphasis placed on peer-reviewed journal articles, while conference papers were retained when they presented an important formulation or computational development. Figure 1 shows the yearly distribution of publications over the principal review period.

1.2.2. Inclusion and Exclusion Criteria

The review included peer-reviewed studies that proposed, evaluated, or reviewed power flow formulations and solution methods relevant to distribution-network analysis. The eligible literature covered conventional and linearized power flow methods, together with approaches addressing numerical robustness, uncertainty, and computational efficiency. Studies were included only when they provided sufficient detail to understand the proposed formulation, its main assumptions, and the conditions used for evaluation.

Publications limited to transmission-system applications were excluded unless they provided a method with clear relevance to distribution networks. Studies centered on OPF, electricity markets, protection, or control were not retained unless the power flow formulation represented a methodological contribution. Duplicate records and non-peer-reviewed material were removed. Publications with insufficient technical information were also excluded.

1.2.3. Literature Selection Process

The records retrieved from the selected databases were combined before screening, and duplicate entries were removed. Title and abstract screening was then used to identify publications consistent with the review scope. The full texts of the remaining studies were assessed according to the predefined eligibility criteria. Publications covering several methodological areas were assigned to the category corresponding to their principal contribution. Their secondary contributions were considered in the relevant comparative discussion. The information extracted from each selected study covered the underlying power flow formulation and its principal assumptions. The reported application domain and numerical behavior were also examined. Computational requirements and stated limitations were used to support the method comparison and the identification of research gaps.

The selected literature was organized according to the main methodological themes addressed in the review, ranging from conventional and linearized formulations to methods focused on numerical robustness, uncertainty representation, and computational acceleration. A total of 205 publications are included in the qualitative synthesis. Moreover, Figure 2 summarizes the identification, screening, eligibility assessment, and final inclusion stages through a PRISMA-based flow diagram. Figure 3 presents the distribution of the selected publications across conference proceedings and the main journal groups.

The rest of the paper is organized as follows. Section 2 discusses the significance of power flow analysis in modern distribution networks and summarizes its main application areas. Section 3 reviews conventional and linearized power flow methods, including their assumptions and validity ranges. Section 4 discusses modern power flow evolution, with emphasis on numerical robustness and convergence enhancement under stressed and ill-conditioned operating conditions. Section 5 reviews uncertainty-based power flow formulations, including probabilistic, interval, fuzzy, and hybrid approaches. Section 6 presents power flow acceleration strategies based on sparse numerical techniques, topology-based methods, parallel computing, and data-driven and physics-informed models. Finally, Section 7 provides the comparative discussion, research gaps, and future directions, while Section 8 provides the conclusions.

Table 1. Comparison of existing review papers on distribution-oriented power flow methods.

Ref	Year	Conventional PF	Linearized PF	Numerical Robustness	Uncertainty Based-PF	Acceleration Strategies	Pros	Cons
[1]	2005	✓					Provides a broad baseline summary of classical distribution load flow formulations and practical modeling considerations.	The discussion is largely descriptive, with limited comparison of accuracy, convergence, and computational effort across methods.
[2]	2005	✓					Provides an overview of distribution load flow techniques and common assumptions for radial networks.	Limited discussion of uncertainty-aware PF and no unified taxonomy across deterministic and uncertainty-based PF.
[3]	2010	✓					Summarizes the evolution of distribution load flow methods and motivates specialized solvers for feeders.	Less emphasis on uncertainty-aware PF and recent acceleration trends.
[7]	2011	✓			✓		Reviews distribution PF with distributed resources and highlights three-phase/unbalanced modeling aspects.	Limited emphasis on formal uncertainty-quantification frameworks and acceleration strategies.
[8]	2015	✓					General review of classical load flow solution families.	Does not focus on modern distribution challenges (DER controls, probabilistic PF, acceleration).
[9]	2016	✓					Reviews multiple distribution load flow methods and compares applicability for radial feeders.	Comparisons are mostly qualitative with limited emphasis on stressed-condition robustness.
[11]	2020				✓		Reviews probabilistic load flow approaches for distribution networks including PV and EV charging uncertainties.	Less emphasis on deterministic PF convergence behavior under stressed and ill-conditioned regimes.
[26]	2023				✓	✓	Distribution-focused probabilistic load flow review with a clear classification of uncertainty modeling.	Does not cover linearized PF and provides limited discussion of acceleration approaches.
[28]	2024				✓		Provides an overview of probabilistic load flow methods for distribution-network planning under photovoltaic uncertainty.	Does not examine deterministic PF solution methods or computational acceleration strategies.
[29]	2019	✓					Clearly explains Newton-based load flow fundamentals and Jacobian-based iterations.	Mainly focuses on Newton-type methods and does not address uncertainty or acceleration.
[30]	2012	✓		✓			Highlights behavior under well-conditioned and ill-conditioned cases and discusses convergence concerns.	Does not cover linearized PF, uncertainty-based PF, or acceleration strategies in a dedicated manner.
[24]	2022	✓			✓		Discusses constrained-network PF under operational limits and includes probabilistic PF aspects.	Does not discuss linearized PF, numerical robustness techniques, or acceleration strategies.
[31]	2017				✓		Structured review of probabilistic load flow with emphasis on PV/renewable uncertainty.	Emphasizes probabilistic frameworks, with limited discussion of deterministic solver robustness and distribution-specific PF methods.
[20]	2023		✓			✓	Reviews OPF formulation, power flow models, and optimization methods, including machine learning.	Briefly discusses linearized PF methods within OPF formulation, without detailed classification and assumptions.
[25]	2008				✓		Surveys probabilistic techniques and uncertainty handling in PF studies.	Limited coverage of robustness mechanisms and acceleration methods for large-scale repeated evaluations.
[32]	2014				✓		Reviews probabilistic and possibilistic uncertainty assessment for PF.	Does not discuss conventional and linearized PF families in depth.
[33]	2025		✓				Provides a detailed review of linear power flow approximations and their applications in power-system analysis.	Limited emphasis on numerical robustness and acceleration strategies for repeated evaluations.
[34]	2025	✓				✓	Reviews load flow methods in electric distribution networks and includes modern computational approaches.	Does not cover linearized PF and provides limited discussion of data-driven PF methods.
[17]	2025	✓					Compact review for distribution systems with dispersed generation.	Limited depth on uncertainty-based PF and robustness-oriented solver design.
This paper	2026	✓	✓	✓	✓	✓	First paper to provide an integrated distribution-focused review covering conventional PF, linearized PF, robustness under stressed operation, and acceleration for time-series/scenario studies, with emphasis on method-selection guidance.	Focuses on power flow analysis and does not provide a detailed review of application-level optimization and market layers that build on power flow models.

Note: ✓ indicates that the corresponding topic is covered or discussed in the reviewed paper.

Table 1. Comparison of existing review papers on distribution-oriented power flow methods.

Ref	Year	Conventional PF	Linearized PF	Numerical Robustness	Uncertainty Based-PF	Acceleration Strategies	Pros	Cons
[1]	2005	✓					Provides a broad baseline summary of classical distribution load flow formulations and practical modeling considerations.	The discussion is largely descriptive, with limited comparison of accuracy, convergence, and computational effort across methods.
[2]	2005	✓					Provides an overview of distribution load flow techniques and common assumptions for radial networks.	Limited discussion of uncertainty-aware PF and no unified taxonomy across deterministic and uncertainty-based PF.
[3]	2010	✓					Summarizes the evolution of distribution load flow methods and motivates specialized solvers for feeders.	Less emphasis on uncertainty-aware PF and recent acceleration trends.
[7]	2011	✓			✓		Reviews distribution PF with distributed resources and highlights three-phase/unbalanced modeling aspects.	Limited emphasis on formal uncertainty-quantification frameworks and acceleration strategies.
[8]	2015	✓					General review of classical load flow solution families.	Does not focus on modern distribution challenges (DER controls, probabilistic PF, acceleration).
[9]	2016	✓					Reviews multiple distribution load flow methods and compares applicability for radial feeders.	Comparisons are mostly qualitative with limited emphasis on stressed-condition robustness.
[11]	2020				✓		Reviews probabilistic load flow approaches for distribution networks including PV and EV charging uncertainties.	Less emphasis on deterministic PF convergence behavior under stressed and ill-conditioned regimes.
[26]	2023				✓	✓	Distribution-focused probabilistic load flow review with a clear classification of uncertainty modeling.	Does not cover linearized PF and provides limited discussion of acceleration approaches.
[28]	2024				✓		Provides an overview of probabilistic load flow methods for distribution-network planning under photovoltaic uncertainty.	Does not examine deterministic PF solution methods or computational acceleration strategies.
[29]	2019	✓					Clearly explains Newton-based load flow fundamentals and Jacobian-based iterations.	Mainly focuses on Newton-type methods and does not address uncertainty or acceleration.
[30]	2012	✓		✓			Highlights behavior under well-conditioned and ill-conditioned cases and discusses convergence concerns.	Does not cover linearized PF, uncertainty-based PF, or acceleration strategies in a dedicated manner.
[24]	2022	✓			✓		Discusses constrained-network PF under operational limits and includes probabilistic PF aspects.	Does not discuss linearized PF, numerical robustness techniques, or acceleration strategies.
[31]	2017				✓		Structured review of probabilistic load flow with emphasis on PV/renewable uncertainty.	Emphasizes probabilistic frameworks, with limited discussion of deterministic solver robustness and distribution-specific PF methods.
[20]	2023		✓			✓	Reviews OPF formulation, power flow models, and optimization methods, including machine learning.	Briefly discusses linearized PF methods within OPF formulation, without detailed classification and assumptions.
[25]	2008				✓		Surveys probabilistic techniques and uncertainty handling in PF studies.	Limited coverage of robustness mechanisms and acceleration methods for large-scale repeated evaluations.
[32]	2014				✓		Reviews probabilistic and possibilistic uncertainty assessment for PF.	Does not discuss conventional and linearized PF families in depth.
[33]	2025		✓				Provides a detailed review of linear power flow approximations and their applications in power-system analysis.	Limited emphasis on numerical robustness and acceleration strategies for repeated evaluations.
[34]	2025	✓				✓	Reviews load flow methods in electric distribution networks and includes modern computational approaches.	Does not cover linearized PF and provides limited discussion of data-driven PF methods.
[17]	2025	✓					Compact review for distribution systems with dispersed generation.	Limited depth on uncertainty-based PF and robustness-oriented solver design.
This paper	2026	✓	✓	✓	✓	✓	First paper to provide an integrated distribution-focused review covering conventional PF, linearized PF, robustness under stressed operation, and acceleration for time-series/scenario studies, with emphasis on method-selection guidance.	Focuses on power flow analysis and does not provide a detailed review of application-level optimization and market layers that build on power flow models.

Note: ✓ indicates that the corresponding topic is covered or discussed in the reviewed paper.

2. Significance and Applications of Power Flow Analysis

Power flow evaluation remains central to distribution engineering, since accurate voltage and branch flow calculations are required in most planning and operational studies [1]. Power flow results are routinely used to assess voltage profiles and voltage regulation performance, quantify network losses, and examine line and transformer loading [29,35]. However, in distribution networks, these tasks are often carried out under feeder characteristics such as high $R/X$ ratios and unbalanced operation, and the quality of the power flow solution directly affects the interpretation of system performance and constraint compliance. Moreover, Figure 4 summarizes the key challenges in modern distribution networks, which motivate the role of power flow analysis, including high DER penetration, variability and forecast errors in load and renewable generation, numerical challenges under stressed conditions, and increasing reliance on time-series studies.

With increasing penetration of distributed energy resources and more dynamic demand behavior, distribution networks experience greater temporal variability in net injections, leading to frequent operating point changes that require repeated power flow evaluation in time-series studies and operational assessments [25,26]. This trend has also strengthened interest in uncertainty-based load flow formulations, where voltages and branch flows are characterized through probabilistic or stochastic descriptions rather than a single operating point, enabling variability and risk to be assessed under renewable and load uncertainty. In this context, computational efficiency and numerical robustness become increasingly important, since large numbers of evaluations may be needed across scenarios, samples, or time periods, and the requirements highlighted in Figure 4 naturally translate into the need for advanced power flow methods that offer robust convergence, fast computation, scalable implementation, and effective uncertainty analysis.

2.1. Key Applications of Power Flow Analysis

The key application areas of power flow analysis are summarized below:

• Operating point evaluation and network feasibility validation: Power flow is used to compute bus voltages and branch flows for a given loading and generation condition, enabling feasibility verification against voltage bounds and equipment thermal limits in distribution networks [35].
• Time-series operating studies: Power flow is repeatedly solved across time steps to evaluate feeder operation under varying load and distributed energy resource conditions, including changing net injections and possible flow direction changes over the day [1,35,36].
• Voltage regulation and inverter-based DER impact studies: Power flow provides the steady-state basis to evaluate voltage regulation needs and the effect of inverter-based reactive power limits and control behavior on voltage profiles and feeder operating margins [36,37].
• Voltage profile and loss assessment: Power flow outputs are used to assess voltage deviation patterns and to estimate network losses, supporting efficiency assessment and monitoring of feeder performance under different operating conditions [37,38].
• Uncertainty and scenario-based evaluation: Probabilistic and scenario-based power flow studies quantify the impact of uncertain injections on voltages and flows through repeated evaluations across samples or scenarios, including Monte Carlo-style assessments used in renewable integration studies [10,25,26].

2.2. Power Flow Methods in Grid Applications

Power flow models are directly embedded in optimization and control applications. Nonlinear PF equations are used as network constraints in OPF to represent nodal power balance, bus voltages, and branch flows, while the objective function determines the optimal operating point [16]. Relaxed or simplified PF formulations are also used to make distribution-network optimization more tractable [19]. Linear branch-flow equations retain the main network constraints while reducing the computational complexity of radial-network optimization [39]. For voltage and reactive-power control, sensitivity-based PF models estimate the effect of inverter outputs, capacitor switching, and transformer taps on bus voltages. These voltage sensitivities support decentralized voltage control in [40], provide analytical relationships for unbalanced radial networks in [41], and enable measurement-based estimation from phasor measurement unit (PMU) data in [42].

Real-time operation and digital twins require PF models that can be updated as network conditions change. The measurement-based approach in [43] updates the PF Jacobian using synchronized measurements, while [44] applies online feedback to improve a linearized PF model for time-varying unbalanced networks. Large batches of PF calculations are accelerated on central processing units (CPUs) and graphics processing units (GPUs) in [45], supporting time-series, contingency, and probabilistic studies. In digital-twin applications, PF models are combined with network data and field measurements to reproduce the operating state of the physical system. OpenDSS-based PF is integrated with state estimation and flexibility assessment for real-time analysis in [46], while unbalanced nonlinear and linear PF models support network-state assessment and voltage-control studies in [47].

