Linear Approximations of Power Flow Equations in Electrical Power System Modelling—A Review of Methods and Their Applications

Abstract

The power system is constantly changing. New elements are being added. The network structure is being changed. Existing network infrastructure is being modernised. This results in the need to perform various types of computational analyses. These analyses are designed to assess the impact of new and existing power system components on its operation. These analyses can be complicated, complex and difficult. These analyses are influenced by the size of the actual grid, technical conditions, and the specific requirements of grid operators. Therefore, there is a constant search for methods that simplify them without significantly compromising the accuracy of the obtained results. In real networks, computation time is also a crucial parameter, which can be excessive when analysing multiple operating variants. Linearisation of the computational model significantly contributes to reducing computational time but also affects their accuracy. Particular attention should be paid to two frequently used and compared linearization methods: the DC method and the first-order Taylor expansion. The DC (Direct Current) method offers relatively simple formulas. These relationships are based on simplifying assumptions. They are well suited for error analysis, at the expense of neglecting reactive power and voltage changes. Taylor linearization preserves the full AC structure around a selected operating point. It takes reactive power and voltage changes into account. Its accuracy is local and depends on the base point. It may require re-linearization for large changes in the power system operating states. This article presents a detailed literature review. It concerns the linearization of the power flow problem in the context of solving various computational problems in the power system. Selected works are grouped and categorised by topic. Linearisation methods used in the literature are presented, and the most frequently used ones are also indicated. Research gaps that may be addressed in future work are highlighted.

1. Introduction

Modern power systems are complex. Their safe, reliable and economical operation requires constant monitoring. Appropriate decisions should be made based on precise technical analyses. One of the key tools supporting power system management is power flow calculations. They allow for the determination of electrical values such as active and reactive power, voltages, and currents in individual network elements. Thanks to them, it is possible, among other things, to forecast the system’s operating status, assess its security and plan repairs. They facilitate decision-making regarding the connection of new customers and energy sources. They support operational and investment activities.

Power flow calculations, also known as load flow or power flow calculations, form the basis for many other networking problems. They enable analysis of the current state of the system, making it easier to predict its behaviour in response to changes in load, network configuration, or failures. In practice, they are performed repeatedly, both in planning mode and in real time. This requires them to be performed quickly and efficiently.

Classical power-flow solvers such as Newton–Raphson achieve excellent accuracy. Nevertheless, their computational cost grows substantially with network scale. In large national or international power systems, encompassing thousands of nodes and transmission lines, performing accurate calculations can require significant computational resources and time. This problem becomes particularly important in applications where the speed of obtaining results is crucial, such as in online analyses, real-time system control, fault analysis or automatic response to network changes.

To reduce computational time, an approach that simplifies real power flow models through linearisation is increasingly being used. Linearisation involves replacing the nonlinear equations describing power flow with their linear approximations. One of the oldest and most well-known methods of this type is the so-called DC method, which assumes constant voltages and ignores reactive power and line resistance, significantly simplifies the equations and allows for very quick calculation of the approximate active power flow.

Although simplified methods, such as the DC model or small increment linearisation, are associated with a loss of accuracy, in many cases they offer a sufficiently good approximation of the actual grid state. This allows them to be used in applications where response time is crucial, such as in algorithms for grid optimisation, overload threat prediction, or operator decision support. With the development of smart grids and the integration of renewable energy sources, which introduce significant variability and uncertainty into system operation, the need for fast and efficient calculations will become increasingly important.

This article is organised in such a way that the Section 1 presents an introduction. The Section 2 contains a literature review covering a broad spectrum of power engineering issues, along with proposed solutions based on linear models. This approach is receiving special attention due to its growing importance in scientific research and proven effectiveness in solving complex problems in large energy systems. Section 3 summarises the methods collected, as well as possible areas for future application of linearisation. Section 4 provides a detailed summary of the research gaps identified by the authors based on an analysis of the available literature. For a clear presentation of the results, a table of research gaps was proposed, allowing for clear identification and reference to specific topic areas. This table serves as a synthetic summary of the most important conclusions drawn from the literature review and can serve as a starting point for future research in this field.

Section 5 provides a comprehensive summary of the conducted research, including both the main findings and conclusions drawn from the analysis of the collected material. The practical and theoretical implications of the identified research gaps are also highlighted, suggesting possible directions for further research.

2. Literature Review on Linear Approximation Methods Used in Power System Analysis

In the following chapter, the authors present research examples from the literature that focus on selected issues related to the application of linearisation-based power flow methods for solving various problems in power systems. This subject is widely explored in the literature, and general issues in power engineering can be categorised into areas such as optimal power flow, overload analysis, dynamic constraints of power sources for overload mitigation, power system planning and development, as well as other selected topics. In their study, the authors have attempted to present the latest publications to familiarise the reader with the current state of knowledge on linearisation methods used.

Section 2.1, Section 2.2, Section 2.3 and Section 2.4 present a literature review covering a wide range of power engineering issues, along with proposed solutions based on linear models. Particular attention is paid to this approach due to its growing importance in research and its proven effectiveness in solving complex problems in large power systems.

2.1. Linear Approximations for the Power Flow Modelling

A fundamental element of power system analysis is the calculation of power flows [1,2,3]. Classical AC power flow equations are nonlinear and, in large networks requiring numerous near-real-time calculations, they are relatively time-consuming to perform [4]. In some cases, they may even fail to converge [5]. For this reason, the use of classical power flow calculation methods can be somewhat limiting in practical analyses [6]. Therefore, linear models have been developed for years [7].

In ref. [7], the authors present a distributed, linearised equivalent active power flow model that achieves high computational efficiency with minimal loss of accuracy. Its practical utility is demonstrated through case studies on radial distribution systems. In ref. [8], Dhople et al. proposed a linear model that captures voltages and reactive power, which the classic DC model did not. The method proposed by the authors satisfies the active power balance, and the method allows for improved accuracy while maintaining computational simplicity, making it useful in planning and market analyses. Li et al. [9] present an approach combining power flow linearisation methods with machine learning methods. The authors focus on presenting a heuristic model that corrects the linear approximation error, which further improves accuracy while maintaining low computational time, which is crucial for real-time overload assessment. In ref. [10], the authors presented a modified version of one of the most widely used linear approximations of the optimal power flow model for power systems. Their approach preserves the original linear structure, ensuring that the model remains easy and fast to compute. Linear approximation is commonly used in power flow models to meet the requirements for power flow calculations. According to the authors of ref. [11], such a conventional approach can cause significant computational error. Therefore, they propose an approach based on neural networks combined with power flow linearisation to obtain the most accurate result possible. According to the results obtained, such an approach can improve the accuracy of linear power flow by approximately 39%. In ref. [12] Liu, Zhang and colleagues propose an approach to power flow linearisation based on partial least squares and Bayesian linear regression. The proposed approach was tested on a number of standard cases, including transmission networks. In ref. [13], the authors use a model based on adaptive linear power flow to address network constraints. Additionally, to reduce computational burden and increase computational speed, they employ a model downscaling method by identifying critical branches. The proposed methodology was tested on multiple closed-loop test systems of different scales, as well as on an actual provincial power grid in China. The results demonstrated that the method effectively decreases power flow linearisation errors.

In recent decades, progress has been made in the approximate linearisation of AC optimal power flow (AC OPF). However, little research has focused on the simultaneous accurate approximation of reactive power and transmission losses. Also in [14], the linearisation method is used for rapid power system analysis. The geometric approach proposed by the authors for deriving a linear approximation of a set of possible power flows in a network is rarely encountered in power flow analysis, but according to Bolognani and Dörfler’s research, it is extremely effective. Paper [15] presents a comparison of approximations for two nonlinear phenomena: power flow and transmission losses in linear problems of expanding the connection capacity of hybrid Renewable energy sources (RES) installations. Also in ref. [16], linearisation of the AC load flow equations was used to optimise the operation of generators subject to constraints due to overloaded power system lines. In ref. [17], the authors propose an optimal linearisation of the power flow equations to improve power system dispatch in order to mitigate the occurring losses. The tests conducted on the IEEE 39 test network showed the flexibility of the method used by achieving a compromise between computational accuracy and task execution speed.

