D1.1 - Steady-state Models of Distribution Grids for Voltage Stability

FLEXDIS project FLEXDIS Deliverable 1.1 Steady-state Models of Distribution Grids for Voltage Stability Evaluation Document information Deliverable nr D1.1 Deliverable Name Steady-state Models of Distribution Grids for Voltage Stability Release date 08/08/2025 Dissemination level Public Status Submitted Authors Alex Blanco Castro, Josep Fanals (eRoots Analytics), Hessam Golmohamadi, Birgitte Bak-Jensen (AAU) and Marc Cheah-Mane (UPC) FLEXDIS project Document history: Version Date of issue Content and changes Edited by 1 08/08/2025 Full deliverable Alex Blanco Castro 2 19/09/2025 Additions Alex Blanco Castro and Hessam Golmohamadi 3 23/09/2025 Review Marc Cheah-Mane 4 24/09/2025 Review Birgitte Bak-Jensen 5 06/11/2025 Final version Marc Cheah-Mane Deliverable beneficiaries: WP and Task WP1– Impact analysis of distributed renewable generation Task 1.2 – Steady state and short-circuit analysis for voltage stability Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 2 of 63 FLEXDIS project Table of contents 1 Introduction 8 1.1 Conventional steady-state analysis . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Steady-state analysis for short-circuit calculations . . . . . . . . . . . . . . . . 9 1.3 Steady-state analysis for unbalanced systems . . . . . . . . . . . . . . . . . . 9 1.4 Objectives ....................................... 10 1.5 Outline......................................... 11 2 Balanced Steady State Analysis 12 2.1 Analysisframework.................................. 12 2.2 Intelligent Control of Flexibility Assets . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Danish Demonstration Site: Hyllegaard Høje . . . . . . . . . . . . . . . . . . . 15 2.3.1 LV Power Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2ThermonetSystem .............................. 17 2.3.3 Integration of Flexibility Assets into LV Network . . . . . . . . . . . . . . 18 2.3.4 CEMS-Based Control Architecture . . . . . . . . . . . . . . . . . . . . . . 19 2.3.5 Aggregation of Flexibility to MV Network . . . . . . . . . . . . . . . . . . 19 3 Unbalanced models for steady state analysis 21 3.1 OverheadLines .................................... 22 3.1.1 πmodel..................................... 22 3.1.2 Carson series impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.3Exampleresults ................................ 25 3.1.4Shuntadmittance ............................... 26 3.1.5Kron’sreduction................................ 26 3.2 Transformers ..................................... 27 3.2.1 Delta-star (Dy) connection . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Loads.......................................... 32 3.3.1 Impedance (Z) definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Current (I) definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.3Power(P)definition.............................. 34 3.4 Generators....................................... 34 3.5 Voltage Source Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Validation of unbalanced models 39 4.1 IEEE13NodeTestFeeder.............................. 39 4.2 IEEE European Low Voltage Test Feeder . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Single Line-to-Ground Fault (SLG) . . . . . . . . . . . . . . . . . . . . . . . . . 48 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 3 of 63 FLEXDIS project 4.4 Line-to-LineFault(LL) ................................ 51 4.5 Double Line-to-Ground Fault (DLG) . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Three-PhaseFault(LLL) ............................... 55 4.7 Three-Phase-to-Ground Fault (LLLG) . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Conclusions 59 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 4 of 63 FLEXDIS project List of Figures 1 Diagram of a part of distribution grid built-up in DIgSILENT with active demands like EVs, heat pumps and PVs . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Steady state analysis in distribution grid in DIgSILENT with results for line loading, transformer loading and nodal voltage . . . . . . . . . . . . . . . . . . 14 3 Distributed control and operation of flexible loads in a power distribution networkinDIgSILENT .................................. 15 4 Model of LV power grids in Hyllegaard Høje built up in PowerFactory . . . . . 16 5 A sample of color coding to indicate voltage levels of power nodes and loading levels of lines and cables in steady-state analysis . . . . . . . . . . . . . . . . 