In engineering practice, power flow studies are implemented through both open-source and commercial software platforms. OpenDSS and GridLAB-D are widely used for detailed unbalanced distribution-network and time-series analysis, while MATPOWER and pandapower provide script-based environments for AC and DC power flow, optimization, automation, and research applications [48,49,50,51]. Commercial platforms such as DIgSILENT PowerFactory, PSS^®E, ETAP, and CYME provide detailed network and equipment models for transmission and distribution planning, operational analysis, protection studies, and DER integration [51,52]. The selection of a platform depends on the required network representation, supported analysis functions, automation capability, and modeling requirements.

3. Classification of Power Flow Methods

Power flow methods can be broadly classified into deterministic and uncertainty-based formulations. Deterministic methods solve the network state under known operating conditions and include both conventional nonlinear methods and linearized approximation methods. Conventional nonlinear methods include backward/forward sweep, Newton–Raphson, Gauss–Seidel, and fast decoupled approaches, whereas linearized approximations include DC power flow, LinDistFlow, sensitivity/Jacobian-based linearization, and fixed-point or secant-type approximations. Uncertainty-based power flow methods extend the analysis by considering variations in load, renewable generation, and other uncertain parameters, and they can be further grouped into probabilistic, interval, and fuzzy-based formulations. Figure 5 summarizes this overall classification, shows the main subcategories discussed in this section, and indicates the number of articles in each main category.

Table 2. Summary of conventional power flow solution methods and their characteristics in distribution studies.

Method	Description	Typical Applicability	Convergence Behavior	Computational Profile	Advantage	Disadvantage
Backward/Forward Sweep	A branch-oriented iterative method that performs an upstream current aggregation followed by a downstream voltage update along the feeder.	Primarily used in distribution networks, especially radial feeders; weakly meshed cases require additional loop handling.	Generally reliable on radial feeders, but convergence can degrade under stressed loading and PV-bus enforcement without dedicated handling.	Low memory demand and low per-iteration cost dominated by simple sweep updates.	Avoids Jacobian construction and factorization, which makes it computationally attractive for radial distribution studies.	Meshed topology and voltage-controlled buses typically require extra modeling steps to preserve convergence and feasibility.
Newton–Raphson	A Jacobian-based method that solves the nonlinear PF equations by iteratively linearizing power mismatches and updating voltage magnitudes and angles.	Widely used for transmission PF and also applied to distribution networks with careful modeling and suitable initialization.	Exhibits local quadratic convergence when the initial point is close to a feasible solution; performance deteriorates under ill-conditioning or poor initialization.	Higher per-iteration cost due to Jacobian assembly and sparse factorization.	Provides strong local convergence and supports detailed device models within a unified nonlinear formulation.	In distribution feeders with high $R/X$ ratios and stressed operation, convergence reliability can weaken and computational effort per iteration remains significant.
Gauss–Seidel	A sequential fixed-point iteration that updates bus voltages one at a time using the latest available values while keeping the slack-bus voltage as reference.	Used in both transmission and distribution mainly as a baseline solver for deterministic PF.	Typically slow with linear convergence and sensitive to initialization; reliability decreases in large-scale or stressed feeders.	Very low per-iteration cost, often offset by a large number of iterations.	Simple implementation without Jacobian construction, which makes it convenient for baseline PF calculations.	Slow convergence and sensitivity to conditioning reduce practicality for large or heavily loaded distribution networks.
Fast Decoupled Load Flow	A Newton-derived method that decouples active-power and reactive-power updates using constant-matrix approximations solved repeatedly.	Common in transmission PF; applied in distribution only when decoupling assumptions remain adequate or when adapted formulations are used.	Converges quickly when decoupling is valid, but accuracy and robustness can degrade in high $R/X$ feeders with stronger coupling.	Moderate per-iteration cost with matrices typically formed once and reused across iterations.	Reduces computational effort relative to full Newton–Raphson by reusing constant matrices, often achieving faster convergence than Gauss–Seidel.	In distribution feeders, weakened decoupling assumptions can reduce robustness and may require adapted formulations to remain reliable.

provides a comparative summary of the main conventional power flow methods in terms of their applicability, convergence behavior, computational profile, advantages, and limitations in distribution-network studies.

3.1. Conventional Power Flow Methods

Conventional power flow (PF) analysis is usually formulated as a deterministic steady-state problem, where network parameters and operating conditions are treated as known and fixed, and the objective is to solve for bus voltages and branch flows that satisfy Kirchhoff’s laws and network equations. In distribution networks, this becomes more challenging because distribution feeders differ from transmission grids in ways that strongly affect numerical behavior, including predominantly radial or weakly meshed topology and relatively high $R/X$ ratios, which can make transmission-oriented solvers less reliable without adaptation.

Consequently, conventional PF methods for distribution networks are often organized into (i) backward/forward sweep method, (ii) Newton–Raphson method, (iii) Gauss–Seidel method, and (iv) fast decoupled method [34]. In this section, we begin with backward/forward sweep methods because they directly exploit distribution-network topology and are widely regarded as the decisive approach for radial distribution PF.

3.1.1. Backward/Forward Sweep Method

Backward and forward sweep power flow is a branch-oriented iterative method that exploits the feeder structure by alternating an upstream current-summation step and a downstream voltage update step. The formulation is aligned with radial feeders and was historically developed around PQ buses with a root slack bus serving as the voltage reference. One of the early and widely cited formulations was proposed in 1988 in [53] for radial and weakly meshed distribution and transmission networks using a compensation-based approach. It is widely adopted for distribution studies because it avoids forming and factorizing a full Jacobian, while its computations are dominated by simple updates of branch quantities along the feeder [9]. This structure also supports extensions to unbalanced networks and distributed resources.

A common current-based implementation converts the specified complex power at each load bus into an equivalent current injection using the previous voltage iterate [9]. For a PQ bus i, this current injection is written as

$$I_i^{\left(k\right)}={\left({\displaystyle \frac{S_i}{V_i^{\left(k\right)}}}\right)}^{*}$$

(1)

which links the nonlinear constant-power model to the network current balance used in the sweep. The backward sweep aggregates downstream currents toward the source using Kirchhoff’s current law, while the forward sweep updates bus voltages using branch voltage drops [34]. A representative voltage update for branch $i\to j$ is

$$V_j^{(k+1)}=V_i^{(k+1)}-Z_{ij}{I}_{ij}^{\left(k\right)}$$

(2)

where $Z_{ij}$ is the branch impedance and $I_{ij}^{\left(k\right)}$ is the updated branch current. The two-sweep loop is repeated until the selected voltage or power mismatch tolerance is satisfied. The method is most straightforward on radial networks, whereas weakly meshed feeders typically require additional handling so that loop constraints are satisfied while preserving sweep efficiency [7]. Handling PV buses is another recurring challenge because voltage control requires adjusting reactive power or equivalent injections to meet voltage setpoints [8]. When PV penetration increases, PV enforcement inside sweep iterations can face convergence degradation, particularly under high $R/X$ ratios [54]. For radial and weakly meshed feeders, the approaches in [55,56] introduce PV-specific corrections or modified sweep updates to satisfy voltage constraints while retaining the basic sweep procedure. Although both approaches improve PV-bus handling, their convergence can still depend on feeder conditions and the treatment of reactive power limits. Beyond PV modeling, the methods in [57,58] improve numerical reliability through adaptive compensation, convergence checking, and modified sweep updates. These refinements preserve the branch-oriented structure and improve numerical reliability under dispersed generation and stressed operating conditions. In islanded microgrids, the absence of a stiff slack reference requires additional modifications, and sweep-based formulations have been proposed to obtain feasible steady-state solutions [59]. A direct-sweep algorithm further incorporates droop regulation so that microgrid control characteristics are represented during load flow computation [60]. Backward/forward sweep methods remain particularly suitable for radial feeders because of their simple structure and low computational burden, while weak meshing, PV-bus enforcement, and islanded operation require additional corrections that increase modeling complexity.

3.1.2. Newton-Raphson Method

Newton–Raphson (NR) power flow is a Jacobian-based iterative method for solving nonlinear PF equations by updating bus voltage angles and magnitudes from linearized active- and reactive-power mismatches at the latest iterate [7]. An early and widely cited application of Newton’s method to large-scale power flow analysis was presented in 1967 [61]. Its practical performance requires a feasible operating point, a sufficiently accurate initial voltage profile, and a nonsingular, reasonably well-conditioned Jacobian, since near-singularity can prevent reliable mismatch reduction [34]. Several techniques can improve NR performance under ill-conditioning. Equation and variable scaling balance mismatch and state-variable magnitudes, thereby reducing numerical disparities in the Jacobian [29,62]. Regularization introduces a small stabilizing term into the Newton system to improve conditioning and limit unstable corrections near singular operating points [63]. Damping or optimal multipliers reduce the Newton step when the full correction increases the mismatch or leaves the convergence region [64]. For severely stressed or ill-posed cases, homotopy methods gradually transform an easier PF problem into the original one and provide a suitable initial estimate for the final NR solution [65]. In distribution networks, NR is often used as a benchmark because of its strong local convergence, although its performance may degrade under high $R/X$ ratios, radial or weakly meshed structures, heavy loading, and ill-conditioning, motivating adapted formulations for distribution studies [7,34]. The standard mismatch equations are

$$\Delta P_i=P_i^{\mathrm{spec}}-P_i(V,\theta )$$

(3)

$$\Delta Q_i=Q_i^{\mathrm{spec}}-Q_i(V,\theta )$$

(4)

and the Newton correction is obtained from $J(V,\theta ){[\Delta \theta \Delta V]}^T={[\Delta P\Delta Q]}^T$, after which $(\theta ,V)$ is updated until the mismatches satisfy the prescribed tolerance. For radial distribution networks, feeder-oriented Jacobian computations reduce computational effort while retaining the Newton update structure [66]. For unbalanced three-phase networks, the decomposed quasi-Newton–Raphson formulation in [67], the complex-form NR formulation in [68], and the enlarged current-injection formulation in [69] improve computational efficiency, phase representation, and numerical conditioning, respectively. These formulations extend NR to more detailed distribution-network models, although convergence still depends on Jacobian conditioning and initialization. NR has also been extended to include detailed device equations. For instance, a static synchronous compensator (STATCOM) model was embedded directly within NR-PF so that the compensator is solved consistently within the iterative process [70].

Newton-type methods are also used in DC grids, where constant-power terminals introduce strong nonlinearities. For DC microgrids, convergence conditions and feasible operating ranges have been studied to identify when Newton iterations remain reliable under different operating modes and loading levels [4,5,6]. Hybrid schemes combining Z-bus Gauss and Newton updates improve convergence in ill-conditioned cases [71], while compressed NR formulations reduce computational effort in DC traction networks [72]. These developments improve convergence reliability or computational efficiency, although performance remains dependent on network conditioning and formulation. Comparative studies have also examined implementation-level differences among NR solvers, supporting the selection of robust numerical settings for large-scale or ill-conditioned PF problems without changing the underlying model [62].

3.1.3. Gauss–Seidel Method

Gauss–Seidel power flow is a sequential fixed-point method that updates bus voltages one at a time using the most recently available values, while keeping the slack-bus voltage fixed as the reference. In deterministic PF studies, it is mainly used because it is simple to implement and does not require forming or factorizing a Jacobian, which makes it convenient for small networks and preliminary studies [7,34]. In practice, Gauss–Seidel converges reliably only when a feasible operating point exists, and the initial voltage guess is reasonable because the iteration is sensitive to initialization and numerical conditioning [7]. A key limitation is slow convergence for larger systems or stressed operating conditions, and the iteration may become unreliable in ill-conditioned cases, which is why Gauss–Seidel is often treated as a baseline reference rather than a default method for large distribution studies [30]. Mathematically, the method is based on the nodal current relation and the sequential voltage update:

$$I_i={\displaystyle \sum _{j=1}^N}{Y}_{ij}{V}_j$$

(5)

together with the complex power–current identity

$$I_i={\displaystyle \frac{P_i-j{Q}_i}{V_i^{*}}}$$

(6)

which leads to the sequential voltage update

$$V_i^{(k+1)}={\displaystyle \frac{1}{Y_{ii}}}\left({\displaystyle \frac{P_i-j{Q}_i}{{\left(V_i^{\left(k\right)}\right)}^{*}}}-{\displaystyle \sum _{j=1,j\ne i}^N}{Y}_{ij}{V}_j^{(·)}\right)$$

(7)

Several studies have considered extensions and comparative evaluations of the Gauss–Seidel framework for distribution-network applications [73,74]. A modified formulation in [75] adapts the sequential voltage update for unbalanced three-phase distribution networks, extending the method beyond its basic single-phase form. In comparison with Newton–Raphson, the practical study in [73] shows that Gauss–Seidel generally requires more iterations, particularly as network size increases. These results confirm that the method remains attractive for its simple implementation and low per-iteration cost, but its slow convergence limits its suitability for large or stressed distribution networks.

3.1.4. Fast Decoupled Load Flow Method

Fast Decoupled Load Flow is a deterministic power flow method derived from the Newton–Raphson formulation by applying decoupling and constant-matrix approximations [76]. This allows the active- and reactive-power updates to be solved separately with lower computational effort than full Newton updates [34].

In contrast to Gauss–Seidel, which updates voltages sequentially bus by bus, the fast decoupled method solves two decoupled linear systems per iteration using coefficient matrices that are typically formed once and reused. This generally provides faster convergence than Gauss–Seidel when the decoupling assumptions remain valid [30].

A compact representation is written in the decoupled mismatch form

$$B^{\prime }\Delta \theta =\Delta P$$

(8)

$$B^{″}\Delta V=\Delta Q$$

(9)

where voltage angles and magnitudes are updated until the mismatches fall below tolerance [34]. The formulation assumes that active power is mainly coupled with voltage angles and reactive power is mainly coupled with voltage magnitudes [30]. In distribution feeders, these assumptions can become less reliable because line resistance is not negligible and the coupling between P–V and Q–$\theta$ is stronger. Therefore, modified fast decoupled formulations are often required for high $R/X$ distribution networks [34,76].

Several extensions adapt the fast decoupled formulation to distribution-network characteristics. For high $R/X$ feeders, the formulations in [76,77] use modified equations and complex normalization to improve convergence under normal and switching configurations. A three-phase formulation in [78] applies complex per-unit normalization to represent unbalance and phase coupling while retaining the computational advantages of decoupled iterations. In contrast, the rectangular-coordinate approach in [79] avoids normalization-based adjustments, showing that performance depends on the selected formulation and constant-matrix approximations. For meshed networks, where stronger coupling further weakens the decoupling assumptions, the fast non-decoupled method in [80] improves solution behavior but sacrifices part of the simplicity of the standard fast decoupled formulation. These studies show that fast decoupled methods can reduce computation when the decoupling assumptions are sufficiently accurate, whereas high $R/X$ ratios, unbalance, and meshing require additional modifications that increase formulation complexity.

3.2. Linearized Power Flow Methods

Conventional nonlinear power flow methods provide accurate steady-state solutions, but their repeated execution can be computationally burdensome in large-scale and multi-interval studies. They may also experience convergence difficulties under stressed or ill-conditioned conditions, including heavy loading, high $R/X$ feeders, and weakly meshed distribution topologies. These challenges have motivated linearized or approximate power flow models that reduce computational effort and simplify numerical solution procedures, making them useful for time-series analysis and planning studies [13,36,81,82]. Table 3 summarizes the main linearized power flow models, their network type, validity range, and contributions.