The interval power flow (IPF) method is a commonly used method for calculating power flows and resolving uncertainties in renewable energy sources. Applications of this methodology can be found, among others, in [18,19,20,21,22,23]. However, these works did not utilise linearisation, so to improve calculation accuracy, the authors of [24] propose a linearised IPF model. Numerical results show that the linearised formula calculates IPF results 6.57 times faster than the nonlinear method, with negligible calculation errors (less than 0.06% for absolute values and 0.02° for angles). IPF linearisation methods can also be found in [25,26]. For example, the authors of [26] tested their methodology on closed IEEE 33-, 69-, and 118-node networks. The results obtained in these cases also confirm that the use of IPF linearisation contributes to increased computational efficiency.

Although linear power flow models are becoming increasingly common, finding a model with minimal error remains a difficult task. In fact, the linear error of power flow is influenced by both the distribution of variables and the correct formulation of the distribution itself. Fan et al. [27] formulated a model for finding linear power flow with minimal linearity error using an optimal choice of variable space. The authors also propose a simplified power loss model and illustrate the difference between choosing variable space and popular hot start methods. The effectiveness of their method was tested on two IEEE test networks—30 and 118 nodes, as well as on the real Polish power system consisting of 2383 nodes. Also in ref. [28], a method using linear power flow to reduce the linearisation error is proposed. Thanks to the error control, the proposed linear optimal power flow (OPF) yields better final solutions. In ref. [29], a method for combining linear power flow models is proposed to reduce the linearisation error. Tests were conducted on an IEEE 30 network, and the obtained results, compared to three existing models in the test system, reduced the linearisation error of active and reactive power by over 60%. Sowa et al. [30] propose the use of linear optimisation of AC optimal power flow to improve the efficiency of power system operation. They used linear programming to significantly reduce the approximation error. The results obtained confirm that the proposed method significantly reduces computational errors and shows strong resilience to challenging real conditions. The authors of [31] propose finding the optimal copula function that best fits the actual distribution of historical data. Figure 1 presents the algorithm proposed by the authors, which allows for linearisation with minimised error.

The inclusion of Figure 1 is intended to clearly illustrate the logical structure and computational flow of the proposed algorithm. By presenting the sequence of operations leading to the optimal copula selection and subsequent linearisation, the figure facilitates a better understanding of the methodological framework adopted by the authors. Moreover, graphical representation enhances the transparency and reproducibility of the approach, enabling readers to trace how error minimisation is achieved within the linearisation process.

In the first step, a set of historical data is used to select a dome function describing the relationships between state variables. On this basis, the common distribution of variables (v_i, v_j), is determined, which is used to calculate the expected value of the linearisation error for five independent candidates of the linear form of the function g(v). The obtained error values are then minimised, which allows both the best linear form and the corresponding parameter to be determined. Finally, based on the candidate form and the estimated parameters, a target linear power flow model is constructed, whose structure is chosen to minimise the error relative to the distribution observed in the data [31].

The use of linearisation to solve the power flow problem in a system is also discussed in [32,33,34]. The results achieved by the researchers in [35] confirm that the proposed method considerably reduces computation time, making it a viable alternative to nonlinear power flow (NPF) approaches. It is also important to note that the non-convex nature of NPF creates substantial difficulties for optimisation and control algorithms in power systems. To address these issues, linear approximations are frequently employed, trading off some modelling accuracy in favour of computational simplicity. The precision of such linearised power flow models largely depends on the specific characteristics of the power system and the operational range within which the linearisation is applied. For this reason, in ref. [36], conservative linear approximations of the power flow equations are used, which aim to overestimate or underestimate the quantity of interest, to enable the use of easy-to-use algorithms that avoid constraint violations. In ref. [37], a new sequential linear programming (SLP) approach is proposed, which, by iteratively constructing hyperplanes, allows for achieving high-quality solutions for non-convex AC OPF using only SLP. Test results show that this method provides fast computation times, high accuracy and stable convergence regardless of the starting point compared to classical nonlinear power flow calculation methods. The authors of [38] applied a similar approach, and their findings also confirm the method’s effectiveness. To enhance the accuracy of the OPF solution, the authors of [39] employed a linear formulation based on the Karush-Kuhn-Tucker (KKT) conditions. By incorporating a deviation factor, they significantly improved the performance of linear OPF optimisation, which was both theoretically demonstrated and validated using various IEEE and Polish test systems. In ref. [40], a linear approximation method for AC-OPF is introduced, enabling a detailed representation of the relationships between active and reactive power flows and voltage profiles, with accuracy comparable to that of nonlinear models. The proposed approach allows for transforming the AC-OPF problem into a mixed integer linear programming (MILP) form, which ensures the possibility of using existing algorithms to achieve global optimality. Numerical results and simulations confirm the effectiveness and practical applicability of the proposed model. Paper [41] introduces an enhanced linearisation model for representing the aggregated flexibility of distribution networks. This approach enables operators to showcase the flexibility potential of their networks without revealing sensitive or confidential data. To achieve this, linear models of standard flexibility providers were created, and a linear optimisation-based aggregation algorithm was implemented. Compared to earlier nonlinear methods, this significantly decreased computation time while preserving a high level of accuracy. The method was validated using two real-world medium-voltage radial networks in Germany, confirming both its effectiveness and practical applicability. In [42], a Customised Sequential Quadratic Programming (CSQP) algorithm is presented, which exploits the physical specificity of the AC OPF problem and its appropriate formulation. Numerical tests show that CSQP outperforms available nonlinear solutions, providing solutions in an industrially acceptable time. The effectiveness of the method was confirmed on a wide set of test cases, covering systems with 500 to 30,000 buses. In ref. [43], a quadratic power flow model is initially trained using a regression-based approach, followed by the development of a linear power flow model derived through Taylor series expansion. Once the initial regression training is completed, the linear power flow (LPF) model does not require retraining, even when system updates occur. Instead, model parameters are updated online using real-time measurement data, enhancing their generalisation ability. Overall, the proposed LPF model offers high accuracy, strong generalisation, and significantly reduces both data usage and computational time. In ref. [44], the authors investigate the causes of errors arising from linearisation and introduce a modular linear power flow model. Simulation results on various IEEE and Polish test systems demonstrate that the proposed approach can reduce linearisation errors by as much as 52.98% compared to existing methods. In ref. [45], the authors propose three versions of an AC Optimal Power Flow approximation based on different Points of Linearisation (PoL), integrated into a fully linear model designed to facilitate the use of ancillary services provided by distributed energy resources (DERs). The models were tested on a modified 34-node distribution network and compared against the nonlinear AC OPF. The results indicate that linearising around the actual power flow solution considerably enhances accuracy, reduces errors in both active and reactive power objective functions, and allows for accurate determination of DER operating points. Paper [46] presents a linear optimal power flow (LOPF) based on modern methods for linearising flow constraints in quadratic branches. An analytical expression defining the number of linearisation segments required to achieve a given error is developed, which allows us to solve the LOPF without iterations using efficient linear methods. An analysis of field reduction methods in thermal boundary linearisation and a modified approach eliminating their shortcomings are also presented. In ref. [47], the authors conclude that the fundamental difference between various linear power flow models lies in the formulation of “independent variables”. Based on this, they propose and investigate general formulations of linear power flow models. Power flow linearisation is also addressed in ref. [48], where the authors use it to account for system constraints. In ref. [49], the authors use the Levenberg–Marquardt (LM) algorithm to linearise power flow problems. In ref. [50,51], network linearisation is used to account for uncertainty in a balanced system. In ref. [52], the authors focused on simultaneous studies of the linear approximation of reactive power and transmission losses. They used the logarithmic transform of voltage magnitudes as the modified voltage magnitude, and the results obtained were compared on 25 test systems. It is also extremely important to take line losses into account in the issue of optimal power flow. In ref. [53], the authors proposed an iterative procedure to solve this problem, in which linear cuts approximating quadratic losses in each transmission line are adjusted iteratively as the optimisation problem is solved.