17 6 Map of thermonet in Hyllegaard Høje . . . . . . . . . . . . . . . . . . . . . . . . 17 7 Integration of EVs and heat pumps into local LV network in Hyllegaard Høje forflexibilityassessment............................... 18 8 Electrical power system network . . . . . . . . . . . . . . . . . . . . . . . . . . 21 9 Line πequivalentcircuit ............................... 22 10 Carson’s geometry data of the tower configuration . . . . . . . . . . . . . . . . 24 11 Example of a power line geometric arrangement . . . . . . . . . . . . . . . . . 25 12 Two-winding transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 13 Single-phase transformer equivalent electrical circuit . . . . . . . . . . . . . . 28 14 Zig-zag transformer connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 15 Delta-starconnection................................ 30 16 SchematicZIPloadmodel.............................. 32 17 Star-connected load (left) and delta-connected load (right) representations . 33 18 Thévenin equivalent circuit scheme . . . . . . . . . . . . . . . . . . . . . . . . . 35 19 Norton equivalent circuit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 36 20 RepresentationofaVSC............................... 36 21 IEEE 13 Node Test Feeder representation . . . . . . . . . . . . . . . . . . . . . 39 22 Voltage magnitude per bus in phase a . . . . . . . . . . . . . . . . . . . . . . . 42 23 Voltage angle per bus in phase a . . . . . . . . . . . . . . . . . . . . . . . . . . 43 24 Voltage magnitude per bus in phase b . . . . . . . . . . . . . . . . . . . . . . . 44 25 Voltage angle per bus in phase b . . . . . . . . . . . . . . . . . . . . . . . . . . 44 26 Voltage magnitude per bus in phase c . . . . . . . . . . . . . . . . . . . . . . . 45 27 Voltage angle per bus in phase c . . . . . . . . . . . . . . . . . . . . . . . . . . 46 28 Single-line diagram of the European Low Voltage Test Feeder . . . . . . . . . 47 29 Phase cvoltage profile comparison for hour 14 . . . . . . . . . . . . . . . . . . 48 30 Phase bvoltage profile comparison for hour 5 . . . . . . . . . . . . . . . . . . . 48 31 Phase-to-ground short-circuit at the bus 634 of the IEEE 13 Node Test Feeder 49 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 5 of 63 FLEXDIS project 32 Single Line-to-Ground Fault (SLG) . . . . . . . . . . . . . . . . . . . . . . . . . 49 33 Voltage magnitude per bus in the faulted phase a during a SLG fault . . . . . 50 34 Voltage magnitude per bus in the health phase b during a SLG fault . . . . . . 50 35 Voltage magnitude per bus in the health phase c during a SLG fault . . . . . . 50 36 Line-to-LineFault(LL) ................................ 51 37 Voltage magnitude per bus in the faulted phase a during a LL fault . . . . . . 52 38 Voltage magnitude per bus in the health phase b during a LL fault . . . . . . . 52 39 Voltage magnitude per bus in the faulted phase c during a LL fault . . . . . . 52 40 Double Line-to-Ground Fault (DLG) . . . . . . . . . . . . . . . . . . . . . . . . . 53 41 Voltage magnitude per bus in the faulted phase a during a DLG fault . . . . . 54 42 Voltage magnitude per bus in the health phase b during a DLG fault . . . . . . 54 43 Voltage magnitude per bus in the faulted phase c during a DLG fault . . . . . 54 44 Three-Phase Fault (LLL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 45 Voltage magnitude per bus in the faulted phase a during a LLL fault . . . . . . 56 46 Voltage magnitude per bus in the faulted phase b during a LLL fault . . . . . . 56 47 Voltage magnitude per bus in the faulted phase c during a LLL fault . . . . . . 56 48 Three-Phase-to-Ground Fault (LLLG) . . . . . . . . . . . . . . . . . . . . . . . . 57 49 Voltage magnitude per bus in the faulted phase a during a LLLG fault . . . . . 58 50 Voltage magnitude per bus in the faulted phase b during a LLLG fault . . . . . 58 51 Voltage magnitude per bus in the faulted phase c during a LLLG fault . . . . . 58 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 6 of 63 FLEXDIS project List of Tables 1 Non-exhaustive list of control combinations for a VSC under balanced conditions 37 2 Transformer data of the IEEE 13-bus Test Case . . . . . . . . . . . . . . . . . . 40 3 Load data of the IEEE 13-bus Test Case . . . . . . . . . . . . . . . . . . . . . . 40 4 Line data of the IEEE 13-bus Test Case . . . . . . . . . . . . . . . . . . . . . . 41 5 Capacitor bank data of the IEEE 13-bus Test Case . . . . . . . . . . . . . . . . 