3.2.1. DC Power Flow Linearization

The DC power flow (DCPF) is a linear approximation of the AC power flow equations in terms of active power and voltage angles, commonly used when fast evaluation of active-power transfers is required and voltage and reactive-power effects are not modeled explicitly [83].

Under the standard approximation, the active power on a line $(i,j)$ is written as

$$P_{ij}\approx {\displaystyle \frac{1}{x_{ij}}}({\theta }_i-{\theta }_j)=B_{ij}({\theta }_i-{\theta }_j)$$

(10)

which leads to the nodal linear system

$$p=B_{\mathrm{bus}}\theta$$

(11)

This approximation is typically justified under the assumptions that (i) voltage magnitudes are near nominal 1 p.u. and treated as constant, (ii) angle differences are small, (iii) transmission losses are neglected, and (iv) reactive power and voltage constraints are ignored [83]. These assumptions are generally acceptable for transmission networks, where reactance dominates and voltage magnitudes are more tightly regulated. However, DCPF accuracy can degrade under large angle differences, non-negligible resistance, and stressed loading conditions [83,84]. In distribution feeders, higher $R/X$ ratios and stronger voltage and reactive-power effects further limit the applicability of the lossless formulation. Error-bounding methods quantify the mismatch between AC and DC solutions over a specified operating region [85], while lossy DC formulations introduce loss-aware corrections to improve accuracy in resistive networks [84]. For cases without a base operating point, modified linear formulations can also include the influence of reactive loads on phase angles while retaining a linear structure [86]. These developments broaden the applicability of DCPF, but they do not fully represent the voltage and reactive-power effects captured by nonlinear AC power flow.

3.2.2. LinDistFlow/Linear Branch-Flow-Based Models

LinDistFlow is a distribution-oriented linearization derived from the DistFlow branch-flow formulation for radial distribution networks [87]. It preserves the feeder branch structure and approximates squared voltage magnitudes as an affine function of power flows by neglecting quadratic terms. This yields a fast, branch-structured model with low computational cost, making it suitable for radial and weakly meshed feeders and convenient for embedding power flow constraints inside multi-period optimization problems [88]. For a radial line $i\to j$, a typical LinDistFlow form combines branch power-balance relations with a linearized squared-voltage drop equation:

$$P_{ij}\approx p_j+\sum _{m\in \mathcal{C}\left(j\right)}{P}_{jm}$$

(12)

$$Q_{ij}\approx q_j+\sum _{m\in \mathcal{C}\left(j\right)}{Q}_{jm}$$

(13)

$$v_j\approx v_i-2\left(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij}\right)$$

(14)

with $p_j,q_j$ denoting net injections at the bus j, $P_{ij},Q_{ij}$ denoting branch flows, and $v_i={\left|V_i\right|}^2$ [87].

This linearization is typically used under assumptions stated as: (i) radial or weakly meshed feeders; (ii) loss-related quadratic terms are small or dropped in the voltage-drop equation; (iii) voltage magnitudes remain close enough to nominal for a first-order approximation to be accurate; (iv) device nonlinearities are represented through simplified linear or piecewise relations rather than full AC equations [88]. These assumptions also indicate failure modes, since errors typically increase with heavy loading and higher losses, while additional modeling is needed when line shunts, unbalance, or strong voltage-control actions become significant [81,88]. Early DistFlow-based studies applied this feeder formulation to radial capacitor placement [87] and capacitor sizing [23]. Later developments extend LinDistFlow beyond the basic radial single-phase formulation. The generalized model in [81] provides a closed-form voltage mapping for broader distribution-network representations, while the shunt-aware formulation in [89] improves voltage estimates when line shunt effects are significant. Accuracy-oriented approaches in [90,91] retain linearity by tuning model coefficients or estimating system-specific parameters from representative operating points. These methods improve voltage estimation over wider loading ranges, although their performance depends on the operating conditions used for tuning or parameter identification.

A closely related branch-flow formulation introduces an additional variable for the squared current magnitude ${\mathcal{l}}_{ij}$, allowing loss terms to be represented explicitly and enabling tractable relaxations of the original equations. For line $i\to j$, the voltage-drop equation retains a loss term through ${\mathcal{l}}_{ij}$:

$$v_j=v_i-2(r_{ij}{P}_{ij}+x_{ij}{Q}_{ij})+(r_{ij}^2+x_{ij}^2){\mathcal{l}}_{ij}$$

(15)

$$P_{ij}^2+Q_{ij}^2=v_i{\mathcal{l}}_{ij}$$

(16)

These equations form the basis for widely used conic relaxations in radial networks, while linear variants are obtained by dropping the ${\mathcal{l}}_{ij}$-dependent loss term or linearizing the coupling relation [39]. The loss-aware formulation in [82] introduces a parameterized loss approximation and extends it to multiphase radial feeders, improving accuracy when resistive losses are significant. Linear models can also include voltage-control devices to evaluate device-driven voltage changes without requiring a full nonlinear PF solution [13]. However, accuracy can still deteriorate under stressed loading, strong unbalance, or meshed operation unless additional treatment is introduced for losses, loops, and phase coupling [39,82].

3.2.3. Sensitivity/Jacobian-Based Linearized Models

Sensitivity-based and Jacobian-based linearizations are first-order power flow approximation methods in which voltage changes are related to active- and reactive-power injection changes through the linearized AC power flow equations. Sensitivity-based linearization uses voltage sensitivity factors to relate slight changes in injections to corresponding changes in voltage variables around a base condition. A generic form writes the incremental voltage response as

$$\Delta V\approx S_P\Delta P+S_Q\Delta Q$$

(17)

where $S_P$ and $S_Q$ are sensitivity matrices. These sensitivities are typically obtained from the inverse Jacobian evaluated at the base point through the incremental AC relation:

$$\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]\approx J\left(x^0\right)\left[\begin{array}{c}\Delta \theta \\ \Delta V\end{array}\right],\left[\begin{array}{c}\Delta \theta \\ \Delta V\end{array}\right]\approx J{\left(x^0\right)}^{-1}\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]$$

(18)

This class of models provides an explicit linear mapping from injection changes to voltage responses, which is convenient for fast screening and control-oriented optimization. It is particularly useful in iterative or real-time settings because sensitivities can be updated from a base operating point without repeatedly solving a full nonlinear power flow [40]. This approximation is typically used when deviations are small around the base point, network parameters and topology remain unchanged over the linearization window, and device operating modes do not switch in a way that introduces strong nonlinear behavior. Prior studies have used voltage-sensitivity factors to update distributed generation (DG) setpoints in a decentralized manner, enabling voltage regulation without continually solving a full nonlinear power flow [40]. Analytical sensitivity formulations have also been developed for unbalanced radial networks, including node-voltage, line-current, and transformer-tap effects [41], while the ABCD-based approach in [92] provides fast voltage sensitivities for distribution-grid voltage control. When detailed network models are unavailable, measurement-driven approaches use localized sensitivity structures [93] or PMU measurements [42] to estimate voltage sensitivities for control applications. These approaches reduce dependence on repeated nonlinear PF solutions, but their accuracy remains local to the operating region and may deteriorate when topology, control modes, or system imbalance change significantly [41,42].

Jacobian-based linearization is closely related to sensitivity-based modeling, since the sensitivity matrices are commonly obtained from the inverse of the power flow Jacobian. However, while sensitivity-based models directly map injection variations to selected voltage responses, Jacobian-based linearization retains the full first-order relationship between power flow mismatches and state-variable updates. Writing the AC equations as $f\left(x\right)=0$, a first-order expansion around $x^0$ gives

$$f\left(x\right)\approx f\left(x^0\right)+J\left(x^0\right)(x-x^0)$$

(19)

$$J\left(x^0\right)\Delta x\approx -f\left(x^0\right)$$

(20)

A unified formulation and error analysis shows that several linear power flow models can be derived from first-order approximations under different variable selections, with different approximation errors [14]. To reduce the computational burden of repeated Jacobian construction, the state-independent model in [94] estimates voltage magnitudes without rebuilding the Jacobian at each operating point. When model-based Jacobians become less reliable, the measurement-based approach in [43] uses synchronized phasor data to adapt the Jacobian to operating point and topology changes. Distribution-oriented extensions also incorporate radial laterals, full $\pi$ line models, ZIP loads, and DER control modes [95], while Taylor-series formulations support rapid analysis with simplified DG models [96]. For bipolar DC distribution networks, the formulation in [97] includes ZIP loads and flexible devices while reducing iterative computation. Online feedback further updates linearization parameters from measurements to maintain accuracy in time-varying unbalanced networks [44]. These developments improve computational efficiency and model adaptability, but a fixed linearization can still lose accuracy when the operating point moves far from the base condition or when nonlinear effects become dominant [14,44].

3.2.4. Fixed-Point/Secant Type Approximations

Fixed-point and secant-type approximations formulate explicit linear or affine surrogate mappings for power flow quantities, enabling fast evaluation without repeatedly solving the full nonlinear power flow equations. Fixed-point linearization methods express the AC power flow equations as an equivalent fixed-point mapping and use this structure to establish existence and uniqueness conditions or derive an explicit approximation. A common formulation for distribution networks assumes PQ buses and a fixed slack bus, and can be written as [98]:

$$\tilde{v}=w+Z\mathrm{diag}{\left(\overline{\tilde{v}}\right)}^{-1}\overline{s}$$

(21)

$${\tilde{v}}^{(k+1)}=w+Z\mathrm{diag}{\left({\overline{\tilde{v}}}^{\left(k\right)}\right)}^{-1}\overline{s}$$

(22)

where $\tilde{v}$ is the complex bus-voltage vector (excluding the slack), w is the no-load voltage profile, Z is an impedance-type matrix, and s stacks complex power injections. Fixed-point formulations provide a structured representation that supports convergence and solvability analysis, while their closed-form linear voltage mappings enable efficient evaluation in planning or control loops. This approach is typically used when (i) a slack/reference is fixed, (ii) loads are represented by PQ or ZIP-type models compatible with the mapping, (iii) the system operates within a domain where the formulation is well-conditioned and convergent, and (iv) network topology and parameters remain constant over the analysis window. Within these conditions, fixed-point formulations differ mainly in their solvability guarantees, network detail, and approximation accuracy. Sufficient solvability conditions and explicit error bounds are developed for distribution networks in [98], while [99,100] extend existence and uniqueness analysis to active distribution networks and unbalanced three-phase systems. Contractiveness of the Z-bus iteration for unbalanced three-phase networks with wye and delta ZIP loads is established in [101], and the broader multiphase formulation in [102] unifies convergence analysis with associated linear load flow models. Practical formulations in [103,104] provide linear voltage mappings for unbalanced multiphase distribution networks and three-phase feeders, supporting fast voltage estimation under typical operating conditions. When first-order approximations are insufficient, higher-order and hybrid fixed-point/Taylor formulations improve voltage accuracy [105,106]. These methods support fast feeder analysis and related applications such as aggregate flexibility assessment [107], but their accuracy and robustness remain dependent on the operating range, voltage deviations, and degree of imbalance.

Fixed-point and secant-type approximations differ mainly in their construction and validity region. Fixed-point models are derived analytically from a fixed-point representation of the PF equations and are commonly linked to a reference or no-load state [98]. Their convergence and solvability are characterized through contraction or existence conditions [101,102]. Secant-type models instead fit affine coefficients from multiple PF samples over a prescribed operating region [36]. They therefore target accuracy or conservative bounds across that region rather than near a single reference point [108,109]. Fixed-point models offer stronger analytical guarantees, whereas secant-type models provide wider-range flexibility but depend on representative sample coverage. Secant-type approximations are typically used when an operating range is specified, network topology and parameters are fixed, and representative operating samples are available, possibly with a prescribed bounding or bias requirement. A representative secant-type affine surrogate is

$$g_i\left(u\right)=a_{i,0}+a_{i,1}^{T}u$$

(23)

where u collects selected inputs such as injections or net powers, and the fitted model is constrained to overestimate or underestimate the true underlying quantity over the sampled region. Sample-based surrogates with prescribed bounding behavior reduce the risk of infeasible decisions when embedded in optimization [36], while piecewise linear and bias-controlled formulations improve accuracy or conservativeness over the selected region [109,110]. Other approaches directly optimize the surrogate or use second-order sensitivity information to reduce worst-case error and guide sample placement [108,111]. Compared with single-point linearizations, these methods provide better accuracy over wider operating ranges, but they require representative samples and may increase model size or conservativeness. Their performance remains dependent on the selected operating region and the quality of the available data. Stronger feasibility guarantees and piecewise linear formulations can further increase model size and computational burden, while accuracy may decrease outside the fitted region.

The approaches discussed so far focus mainly on deterministic power flow computation through nonlinear equations or linear and region-valid surrogates. In DER-rich distribution networks, however, power flow analysis must also remain reliable under stressed conditions and frequent control switching, account for uncertainty in load and renewable injections, and support large-scale time-series studies, which motivates the modern power flow developments examined in Section 4, Section 5 and Section 6 for improved robustness, uncertainty handling, and computational scalability.

3.3. Illustrative Power Flow Analysis on a Distribution Network

An illustrative power flow analysis is conducted using the CIGRE 15-bus medium-voltage distribution network [112] under radial and meshed configurations, as shown in Figure 6. Three representative methods from the families reviewed in Section 3 are considered: Newton–Raphson [61], a branch-flow linearization [39], and a fixed-point linear approximation [98]. The Newton–Raphson solution is taken as the nonlinear reference, and the linearized methods are compared using maximum errors in voltage magnitude, voltage angle, active branch flow, and reactive branch flow.

Table 3. Summary of linearized power flow models.