The attentive reader should also refer to [54,55,56,57,58,59,60,61,62,63], which also use linearisation to solve power flow problems in a power system. These papers are particularly valuable due to their timeliness. Article [64] extends the formulation of branch flows by explicitly modelling line shunt admittances and transformer effects while maintaining strict linearity; the resulting model supports MILP-based voltage/reactive power control and provides accurate voltage estimates without iterative AC solutions. In ref. [65], the authors derive a linear steady-state method for radial networks through local linearisation based on the Jacobian matrix, which yields closed-form sensitivities between active/reactive injections and square-law voltages/flows, enabling fast non-iterative evaluation. In ref. [66], a data-driven robust linear OPF is developed, where power flow relationships are learned from data and embedded in linear optimisation with uncertainty sets to handle prediction/measurement errors. In ref. [67], an improved linear power flow is proposed by decomposing fundamental and incremental components, increasing accuracy over conventional LPF while maintaining linear computational complexity for planning and operations. For islanded microgrids, ref. [68] presents a tertiary control scheme based on a linearised AC-OPF with explicit loss compensation, mitigating the linearisation error and enabling efficient dispatch. Power flow linearisation for unbalanced three-phase distribution systems is discussed in [69], where a linear load flow with current injection is formulated, suitable as a fast linear layer within optimisation. Furthermore, in ref. [70], two three-phase linear formulations for autonomous voltage-drop-controlled microgrids were introduced, enabling linear OPF with approved voltage errors below one percent.

The reader should also consult the book by Bazara, Jarvis, and Sherali [71], which discusses various linear programming techniques and their application in network flow analysis. The authors pay particular attention to the modelling, design and analysis of linearisation algorithms in the context of power engineering problems. This work is a valuable resource for those wishing to explore linearisation methods in power engineering problems, serving as a significant reference in this field.

2.2. Linear Approximations for Assessing the Capacity of the Power System and Planning Network Development

Total transmission capacity (TTC) is essential for maintaining the reliability of power systems. As power grids integrate more variable renewable energy sources, operators face the challenge of solving complex, multidimensional optimisation problems within short time frames. To reduce the computational load, it is common practice to linearise AC power flow constraints.

In ref. [72], the authors introduce a transmission capacity assessment method based on Linearised AC power flow (LACPF), which delivers information on TTC limits without requiring any prior distribution assumptions. Figure 2 illustrates the block diagram of the proposed approach. Figure 2 clarifies the overall workflow of the proposed methodology and highlights the sequential relationships between its analytical stages. By visualising the structure of the LACPF-based assessment process, the figure enhances the reader’s understanding of how linearisation is applied within the broader context of transmission capacity evaluation. This graphical representation also supports methodological transparency and facilitates replication of the approach in future studies, which is particularly relevant given the increasing reliance on linearised models in grid planning and operational analyses.

The algorithm begins with the input of data (generator power, loads, network parameters and other variables). On this basis, a TTC model is constructed, which is divided into two complementary sub-problems: an ‘optimistic’ and a ‘pessimistic’ variant, each with its own objective functions and constraints. The optimistic part is solved as a maximisation task using the simplex method, which gives the upper limit U. The pessimistic part is transformed into a dual form for the inner layer of the problem; binary variables and a large constant M, are introduced, which allows the problem to be written in MILP form and solved, leading to the lower limit L. After combining the results of both paths, we obtain the interval [L, U], interpreted as the solution to the original TTC model, which ensures both compliance with the constraints and a transparent interpretation of uncertainty [72].

The authors validated their proposed method on the IEEE 118 test network as well as on an actual power system in northwestern China. The results demonstrated the high computational efficiency of the approach.

Also in Ejebe et al. [73] focused on presenting a program for quickly calculating the available connection capacity of a power system based on linear incremental power flow. The presented test results on practical power system models confirm the effectiveness of the presented program. In ref. [74], linearisation is used to manage overloads in transmission lines while simultaneously reducing generation costs in the power system. The proposed model was applied to test systems following the IEEE 9-node and IEEE 118-node standards. Additionally, to evaluate system reliability both before and after switching operations, two approaches were employed: contingency analysis and calculation of the loss of load probability (LOLP) index. The obtained results show that the methodology proposed by the authors can be applied to real systems. In ref. [75], the authors used linearisation to determine the buffer capacity for transmission lines, thus ensuring safe system operation. They evaluated the performance of their method on several test networks of different sizes, and the obtained results are satisfactory. The algorithm presented in [76] linearises the optimal power flow formula and uses the Gram-Charlie expansion to find the maximum available transmission capacity. The key advantage of the proposed method is its short computational time, verified on an IEEE test system. In ref. [77], the authors proposed the use of a novel MILP for transmission expansion planning (TEP) frameworks, taking into account the role of compressed air energy storage (CAES) integration, to address the growing problem of decreasing connection capacity. The authors conducted studies on various test networks, and the numerical results demonstrate the positive impact of CAES units on the power system. Also in ref. [78], the authors proposed a methodology for increasing system flexibility based on MILP. Their proposed linearisation approach is based on a second-order Taylor series expansion of trigonometric expressions. It does not rely on the assumptions of flat voltage, near-voltage and small-angle voltage, and formulates linear approximations using the squared voltage and voltage angle difference variables. The second novelty is an advanced heuristic sequential linearisation algorithm (SLA), which iteratively considers and improves the accuracy of the MILP model. Test results show that the proposed methodology outperforms alternative methods in terms of solution accuracy and computational efficiency under normal and loaded operating conditions. In [79] the authors propose using linearisation in combination with dynamic line estimation (DLR), which enables better system expansion planning and provides greater flexibility. They tested their proposed methodology on two test networks. Also in [80] the authors use MILP in combination with DLR to provide improved system flexibility. Tests conducted on an IEEE test system confirm the positive impact of the methodology on the obtained results.

Liao, Wu and Lin [81] propose an active planning model for distributed networks that relies on linear constraints. They begin by outlining the fundamental principles and methods for linearising a distributed network model, approaching the problem from two angles: power flow linearisation and linearisation of distributed network connectivity. The results obtained prove the method has a small error of only 3.3%, reducing the calculation time to below 0.2 s.

In ref. [82] the authors propose a power flow analysis method based on the interval AC-linear power flow (AC-LPF) model and affine arithmetic. By linearising the power flow and introducing power injection intervals, it is possible to determine the ranges of network state variables while taking into account the uncertainties associated with Available Transfer Capacity (ATC). Deng et al. [83] focus on assessing the TTC between asynchronous areas and ensuring stability between systems. In their paper, a linear model is used to assess transmission capacity. In ref. [84] the authors present the AC power flow-constrained robust unit commitment (ACRUC), which linearise the quadratic constraints of AC power flow using a circular method. This allows for the transformation of nonlinear dependencies into a linear form, which can be effectively used in optimisation. Test results indicate that the model faithfully represents actual system operating conditions and enables a more accurate assessment of network capacity with high RES penetration.

Jakus et al. [85] developed an optimisation model based on mixed integrated linear programming for automatic planning of distribution network development and determining the optimal power supply solution for network users, taking into account network development costs as well as power quality and security criteria. In ref. [86] an optimal network planning model was developed to maximise the system operator’s profit. To ensure its accuracy, the model was linearised using the network loss coefficient method combined with circular constraints and the McCormick convex envelope method. A 15-node system in Jiangsu Province was considered as a test network. In ref. [87] the authors aimed to reduce CO₂ emissions in the power generation process on Jurong Island in Singapore. For this purpose, they used the linear loss linearisation method with modified DC power flow to optimise network development with high saturation of distributed generation sources. In [88], in order to improve system flexibility and distribution network planning, a method for optimal allocation of electricity storage facilities based on interval linear programming is proposed. The conducted studies proved that this method ensures good network balance and efficiency. The reader can also refer to works [89,90,91,92,93,94,95,96], which also address the issue of power network planning using linear programming. Work [97] focuses on planning transmission network expansion, and linearisation is used to create an OPF model. Tests were conducted on closed IEEE 12 and IEEE 118 test networks. In ref. [98] the authors propose planning an orderly charging strategy for electric vehicles based on the reconstruction of the distribution network. Linearisation combined with proprietary soft normally open point (SNOP) software allowed for the development of an optimisation model for a reconfigurable distribution network. The proposed methodology was verified on a 94-node system. In ref. [99] a dynamic linear programming model is developed, as well as a simulation of optimal generation capacity development paths. Babonneau et al. [100] propose employing a linear programming model to capture the characteristics of the distribution network, alongside market balancing processes for flexible loads and distributed energy resources that provide reserve capacity and reactive power compensation.