41 6 Phase a Voltage Profile of the IEEE 13-bus Test Case . . . . . . . . . . . . . . 42 7 Phase b Voltage Profile of the IEEE 13-bus Test Case . . . . . . . . . . . . . . 43 8 Phase c Voltage Profile of the IEEE 13-bus Test Case . . . . . . . . . . . . . . 45 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 7 of 63 FLEXDIS project 1 Introduction 1.1 Conventional steady-state analysis Steady state analysis in distribution grids is a set of calculations in unchanged (normal) conditions where the dynamic transients of the electrical equipment died out and the system works in a constant manner. Therefore, it makes it possible for the power system operator to check the technical status of the power grid equipment, e.g., power generators, demands, overhead lines, underground cables or transformers. Generally, in steady state analysis, the following variables are calculated: • Voltage magnitude and angles at all buses of the distribution grid. • Active and reactive power flows in branches (lines, cables or transformers). • Overloading of these branches (based on apparent powers or currents) and over/under voltage of buses. Therefore, by calculation of the above-mentioned variables, the Distribution System Operator (DSO) can identify key technical issues of the power grids, including: • Voltage violation: under-voltage and over-voltage of power nodes, which can affect the lifespan of power equipment, damage critical demands or lead to misoperation of equipment (in case of under-voltages). • Reactive power imbalance: detection of nodes with deficit/excess of reactive power which causes voltage instability and lower the total grid efficiency. • Overloading: identification of overloading of transformers and lines that affects the safe operation of the distribution grids. • Hosting capacity limits: calculation of hosting capacity of renewables, e.g., PV and wind, in the distribution grids without violation of voltage or thermal limits. Also, this can be related to hosting capacity for the electrification of transportation and heating, e.g. with Electric Vehicles (EV) and heat-pumps. • Power loss: quantification of power loss in different parts of the distribution grids that can be used further for energy efficiency improvement measures. • Grid reinforcement and reconfiguration: identify weak lines or nodes where further reinforcement or reconfiguration are required to keep the system’s stability and reliability. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 8 of 63 FLEXDIS project 1.2 Steady-state analysis for short-circuit calculations Steady state analysis for short-circuit calculations is motivated by the rapidly evolving nature of modern power systems, which are increasingly dominated by power electronics converters rather than conventional synchronous generators. The deployment of converters for renewable grid integration, energy storage, HVDC transmission, and other applications fundamentally changes system behavior during faults, requiring new approaches for reliable short-circuit assessment. In particular, converters feature fast dynamic responses and full controllability under disturbances, but they are also subject to current saturation during fault events, which alters their contribution to fault currents. Traditional short-circuit calculation methods, typically based on linear models and the use of Thevenin equivalents for synchronous generators, are not capable of capturing the nonlinear behavior, control modes, and current-limited operation of converters. Standards such as IEC 60909 recommend modeling converters as current sources with maximum injection capability, yet do not address how to determine the current angle or incorporate specific control modes during fault scenarios [1]. Steady state short-circuit analysis encompasses the control and saturation behavior of converters without requiring detailed dynamic models, which enables more accurate and efficient assessment of system responses during faults. This kind of analysis is essential for proper equipment sizing, tuning protection systems, and optimizing the grid-support functions expected of converters in the context of grid codes. Furthermore, while dynamic simulations remain a valuable reference for accuracy, they are computationally expensive and time-consuming, rendering steady state approaches preferable for large-scale or routine studies. By explicitly including the steady-state control equations governing each converter and their saturation states, steady state analysis enables the identification of realistic fault equilibrium points, thus providing results that are more closely aligned with how modern, converter-rich power systems behave in practice [2]. 