Ref	Method	Network	Validity Range	Contribution
[81]	LinDistFlow	Radial and meshed	Accuracy is highest near the selected linearization point; it decreases as operating conditions depart.	Proposes a generalized LinDistFlow model for multiphase networks with generic topologies, improving voltage-magnitude estimation relative to common linear baselines.
[87]	LinDistFlow	Radial	Most reliable in radial feeders where loss terms are not dominant	Uses DistFlow-style feeder power flow equations to represent radial distribution networks and formulates capacitor placement within the framework.
[89]	LinDistFlow	Radial with extensions to weakly meshed and unbalanced three-phase	Designed for feeders where line shunt charging effects are non-negligible; validity is tied to the model’s linearization assumptions and neighboring-node voltage differences	Proposes an enhanced LinDist model that incorporates line-shunt effects into the voltage-drop equations, and extends the framework to handle ZIP loads, weakly meshed networks, and unbalanced three-phase feeders.
[90]	LinDistFlow	Typically radial feeders	Valid over the operating range captured by the tuning data, with accuracy depending on the representativeness of the sampled loading conditions.	Tunes LinDistFlow parameters to reduce approximation error across multiple operating points, improving voltage estimates compared with fixed-parameter LinDistFlow.
[91]	LinDistFlow	Radial distribution test feeders	Valid over a wide range of operating points targeted by the learned parameter and designed to remain accurate under varying loading levels	Introduces a parameterized linear PF model with a system-specific learned parameter to improve voltage accuracy over simplified DistFlow and other linear models.
[39]	Branch-flow model	Radial	Valid as a modeling framework for radial branch-flow representations; used for tractable relaxations rather than local linear accuracy	Develops branch-flow model relaxations/convexification foundations that enable tractable analysis and optimization using branch-flow variables.
[82]	Lossy DistFlow	Radial, single, and multiphase	Intended for radial feeders where loss modeling is important; validity follows the DistFlow assumptions and feeder structure	Provides a lossy DistFlow formulation for single-phase and multiphase radial feeders to improve representation of losses and voltage behavior.
[13]	Jacobian-based linear method	Radial	Valid around the operating conditions used for linearization and device-mode assumptions; evaluated across multiple operating conditions	Extends a constrained Jacobian-based linear PF method to include common voltage control devices and DER controls, using a non-iterative solution procedure for discrete device actions.
[14]	Jacobian-based	Meshed	Local around the linearization point	Develops a unified formulation of linear power flow models via first-order Taylor expansion and provides theoretical error analysis, including how variable choice affects linearization error.
[94]	Jacobian-based	Radial and meshed	Broad across typical operating conditions without rebuilding the Jacobian each time	Proposes a state-independent linear model that estimates voltage magnitudes accurately while avoiding repeated Jacobian reconstruction for time-series studies.
[41]	Sensitivity-based	Radial	Local around a base operating point, extended to multiphase unbalanced networks	Derives analytical voltage and line-current sensitivity coefficients as functions of nodal injections and tap positions using a sparse compound admittance matrix, validated on IEEE 13- and 34-node feeders.
[43]	Jacobian-based	Meshed	Local, updated online from measurements	Estimates the power flow Jacobian from synchronized phasor measurements using a data-driven approach so the Jacobian can adapt to operating point and topology changes.
[44]	Jacobian-based	Radial	Maintains accuracy over time by updating parameters through online feedback.	Proposes an online feedback-based linearized power flow model that updates linearization parameters from measurements to maintain accuracy in time-varying unbalanced distribution networks.
[98]	Fixed-point linear approximation	Radial and meshed	Valid when voltages and loading remain within the operating region assumed by the fixed-point approximation, so the no-load-based linearization remains accurate.	Provides solvability conditions and a linear approximation of the PF solution for distribution networks.
[102]	Fixed-point method	Radial and meshed	Valid within the region where existence and uniqueness hold and the Z-bus fixed-point map remains contractive.	Develops multiphase existence, uniqueness, and non-singularity conditions using a Z-bus fixed-point formulation and derives associated linear models.
[101]	Fixed-point method	Radial and meshed	Valid within the contraction region defined by the model, where Z-bus iterations converge to a unique solution under ZIP load representation.	Proves convergence of Z-bus for three-phase distribution PF with ZIP loads and characterizes a convergence region.
[105]	Fixed-point method	Radial and meshed	Valid when higher-order terms remain bounded over the relevant operating range, so accuracy improves beyond first-order approximations.	Develops higher-order approximate PF solutions to improve accuracy compared with first-order linear models.
[109]	Secant-type region	Radial and meshed	Valid over the sampled operating region used to fit the surrogate, with accuracy that depends on how well samples cover the intended operating range.	Develops sample-based piecewise linear PF approximations using second-order sensitivities to improve region-wide accuracy.
[36]	Secant-type conservative linear approximation	Radial and meshed	Valid over the operating range represented by the selected samples and designed to remain conservative for the fitted quantities.	Proposes a sample-based approach to compute conservative linear power flow approximations through constrained fitting over an operating region.
[110]	Secant-type conservative biased approximation	Radial and meshed	Valid over the operating range used for training, with conservativeness enforced through bias design in the approximation.	Develops conservative biased linear power flow approximations and demonstrates their use in unit commitment to reduce infeasible decisions.

Table 3. Summary of linearized power flow models.

Ref	Method	Network	Validity Range	Contribution
[81]	LinDistFlow	Radial and meshed	Accuracy is highest near the selected linearization point; it decreases as operating conditions depart.	Proposes a generalized LinDistFlow model for multiphase networks with generic topologies, improving voltage-magnitude estimation relative to common linear baselines.
[87]	LinDistFlow	Radial	Most reliable in radial feeders where loss terms are not dominant	Uses DistFlow-style feeder power flow equations to represent radial distribution networks and formulates capacitor placement within the framework.
[89]	LinDistFlow	Radial with extensions to weakly meshed and unbalanced three-phase	Designed for feeders where line shunt charging effects are non-negligible; validity is tied to the model’s linearization assumptions and neighboring-node voltage differences	Proposes an enhanced LinDist model that incorporates line-shunt effects into the voltage-drop equations, and extends the framework to handle ZIP loads, weakly meshed networks, and unbalanced three-phase feeders.
[90]	LinDistFlow	Typically radial feeders	Valid over the operating range captured by the tuning data, with accuracy depending on the representativeness of the sampled loading conditions.	Tunes LinDistFlow parameters to reduce approximation error across multiple operating points, improving voltage estimates compared with fixed-parameter LinDistFlow.
[91]	LinDistFlow	Radial distribution test feeders	Valid over a wide range of operating points targeted by the learned parameter and designed to remain accurate under varying loading levels	Introduces a parameterized linear PF model with a system-specific learned parameter to improve voltage accuracy over simplified DistFlow and other linear models.
[39]	Branch-flow model	Radial	Valid as a modeling framework for radial branch-flow representations; used for tractable relaxations rather than local linear accuracy	Develops branch-flow model relaxations/convexification foundations that enable tractable analysis and optimization using branch-flow variables.
[82]	Lossy DistFlow	Radial, single, and multiphase	Intended for radial feeders where loss modeling is important; validity follows the DistFlow assumptions and feeder structure	Provides a lossy DistFlow formulation for single-phase and multiphase radial feeders to improve representation of losses and voltage behavior.
[13]	Jacobian-based linear method	Radial	Valid around the operating conditions used for linearization and device-mode assumptions; evaluated across multiple operating conditions	Extends a constrained Jacobian-based linear PF method to include common voltage control devices and DER controls, using a non-iterative solution procedure for discrete device actions.
[14]	Jacobian-based	Meshed	Local around the linearization point	Develops a unified formulation of linear power flow models via first-order Taylor expansion and provides theoretical error analysis, including how variable choice affects linearization error.
[94]	Jacobian-based	Radial and meshed	Broad across typical operating conditions without rebuilding the Jacobian each time	Proposes a state-independent linear model that estimates voltage magnitudes accurately while avoiding repeated Jacobian reconstruction for time-series studies.
[41]	Sensitivity-based	Radial	Local around a base operating point, extended to multiphase unbalanced networks	Derives analytical voltage and line-current sensitivity coefficients as functions of nodal injections and tap positions using a sparse compound admittance matrix, validated on IEEE 13- and 34-node feeders.
[43]	Jacobian-based	Meshed	Local, updated online from measurements	Estimates the power flow Jacobian from synchronized phasor measurements using a data-driven approach so the Jacobian can adapt to operating point and topology changes.
[44]	Jacobian-based	Radial	Maintains accuracy over time by updating parameters through online feedback.	Proposes an online feedback-based linearized power flow model that updates linearization parameters from measurements to maintain accuracy in time-varying unbalanced distribution networks.
[98]	Fixed-point linear approximation	Radial and meshed	Valid when voltages and loading remain within the operating region assumed by the fixed-point approximation, so the no-load-based linearization remains accurate.	Provides solvability conditions and a linear approximation of the PF solution for distribution networks.
[102]	Fixed-point method	Radial and meshed	Valid within the region where existence and uniqueness hold and the Z-bus fixed-point map remains contractive.	Develops multiphase existence, uniqueness, and non-singularity conditions using a Z-bus fixed-point formulation and derives associated linear models.
[101]	Fixed-point method	Radial and meshed	Valid within the contraction region defined by the model, where Z-bus iterations converge to a unique solution under ZIP load representation.	Proves convergence of Z-bus for three-phase distribution PF with ZIP loads and characterizes a convergence region.
[105]	Fixed-point method	Radial and meshed	Valid when higher-order terms remain bounded over the relevant operating range, so accuracy improves beyond first-order approximations.	Develops higher-order approximate PF solutions to improve accuracy compared with first-order linear models.
[109]	Secant-type region	Radial and meshed	Valid over the sampled operating region used to fit the surrogate, with accuracy that depends on how well samples cover the intended operating range.	Develops sample-based piecewise linear PF approximations using second-order sensitivities to improve region-wide accuracy.
[36]	Secant-type conservative linear approximation	Radial and meshed	Valid over the operating range represented by the selected samples and designed to remain conservative for the fitted quantities.	Proposes a sample-based approach to compute conservative linear power flow approximations through constrained fitting over an operating region.
[110]	Secant-type conservative biased approximation	Radial and meshed	Valid over the operating range used for training, with conservativeness enforced through bias design in the approximation.	Develops conservative biased linear power flow approximations and demonstrates their use in unit commitment to reduce infeasible decisions.

The results in

Table 4. Power flow results for representative methods on the CIGRE 15-bus distribution network.

Method	Family	Topology	Min. $\left|\mathit{V}\right|$ (p.u.)	Max. $\left|\mathit{\theta }\right|$ (deg.)	Max. $|{\mathit{P}}_{\mathit{ij}}|$ (MW)	Max. $|{\mathit{Q}}_{\mathit{ij}}|$ (MVAr)	Runtime (s)	Solution/ Convergence
Newton–Raphson [61]	Conventional nonlinear PF	Radial	0.984555	0.430135	16.2843	3.5327	0.110	Iterative
Branch-flow linearization [39]	Branch-flow	Radial	0.983687	0.339822	16.2600	3.6900	0.00234	Non-iterative
Fixed-point linear approximation [98]	Fixed-point approximation	Radial	0.984892	0.431590	16.2583	3.4711	0.00195	Non-iterative
Newton–Raphson [61]	Conventional nonlinear PF	Meshed	0.990252	0.311754	15.9020	3.3699	0.132	Iterative
Branch-flow linearization [39]	Branch-flow	Meshed	0.989570	0.251166	15.8893	3.4798	0.0021	Non-iterative
Fixed-point linear approximation [98]	Fixed-point approximation	Meshed	0.990378	0.313021	15.8902	3.3213	0.00183	Non-iterative

show that all three methods provide comparable voltage and branch-flow estimates for the considered operating conditions. The fixed-point approximation remains close to the Newton–Raphson solution for minimum voltage, voltage angle, and branch flows in both network configurations. The branch-flow formulation also provides similar active-power flows, although larger differences are observed in voltage angle and reactive power. The comparison further shows that the numerical behavior of the linearized formulations varies between radial and meshed configurations, highlighting the importance of validating the selected power flow model for the network topology and operating condition under study.

4. Modern PF Evolution

This section reviews modern developments that extend power flow analysis beyond classical deterministic computation. It summarizes techniques that strengthen the numerical robustness and convergence of deterministic solvers under stressed operating points and control switching, as discussed in Section 4.1 and Section 4.2. Modern PF evolution also includes uncertainty-based formulations that represent variability in load and renewable injections, as discussed in Section 5, and acceleration strategies for large-scale repeated evaluations, as discussed in Section 6.

Table 5. Historical evolution of deterministic power flow methods.

Period	Key Development	Main Advantage	Main Limitation	Research Trend
1960s–1970s	Newton-based solution and sparse factorization established the computational basis for large-scale nonlinear power flow analysis [61,113].	Accurate solution with rapid local convergence.	Sensitive to initialization and Jacobian conditioning.	Research focused on improved conditioning and more reliable nonlinear solution procedures.
1980s–1990s	Sweep-and-branch-flow formulations adapted power flow analysis to radial distribution feeders [53,87].	Low computational burden and direct use of feeder topology.	Meshing, unbalance, and voltage-controlled buses required additional treatment.	Later studies extended these formulations to weakly meshed and unbalanced networks.
2000–2009	Modified sweep, continuation, and step-control methods improved convergence under difficult operating conditions [55,57,114].	Greater numerical reliability than basic iterative methods.	Performance still depended on initialization and network condition.	Research progressed toward systematic robust and multiphase formulations.
2010–2015	Sensitivity-based and fixed-point formulations enabled faster evaluation of detailed distribution-network models [41,115,116].	Supported unbalanced analysis, solvability assessment, and rapid local evaluation.	Accuracy remained dependent on the reference operating point and model assumptions.	Research moved toward generalized linear models with wider validity regions.
2016–2020	Generalized linearizations and robust nonlinear formulations improved repeated evaluation and stressed-condition analysis [14,104,117].	Reduced computational effort with broader network applicability.	Linear-model accuracy decreased outside the selected operating region.	This limitation encouraged adaptive and online-updated approximations.
2021–present	Secant-based, sample-based, and conservative approximations expanded model validity across wider operating regions [36,44,108].	Improved regional accuracy with low online computational cost.	Performance depends on calibration data and operating-region coverage.	Current research emphasizes validity monitoring, feasibility guarantees, and automatic recalibration.

summarizes the historical evolution of conventional and linearized power flow methods, together with robust solution techniques developed to improve convergence under stressed and ill-conditioned operating conditions. The table highlights their main contributions, advantages, limitations, and subsequent research directions.

4.1. Robust Power Flow

Robust power flow refers to solution methods and modeling enhancements that aim to compute power flow solutions reliably when conventional Newton-type solvers converge slowly or fail. Power flow computation becomes challenging when the problem is ill-conditioned due to stressed operation or proximity to voltage instability, and when device limits and control actions introduce switching behavior such as PV-to-PQ transitions. These effects are amplified in distribution feeders with high $R/X$ ratios and phase imbalance, where standard Newton iterations can become highly sensitive to initialization and Jacobian conditioning. As a result, robust PF techniques are used to improve numerical stability [117] and support operational studies, including voltage-stability analysis under stressed conditions [118]. In this section, these techniques are broadly classified into damped or step-size Newton approaches [114], staged initialization methods [117], continuation or homotopy-based approaches [119], and sweep-based robust backward/forward or multiphase distribution-flow formulations [37].

4.2. Robust Solution Approaches

The robustness of power flow computation can be improved through numerical approaches that modify the deterministic iteration to maintain stable progress under stressed conditions. The numerical robustness strategies for power flow analysis are depicted in Figure 7.

A widely used approach is damped Newton power flow, in which the Newton correction is scaled by an iteration-dependent step size selected to ensure sufficient reduction in the power mismatch, thereby avoiding unstable updates when the local linearization is poor. This strategy enlarges the convergence region under stressed and near-limit operating conditions; however, selecting the step size may require additional calculations, and damping can slow convergence in well-conditioned cases. Step-size optimization and optimal-multiplier approaches improve convergence by selecting suitable correction scaling, although their effectiveness depends on the coordinate formulation and operating condition [64,114]. Nondivergent fast power flow variants retain the reduced-computation structure while lowering divergence risk through an optimal multiplier [120]. These methods improve robustness under difficult conditions, but the additional step control may increase iteration counts or reduce efficiency in well-conditioned systems.