In ref. [101] the focus is on a MILP-based approach for optimal planning and operation of the urban power grid. Similarly, in ref. [102], the authors focus on optimal management of the urban power grid. In ref. [103], linear programming methods are used to solve the problem of proper power system design. This problem becomes difficult to solve with the inclusion of a large number of variable power generators, such as wind farms or photovoltaic plants. Study [104] presents a novel approach to energy planning that takes into account the greater penetration of variable energy sources and the growing interaction between energy sectors. The presented method combines energy planning models with power flow models, enabling faster and more reliable solutions of optimisation problems. A linear continuous optimisation model for the power system, combined with a nonlinear model for power flow analysis, is employed. This approach is used to assess different energy planning scenarios for interconnected islands. The results show a 26.7% increase in system costs, a 3.3-fold decrease in battery capacity, and a 14.9 MW increase in renewable power compared to general spatial planning methods. Additionally, the power flow analysis revealed that the maximum voltage deviation reached 16% above the nominal level, underscoring the importance of evaluating the feasibility of the proposed scenarios before implementation.

Article [105] proposes the use of a mixed integer programming formulation for locating and sizing aggregated models of hybrid distributed energy sources in radial power distribution systems. The goal is to minimise errors in estimations based on available field measurements, which allow for improved system visibility. Aggregated models capture the collective impact of multiple unseen DERs, enabling the reconstruction of voltages at unobserved nodes and power flows in branches, which increases monitoring efficiency. The paper focuses on the linearisation method, which simplifies the calculations related to modelling nonlinear relationships between system variables, thus enabling the use of MILP methods. The use of linearisation makes the problem more computationally tractable, as verified in tests conducted using the OpenDSS (EPRI (Electric Power Research Institute), Palo Alto, California, USA) tool. Examples from the studies show average errors below 5% in estimating power flows in branches, despite limited voltage measurements. The results suggest high accuracy and efficiency of the proposed method, even under uneven system load conditions. In ref. [106] a new approach to solving the capacity development planning problem is proposed, which has three objectives: maximising total energy production, minimising system costs and reducing CO₂ emissions. Instead of three objectives, the problem was transformed into two objectives: maximising the ratio of energy production to system costs and the ratio of energy production to CO₂ emissions. This allows for obtaining better solutions without the need to specify preferences. A novel linearisation and parameterisation technique, based on Dinkelbach’s theorem and Güzel’s method, was employed to address this problem. This approach transformed the objective functions into a single linear programming problem. When applied to capacity development planning, the new method demonstrated improved handling of trade-offs between energy production, costs, and CO₂ emissions compared to the traditional weighted sum method. Paper [107] discusses the optimal scheduling of prosumers who own energy storage in communities using renewable energy sources. It proposes an innovative strategy for optimising storage utilisation in a demand response model that depends on price and volume. The problem is formulated as a scalable, low-complexity linear mixed-integer program. Additionally, a heuristic procedure is proposed to fairly redistribute demand response rewards among participants based on their contribution to the demand response goals. The aim of this approach is to enhance the benefits for prosumers operating within a shared community compared to acting individually.

Readers seeking information on the use of linearisation in distribution or transmission network planning should also refer to [108,109,110,111,112,113].

2.3. Linear Approximations for Solving the Problem of Voltage Regulation in the Power System

Linearisation methods in power systems can also be applied to voltage regulation [114,115,116,117,118]. For example, in [116], the authors proposed a coordinated voltage regulation scheme for transmission systems that uses linearised power flow definitions to determine setpoints for voltage regulation devices. They tested the effectiveness of this method on two closed-loop test systems: IEEE 30 and IEEE 118.

In ref. [119], a mixed-integer linear programming method for optimising voltage profiles in active distribution networks is presented. The effectiveness of the proposed method was tested on a reference network. In ref. [120] the authors propose a linear power flow model for a distribution system with accurate voltage estimates. The model can be viewed as an extension of the LinDistFlow model. The proposed methodology was tested on a test network. Modifications of the linear DistFlow model for voltage regulation are also used in [117,121,122]. The increasing share of photovoltaic installations in distribution networks (DN) leads to voltage fluctuations that require effective mitigation. One of the basic devices used for this purpose is a step voltage regulator (SVR). In ref. [121], the authors focus on the problem of optimal selection of the regulation stage (OPTS) for SVRs operating in a star configuration, using a linear approximation of the power flow equations. In particular, the LinDist3Flow model was used, assuming a continuous nature of the effective SVR coefficient. This allows for the formulation of the LinDist3Flow-OPTS problem, which takes the form of a linear model. In ref. [123], a new centralised voltage regulation method is introduced, designed to enhance the security of active distribution networks. This approach relies on sensitivity analysis to optimally allocate control variables. The effectiveness of the proposed sensitivities and their use in voltage regulation was successfully validated on an 11 kV distribution network. In ref. [124], the authors propose a new control method based on Linear Active Disturbance Rejection Control (LADRC) integrated with the hybrid Salp Particle Swarm Algorithm (SPSA). This solution enables simultaneous tracking of Global Maximum Power (GMP), DC-Link voltage regulation and improved active and reactive power control. The test results show that LADRC provides faster response and lower THD (2.25%) compared to PI, PID and SMC controllers, and the effectiveness of the method was confirmed in Hardware-in-the-Loop (HIL) tests. In ref. [125] a DM-PI-DC controller tuned with a sine-cosine algorithm (SCA) is proposed, which limits the frequency (LFC loop) and voltage (AVR loop) fluctuations, taking into account communication delays and real system constraints. The model also incorporates an interline power flow controller (IPFC) to manage power flows and redox flow batteries for increased stability. Analyses show that the proposed controller achieves oscillation-free response in 3.3 s, and 33% better overshoot performance compared to other advanced methods, effectively improving power quality.

Matallana et al. [126] propose a linear sensitivity modelling algorithm that is useful for controlling voltage regulation in systems with a large number of distributed energy sources. The proposed algorithm was tested on an IEEE 33 network, and its application demonstrated enhanced performance in voltage monitoring and control scenarios. Its application allows for better estimation and compensation of voltage deviations with respect to operating voltage constraints.

In ref. [127] a new distribution LMP (DLMP) is proposed, which takes into account both reactive power and voltage constraints. For this purpose, three tools were developed: linearised power flow for distribution (LPF-D), loss factors for distribution (LF-D), and linear optimal power flow for distribution (LOPF-D). Integrating LPF-D and LF-D allows for reformulating the classical OPF into a linear form while maintaining high accuracy compared to the full AC model. Tests have shown that the proposed approach enables precise DLMP determination and provides valuable pricing information, and the tools themselves can also be applied to other distribution network analyses. In ref. [128] a new mixed integer linear programming formulation for AC-OPF in three-phase, unbalanced distribution networks is presented. The model aims to minimise energy production costs while maintaining voltage limits. New Euclidean norm linearisations in voltage and current calculations were introduced using a linear transformation of weighted norms and intersecting planes. Tests on two unbalanced systems showed that the proposed method is computationally more efficient, accurate and conservative than traditional approaches, with maximum errors below 0.1%. In ref. [129] an Under-Voltage Load Shedding (UVLS) model based on mixed integer programming (MIP) is proposed, aiming to maintain a given load margin with minimal power shedding. The complete nonlinear AC power flow equations are approximated through a piecewise linearisation approach and incorporated into the UVLS formulation. The model accounts for load characteristics and was evaluated under different operating and fault scenarios using the IEEE 14-node and 118-node test systems, showing superior performance compared to conventional nonlinear approaches. In [130], linearisation is used to optimise system voltages within an acceptable range by optimally utilising reactive power generated by distributed generation (DG). Also in ref. [131] the authors use linearisation methods for voltage regulation in power systems. In ref. [132], a type-2 interval fuzzy controller is introduced to enhance transient stability and voltage regulation in multi-machine power systems. The controller is developed using direct feedback linearisation (DFL), which converts the compensated DFL model into a type-1 Takagi–Sugeno (T–S) fuzzy representation. The stability assessment and design of stabilising controllers for type-2 fuzzy control systems (IT2 T–S FLCS) rely on linear matrix inequalities (LMIs) to ensure compliance with stability criteria. Based on local measurements, the controller provides transient stability, voltage regulation, and desirable dynamic performance. It was implemented in a two-generator system connected to an infinite bus, with simulation results demonstrating its effectiveness under various operating conditions. In [133], an adaptive control strategy for voltage and frequency regulation in a stand-alone hybrid power system combining a wind turbine and a diesel generator is proposed, employing constrained linear model predictive control (MPC). The system configuration includes a wind turbine with a self-excited induction generator (SEIG), connected through a DC link to a diesel engine-driven synchronous generator.