1.3 Steady-state analysis for unbalanced systems The transmission and distribution of electrical energy in alternating current (AC) form has historically favoured the use of three-phase systems due to their superior efficiency, power handling capability, and infrastructure compatibility. In ideal conditions, these systems are perfectly balanced, with each phase carrying identical sinusoidal waveforms equally spaced by 120 electrical degrees. Under such assumptions, the network impedance is symmetrical, and positive-sequence modelling suffices for most steady-state analyses. However, real-world power systems are rarely perfectly balanced. Several factors, such as untransposed transmission lines, uneven distribution of loads, single-phase connections Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 9 of 63 FLEXDIS project To analyse the system dynamics and flexibility potentials, two models are developed: • A LV power distribution network model, built in DIgSILENT PowerFactory. • A thermal system model, representing the Thermonet configuration. 2.3.1 LV Power Network Model The local grid model simulates the full low-voltage electricity distribution system in PowerFactory, capturing the integration of renewables and flexible loads. The site is fed through two 0.63 MVA transformers (IDs 41542 and 41543) and operates as a radial feeder network. Lumped loads are used to represent combined household and heat pump consumption at each node. PV generation, EV charging, and future battery units are connected at defined points of common coupling (PCCs). Figure 4 shows the network model layout. Load flow simulations are performed to assess transformer and cable loading, voltage levels, and hosting capacity under different flexibility scenarios. Color coding in Figure 5 reflects operational stresses (e.g., overvoltage or transformer overload) under peak and flexible conditions. Figure 4: Model of LV power grids in Hyllegaard Høje built up in PowerFactory Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 16 of 63 FLEXDIS project Figure 5: A sample of color coding to indicate voltage levels of power nodes and loading levels of lines and cables in steady-state analysis 2.3.2 Thermonet System The thermonet at Hyllegaard Høje is a shared, low-temperature district heating system tailored for compact residential developments. It consists of approximately 30 km of uninsulated piping looped underground, circulating ambient-temperature water. Each building has a small heat pump that upgrades the temperature for space heating and domestic hot water. This design addresses issues commonly seen with individual air-to-water heat pumps, such as noise problems, and avoids the cost and spatial demands of separate ground-source heat pumps. Thermonet centralizes the underground infrastructure, making it more costefficient while reducing thermal interference risks. The Thermonet concept aligns with integrated energy management. It enables dynamic operation of heat pumps based on electricity market signals and DSO flexibility requests. This makes it possible to unlock significant power-to-heat flexibility without compromising end-user comfort. Figure 6 shows a diagram of Thermonet in Hyllegaard Høje demo site. Figure 6: Map of thermonet in Hyllegaard Høje Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 17 of 63 FLEXDIS project 2.3.3 Integration of Flexibility Assets into LV Network To simulate and analyze the interplay of DERs and flexibility services, all energy assets, i.e., heat pumps, PVs, EVs, and batteries, are embedded in the LV grid model. Technical specifications for each asset are used to place and configure their connections accurately within the PowerFactory environment. Residential demand is modeled accordingly. Heat pumps, which are boosted by the Thermonet’s ambient loop, are key to unlocking flexibility. Their operation depends on both thermal demand and available electrical capacity. The interaction between the Thermonet and the local grid determines how much demand can be shifted without violating technical limits, e.g., voltage drops or transformer loading. Therefore, the information about booster heat pumps and heat demands are extracted from the thermonent map and integrated into the LV network. After the integration of flexibility assets into the LV network, intelligent local controllers are being