Another approach is staged initialization [117], which first computes a stable approximate solution and uses it as a warm start for the full AC power flow. The two-stage formulation in [117] uses a simplified model to obtain a stable initial point before refining the solution with the full AC equations. This reduces sensitivity to initialization in large and heavily loaded systems, although the final refinement can remain difficult near voltage-instability conditions. Robustness has also been pursued through step-control and extrapolation-based methods [121], where convergence is supported by adaptive step-size selection together with extrapolation. The Bulirsch–Stoer-based method in [121] combines repeated extrapolation with adjustable step size to maintain convergence under ill-conditioning. Compared with staged initialization, this method improves convergence during the iterative solution, but repeated extrapolation increases computational effort and may limit its use in larger networks.

For DER-rich feeders, robustness is achieved using continuation or homotopy-based methods [119], which track a sequence of gradually transformed problems to maintain convergence under stressed operating conditions and control-mode switching. An equivalent-circuit representation and homotopy techniques, such as Tx-stepping and dynamic power stepping, are adopted in [119] to obtain stable solutions in large and unbalanced networks, at the cost of additional computation and modeling complexity. Within the same framework, continuation power flow is used for voltage-stability assessment by tracing the solution as the loading level changes to identify collapse points and stability margins. Three-phase continuation formulations extend this analysis to DER-integrated and unbalanced distribution networks, including PV–PQ switching and voltage-stability assessment along a loading path [118,122]. The sequence-component approach in [123] further extends continuation analysis to integrated transmission and distribution systems. For radial networks, the formulation in [124] traces the P–V curve and determines the maximum loading point, while [22,125] incorporate protection events, transformer regulation, and distributed generation into the loading-path analysis. The loop-analysis-based formulation in [116] extends the method to radial and weakly meshed networks by accounting for network loops. These developments extend the applicability of continuation power flow from conventional radial loading studies to unbalanced, integrated, and event-driven network analysis, although the increased formulation complexity can raise computational effort and numerical sensitivity near switching points.

Robust PF is also supported by backward/forward sweep variants and multiphase extensions [37,115], which are effective for radial feeders and high $R/X$ conditions. The three-phase formulation in [37] retains the sweep structure while representing detailed phase behavior in radial feeders, whereas the multiphase method in [115] introduces load stepping to improve convergence under transformer loops and heavy loading. These methods are well suited to radial and weakly meshed distribution networks, but their modeling and sensitivity-matrix requirements become more complex as network meshing increases. These robust techniques primarily improve convergence and numerical stability for a specified operating condition; however, practical distribution operation is also affected by uncertainty in load and renewable injections across time, which motivates uncertainty-based power flow formulations.

5. Uncertainty-Based Power Flow

Modern distribution networks increasingly face uncertainty due to renewable generation variability, widespread distributed resources, and continuously changing load and prosumer behavior. As a result, operating conditions can vary significantly over time, and deterministic power flow solutions may not capture the full range of realistic outcomes. Uncertainty-based power flow methods incorporate uncertainty in the input data and evaluate its effect on voltages, branch flows, and losses. In this section, we review three main approaches: probabilistic power flow, interval power flow, and fuzzy power flow.

5.1. Probabilistic Power Flow

Deterministic power flow is formulated with fixed values of loads and generations, and it evaluates the network state for the assumed operating condition. In modern distribution networks, net injections can vary due to load variation and renewable generation. Probabilistic power flow therefore describes system states under uncertainty using probability distributions for the inputs and statistical descriptions for the outputs, such as the mean, variance, and related measures of system response [25,31,32]. Probabilistic power flow is commonly carried out using numerical, analytical, or approximate methods. Numerical methods generate values for the uncertain inputs and solve a deterministic load flow for each set, while analytical methods work directly with the input probability density functions to obtain output statistics in a more explicit mathematical form [25]. Approximate methods are adopted when only a limited number of deterministic evaluations is feasible, providing estimates of key statistical measures with reduced computational effort [31,32]. Figure 8 presents the article classification based on probabilistic power flow methods and highlights the relative share of numerical, analytical, and approximate approaches within the reviewed literature.

5.1.1. Numerical Methods

Numerical methods perform repeated deterministic load flow simulations by sampling uncertain input variables from their probability density functions. Each deterministic solution provides one output sample, and the resulting samples are used to characterize the probabilistic behavior of the load flow results. Because these methods rely on repeated deterministic analysis, they are generally robust and accurate and are commonly used as a reference for assessing newer probabilistic methods. Their main limitation is computational time, since a large number of simulations may be required to obtain stable output distributions, particularly as the number of uncertain inputs increases [31,32]. The numerical techniques discussed in this subsection include Monte Carlo simulation and structured sampling variants such as quasi-Monte Carlo and Latin hypercube sampling [31].

• Monte Carlo Sampling:
  Monte Carlo simulation evaluates probabilistic power flow by generating random samples of uncertain inputs from their probability distributions, solving a deterministic load flow for each sample, and estimating the output statistics and distributions from the ensemble of solutions [25,31]. Because it relies on repeated deterministic solutions, it is often used as an accuracy benchmark and can represent uncertainty in both load demand and renewable generation [31,32]. Applications in distribution networks have used Monte Carlo-based probabilistic power flow to evaluate voltage and branch-flow variation under photovoltaic uncertainty [126] and time-series-based renewable and load uncertainty using realistic input samples [127]. These studies show that Monte Carlo simulation provides flexible uncertainty representation, although its computational burden increases with the number of samples, uncertain variables, and time periods.
  The main limitation of Monte Carlo simulation is the large number of deterministic load flow runs needed to achieve reliable accuracy, particularly when low-probability events must be captured, which can make large networks and multi-period studies computationally expensive [12,32]. This has motivated improved Monte Carlo-style formulations that reduce computational cost while maintaining acceptable accuracy. For example, the multi-linear Monte Carlo method in [128] combines random sampling with a multi-linear approximation to accelerate probabilistic assessment in distribution systems with wind and photovoltaic generation. However, such acceleration depends on the accuracy of the adopted approximation, while standard Monte Carlo remains more general but computationally demanding. The high computational cost of standard Monte Carlo has also motivated alternative uncertainty-propagation techniques, including polynomial-based expansions and other approximate approaches when many scenarios must be evaluated [12].
• Quasi-Monte Carlo Sampling:
  Quasi-Monte Carlo sampling replaces random sampling with deterministic sequences designed to sample the input space more uniformly than standard Monte Carlo. This typically reduces the number of deterministic load flow simulations needed to reach a similar level of accuracy, making it a practical acceleration of Monte Carlo in probabilistic load flow studies [24,25,31,32]. Common sequences include Sobol and Halton. Sobol sequences are widely used because they often provide reliable convergence with fewer samples, while Halton sequences can lose uniformity as the dimension increases [129]. The main trade-off is that quasi-Monte Carlo requires more careful sequence generation and distribution mapping, and its advantage can decrease in high-dimensional problems [129]. In probabilistic power flow applications, quasi-Monte Carlo has been combined with distribution transformations and correlation handling, while Sobol samples have shown improved efficiency relative to Latin hypercube sampling at the same sample size [130]. This advantage remains dependent on the dimensionality of the uncertainty space and the treatment of correlated inputs.
• Latin-hypercube Sampling:
  Latin hypercube sampling evaluates probabilistic power flow by selecting input samples in a stratified manner, so each uncertain input is sampled more evenly across its probability distribution than under simple random sampling. Each input distribution is divided into N equal probability intervals, one value is selected from each interval, and the selected values are combined across the inputs through a permutation step [24,32]. This sampling structure typically reduces the number of simulations needed to reach comparable accuracy, leading to lower computational effort than standard Monte Carlo. Latin hypercube sampling is straightforward for independent inputs, but additional steps are required when correlations among inputs must be preserved [24,32].
  Applications have used Latin hypercube sampling to reduce the number of simulations in probabilistic carbon-emission analysis [131] and have combined it with multiple linear regression to preserve dependence among correlated inputs [132]. These extensions improve sampling efficiency and correlation representation, but repeated deterministic solutions are still required, and performance remains sensitive to sample size, dependence modeling, and system nonlinearity [24,32,131].

5.1.2. Analytical Methods

Analytical methods derive mathematical relationships that relate the output probability distributions to the input probability distributions. Since load flow equations are nonlinear and uncertain inputs can be correlated, these methods usually rely on simplifying assumptions, such as linearized load flow equations or simplified correlation models, to keep the formulation tractable. Compared with numerical methods, analytical methods generally offer lower computational effort, but their accuracy and robustness depend on the adopted assumptions. In this subsection, we discuss two widely used analytical techniques, the convolution method and the cumulant method [24,133].

1.

Convolution Method:

The convolution method derives the output probability distributions by combining the input probability distributions through convolution, so the probabilistic load flow results are obtained without repeated sampling. This method is simple for a small number of uncertain injections, but the computational effort increases quickly as the number of uncertain inputs grows, and it becomes harder when correlations must be preserved. In distribution systems with uncertain PV generation, conventional convolution can suffer from a high computational burden, which motivates more efficient alternatives for uncertainty analysis [134]. Recent formulations address these limitations through dependent discrete convolution, copula-based dependence modeling, and dimension-reduction techniques that reduce computational effort while preserving correlations among uncertain inputs [133,135]. However, their efficiency still depends on the number of uncertain variables and the complexity of the adopted dependence model.

2.

Cumulant Method:

The cumulant method propagates uncertainty using cumulants of the input random variables and maps them through a linearized load flow relation to obtain cumulants of the output quantities. The output probability distributions are then approximated through a series expansion, with Gram-Charlier expansion being a common choice in probabilistic load flow [136]. This approach is usually faster than the convolution method, but its accuracy depends on the linearization quality and the series approximation. Alternative formulations apply cumulant-based probabilistic power flow to renewable-integrated voltage and reactive-power control [137], or compare sequence-operation methods with cumulant and Monte Carlo approaches under PV uncertainty [138]. These developments improve flexibility and computational performance, but the resulting accuracy remains dependent on the adopted expansion, linearization, and representation of uncertainty.

5.1.3. Approximate Methods

Approximate methods estimate probabilistic load flow results using a small set of representative input points rather than a large number of simulations. They typically rely on limited statistical information of the uncertain inputs, such as mean and variance, without requiring a complete description of the input probability density functions. Deterministic load flow is solved only for these selected points, and the resulting outputs are used to estimate statistical measures of the network response [24]. Approximate methods are computationally efficient and simple to implement, but their accuracy can decrease as the number of uncertain inputs increases, especially when inputs are strongly non-Gaussian or correlated.

1.

Point Estimation Method:

Point estimation methods select a small set of representative input points with associated weights to match key statistical quantities of the uncertain inputs, and the resulting weighted solutions are used to estimate the probabilistic load flow outputs. Different schemes balance accuracy against the number of simulations, and comparative results indicate that some multi-point schemes remain reliable even when many continuous and discrete uncertain inputs are present [139]. Applications of point estimation include two-point schemes for quantifying uncertainty effects with a limited number of simulations [140], three-point schemes combined with correlation transformations to represent dependence among nodal injections [141], and probabilistic studies of aggregated plug-in hybrid electric vehicle charging demand [142]. These approaches reduce the number of deterministic evaluations, although their performance depends on the selected points, weighting scheme, and treatment of correlated inputs.

2.

Unscented Transformation Method:

The unscented transformation uses a small set of selected input points, often called sigma points, to propagate uncertainty through nonlinear power flow mappings while preserving the mean and covariance of the inputs. It is attractive when correlation is important, since correlated uncertain inputs can be represented directly through the covariance structure, and the number of simulations remains far below the Monte Carlo approach [143]. For distribution systems with wind production, the unscented transformation has been used to estimate voltage and line flow statistics with much lower computational time than large-scale sampling by solving the load flow only for a limited set of representative points [144]. For unbalanced three-phase islanded microgrids, an unscented transformation-based probabilistic PF with droop control was applied to reduce power flow evaluations compared with Monte Carlo simulation [145].

3.

Polynomial Chaos Expansion:

Polynomial chaos expansion represents uncertain power flow outputs by using orthogonal polynomial basis functions of the uncertain input variables. Instead of solving a very large number of random scenarios, the method constructs an approximate input–output relationship between the uncertain injections and the resulting power flow quantities, such as voltage magnitude, branch flow, losses, and substation power. Generalized polynomial chaos has been used for probabilistic power flow by selecting polynomial bases according to the probability distributions of the input uncertainties [146]. To improve computational efficiency and applicability across different uncertainty descriptions, basis-adaptive and sparse polynomial chaos formulations reduce the number of required expansion terms [147,148], while data-driven and arbitrary polynomial chaos methods construct basis functions from actual data or non-standard probability distributions [149,150]. Recent non-intrusive formulations further apply deterministic AC power flow evaluations to train the polynomial surrogate and use Monte Carlo simulation as an accuracy benchmark in renewable-rich distribution networks [151]. The main limitation of polynomial chaos expansion is that its accuracy depends on the selected polynomial order, the number of uncertain variables, and the smoothness of the power flow response. In high-dimensional systems or cases with strong nonlinearities, correlations, discrete controls, or non-smooth operating behavior, the number of basis terms can increase significantly, and adaptive or sparse formulations may be required to maintain computational efficiency [147,148,151].

5.2. Interval Power Flow

Interval power flow represents uncertain inputs as bounded intervals and computes corresponding lower and upper bounds on the power flow state. This is suitable when uncertainty is specified through bounded ranges rather than probability distributions, and it yields explicit lower and upper bounds for network quantities [152,153]. A main advantage is that it avoids scenario sampling and distribution fitting, so it can work with limited data and still provide guaranteed ranges under the assumed input bounds [152]. The main limitation is conservativeness, since basic interval arithmetic can widen output ranges as the number and width of uncertain inputs increase, and correlation among uncertain injections is not automatically preserved [152,153].

Early interval arithmetic-based formulations established interval power flow as a bound propagation problem and showed that the resulting bounds can become wider when uncertainty is large or strongly dependent [152]. To reduce this conservativeness, the correlated interval algorithm in [154] incorporates correlation information directly into the distribution power flow calculation. The affine arithmetic-based method in [155] further tracks dependence through affine forms and convex set representations. The ellipsoid-based formulation in [156] reduces redundant combinations of correlated uncertain injections, thereby lowering the computational effort required for bound calculation. For wind-dominated systems, the method in [153] represents correlation through a correlation angle and applies an affine transformation before solving the interval power flow, producing tighter bounds than treating wind-farm uncertainties as independent.