In refs. [134,135] the authors use the assumption of small angular differences between lines, which simplifies the calculations of bus voltage values. In ref. [136] it is assumed that voltages tend to approach 1.0 p.u., which facilitates the calculation of power flows in branches. In these works, linearisation is crucial. In refs. [137,138,139] the authors discuss the use of linear approximations to model unbalanced three-phase power flows, which enable voltage forecasting as a linear representation of load forecasts using Gaussian processes. In ref. [137] a privacy-preserving short-term voltage forecasting method is proposed, where federated learning is used to preserve the privacy of user data without revealing local information. In ref. [138] a Bayesian approach is used to solve the system state estimation (DSSE) problem with limited measurements, learning pseudo-measurement distributions to support DSSE. It should be emphasised, however, that these methods either do not fully exploit the available data from distribution systems or rely on assumed model accuracy, which limits the precision of voltage estimation. Moreover, they generally do not account for uncertainty in voltage estimation, particularly when measurement data are sparse. In ref. [139] proposes an approach using Gaussian processes to recover missing data and an approach based on Bayesian matrix completion for state estimation. However, the use of linear power flow constraints in this approach can lead to distorted results, especially with imprecise power flow models, and further limit the scalability of Gaussian processes.

Other works that provide readers with insight into the linearisation of voltage problems in power systems include [140,141,142,143,144,145,146,147,148,149,150,151,152,153,154].

2.4. Linear Approximations for Solving the Problem of Redistributing the Capacity of Renewable Energy Sources

Eliminating overloads in high-voltage lines and maintaining a power balance in the system under conditions of high renewable energy generation pose significant challenges for grid operators [155]. In 2024 alone, the use of nearly 1 TWh of energy was reduced in Poland in this way, while in Germany, this figure exceeded 20 TWh [156].

In ref. [157] the authors linearise the flow equation around the operating point and formulate the congestion management problem as a linear program with so-called “strict constraints,” determining sensitivity coefficients (also referred to as participation indices), that are used to select the optimal units for redispatch. In ref. [158] a local network flexibility market with single-line advance, taking into account network constraints, is presented using Successive Linear Programming. The authors demonstrated that the proposed algorithm is robust and capable of producing high-quality, feasible AC solutions with satisfactory computational efficiency. Its effectiveness was validated through applications to multiple distribution networks and a range of load scenarios. The authors of [159] also use Successive Linear Programming to determine the optimal power flow. Paper [160] proposes a new formulation based on mixed integer linear programming, in which various linearisation methods are used to describe the voltage stability index (VSI). Network reconfiguration in the MILP approach allows for increased accuracy and speed of calculations while maintaining the optimality of the solution. The main goal is to minimise active power losses and improve the VSI, while taking into account constraints on branch currents, node voltages, network radiality and distributed generation penetration. The effectiveness of the method was validated on 33- and 69-node radial test networks. In ref. [161] a new model for unit engagement with network constraints was presented, formulated as MILP with linear AC-OPF equations. The model allows limiting the unit dispatch schedule within the allowable line capacity and voltage deviations without the need for additional iterations. The effectiveness of the proposed approach was validated using benchmark test systems. In ref. [162] the authors present a novel Network Functions Virtualisation (NFV) algorithm that enables efficient implementation of planned network services while reducing load and limiting congestion.

Article [163] addresses the allocation planning problem for large-scale distributed renewable energy systems in distribution networks. It focuses on the integration of hybrid renewable sources such as photovoltaic, wind, and biomass energy, and examines the effects of PV and wind generation variability on long-term voltage dynamics and stability in distribution networks. Linearisation is used to account for variability in RES power generation, assess its impact on voltage stability, and maximise renewable energy penetration while minimising costs. The variability of RES power (especially PV and wind) is simplified to linear functions, allowing these variables to be effectively incorporated into the MILP model. Furthermore, the stochastic nature of the problem, related to forecasting variable RES energy production, is also addressed in the form of linear constraints, allowing for a scalable and optimised solution.

In refs. [164,165] the authors use piecewise linear approximations to calculate the squared active and reactive flows in the branches. In the context of optimisation applications, McCormick envelopes are used to linearise current calculations in a power flow model along a line. This method relaxes the nonlinear expression by introducing a set of appropriate linear constraints, facilitating the solution to the optimisation problem [166,167]. In ref. [168] Yang, Guo and co-authors propose applying a linearisation method to the optimal power flow problem to achieve optimal network reconfiguration. By introducing an error-control mechanism, the proposed linear OPF allowed for more accurate results. The method was verified on several standard test systems, and the results obtained confirmed its effectiveness. The paper constitutes an interesting addition to the literature on system reconfiguration, especially in the context of works that did not employ linearisation methods [169,170,171]. Linearisation in network reconfiguration was also applied in [172].

The problem of the need to use system reconfiguration to mitigate current overloads is a relatively new phenomenon that has emerged on a large scale in recent years. Readers wishing to delve deeper into this topic should also consult [173,174,175].

Linearisation is also used in wind farm analyses [176,177,178,179,180,181,182,183,184,185,186], e-mobility [187,188,189,190,191,192,193,194,195,196], energy management in multi-energy microgrids [197,198,199,200,201,202,203,204], and on-load tap changer modelling to reduce losses [205,206,207,208,209,210,211].

For instance, ref. [180] proposes a control strategy for wind energy conversion systems (WECS) connected to distribution networks, with a focus on providing both energy supply and ancillary services. A typical WECS includes variable-speed wind turbines coupled with a direct-driven permanent magnet (DDPM) generator, which offers significant operational flexibility as its output is delivered to the grid through a fully controlled frequency converter. The proposed control strategy aims to regulate the electrical and mechanical parameters of the generating unit, ensuring optimal power delivery to the grid while providing voltage support at the point of common coupling. The approach is based on feedback linearisation (FBL), which enables decomposition and linearisation of a nonlinear multiple-input, multiple-output (MIMO) system. FBL allows independent control of each system variable, simplifying controller design and system operation. Numerical simulations demonstrate that this method fully exploits the flexibility of the DDPM generator, supporting the integration of WECS into modern microgrids, where operational modes (grid-connected or islanded) and the charging or discharging states of energy storage systems can be represented as binary or integer variables. In ref. [202], an energy management system (EMS) for a power node was presented using MILP, which aims to reduce operating costs through optimal daily advance scheduling. The objective function that describes this problem is presented by the following formula:

$$\mathit{min} {\displaystyle \sum _{t=1}^T{C}_t^{CHP}+} C_t^{ST}+ C_t^{shed}+ C_t^{Heat}$$

(1)

where

• $C_t^{CHP}$—represents the cost related to the operation of Combined Heat and Power units;
• $C_t^{ST}$—refers to the cost or revenue resulting from electricity purchases from or sales to the grid;
• $C_t^{shed}$—indicates the cost associated with load shedding;
• $C_t^{Heat}$—corresponds to the cost of meeting heat demand.

Additionally, the paper applies a linearisation technique to the AC network model’s line flow equations, as illustrated in Equations (2) and (3):

$$l_{(i,k)t}^P=g_{(i,k)}^L(V_{it}-V_{kt}-{\mathrm{\Psi }}_{ikt}+1)-b_{(i,k)}^L{\mathrm{\Phi }}_{ikt}$$

(2)

$$l_{(i,k)t}^Q=-b_{(i,k)}^L(V_{it}-V_{kt}-{\mathrm{\Psi }}_{ikt}+1)-g_{(i,k)}^L{\mathrm{\Phi }}_{ikt}$$

(3)

where

• $P_{(i,k)t}$ and $Q_{(i,k)t}$—are line power flow equations from bus i to bus k;
• $g_{(i,k)}^L$ and $b_{(i,k)}^L$—are the conductance/susceptance of the lines;
• V and $\mathrm{\Phi }$—are the voltage and angle difference between the buses.