developed in DIgSILENT using DSL and DPL to manage asset-level behavior. These controllers are implemented in the simulation environment and interfaced with the grid model to test response to dynamic conditions and external signals. These controllers optimize the operations of responsive demand, such as EVs and heat pumps, in response to technical problems of the local grid, like undervoltage and transformer overloading, or dynamic electricity prices. Figure 7 shows part of the local network with EV chargers and heat pumps mapped to specific nodes. Figure 7: Integration of EVs and heat pumps into local LV network in Hyllegaard Høje for flexibility assessment Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 18 of 63 FLEXDIS project 2.3.4 CEMS-Based Control Architecture Neogrid leads the development of the CEMS for this site. The system integrates forecasting, optimization, and real-time control features aimed at improving the operational efficiency of the local energy community. Its main functionalities include: • Dynamic control of heat pumps in response to real-time electricity prices. • Predictive scheduling to align PV generation with heating needs. • Automated load shifting based on grid constraints, e.g., transformer loading, voltage profiles. • User interfaces to provide household-level feedback on cost savings, carbon impact, and consumption profiles. The CEMS is tested in two living lab configurations: one community-scale and one compact centralized setup. Both demonstrate scalability and adaptability to diverse residential configurations. The expected outcomes are as follows: • Reduction in household electricity costs. • Improved voltage stability through demand response. • Increased local energy autonomy and use of renewables. • Enhanced user engagement due to visibility and control of energy flows. 2.3.5 Aggregation of Flexibility to MV Network In order to study the impacts of energy flexibility on higher voltage levels, the local LV model of Hyllegaard Høje will be aggregated to the 50 kV network developed by Chalmers University of Technology. In this way, the flexibility potentials of assets like heat pumps, EVs and PVs are studied in the MV network. To enable the aggregation model, the following steps are suggested: • Net power profiling: the LV network is connected to the higher voltage level through 0.63 MVA transformers. The net load and generation of the LV network are determined to give a time-varying profile of active (P) and reactive power (Q) seen by the MV network. • Time-series exporting: after the determination of a P and Q, these are exported to the MV network as time series data in appropriate time resolution, e.g., 15-minutes or hourly basis. In this step, the time series includes base scenarios and different flexibility-activated scenarios for heat pumps, EVs and PVs. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 19 of 63 FLEXDIS project • Flexibility scenarios scaling: then the flexibility-activated scenarios can be scaled up to simulate a larger area (than the baseline scenario) that can be connected to the MV network. In the base scenario, 13 residential buildings are modeled. In the scaling scenarios, extra residential buildings are considered with different diversity factors, simultaneity factors, and penetration levels of DERs. Therefore, it will be studied how the combined similar communities impact the operation of the MV grid. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 20 of 63 FLEXDIS project 3 Unbalanced models for steady state analysis The conventional elements of the electrical power system are generators, loads, transformers, and lines, as well as the necessary reactive power compensation equipment. These elements are suitably interconnected to enable the generation, transportation, and consumption of electricity to meet system demand at any given point in time [6]. Traditionally, the electrical power network has been then divided into generation, transmission, distribution, and consumer subsystems, as represented in Figure 8. Transmission networks operate at high voltages (HV) to minimise transport losses. Conversely, electricity is generated at medium voltages (MV), and step-up transformers are employed at the generator substation to raise the voltage to transmission levels. In contrast, step-down transformers are used to reduce the high transmission voltages to levels suitable for distribution (MV), and eventually for industrial, commercial, and residential applications at low voltages (LV) [7]. Generation Transmission Distribution Consumers MV HV MV LV Figure 8: Electrical power system network Three-phase synchronous generators has been used as the main source to produce electrical power, which is delivered to demand points via alternating current (AC) three-phase transmission and distribution lines. The geometrical arrangement of these lines introduces some impedance unbalance between phases. Often, long-distance transmission circuits consist of more than one three-phase circuit and include series and shunt compensation to enable stable operation [8]. It should also be noted that the windings of three-phase transformers can be connected in various ways to suit specific requirements, and that transformer connections must be modelled in detail when system imbalances cannot be neglected in power system studies [9]. Additionally, distribution load points may be highly unbalanced due to the prevalence of individual single-phase loads. An increasing share of photovoltaic (PV) generation is now connected at the LV level, with part of this output is directly consumed there or injected into distribution. The application tool used to assess the steady-state operation of power systems exhibiting a considerable degree of unbalance is known as three-phase power flow [10]. In this application, all operations are carried out on a per-phase basis, and all power system components are modelled in the phase reference frame [11]. This project considers that the system’s geometrical unbalances are significant and that the system load is unbalanced. Therefore, all power system components will be represented in the phase domain. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 21 of 63 FLEXDIS project 3.1 Overhead Lines Power lines are fundamental components of power systems, responsible for transmitting electrical energy from generation sources to loads. They consist of a group of phase conductors located at a finite distance from the earth’s surface and may use the ground as a return path. Accordingly, it becomes necessary to account for this effect when calculating the line parameters [12]. High-voltage transmission lines may contain several conductors per phase (known as bundled conductors) and ground wires, while distribution lines may include a neutral wire as a return path. Both transmission and distribution circuits may introduce considerable geometric unbalances, hence electrical unbalances, depending on their layout [13]. The main objective is to develop an overhead power line model that enables precise calculation of voltage drops and losses during the power transmission process. The basis of power line modelling is to determine the resistance R, inductance L, conductance G, and capacitance Cper unit length [14]. 3.1.1 πmodel Lines can be represented using mathematical models that describe their electrical behaviour. It is a current practice to model the inductive and resistive effects of multiconductor transmission lines as a series impedance matrix, and the capacitive effects as a shunt admittance matrix. The overall transmission line model then can be represented by the πequivalent model, as shown in Figure 9. It consists of: • Series impedance: ~ Zseries =R+jX • Shunt admittance: ~ Yshunt =G+jB Where Ris the series equivalent resistance of the conductors, Xis the series self and mutual inductive reactances resulting from the magnetic fields surrounding the conductors, Gis the shunt conductance and Brepresents the shunt susceptance derived from the line capacitance due to the induced electric fields. ~ Uf ~ If G+jB 2 RjX G+jB 2~ Ut ~ It Figure 9: Line πequivalent circuit Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 22 of 63 FLEXDIS project The line capacitance is split between the two connection points with the resistance and the reactance in the middle, whose shape gives the so-called π-model. Although this singlephase simplification is a widely used technique, when modeling unbalanced systems it is needed to consider a general 5-wire approach, that later becomes a 3-wire equivalent. This 5-wire model is that including the three phases, the neutral and the ground: • The first three wires are the three-phase abc conductors. These carry the alternating current (AC) electricity and are used to distribute power efficiently. • The fourth wire is the neutral conductor n. It provides a return path for unbalanced current and is often grounded at substations to stabilize the system voltage. • The fifth wire is typically a ground wire g, which