5.3. Fuzzy Power Flow

Fuzzy power flow is used when uncertain inputs are better described by imprecise ranges rather than by detailed probability distributions. In this approach, loads, generations, and sometimes network parameters are represented as fuzzy numbers, and the power flow solution becomes a possible description of the system states. This makes fuzzy power flow attractive when statistical data are limited, while still providing a structured way to reflect uncertainty in planning and operational analysis [157]. The basic idea of fuzzy logic is to describe uncertainty through degrees of membership instead of exact probabilities. Each uncertain value is assigned a membership degree between 0 and 1, which indicates the extent to which that value belongs to the uncertain set. Most fuzzy power flow formulations rely on the extension principle through interval calculations across $\alpha$-cuts, where an $\alpha$-cut represents the interval of all values whose membership degree is greater than or equal to a selected level $\alpha$ [158]. Therefore, uncertainty is propagated by solving a sequence of interval-type power flows at different $\alpha$ levels and combining the results into fuzzy outputs. The main challenges are the growth in computations as the number of fuzzy inputs increases and the overestimation that can arise from interval operations when the same variables appear repeatedly in nonlinear expressions [158]. In addition, correlations among uncertain inputs are not automatic and usually need explicit handling [159].

Representative developments incorporate practical constraints and multiple uncertainty sources within fuzzy power flow while proposing techniques that reduce computational effort and limit overestimation. Reactive power limits at PV buses are enforced in [160], while uncertainty in load forecasts, voltage-dependent load model parameters, and selected network parameters is represented through fuzzy inputs to obtain breakpoint ranges of system states and outputs. Distribution-network extensions include a fuzzy power flow method for weakly meshed systems [161] and fuzzy set-based load estimation integrated into radial distribution power flow [157]. These formulations extend fuzzy power flow to different network structures and practical uncertainty sources, although their computational effort still increases as the number of fuzzy inputs and $\alpha$-cut levels grows. To reduce conservativeness and improve tractability, the fuzzy transformation method combines structured fuzzy arithmetic with backward/forward sweep and validates the resulting fuzzy intervals against true interval bounds and Monte Carlo ranges, aiming to avoid global optimization and heavy randomized simulation while keeping sharper uncertainty envelopes [159]. For radial networks with PV generation, a fuzzy backward/forward sweep method represented uncertain inputs through fuzzy values to evaluate their effects on voltage and losses [162]. Both sweep-based approaches reduce the computational burden of fuzzy power flow in radial networks. However, their accuracy and efficiency remain dependent on the adopted fuzzy representation and the number of uncertainty evaluations. When both probabilistic randomness and fuzzy imprecision are present, a random fuzzy power flow framework models wind, PV, and loads with two-stage processing and reports both expected values and risk-related indices for voltage limit violations [163].

Hybrid approaches combine two uncertainty descriptions, typically probability-based randomness with interval or fuzzy imprecision, to capture mixed data quality within one power flow framework [164]. The hybrid stochastic and interval formulation in [164] applies probabilistic sampling to wind and PV while retaining selected inputs as bounded intervals. The resulting outputs therefore reflect both probabilistic variability and guaranteed bounds. The probabilistic interval formulation in [165] further incorporates correlations among uncertain injections to obtain tighter bounds. Extensions also cover distribution-specific features, such as three-phase power flow under mixed probability and interval uncertainty [166], and evidence theory combined with affine arithmetic to represent model uncertainty alongside bounded propagation without relying on a single assumed probability model [167]. These hybrid formulations can represent uncertainty sources with different levels of available information, but they require more complex uncertainty propagation than methods based on a single uncertainty description.

Table 6. Summary of uncertainty-based methods for power flow analysis.

Method	Uncertainty Input	Strength	Limitation
Monte Carlo simulation	Probability density functions for uncertain loads and DER injections	General and widely accepted benchmark when a sufficiently large number of samples are required	High computational cost due to a large number of deterministic power flow runs
Quasi Monte Carlo sampling	Probability density functions mapped to low-discrepancy sequences such as Sobol or Halton	Often achieves comparable accuracy with fewer simulations than random Monte Carlo	Sequence generation and distribution mapping must be handled properly, and efficiency can degrade in high-dimensional problems
Latin hypercube sampling	Stratified sampling of each input probability distribution with a permutation-based combination	Improved space filling typically reduces variance relative to simple random sampling	Correlation handling requires additional processing, and repeated deterministic runs remain necessary
Convolution method	Input probability distributions, often discretized for a tractable combination	Avoids repeated sampling when the uncertainty dimension is small	Computational effort grows quickly with the number of uncertain inputs, and correlation modeling is challenging
Cumulant method	Input statistics and cumulants propagated through linearized relations with series-based reconstruction	Lower computational effort than sampling-based methods and suitable for repeated evaluation	Accuracy depends on the linearization quality and the truncation used in the distribution approximation
Point estimation methods	Limited input statistics represented through weighted representative points	Requires only a small number of deterministic evaluations	Accuracy can deteriorate under strong nonlinearity, non-Gaussian inputs, or complex dependence
Unscented transformation	Input mean and covariance represented through sigma points	Captures nonlinearity more effectively than simple cumulant-based approximations with few evaluations	Tail behavior and rare events may be missed due to a limited set of points being propagated
Interval power flow	Bounded intervals for uncertain inputs without assuming probability distributions	Does not require probability models and can operate with limited statistical data	Can be conservative due to bound widening and dependence issues as the number of uncertain inputs grows
Fuzzy power flow	Fuzzy numbers for imprecise inputs propagated through $\alpha$-cut-based interval calculations	Appropriate when inputs are imprecise and statistical characterization is not credible	Computational burden increases with the number of fuzzy inputs, and overestimation can arise from repeated interval operations

summarizes uncertainty-based methods for power flow analysis by comparing their uncertainty inputs, main strengths, and key limitations.

Uncertainty-based power flow formulations capture variability in loads and renewable injections, but they also require repeated network evaluations across many samples, scenarios, or time steps. As the number of evaluations increases, computational runtime can become the dominant limitation. This motivates power flow acceleration strategies that reduce the cost of repeated PF evaluations without sacrificing numerical robustness.

6. Power Flow Acceleration Methods

Recent acceleration strategies can be grouped into three main categories, illustrated in Figure 9: (i) solver-level methods based on sparse numerical linear algebra and factorization; (ii) topology-aware methods that exploit network and graph structure for fast updates; (iii) model-level methods using data-driven and physics-informed surrogates to accelerate repeated power flow evaluations through learned mappings and physics-based representations. Together, these strategies enable scalable deterministic time-series studies, accelerate uncertainty-driven analyses, and support large-scale repeated power flow evaluation across many operating points.

Table 7. Historical evolution of uncertainty-based and accelerated power flow methods.

Period	Key Development	Main Advantage	Main Limitation	Research Trend
1980s–1990s	Cumulant- and interval-based formulations established analytical and bounded representations of uncertain PF inputs and outputs [152].	Reduced dependence on repeated random simulations and enabled uncertainty bounds.	Accuracy depended on linearization assumptions, while interval estimates could become conservative.	Improved uncertainty propagation and more accurate representation of correlations among uncertain variables.
2000s	Monte Carlo, point-estimation, and fuzzy methods expanded uncertainty analysis using probabilistic samples, representative points, and imprecise input descriptions [126,136,139,160].	Supported different forms of uncertainty with simple implementation.	Monte Carlo required many PF evaluations, while approximate and fuzzy methods could lose accuracy or widen output ranges.	Structured sampling and reduced-evaluation techniques to lower the computational burden.
2010–2015	Quasi-Monte Carlo, Latin hypercube, and unscented transformation improved probabilistic PF efficiency [130,132,143,144].	Required fewer deterministic evaluations than conventional Monte Carlo.	Performance remained sensitive to dimensionality, correlations, and nonlinear operating behavior.	Sparse expansions and correlation-aware models for high-dimensional uncertainty analysis.
2016–2020	Sparse polynomial-chaos and correlation-aware models improved uncertainty representation [147,148,155], while data-driven mappings supported faster PF evaluation [164,168].	Improved scalability and representation of correlated uncertainty.	Model construction required representative data, suitable basis functions, or information on variable dependence.	Adaptive surrogate models and physics-guided learning for improved scalability and reliability.
2021–2023	Deep-learning models reduced online PF computation, while GPU-based platforms enabled parallel evaluation of large scenario sets [45,169,170,171].	Provided rapid evaluation and efficient parallel processing.	Reliability depended on training coverage and available computational resources.	Greater integration of physical constraints and network-topology information into accelerated PF models.
2024–present	Graph-based and physics-informed models improved adaptation to changing operating conditions and network structures [172,173,174,175,176].	Improved generalization with less dependence on extensive labeled data.	Performance remains sensitive to unseen conditions and topology changes.	Current research seeks reliable real-time PF models that remain physically feasible under changing operating conditions and network topologies.

summarizes the historical evolution of uncertainty-based and accelerated power flow methods, highlighting their main developments, limitations, and the research directions that emerged in response.

6.1. Fast PF Using Numerical Linear Algebra and Network Structure

Fast PF evaluation approaches accelerate repeated computations by exploiting sparse numerical linear algebra and network structure. Matrix-based methods formulate power flow in sparse matrix form and improve efficiency through sparse factorization followed by repeated triangular substitutions across operating points under a fixed network configuration. However, graph-based methods express the equations through incidence and loop representations derived from connectivity and use these structures to evaluate power flow efficiently in radial and weakly meshed networks, especially under operating-point variations.

• Matrix-based formulations:
  Matrix-based formulations are attractive in distribution networks because they express repeated evaluations as sparse linear solves whose structure is determined by network topology. For a fixed network configuration, the sparse system matrix can be factorized once, and the stored factors can be reused efficiently as injections vary across time steps or scenarios, thereby reducing total computation in large time-series and probabilistic studies. This relies on classical sparse triangular factorization techniques with careful ordering that keeps the factors sparse and limits extra nonzero terms, improving both runtime and memory requirements for large network equation sets [113,177].
  In distribution feeders, several formulations construct constant network matrices directly from feeder connectivity so that the same matrices can be applied repeatedly as operating conditions change. For radial networks, an upper triangular branch-to-node mapping derived from the connectivity avoids explicit matrix inversion, leading to faster computation and reduced memory usage [178]. Radial DC resistive grids with constant-power loads further exploit this structure through a triangular formulation and a primitive impedance representation that avoids costly inversions inside the iteration [179]. For LVDC distribution grids, Laplacian and connectivity matrices support efficient repeated solutions in radial and weakly meshed networks under stated convergence and uniqueness conditions [180]. A related formulation for unbalanced three-phase feeders builds constant matrices that link bus injections to branch currents and branch currents to bus voltages [181]. Each iteration therefore relies mainly on matrix multiplication instead of repeated Jacobian or admittance factorization. Beyond connectivity-driven matrix constructions, implicit linearization of the power flow manifold yields a structure-preserving linear approximant whose sparsity pattern follows the network [182]. Although these formulations use different network representations, each improves repeated evaluations through the reuse of constant or sparse matrices. Their efficiency decreases when frequent topology changes, stronger meshing, or ill-conditioning require matrix updates, additional iterations, or damping [113,177,181].
• Graph-based formulations:
  Graph-based formulations model a distribution network as a sparse graph and use branch-to-node incidence information to assemble the power flow equations directly from connectivity. Their main advantage is computational efficiency in probabilistic studies, since most operations involve sparse graph traversals and structured matrix-vector products rather than repeated Jacobian or admittance factorization. The balanced formulation in [183] derives a unified current–voltage relationship for both radial and meshed feeders, avoiding mesh breaking and special node-layering rules. For unbalanced feeders, the incidence-matrix formulation in [21] develops a compact power flow model in the $\alpha \beta 0$ frame using KCL and KVL with embedded transformer modeling. Compared with the balanced formulation, it provides a more detailed representation of phase unbalance, but requires additional transformation and device modeling.
  For weakly meshed networks, loop information is added to represent tie lines and mesh paths while preserving the connectivity-based structure. The method in [184] uses active and reactive branch-power flows with tree labeling to reduce mismatch calculations in large weakly meshed networks. Graph-based methods are therefore suitable for radial, unbalanced, and weakly meshed feeders, but their computational benefit depends on network topology. Switching actions and reconfiguration may require rebuilding incidence and loop structures, while stronger meshing increases loop handling and can slow convergence [183,184].

6.2. Data-Driven and Physics-Informed PF Methods

Data-driven and physics-informed power flow methods accelerate repeated evaluations by learning surrogate mappings from operating conditions to power flow solutions, reducing the need for full numerical solves in studies that require many evaluations across time steps and operating points. Data-driven approaches construct regression-based or neural network-based predictors from historical or simulation-generated samples to estimate outputs such as bus voltages, voltage angles, and line flows [168], while graph neural networks are particularly effective because they align with the network graph structure and can generalize better across operating points [171]. Moreover, physics-informed approaches improve feasibility and robustness by embedding physical principles and network constraints directly into the learning process, either by enforcing power flow equation residuals during training through physics-informed neural networks or by using physics-guided learning that constrains model structure and outputs using domain knowledge, thereby providing fast yet physically consistent power flow approximations [185]. Table 8 summarizes data-driven and physics-informed power flow surrogate methods by highlighting their study type, surrogate category, main approach, and key limitations.

(1)

Data-driven and regression-based PF surrogates:

Data-driven power flow surrogates reduce the computational cost of repeated evaluations by learning a direct mapping from operating conditions to power flow outputs. They are particularly useful for time-series simulation, probabilistic analysis, and uncertainty-based studies, where repeated iterative solutions can become computationally demanding. However, their accuracy and physical consistency depend on the quality and representativeness of the training data. Their performance may therefore deteriorate under measurement noise, unseen operating conditions, or topology changes.

Several studies develop linear or piecewise linear regression models for repeated power flow evaluation [168,186,187,188,189,190]. The framework in [168] applies forward and inverse regression, while partial least squares and Bayesian linear regression improve numerical conditioning and reduce overfitting. The noise-aware method in [186] introduces Jacobian-guided constraints to reduce sensitivity to errors in injections and voltage measurements. Generalization bounds in [187] further relate estimation error to sample size and model complexity. To represent wider operating ranges, the model in [188] applies improved K-plane regression to three-phase stochastic power flow. Other studies combine regression with physical information to improve accuracy and robustness. Hybrid formulations retain a physical linear model and learn only its residual error, improving branch-flow accuracy while preserving network structure [189,191]. When training data are incomplete, the model-aided approach in [190] incorporates parameter information and distributionally robust chance constraints. Regression-based models offer low computational cost and good interpretability, but a single linear mapping may lose accuracy over wide nonlinear operating ranges. Piecewise and hybrid formulations improve accuracy under changing conditions, although they require additional training and model design.

(2)

Graph neural network-based PF surrogates:

Graph neural networks use bus and branch connectivity directly, making them suitable for topology-dependent and multiphase power flow relationships. For unbalanced three-phase distribution grids, the model in [192] estimates network states with lower computational time than iterative power flow methods. Deep neural networks have also been applied to probabilistic power flow acceleration and multiple network configurations [169,193]. The multigraph neural network in [194] represents phase and branch coupling in unbalanced three-phase networks. More recent graph-based methods use multi-fidelity training or physics-guided message passing to improve topology transferability and probabilistic power flow accuracy [172,173]. Compared with regression-based models, graph and neural-network surrogates can represent stronger nonlinearities and network dependencies. Their main limitations are higher data requirements and weaker interpretability. Their generalization also depends on whether the training data represent the expected operating conditions and topology changes.