Lima et al. [212] used MILP to perform a design analysis for the optimal deployment of Thyristor Controlled Phase Shifter Transformers (TCPSTs) in large power systems. They investigated how the number, location, and alignment of these transformers in the network affect the system’s load capacity maximisation using a DC load flow model. Constraints related to installation costs and the total number of TCPSTs were considered. The computational time of this method is considerably shorter compared to other similar approaches reported in the literature.

Research on IEEE test networks indicates that the linear model can very well replicate the results of the nonlinear model with shorter computation times. In the case of AC-OPF with Taylor (fractional-interval) linearisation for the IEEE-14 system, a maximum voltage module error of <2.2% (average 1.35%), an average phase angle error of ≈ 2.27° and a target gap of ≈ 0.87% were recorded. For IEEE-118 (three load levels), the cost error was ≈0.56–0.57%, the average generation error ≈ 0.57–0.75%, the voltage module error ≈ 3.5–3.8%, and the angle error was ≈0.28–0.45°; calculation times: 1.2 s (linearised model) vs. 1.76 s (nonlinear model) on the same platform, which confirms comparable or shorter calculation times while maintaining AC sensitivity [213].

In contrast, DC-OPF can significantly distort redistribution results in conditions with a high share of RES. A broad comparison (from 9 to 2383 nodes) shows that DC and AC models may indicate different sets of congested lines, different total redistribution costs and different cost allocations; for the 118-node system, both the total costs and the constraints that were binding changed. These results caution against relying solely on the DC approximation in the quantitative assessment of RES redistribution [214].

2.5. Summary of the Literature Review Conducted on the Subject in Consideration

The analysis of complex power systems (as nonlinear objects in which steady-state and transient processes occur) often requires the use of simplified methods or model approximations that enable effective analysis and design of control algorithms. In the context of power systems, and, in particular, the problem of power flow, nonlinear models describing the relationships between voltages and powers make direct calculations difficult and time-consuming. In practice, linear or partially linear approaches are often used, which preserve the essential properties of the system. Linearization significantly reduces calculation time and allows the use of standard optimisation tools, which would be computationally very costly in their full nonlinear form. However, the selection of an appropriate linearization method requires a critical assessment of a number of factors, including theoretical assumptions, trade-offs between accuracy and computational efficiency, and the characteristics of the system to which the method is to be applied

The most common approaches in the literature are linearization based on Taylor expansion (Newton-Raphson linearization) and DC linearization. The literature also mentions the following methods: Fast Decoupled Load Flow (FDLF) and Linearized AC Power Flow (Linearized AC PF). Each of the above methods has different assumptions and limitations that determine their suitability in specific application scenarios.

The DC linearization method is based on the assumption that the voltage angles at the nodes are small and that line losses can be ignored or assumed to be minimal. As a result, active power equations become linear, which allows for a quick solution to the power flow problem. The main advantage of this approach is a significant reduction in calculation time and the possibility of applying it in large systems where a full AC analysis would be too time-consuming. For this reason, the computational load of the method remains relatively low. This method also introduces limitations in terms of accuracy, especially in systems with large voltage deviations, significant losses, or in the event of critical situations such as overloads of power system components. As shown in the literature, under normal system operating conditions, this method can produce results with an error of up to several percent compared to the nonlinear AC model. However, in critical situations, this error can increase significantly.

The linearization method based on Taylor expansion allows for a more flexible consideration of the impact of voltage changes and power losses than the classic DC method. In this case, a specific operating state of the system is taken as a reference point, and linearization is performed relative to this point. This method allows for corrections to be made in the event of moderate changes in system parameters and provides better accuracy than the traditional DC method. On the other hand, it requires a preliminary estimate of the operating point and may be less stable in the event of sudden system changes. This method allows for a significant increase in calculation accuracy compared to the DC method. However, a significant increase in calculation costs should also be taken into account.

The Fast Decoupled Load Flow (FDLF) method is an intermediate method between full Newton-Raphson and simplified DC Power Flow. It assumes that active power depends mainly on voltage angles, and reactive power on voltage amplitudes, which allows the Jacobian matrix to be simplified into two separate matrices. This makes the method very fast and maintains decent accuracy under typical transmission conditions. Its limitation is lower precision in systems with strong nonlinearities or large voltage fluctuations. The Linearized AC Power Flow method, on the other hand, is a more modern approach that attempts to linearly approximate the full AC equations while maintaining greater accuracy than simplified DC PF methods. It usually involves expanding the power equations around the operating point while retaining first-order terms or using other approximations that take into account both angles and voltages. This makes the method suitable for both transmission and distribution systems, as well as for optimisation tasks. It should be noted that this method is much more computationally complex than the FDLF method.

Less common methods include: Modified voltage magnitude method, Quadratic form of line loss, and Proportional to power flow. Although less frequently used, these methods also find application in linearization.

The modified voltage magnitude method is characterised by very high accuracy in power flow mapping thanks to logarithmic voltage transformation, which takes into account both amplitude and phase angle changes. Compared to the classic DC model, it significantly reduces errors in active power calculations. However, the disadvantage of this method is the significantly longer calculation time and increased sensitivity, especially in large networks with strongly nonlinear voltage dependencies. Quadratic form of line loss is characterised by moderate accuracy at low computational cost, as it introduces only a simple quadratic term dependent on phase angle differences. It enables fast solutions with improved loss mapping, but its effectiveness decreases in high-resistance networks where voltage relationships dominate. The proportional to power flow method is a compromise between accuracy and computation time. Its iterative nature allows for gradual improvement of results while maintaining a short convergence time. This method is considered one of the most practical modifications of DC-OPF, but its effectiveness depends on the proper selection of the proportionality coefficient. It may also require several iterations, which limits its effectiveness in very large systems.

The literature also describes methods based on available measurements, e.g., power, current, voltage, and power flows. However, this type of approach requires data availability, good quality, and consistency of measurements. Although modern power grids are increasingly equipped with measuring devices that enable data acquisition, in practice, there are often gaps in the measurement data. It is therefore necessary to use further measures to fill these gaps. These include state estimates [215] and the use of machine learning [216,217]. These tools improve the quality of measurements but introduce further simplifications and certain errors in subsequent analyses. Nevertheless, they enable the creation of a linear model for calculations. This model is slightly more complicated, but the calculation time is relatively short. The accuracy, on the other hand, is better than in the case of the DC method and comparable to other simplified methods. Another advantage is that it is based on real measurements.

As part of an example case study, the authors presented a comparison of the accuracy of the results obtained and the calculation time, based on a specific example taken from the literature.

In order to critically compare different linearization methods, the authors analysed [58], in which Li and his colleagues analysed seven approaches to power flow linearization relative to the full AC–OPF model. The accuracy of the methods, the optimality of the solutions obtained, and the calculation time were evaluated. The results showed that the highest accuracy was achieved for the method based on logarithmic voltage transformation, which was characterised by minimal errors even in large test systems. In turn, the classic DC model was distinguished by the shortest calculation time and high computational robustness, but its accuracy remained relatively low. Iterative methods that take transmission losses into account achieved better quality parameters than simple linearizations, but their calculation time increased significantly with the size of the system—in the case of a 2000-node system, it was even about 70 times longer than for the classic DC model. The authors of the paper emphasised that the choice of linearization method should be a conscious compromise between the accuracy of the approximation and its feasibility and computational efficiency. In practice, the best balance between the quality of results and the speed of calculations is provided by iterative DC models that take line losses into account (Methods 6–7), which retain almost full feasibility with only a few times increase in calculation time compared to the classic DC model.

To more fully illustrate the issue under analysis, the authors have included a sample tabular summary (

Table 1. Comparative summary of methods and their results based on the results of the analyses presented in [57].