is connected to earth to protect equipment and people by providing a low-resistance path for fault currents, ensuring safety and system stability. It is also known as the shield wire. Not all physical lines will have neutral, and the ”ground return” is certainly not a wire, but it is represented as one in order to close the electrical circuit. Each wire has its own impedance, however, other impedances arise from the electromagnetic coupling of the wires, such that it is necessary to use a 5x5matrix to represent the total line impedance or admittance. ~ Z=            ~ Zaa ~ Zab ~ Zac ~ Zan ~ Zag ~ Zba ~ Zbb ~ Zbc ~ Zbn ~ Zbg ~ Zca ~ Zcb ~ Zcc ~ Zcn ~ Zcg ~ Zna ~ Znb ~ Znc ~ Znn ~ Zng ~ Zga ~ Zgb ~ Zgc ~ Zgn ~ Zgg            (1) Most transmission power lines do not carry the neutral, since that one is normally earthed at both ends of the line. Also, the ground return effect can be incorporated into the phase impedance as it will be detailed later. Then, it be can simply expressed as a 3x3 matrix: ~ Z=     ~ Zaa ~ Zab ~ Zac ~ Zba ~ Zbb ~ Zbc ~ Zca ~ Zcb ~ Zcc      (2) 3.1.2 Carson series impedance Carson’s equations are used in power system analysis to calculate the series self and mutual impedances of overhead transmission lines, taking into account the effects of the ground return path [15]. They are based on the following assumptions: Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 23 of 63 FLEXDIS project • The conductors are perfectly horizontal above ground and are long enough so that three-dimensional end effects can be neglected. This makes the field problem twodimensional. The sag is taken into account indirectly by using an average height above ground. • The free space is homogeneous and lossless, with permeability µ0and permittivity ε0. • The earth is homogeneous, with uniform resistivity ρ, permeability µ0, and permittivity ε0, bounded by a flat plane of infinite extent. • The spacing between conductors is at least one order of magnitude larger than the conductor radius, so that proximity effects can be ignored. i j i0 j0 dij Dij xij hi hj θij Figure 10: Carson’s geometry data of the tower configuration The elements of the series impedance matrix can then be calculated from the geometry of the tower configuration (see Figure 10) and from characteristics of the conductors. The self and mutual impedances can be computed using equations (3) and (4) respectively [16]. ~ Zii = (Ri+Rc ii) + jωµ0 2πln 2hi ri +Xi+Xc ii(3) ~ Zij =~ Zji =Rc ij +jωµ0 2πln Dij dij +Xc ij(4) Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 24 of 63 FLEXDIS project Where: •Riand Xiare the internal resistance and reactance of conductor iin Ω/km. •Rcand Xcare the Carson’s correction terms for earth return effects in Ω/km. •ω= 2πf is the angular frequency in rad/s. •hiis the average height above ground of conductor iin m. •riis the radius of conductor iin m. •dij is the distance between conductors iand jin m. •Dij is the distance between conductor iand the image of conductor jin m. 3.1.3 Example results Carson’s exact formulation for calculating the series impedance of overhead lines has been implemented and validated. To verify its correctness, a three-phase transmission line is analysed. The line operates at 50 Hz, with an earth resistivity of 100 Ω·m. Each phase consists of a single conductor with a radius of 10.5 mm and an internal resistance of 0.1363 Ω/km. The geometric configuration of the conductors is illustrated in Figure 11. c b a 27,5 m 12,65 m 12,65 m Figure 11: Example of a power line geometric arrangement Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 25 of 63 FLEXDIS project ~ Y=                                   2~ Ys+~ Ysh 3m2m2 f −~ Ys− ~ Ysh 2 3m2m2 f −~ Ys− ~ Ysh 2 3m2m2 f −~ Ys √3~m∗mfmt 0~ Ys √3~m∗mfmt −~ Ys− ~ Ysh 2 3m2m2 f 2~ Ys+~ Ysh 3m2m2 f −~ Ys− ~ Ysh 2 3m2m2 f ~ Ys √3~m∗mfmt −~ Ys √3~m∗mfmt 0 −~ Ys− ~ Ysh 2 3m2m2 f −~ Ys− ~ Ysh 2 3m2m2 f 2~ Ys+~ Ysh 3m2m2 f 0~ Ys √3~m∗mfmt −~ Ys √3~m∗mfmt −~ Ys √3~mmtmf ~ Ys √3~mmtmf 0 ~ Ys+~ Ysh 2 m2 t 0 0 0−~ Ys √3~mmtmf ~ Ys √3~mmtmf 0 ~ Ys+~ Ysh 2 m2 t 0 ~ Ys √3~mmtmf 0−~ Ys √3~mmtmf 0 0 ~ Ys+~ Ysh 2 m2 t                                   (23) 3.3 Loads Owing to the large number and diversity of loads present in power networks, it is preferable to group loads and treat them as bulk consumption points, rather than employing distinct models for rotating, static, and converter-based loads [25]. The model used for the loads is the ZIP model, meaning it is considered as a combination of impedance, current, and