(3)

Physics-informed and hybrid AI approaches:

Physics-informed and hybrid power flow surrogates aim to retain the speed of learned models while improving physical consistency. They embed network equations or physical constraints into the learning process. These methods are particularly relevant when purely data-driven models are affected by unseen operating conditions or strong nonlinearities [185,195]. Recent developments also include physics-informed neural networks with adaptive activation that incorporate power flow residuals and partial topology information [176]. These approaches can improve generalization and reduce physically infeasible predictions. However, their performance depends on the formulation and weighting of the physical constraints.

The physics-guided deep learning model in [170] combines voltage prediction with power-mismatch regularization, reducing overfitting and improving out-of-sample performance. The physics-embedded graph convolution method in [171] incorporates a linearized power flow model to improve prediction under uncertain injections and topology variations. The graph-based surrogate in [185] assigns adaptive weights to neighboring nodes and penalizes violations of Kirchhoff’s laws. The loss function in [195] combines nodal power imbalance with real-power-loss terms to preserve meaningful voltage profiles under unseen conditions. These methods use different forms of physical information, but all aim to improve consistency beyond purely data-driven training. Their main limitation is the additional design and training complexity required to represent network equations effectively.

Hybrid AI approaches combine learned models with physical information to reduce data requirements and improve adaptability under changing operating conditions. The framework in [196] uses physical constraints and data augmentation when training data are limited. The transfer-learning method in [174] embeds power-balance equations in the fine-tuning loss and adapts to new configurations using unlabeled data. Self-supervised methods also use unlabeled operating points through physical-consistency objectives [175]. For probabilistic power flow acceleration, the method in [173] incorporates AC power flow sensitivities into a graph neural network and applies the trained surrogate within Monte Carlo sampling. Transfer and self-supervised approaches are particularly useful when labeled data are limited or the network configuration changes. Their reliability still depends on the quality of the physical constraints and the coverage of the available operating data.

Table 8. Summary of data-driven and physics-informed PF surrogate methods.

Ref	Study Type	Surrogate Category	Approach	Limitations
[168]	Deterministic PF	Regression-based	Forward and inverse regression to learn linear PF relations, with partial least squares and Bayesian linear regression to reduce overfitting	Accuracy depends on the operating range represented in the data, and a single linear mapping can degrade under large regime shifts.
[186]	Deterministic PF	Regression-based	Data-driven linearization with Jacobian-guided constraints, solved via constrained quadratic programming to reduce sensitivity to measurement noise	Requires consistent noisy samples and careful constraint scaling, and performance can drop under unseen regimes.
[187]	Deterministic PF	Regression-based	Generalization error bounds for regression-based steady-state models, linking error behavior to sample size and model complexity	Bounds can be conservative and depend on assumptions on the data and model class.
[188]	Stochastic/ probabilistic PF	Regression-based	Improved K-plane regression for piecewise linearization in three-phase stochastic PF, enabling fast evaluation in sampling loops	Piecewise partitioning requires careful regime selection, and complexity increases with operating diversity.
[189]	Deterministic PF	Hybrid physics-guided regression surrogate	Physical linear baseline retained with regression-based residual correction to improve branch-flow accuracy	Benefit depends on baseline quality and residual structure.
[190]	Deterministic PF	Hybrid physical-model-aided regression	Physical-model-aided linear surrogate with parameter guidance and distributionally robust chance constraints to address limited training data	Requires partial parameter knowledge and tuning of chance constraints, which can affect conservativeness.
[171]	Deterministic PF	Physics-embedded graph neural network	Graph convolution surrogate with embedded linearized PF representation to improve robustness under uncertainty and topology variation	Training still depends on scenario diversity, and generalization can degrade under strongly unseen topology patterns.
[170]	Deterministic PF	Physics-guided deep neural network	Deep neural network trained with voltage prediction loss and power mismatch regularization to improve consistency and generalization	Performance depends on the weighting between data loss and physics regularization, and training can be sensitive to tuning.
[169]	Probabilistic PF	Deep neural network with physics-based loss	Model-based deep learning for probabilistic PF with branch-flow physics embedded in the training objective to accelerate Monte Carlo evaluation	Requires representative probabilistic training scenarios.
[193]	Probabilistic PF	Deep neural network	Surrogate extended across multiple topology configurations through learned mappings and transfer across configurations	Needs diverse topology states in training, and accuracy depends on topology representation quality.
[172]	Deterministic PF	Graph neural network	Multi-fidelity training combining low-fidelity and high-fidelity data to reduce labeling costs while retaining accuracy	Requires careful design and alignment of fidelity levels, and depends on the quality of high-fidelity reference data.
[173]	Probabilistic PF	Physics-guided graph neural network	Physics-guided graph learning using sensitivity guidance, with the surrogate deployed inside Monte Carlo sampling for fast probabilistic PF	Sensitivity guidance and training distribution influence robustness, and performance can drop under unseen uncertainty structure.
[185]	Deterministic PF	Physics-informed graph-based model	Graph-based physics-informed surrogate with adaptive neighbor weighting and Kirchhoff consistency penalty to support inductive generalization	Constraint penalty calibration is needed for stable learning, and generalization still depends on training diversity.
[195]	Deterministic PF	Physics-informed neural model	Improved physics-informed loss with nodal mismatch and line-loss terms to preserve voltage profiles under unseen conditions	Effectiveness depends on loss formulation and scaling, and optimization stability can be sensitive to tuning.
[176]	Deterministic PF	Physics-informed neural network	Adaptive activation with power flow residuals and partial topology information for PF prediction across systems of different sizes	Performance depends on training-data coverage, physics-loss design, and the available network information.
[174]	Deterministic PF	Physics-informed transfer learning	Unsupervised physics-informed transfer learning using power-balance residual in the fine-tuning loss to adapt to new configurations without labels	Adaptation quality depends on initialization and tuning of the physics-loss strength.

Table 8. Summary of data-driven and physics-informed PF surrogate methods.

Ref	Study Type	Surrogate Category	Approach	Limitations
[168]	Deterministic PF	Regression-based	Forward and inverse regression to learn linear PF relations, with partial least squares and Bayesian linear regression to reduce overfitting	Accuracy depends on the operating range represented in the data, and a single linear mapping can degrade under large regime shifts.
[186]	Deterministic PF	Regression-based	Data-driven linearization with Jacobian-guided constraints, solved via constrained quadratic programming to reduce sensitivity to measurement noise	Requires consistent noisy samples and careful constraint scaling, and performance can drop under unseen regimes.
[187]	Deterministic PF	Regression-based	Generalization error bounds for regression-based steady-state models, linking error behavior to sample size and model complexity	Bounds can be conservative and depend on assumptions on the data and model class.
[188]	Stochastic/ probabilistic PF	Regression-based	Improved K-plane regression for piecewise linearization in three-phase stochastic PF, enabling fast evaluation in sampling loops	Piecewise partitioning requires careful regime selection, and complexity increases with operating diversity.
[189]	Deterministic PF	Hybrid physics-guided regression surrogate	Physical linear baseline retained with regression-based residual correction to improve branch-flow accuracy	Benefit depends on baseline quality and residual structure.
[190]	Deterministic PF	Hybrid physical-model-aided regression	Physical-model-aided linear surrogate with parameter guidance and distributionally robust chance constraints to address limited training data	Requires partial parameter knowledge and tuning of chance constraints, which can affect conservativeness.
[171]	Deterministic PF	Physics-embedded graph neural network	Graph convolution surrogate with embedded linearized PF representation to improve robustness under uncertainty and topology variation	Training still depends on scenario diversity, and generalization can degrade under strongly unseen topology patterns.
[170]	Deterministic PF	Physics-guided deep neural network	Deep neural network trained with voltage prediction loss and power mismatch regularization to improve consistency and generalization	Performance depends on the weighting between data loss and physics regularization, and training can be sensitive to tuning.
[169]	Probabilistic PF	Deep neural network with physics-based loss	Model-based deep learning for probabilistic PF with branch-flow physics embedded in the training objective to accelerate Monte Carlo evaluation	Requires representative probabilistic training scenarios.
[193]	Probabilistic PF	Deep neural network	Surrogate extended across multiple topology configurations through learned mappings and transfer across configurations	Needs diverse topology states in training, and accuracy depends on topology representation quality.
[172]	Deterministic PF	Graph neural network	Multi-fidelity training combining low-fidelity and high-fidelity data to reduce labeling costs while retaining accuracy	Requires careful design and alignment of fidelity levels, and depends on the quality of high-fidelity reference data.
[173]	Probabilistic PF	Physics-guided graph neural network	Physics-guided graph learning using sensitivity guidance, with the surrogate deployed inside Monte Carlo sampling for fast probabilistic PF	Sensitivity guidance and training distribution influence robustness, and performance can drop under unseen uncertainty structure.
[185]	Deterministic PF	Physics-informed graph-based model	Graph-based physics-informed surrogate with adaptive neighbor weighting and Kirchhoff consistency penalty to support inductive generalization	Constraint penalty calibration is needed for stable learning, and generalization still depends on training diversity.
[195]	Deterministic PF	Physics-informed neural model	Improved physics-informed loss with nodal mismatch and line-loss terms to preserve voltage profiles under unseen conditions	Effectiveness depends on loss formulation and scaling, and optimization stability can be sensitive to tuning.
[176]	Deterministic PF	Physics-informed neural network	Adaptive activation with power flow residuals and partial topology information for PF prediction across systems of different sizes	Performance depends on training-data coverage, physics-loss design, and the available network information.
[174]	Deterministic PF	Physics-informed transfer learning	Unsupervised physics-informed transfer learning using power-balance residual in the fine-tuning loss to adapt to new configurations without labels	Adaptation quality depends on initialization and tuning of the physics-loss strength.

6.3. Computational Platforms and Specialized Network Applications

PF methods require different computational implementations and network models depending on the study scale and system configuration. The following applications summarize GPU acceleration, parallel and cloud-based computation, together with formulations developed for isolated microgrids and hybrid AC/DC networks.

• GPU-based PF computation: GPU implementations accelerate PF by assigning repeated numerical operations or independent PF cases to multiple parallel processing units. A Chebyshev-preconditioned conjugate-gradient solver is implemented for linearized DC PF in [197], while an OpenCL implementation evaluates Monte Carlo samples concurrently for probabilistic PF in [198]. Batched Newton–Raphson calculations are parallelized across CPUs and GPUs in [45], where reusable sparse structures reduce the computation required for time-series, contingency, and probabilistic studies. These implementations are particularly useful when large numbers of PF cases must be evaluated repeatedly.
• Cloud-based PF computation: Cloud environments distribute PF data and calculations across remote or clustered computing resources. The InterPSS Cloud Edition in [199] provides remotely accessible load flow, contingency-analysis, and network-data services through Google App Engine. The Spark-based framework in [200] stores network data in resilient distributed datasets and organizes Newton–Raphson operations through a directed acyclic graph, reducing repeated data transfer and improving scalability for large-scale PF calculations.
• PF for isolated microgrids: Isolated microgrids require PF formulations that determine voltage and frequency without support from an upstream slack bus. A modified backward–forward sweep method represents islanded radial operation in [59], while a direct-sweep-based formulation incorporates droop-controlled sources in AC microgrids in [60]. Newton-based PF for DC microgrids is examined in [4], with emphasis on convergence under converter-controlled sources and constant-power devices. Uncertainty in unbalanced three-phase islanded microgrids is further represented through an unscented probabilistic PF formulation in [145].
• PF for hybrid AC/DC networks: Hybrid AC/DC systems require coordinated solution of the AC and DC subsystems through converter models and control equations. Newton–Raphson and Gauss–Seidel methods are applied to a hybrid AC/DC system in [201], while the sequential unbalanced formulation in [202] couples three-phase AC equations with DC-network, voltage-source converter (VSC), and DC/DC converter models under different control modes.
• Real-time state estimation and situational awareness: PF models support distribution-system state estimation by reconstructing voltages and branch flows from available measurements. These estimates provide the network states required for real-time situational awareness. Digital-twin deployment studies show that reliable operation depends on measurement quality, network-model accuracy, and bad-data handling, in addition to PF solution speed [47]. End-to-end latency must also include data acquisition, communication, synchronization, topology processing, and estimator execution. Mixed-frequency distributed estimation can reduce the computational and communication burden associated with heterogeneous measurement rates [203]. Learning-based state estimation can improve network visibility when only limited real-time measurements are available [204]. Power-flow-informed learning models further support state estimation in partially observable distribution networks by embedding network physics into the estimation process [205]. The PF or surrogate model should therefore be completed within the measurement-update interval, leaving sufficient time for data processing and subsequent operator or control actions.

7. Discussion and Future Research Directions

This section synthesizes the main findings across the reviewed power flow literature and identifies the principal limitations and future research needs. Since Table 1 compares the coverage of previous review papers with the present review, the discussion focuses on the power flow method categories examined in Section 3, Section 4, Section 5 and Section 6. Particular attention is given to the practical trade-offs among modeling accuracy, convergence reliability, uncertainty representation, and computational performance.

7.1. Synthesis and Method-Selection Criteria

The reviewed methods differ mainly in the balance they provide among modeling accuracy, convergence reliability, and computational performance. Conventional nonlinear methods retain detailed voltage, reactive-power, and control representations [61,66]. This makes them suitable for detailed operational studies and benchmark analysis [29,62]. Backward/forward sweep methods are computationally attractive for radial feeders because they exploit the feeder structure and avoid full Jacobian construction [53,58]. Newton-type formulations provide greater flexibility for meshed, unbalanced, and component-detailed networks [68,80]. However, nonlinear methods remain iterative and may experience convergence difficulties under stressed or ill-conditioned operating conditions [30,117].

Linearized methods reduce the computational burden of repeated nonlinear solutions and are widely used in time-series and optimization studies [14,81]. Their reliability depends on the network characteristics, operating range, and assumptions used in their derivation [33,98]. Accuracy can decrease under large voltage deviations, topology changes, or operation outside the approximation region [36,91]. Therefore, linearized models should be treated as application-dependent approximations rather than general replacements for nonlinear power flow.

When convergence reliability becomes the main concern, robustness-oriented methods are more appropriate. These methods improve numerical reliability under heavy loading, weak-grid conditions, control switching, and poor initialization [117,119]. Damping and optimal-multiplier approaches improve convergence by modifying the iterative update process [114,120]. Continuation and homotopy-based methods enlarge the convergence region by gradually moving from an easily solvable initial case toward the required operating condition [123,124]. Their main limitation is the additional computation or algorithmic tuning required to improve convergence reliability.