Method	Average Approximation Error	Average Optimality Error	Computation Time (14–2000 Nodes) [s]	General Remarks
DC power flow model	High (large errors for a 2000-node network)	Low	0.9–14	Fastest but not very accurate
First-order Taylor series	Average	High	2.6–1009	More accurate than DC, but unstable for large networks
Modified Phase Angle	Varies depending on network size	High	2.6–1006	Better than 2 for small networks, poor scalability
Square of Vol-tage	Average	High	2.5–1012	Similar accuracy to 2, longer time for large networks
Voltage magnitude method	Lowest error rate	Medium	2.8–1004	Most accurate, but unstable and time-consuming
Quadratic form of line loss	-	Very low	5.3–59	Improves the accuracy of DC, maintains speed
Proportional to power flow	-	Very low	5.2–59	Similar results to 6; slightly faster

), enabling the reader to gain a comprehensive understanding of the methods being compared. Table 1 was prepared based on the results of the analyses presented in [57]. The tables are included for comparative purposes and the results are presented in a collective and summary form. They provide a valuable comparison of the effectiveness of individual methods in solving an example task of optimal power flow.

Despite their extensive use, all of the methods discussed have their limitations. Linearizations around the operating point are only accurate for small deviations from that point, which limits their use in systems with high generation saturation (e.g., with a high share of renewable sources). Fast Decoupled Load Flow, although fast, simplifies the relationships between active and reactive power, which can lead to errors in low-voltage distribution systems with high losses. The linearized AC PF function increases accuracy but is more difficult to analyse and is still based on the operating point. The DC method is the least accurate of those discussed. The measurement-based method is more accurate but more complicated in terms of obtaining good-quality data. The other methods are less commonly used but also worth considering.

Potential areas for improvement include:

• Use of higher-order nonlinear approximations within Linearized AC PF, which can improve accuracy with moderate changes in voltage and angle.
• Integration with probabilistic and stochastic methods, especially in systems with a high proportion of unstable energy sources, which would allow uncertainty in power flows to be taken into account.
• Optimisation of the Jacobian matrix structure for computational efficiency, which could combine the accuracy of Newton-Raphson with the speed of FDLF.

Such improvements could significantly increase the scope of application of linearized power flow methods, allowing for faster and more accurate analysis of both transmission and distribution systems.

Although linearisation methods offer significant computational benefits, their effectiveness may be reduced in some situations and may decrease with distance from the operating point, especially near operating limits or after topology changes. The diversity of test networks and reporting conventions in the literature further complicates direct cross-sectional comparisons, prompting cautious interpretation of results and emphasis on conditions of applicability. Linearizations based on Taylor expansions provide useful sensitivities, but require frequent updates and lose accuracy with larger load deviations or transitions between node operating modes. Fast-coupled methods offer high speed for large transmission networks, but their accuracy decreases in the presence of strong transverse admittances and in radial systems. DistFlow variants provide very fast calculation in distribution networks, but with simplified treatment of losses, they may underestimate voltage drops and branch loads. In three-phase systems, there is also the issue of interphase couplings and unbalance, which requires careful parameterization. In practical applications, it is therefore recommended to combine linearisation with post-validation of AC power flow, to apply slight narrowing of voltage and current constraints, and to periodically recalibrate the model, which together allows the speed of calculations to be balanced with the predictability of the quality of results.

3. An Overview of the Approximation Methods and Their Potential Future Applications

Based on the literature review, it can be concluded that linear approximations for power flow are frequently used to solve complex problems arising in modern power systems worldwide. Research results indicate that both distribution and transmission system operators expect new, increasingly accurate tools to support grid management. The use of linearisation-based methods should be considered not only during the design and modernisation stages of infrastructure, but also during its ongoing operation.

As evidenced by the extensive literature review discussed in Section 2, linearisation methods are used more frequently in some areas than others. This variation stems from, among other factors, their effectiveness, computational efficiency, degree of popularisation and computational time.

Table 2. Summary of linearisation applications in solving power system problems.

The Issue in the Power System	Reference in Literature
Linearisation in power flow modelling	[7,8,9,10,11,12,13,14,15,16,17,18,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]
Linearisation in the assessment of transmission capacity and network development planning	[72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113]
Linearisation in the problem of voltage regulation in the power system	[114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154]
Linearisation in the problem of redistributing the capacity of renewable energy sources	[157,158,159,160,161,162,163,164,165,166,167,168,172]
Remaining linearisation implementations	[176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214]

and Figure 3 summarise the main types of problems for which linearisation methods have been used, along with their frequency of use.

Figure 3 illustrates the overall trends in the problems tackled using linearisation methods. Analysis of this graph allows us to identify the areas where these techniques prove most effective. The data presented in Table 2, in turn, allows us to determine which issues are most frequently addressed in the literature. On this basis, it can be concluded that one of the most frequently addressed problems using linearisation methods is optimal power flow. Linearisation in the context of optimal power redispatching is still an area requiring further research, as reflected in the limited number of published works on this topic. This issue, a significant challenge for modern power systems, will gain in importance in the coming years, and the difficulties associated with solving it may intensify as the complexity and operating conditions of power grids increase. Optimising power redistribution in such systems requires consideration of numerous variables, both technical and economic, which makes traditional approaches not effective enough. In this context, linearisation is a key tool for simplifying calculations and accelerating the process of solving optimisation problems. However, due to the specific properties and complexity of energy systems, full implementation of this method requires further research.

4. Possible Research Gaps in the Literature

It should be emphasised that the articles included in this review do not exhaust the entire spectrum of research, but rather reveal certain trends, indicating which problems are frequently investigated and which remain insufficiently addressed in the literature.

This review also identifies significant research gaps that can inspire further scientific development. Key areas requiring in-depth analysis include:

• Analyses conducted to assess the potential role of renewable energy sources and energy storage systems in restoring power plant generation capacity following a major system failure [218,219];
• Optimal choice of a compensation device for a wind or solar farm connected to the power grid through a cable line [134];
• Cable pooling—efficient utilisation of shared grid infrastructure by different types of renewable energy sources [220,221];
• Insufficient number of studies examining computational errors resulting from the use of linear methods, particularly in the context of comparing their accuracy with nonlinear methods for various types of computational problems [222];
• A limited number of studies comparing the effectiveness of various linear methods available in the literature in the context of specific computational problems. There are no comprehensive analyses presenting a comparison of the errors characteristic of available linear methods within a single problem [223,224];
• An insufficient number of studies comparing the computational efficiency of linear and nonlinear methods, especially in terms of computational time and real-time applicability. Such analyses are important for implementing practical solutions in large power systems [225,226];
• An insufficient number of studies analysing the computational error for power grids of various sizes, not only for test networks but also for real networks [227,228];
• Limited consideration of reactive power flows in the analyses of computational methods, especially in the context of large and complex power grids. There is a need for more accurate models that allow for a reliable representation of this aspect [229];
• Insufficient consideration of probabilistic models in power system analysis, particularly with respect to uncertainty related to measurements, load and generation. These models could significantly improve the reliability of forecasts and system analyses [230,231];
• Insufficient number of studies incorporating actual measurement data to create linear models [232,233];

Table 3. Key research gaps and proposed directions for further research on linearisation methods in the power industry.

ID Gap	Description of the Research Gap	The Proposed Research Approach	Priority	Position in the Research Track
G4	Lack of error analysis of linear vs. nonlinear methods	Comparative simulations, error analysis (MAPE, RMSE) on a common model	High	Assessment and verification of methods
G5	Lack of comprehensive comparisons of linear methods	Benchmarking multiple methods on the same problem	High	Assessment and verification of methods
G6	Lack of computational efficiency in analyses of methods	Measurement of computation time, resource consumption, real-time usability analysis	Medium	Assessment and verification of methods
G7	No studies on the relationship between error and network size	Tests on real and test networks of various sizes	Medium	Assessment and verification of methods
G8	Neglecting reactive power flows	Extension of linearisation models with a passive component, accuracy analysis	Medium	Extension of calculation models
G9	Lack of a probabilistic approach	Introduction of uncertainty (statistical distributions), sensitivity analysis	Low	Extension of calculation models
G10	Insufficient use of actual data	Validation of linear models using SCADA and PMU data	Medium	Validation on real data
G1	The role of RES and storage facilities in power restoration	Dynamic modelling, optimal control using linearisation	Low	New applications in electrical power engineering
G2	Selection of compensation devices	Compensation optimisation (MILP, heuristics) based on linearised models	Low	New applications in electrical power engineering
G3	Cable pooling	Linearisation of capacity models, optimisation of power allocation	Low	New applications in electrical power engineering

summarises potential areas identified in the field of linearisation methods in the power industry, which could provide a basis for further, in-depth scientific analyses. The areas identified address both fundamental issues related to the accuracy and reliability of linear methods compared to nonlinear methods, as well as more specific issues such as computational efficiency, validation on actual data and extending models to include new aspects.