Beyond convergence issues, uncertainty-based formulations are required when load and renewable injections cannot be represented by fixed values. Probabilistic methods are appropriate when probability distributions or representative samples are available [31,32], while interval methods are suitable when uncertainty is described through bounded ranges [152,153]. Fuzzy methods are more appropriate when the available information is imprecise and cannot be represented reliably through probability distributions [158,160]. Hybrid formulations combine different uncertainty descriptions when probabilistic variability and bounded or fuzzy information are both present [164,166]. However, uncertainty-based methods can require substantial computation when many scenarios, uncertain variables, and time steps are considered [12,127].

This computational burden motivates acceleration strategies for repeated power flow evaluations. Sparse numerical methods exploit the structure of the network equations [113,177], while graph-based formulations use network topology to simplify the solution process [183]. Parallel and GPU-based implementations reduce runtime for large numbers of power flow calculations [45,197]. Data-driven and physics-informed models also provide fast mappings from operating conditions to power flow outputs [168,171]. However, their reliability depends on training-data coverage [187], robustness to measurement noise [186], and generalization to unseen operating conditions or topology changes [172,193].

The temporal distribution of the reviewed studies further reflects this methodological development, as shown in Figure 10. Early studies focused mainly on conventional deterministic methods. Later research increasingly considered linearized, uncertainty-based, and acceleration-oriented approaches. Robust power flow has remained relevant because stressed conditions, control limits, and topology changes continue to affect convergence. These comparisons indicate that no power flow method category is uniformly superior. The practical choice depends on network topology, required accuracy, convergence conditions, available uncertainty information, and computational requirements, as summarized in

Table 9. Practical criteria for selecting power flow methods in modern distribution-network studies.

Selection Criterion	Recommended Method	Accuracy and Performance	Complexity and Scalability
Radial feeder under normal operation	Backward/forward sweep	Provides detailed AC solutions with reliable convergence; loops and voltage-controlled buses require additional treatment.	Low cost and memory demand; highly scalable for radial feeders.
Meshed, unbalanced, or component-detailed network	Newton-type or multiphase nonlinear PF	Retains voltage, reactive-power, loss, and phase-coupling detail, but requires suitable initialization and may be affected by ill-conditioning.	Jacobian formation and factorization increase computation and memory demand; scalability is moderate.
Large-scale planning and repeated time-series or scenario analysis	Linearized PF	Accuracy and feasibility may deteriorate outside the validity range. Detailed guidance is given in Table 10.	Low solution cost and high scalability within the intended operating range.
Stressed or ill-conditioned network	Robust PF technique	Improves convergence under difficult operating conditions but requires algorithmic tuning.	Damping, continuation, or homotopy steps increase cost and reduce scalability for large scenario sets.
Probability-based uncertainty	Probabilistic PF	Provides statistical information on voltages and flows, with accuracy depending on the uncertainty model.	Computational effort increases with the number of samples, uncertain variables, and time steps.
Bounded or imprecise uncertainty	Interval or fuzzy PF	Interval PF gives bounded ranges, whereas fuzzy PF represents graded imprecision without complete probability distributions.	Complexity and conservativeness increase with the number of uncertain inputs and their correlations.
Real-time screening or repeated evaluation	PF acceleration strategy or physics-informed surrogate	Reliability depends on physical consistency, training-data coverage, and generalization.	Low online cost and high scalability, but offline training and validation may be significant.

.

Among these categories, linearized power flow methods require more specific selection criteria because their accuracy depends strongly on network characteristics and the operating range.

Table 10. Quantitative guidance for selecting linearized power flow methods.

Method	Reported Threshold, Range, or Condition	Selection Guidance	Computational Profile
Conventional DCPF	An indicative boundary of $X/R=4$ $(R/X=0.25)$ was associated with an average active-flow error of about $5\%$ [206].	Use for predominantly inductive networks with nearly flat voltages. Avoid for high-$R/X$ feeders.	Very low solution cost and high scalability due to its sparse linear formulation.
Classical LinDistFlow	Tested for $k\in [-2.5,2.5]$, minimum voltages of $0.908$–$0.946$ p.u., loads up to $150\%$, and power factors of $0.7$–$1.0$ [81].	Use for radial feeders near the reference condition. Validate under heavy loading, low voltages, or reverse flow.	Very low solution cost and good scalability for radial feeders.
Calibrated or optimized LinDistFlow	Parameterized models were tested over wide loading ranges [91]. Optimized parameters reduced $L_1$- and $L_{\infty }$-norm voltage errors by up to $92\%$ and $88\%$ [90].	Use when representative calibration scenarios are available. Recalibrate after major changes in the operating range or topology.	Low online cost, with additional offline calibration or parameter optimization.
Sensitivity/ Jacobian-based	Online parameters were updated using measurements from the preceding operating point [44].	Use near the current operating point. Increase the update frequency when operating conditions vary rapidly.	Low online cost after sensitivity computation, but repeated updates may be required.
Fixed-point linearization	A practical solution is guaranteed when $V_0^2{>4\parallel Z\parallel }^{*}\parallel s_{\mathcal{L}}\parallel$, with an explicit approximation-error bound [98].	Use within the stated solvability region. Revalidate under stressed loading or near voltage limits.	Low evaluation cost after constructing the fixed-point mapping.
Sample-based adaptive	For loads from $30\%$ to $170\%$ of nominal values, rational approximations reduced voltage errors by $14.51$–$78.56\%$ relative to linear approximations [108].	Use within the sampled range. Extend the samples when the expected operating range changes.	Low online cost, with additional offline sampling and model fitting.

summarizes quantitative conditions and model-specific recommendations reported in the literature. The reported values represent tested ranges or study-specific thresholds and should not be interpreted as universal validity limits.

7.2. Limitations and Open Research Gaps

Although the reviewed literature has advanced several power flow approaches for modern distribution-network analysis, methodological challenges remain due to differences in model assumptions, operating conditions, uncertainty representation, and computational requirements.

7.2.1. Lack of Standardized Benchmarking and Comparative Evaluation

A major limitation in the existing literature is the lack of consistent benchmarking across power flow studies. Although many papers evaluate accuracy, computation time, and stressed operating conditions, they use different test feeders, loading scenarios, DER penetration levels, solver settings, and performance metrics [26,34]. Direct comparison among methods therefore remains difficult, and the reported performance may depend strongly on the selected network and operating conditions. Consequently, it is often unclear whether a method is generally reliable or effective only under the assumptions of a particular study.

7.2.2. Limited Validity Assessment of Linearized Models

Linearized power flow models are often evaluated over limited operating ranges. Their accuracy can deteriorate when the system operates far from the linearization point or encounters conditions not represented during model development [14,98]. This limitation is evident in the OPF tests reported in [36], where the DC approximation produced a maximum voltage violation of $0.029$ p.u. in the 6-bus system, while the non-conservative linear approximation resulted in a $0.004$ p.u. violation in the 14-bus system. These results show that solutions considered feasible by an approximate model may still violate the nonlinear power flow constraints. Many studies emphasize average voltage or flow errors [95,96], while worst-case errors and validity-region boundaries receive less consistent attention. These models should therefore be validated over the intended operating range before use in planning or operational studies.

7.2.3. Robustness Under DER-Rich and Stressed Operating Conditions

Robust power flow methods improve convergence reliability, but their performance is not always evaluated under the full range of conditions expected in active distribution networks. High DER penetration introduces inverter reactive-power limits and PV-to-PQ switching [54,57]. Islanded operation adds voltage- and frequency-control requirements [59], while topology changes alter network characteristics [118]. In the IEEE 118-bus tests reported in [22], traditional continuation power flow diverged for DG penetration levels of $10\%$, $20\%$, and $30\%$ when undervoltage protection triggered DG disconnections. The resulting error in the estimated disconnected DG capacity reached $27\%$, indicating limited reliability near discontinuous operating points. Existing robust methods are often assessed mainly through convergence behavior [117,121], whereas runtime, scalability, initialization sensitivity, and device-control switching are not always evaluated together.

7.2.4. Computational Burden in Uncertainty-Based Power Flow

Uncertainty-based power flow methods provide information about variability and risk, but their computational cost remains a major limitation. Sampling-based methods require repeated deterministic PF solutions, while interval and fuzzy formulations may become conservative or computationally expensive as the number of uncertain inputs increases [126,152]. Correlations among uncertain injections are also difficult to represent accurately [135,143]. Ignoring these dependencies may distort voltage-risk and branch-loading estimates. Uncertainty-aware PF methods therefore require improved scalability and more realistic representation of dependent variables.

7.2.5. Generalization of Data-Driven and Physics-Informed Surrogates

Data-driven and physics-informed PF surrogates enable fast repeated evaluation, but their generalization beyond the training domain remains limited. Many models are trained and tested under similar operating conditions [168,188], whereas practical networks may experience topology changes [193], device-control switching [54,59], and measurement noise [186]. At a measurement-noise level of 40 dB, the conventional data-driven linearization in [186] produced errors one to three orders of magnitude larger than its noiseless counterpart, indicating that accuracy obtained from noise-free data may not be maintained with field measurements. Training-data coverage therefore directly affects surrogate reliability [187]. Physics-informed learning improves consistency by embedding PF equations or network constraints [170,171], but its performance depends on how these constraints are formulated and weighted. Without sufficient validation, predictions may violate power balance, voltage limits, or branch-flow consistency outside the training domain.

Table 11. Key research gaps in power flow methods for modern distribution networks.

Gap Theme	Key Research Gap	Relevance to PF Studies	Priority
Benchmarking and comparison	Existing studies use different feeders, loading levels, DER scenarios, solver settings, and performance metrics.	Fair comparison among power flow methods remains limited by differences in benchmark systems, operating scenarios, and reporting criteria.	High
Validity of linearized PF	Linearized models are often tested within limited operating ranges and selected loading conditions, with less consistent assessment in OPF and uncertainty-based applications.	Accuracy may deteriorate under high R/X ratios, unbalance, topology changes, voltage-control actions, and stressed loading.	High
Robust convergence	Robustness is often evaluated without consistently reporting initialization sensitivity, device-control switching, and solver failure behavior.	Practical reliability under DER-rich and ill-conditioned operation remains uncertain.	High
Uncertainty representation	Correlation, rare events, mixed uncertainty types, and dependency among uncertain variables remain difficult to model.	Voltage-risk and branch-loading estimates may be under- or over-estimated when uncertain injections are not represented realistically.	Medium–High
Computational scalability	Scenario-based and probabilistic methods often require many repeated PF evaluations.	Large feeders, three-phase models, and long time-series studies can become computationally expensive.	High
Surrogate generalization	Data-driven and physics-informed PF models may not generalize when training data do not cover unseen loading regimes, topology changes, rare operating conditions, or noisy measurements.	Fast predictions may lose accuracy or feasibility outside the training domain.	High
Physical feasibility	Some learning-based models may violate power balance, voltage limits, branch-flow consistency, or loss relationships.	Operational use becomes difficult without feasibility checks, correction layers, or physics-based constraints.	High

summarizes the key research gaps identified from the reviewed literature.

7.3. Future Research Directions

The reviewed literature identifies several unresolved challenges and emerging research topics that require further investigation for the reliable application of power flow methods in modern distribution networks.

• Validity assessment of linearized power flow models: A major unresolved challenge is the lack of a reliable criterion for identifying when changes in loading, topology, or voltage-control conditions make a linearized PF model inaccurate [44,90,108]. Future work should establish computable error indicators that compare the estimated approximation error with a specified tolerance and initiate model recalibration when this tolerance is exceeded. Conservative formulations should also be extended to optimization problems so that approximation errors do not produce infeasible operating decisions [110].
• Power flow under changing control states: The interaction of inverter controls with tap changers and capacitor switching can alter the governing equations during the solution process and lead to discontinuities or convergence difficulties [13,22,54]. Future formulations should update the relevant equations when control modes change while preserving numerical stability throughout the solution. Their performance should also be examined when several control actions occur sequentially. Self-adaptive methods developed for changing power flow directions provide a useful basis for this direction [35].
• Scalable treatment of correlated uncertainty: Representing dependence among uncertain injections becomes increasingly difficult when many resources and time periods are considered [135,154,155]. Simplified dependence models may distort voltage-risk and branch-loading estimates, whereas detailed formulations can require excessive computation. Future research should develop reduced representations that preserve the dominant correlation structure and retain critical voltage or loading outcomes while limiting repeated PF evaluations. Sparse polynomial-chaos formulations [148], correlation-based expansions [149], and reproducible correlation-aware frameworks [151] provide relevant foundations for this development.
• Reliable surrogates beyond the training domain: A data-driven surrogate may produce a plausible output even when the current operating condition is poorly represented by the training data [170,171,173,195]. Future research should develop methods for assessing prediction reliability and detecting operating conditions outside the training range before surrogate outputs are used in operational studies. A correction stage should also restore nodal power balance and branch-flow consistency when the predicted state violates the network equations. Transfer learning [174], self-supervised training [175], and adaptive-activation physics-informed models [176] provide relevant directions for improving adaptability and consistency.
• Real-time power flow in digital-twin applications: A key unresolved issue is preserving the consistency of the digital model with the physical network when measurements are delayed or received at different rates [46,47]. Changes in topology or device status can further reduce model accuracy between successive updates. Future work should integrate fast PF calculation with systematic model correction and define when the network model must be updated using new measurements. The complete analysis time should also be evaluated against the available measurement interval, since solver speed alone does not determine real-time applicability [45,200].

8. Conclusions

Power flow analysis remains fundamental to distribution system planning and operation because it provides the steady-state voltages and branch flows required for network assessment and operating-limit verification. This paper presented a comprehensive review of distribution-oriented power flow methods within a unified framework that brings together conventional nonlinear formulations and linearized models while also examining numerical robustness, uncertainty representation, and computational acceleration techniques. The reviewed methods were compared according to their modeling assumptions, convergence behavior, accuracy, and computational requirements. This comparison was used to support method selection and to identify the main limitations of existing approaches.

The reviewed studies show that the suitability of a power flow method depends on the network characteristics and the requirements of the intended analysis. Conventional nonlinear methods remain necessary for detailed and physically consistent studies, although their convergence can become difficult under high R/X ratios, unbalanced operation, stressed loading, and control-mode switching. Linearized models reduce the computational burden of repeated evaluations, but their accuracy depends on the assumptions used in their derivation and the operating range over which they are applied. Their use should therefore be supported by validity assessment against nonlinear power flow solutions.

The review further indicates that convergence reliability, uncertainty representation, and computational efficiency directly influence the validity of distribution-network studies. Uncertainty in renewable generation and demand can significantly alter the calculated operating states, while faster computation is useful only when sufficient accuracy and consistency with network limits are preserved. This consideration is particularly important for time-series and scenario-based studies, where a large number of power flow evaluations must be completed without reducing physical reliability.

Finally, no single power flow formulation is suitable for all modern distribution-network studies. Method selection should therefore be guided by the network conditions and the purpose of the analysis, while an appropriate balance between accuracy and computational effort is maintained. The proposed review framework supports this selection by relating the assumptions and numerical behavior of each method to its practical applicability. Future research should focus on power flow frameworks that can assess their validity, adapt to evolving distribution-network conditions, and maintain reliable results across a wider operating range.