Research gaps are prioritised based on their significance for power system operation and the practical utility of linearisation methods. High priority was assigned to gaps related to error analysis, accuracy, and method validation, as these constitute the basis for their reliable implementation in real networks. Medium priority was assigned to issues such as validation on real data and computational efficiency analyses, which are essential for further development but naturally follow error analyses and benchmarking. Low priority, on the other hand, was assigned to new, innovative application areas, such as the contribution of renewable sources to power restoration, compensation, and the concept of cable pooling. Although these areas have significant development potential, they do not represent critical shortcomings in existing methods, and therefore their importance for system stability and security was rated lower.

Research path position refers to the stage of development at which a given research gap is situated in the process of improving linearisation methods. The gaps identified in the “Evaluation and Verification of Methods” section address fundamental issues related to their correctness, accuracy and practical implementation. The “Extension of Computational Models” section encompasses a further stage in which the basic methods are expanded to include additional aspects, such as reactive power flows and probabilistic modelling. The next level is “Validation on Real-World Data,” which emphasises empirically verifying the correctness of models in real-world power grid conditions. The most advanced stage is “New Applications in Power Engineering,” encompassing innovative approaches to linearisation methods, such as power restoration using renewable sources, compensation selection and cable pooling. This structure reflects the sequential nature of the research process, from analysis of the basic properties of the methods, through their extension and validation, to the development of new application areas.

To more fully illustrate the identified research gaps, a diagram of their interrelationships was developed. This diagram presents not only the hierarchy of individual areas but also the logical connections between them. This makes it possible to capture the sequence of research activities, from error analysis and method validation, through the assessment of their effectiveness and scalability, to potential innovative applications in practical power systems.

The diagram in Figure 4 illustrates the hierarchy and interconnections between the identified research gaps. It shows that individual gaps do not exist independently but rather maintain logical cause-and-effect relationships, creating a coherent framework for research development.

In particular, the gap related to the lack of method validation (G10) is a consequence of deficiencies in error analyses and comparisons between linear and nonlinear methods (G4, G5). This means that until robust research on the accuracy and limitations of linearisation methods is conducted, full validation of their performance based on real-world data remains unattainable. Similarly, the gap related to the underestimation of error depending on network size (G7) directly impacts practical implementation issues (G6), as the lack of knowledge about the scalability of computational errors limits the applicability of models in operational settings.

Based on this, it can be seen that the identified research gaps (G4–G7, G10) provide a foundation for further, more innovative research directions, such as the contribution of renewable sources and storage to power restoration (G1), the optimal selection of compensation devices (G2) and the concept of cable pooling (G3). This indicates that new applications of linearisation models can only be fully credible and useful when grounded in solid foundations of error analysis, validation and method benchmarking.

Both the table and the diagram above constitute an original proposal developed as part of this article, the aim of which is not only to systematically organise the identified research gaps but also to inspire future authors to undertake in-depth research in this area. Their role is to identify priority research directions that may be crucial for the further development of linearisation methods in power engineering and their practical implementation in real-world energy systems. The table and diagram should be treated as a point of reference and an impulse to generate new ideas that, in the long run, can contribute to both improving the quality of analyses and developing modern, reliable solutions for the future energy sector.

5. Summary

The literature review demonstrates that linearisation methods play a significant role in power system research. Distribution and transmission system operators require modern computing tools that combine high accuracy, efficiency, and practical application. This is due to the ongoing energy transformation, the dynamic development of renewable energy sources and the greater complexity of the grid management process. Linearisation meets these requirements, enabling significant simplification of complex models while maintaining an acceptable level of accuracy. This allows these methods to be used in both planning analyses and system operations.

The reviewed literature demonstrates that linearisation has found particularly widespread application in optimal power flow problems, as well as in grid reconfiguration and renewable energy integration optimisation. However, its popularity and effectiveness vary depending on the specific problem, indicating the need for further research on adapting linearisation methods to new challenges. In some cases, simple simplifications and engineering logic prove to be fully sufficient to find a solution. For complex problems, the use of advanced algorithms becomes necessary.

The literature review also identified significant research gaps. Areas requiring further development include research on the role of RES and energy storage in system restoration after failures, the optimal selection of compensation devices for wind and photovoltaic farms connected via cables, and the development of the cable pooling concept, enabling the efficient use of shared infrastructure by various renewable sources. These issues can be largely addressed using linearisation methods, yet they have not been adequately explored in the literature so far.

An important direction of development is combining classical linearisation methods with modern artificial intelligence and advanced optimisation techniques. Hybrid approaches can enable better representation of complex phenomena while maintaining the required computational speed. Further improvement of programming tools and algorithms enabling real-time linearisation is also crucial.

It is also important to emphasise the need for an interdisciplinary approach. The development of linearisation methods requires the collaboration of experts in the fields of energy, mathematics, optimisation and artificial intelligence. Close collaboration with power system operators, who possess practical knowledge, data and experience in grid monitoring and management, is also crucial. This allows for the creation of methods that not only meet academic criteria but also have real-world applications.

In the context of combining linearisation with artificial intelligence methods, it is important to clarify the role of the ‘learning’ component. It should be seen as an addition that works alongside physical models, not instead of them. In practice, this means using predictors to: correct systematic linearisation deviations (e.g., compensation for typographical errors in voltages or losses), quickly ‘warm-start’ the solver (by predicting active constraints), and adaptively select the complexity of constraint approximations and inactive scenarios. Integration understood in this way reduces the risk of over-reliance on models and allows key decisions to remain interpretable.

At the same time, the relationship between time and accuracy should be considered. Training machine learning models is a process that is performed infrequently and offline, usually when there are significant changes in topology or workload profile. This cost is then amortised over time: inference in operational mode is very fast and accounts for a fraction of the total time spent solving a linear problem. In operational applications, decisions are made repeatedly over successive time horizons and for multiple scenarios. Under such conditions, the cost of training is quickly recouped, and adding a lightweight AI component to LP/MILP does not degrade the system’s response time. Moreover, the hybrid can speed up calculations by reducing the number of solver iterations or the size of the search space through better initialisation and filtering.

The issue of accuracy is addressed through a layer of security and validation embedded in the decision-making process. Firstly, conservative margins are applied in the constraints (e.g., slight ‘tightening’ of acceptable ranges), which reduces the risk of violations resulting from imperfect approximations. Secondly, predictions outside the training range are rejected and the system automatically reverts to a purely linear procedure if quality signals (e.g., uncertainty measures) indicate an increased risk of error. The importance of cooperation with power system operators should also be emphasised. Access to measurement data, operational experience and ‘digital twin’ platforms allows for the design of solutions that address real constraints from the outset: data quality and availability, hardware limitations, security and compliance requirements. Joint pilot projects, from the station level, through selected areas of the network, to larger-scale implementations, enable iterative refinement of methods and the definition of transparent procedures.

In summary, the integration of linearisation with AI is justified both computationally and qualitatively, as long as the learned component plays a supporting role rather than replacing physical models, and the whole is embedded in a clearly defined chain of security, validation and monitoring. This direction of development, supported by mature programming tools and interdisciplinary and operational cooperation, allows the benefits of computational speed to be combined with improved representation of phenomena, resulting in methods that are both scientifically sound and ready for practical application.

Linearization methods are an important tool supporting the development and operation of modern power systems. It should be noted that there is no universal method that can solve all the problems facing the energy sector. However, thanks to its flexibility and computational efficiency, linearization plays a significant role in various issues in the field of power engineering. The identified research gaps and potential development directions should inspire future studies focused on enhancing the reliability, security, and efficiency of power system operation.