power loads, as depicted in Figure 16. G+jB Ire +jIim P+jQ Figure 16: Schematic ZIP load model In steady-state applications, most system loads can be effectively represented as threephase power sinks, connected either in star or delta configuration depending on system requirements [23]. Figure 17 illustrates a star-connected load with the neutral point solidly grounded, alongside a delta-connected load. As will be shown later, the power flow formulation operates using phase-to-neutral voltages and line currents, since the per-unit normalization is based on these quantities. For this reason, the developed tool models all loads as star-connected. Consequently, a preliminary conversion is required for deltaconnected loads. The software must therefore be capable of transforming impedance, current, and power from delta to star equivalents. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 32 of 63 FLEXDIS project ~ Ua ~ Ia ~ Ub ~ Ib ~ Uc ~ Ic ~ Za ~ Zb~ Zc ~ Ua ~ Ia ~ Ub ~ Ib ~ Uc ~ Ic ~ Zab ~ Zca ~ Zbc Figure 17: Star-connected load (left) and delta-connected load (right) representations 3.3.1 Impedance (Z) definition Three-phase loads defined as constant impedance can be connected either in star or delta configurations. If the impedance load is defined in star, the diagonal of the 3x3 matrix shown in (24), is simply filled with these values. ~ Y0=     ~ Ya0 0 0~ Yb0 0 0 ~ Yc      (24) However, if the load is defined in delta connection, the 3x3 matrix shown of (25) will be used to mathematically represent the load. ~ Y0=1 3      ~ Yab +~ Yca −~ Yab −~ Yca −~ Yab ~ Yab +~ Ybc −~ Ybc −~ Yca −~ Ybc ~ Ybc +~ Yca      (25) These matrices are also applicable for other shunt elements like capacitor banks. 3.3.2 Current (I) definition Traditionally, in positive sequence power flow analysis, constant current loads are directly stored in the ~ I0vector. However, since we are performing a three-phase power flow, the voltage angles of phases b and c are not zero, so it must be taken into account for both star (26) and delta-connected (27) current loads. ~ I0=     ~ I∗ aej δa ~ I∗ bej δb ~ I∗ cej δc      (26) Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 33 of 63 FLEXDIS project ~ I0=          ~ I∗ ab ej(δa−δb)−~ I∗ ca ej(δc−δa) √3 ~ I∗ bc ej(δb−δc)−~ I∗ ab ej(δa−δb) √3 ~ I∗ ca ej(δc−δa)−~ I∗ bc ej(δb−δc) √3          (27) 3.3.3 Power (P) definition Finally, if the load is defined as constant power, the values are stored in the ~ S0vector for star loads (28), while the corresponding delta-to-star transformation must be applied (29). Note that the founded transformation also involves the voltage, here the complete phasor. In both cases, the power and current transformation vector to star, will be updated in each iteration, adding significant complexity compared to the traditional power flow. ~ S0=     ~ Sa ~ Sb ~ Sc      (28) ~ S0=          ~ Ua·~ Sab ~ Ua−~ Ub− ~ Ua·~ Sca ~ Uc−~ Ua ~ Ub·~ Sbc ~ Ub−~ Uc− ~ Ub·~ Sab ~ Ua−~ Ub ~ Uc·~ Sca ~ Uc−~ Ua− ~ Uc·~ Sbc ~ Ub−~ Uc          (29) Note that only three-phase load models have been detailed in this deliverable. For a more in-depth explanation for three-phase but also for two-phase and single-phase load models, the reader is kindly referred to the master thesis of Alex Blanco [24]. 3.4 Generators For the power flow simulations, generators had been modelled as simple power injections into the system, which was completely valid. However, this is not sufficient when performing the short-circuit analysis, as the impedance of the generator must also be taken into account. VeraGrid has been programmed to accept a 3×3impedance matrix, which includes both the self-impedances and the couplings between the abc phases. Furthermore, as this three-phase analysis is still under development, it is common to encounter generator impedances in the sequence domain. Therefore, Fortescue’s theorem must be applied to obtain the equivalent values for the three phases: Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 34 of 63 FLEXDIS project ~ Zgenabc =     ~ Z0+~ Z1+~ Z2~ Z0+~a~ Z1+~a2~ Z2~ Z0+~a2~ Z1+~a~ Z2 ~ Z0+~a2~ Z1+~a~ Z2~ Z0+~ Z1+~ Z2~ Z0+~a~ Z1+~a2~ Z2 ~ Z0+~a~ Z1+~a2~ Z2~ Z0+~a2~ Z1+~a~ Z2~ Z0+~ Z1+~ Z2      (30) Where the transformation eigenvector ~a =ej2π/3 is used. A key parameter that must be transferred from the power flow to the short-circuit analysis is the induced electromotive force (EMF) in the generators ~ E, as this is the only voltage that will not change during the fault. The electromotive force depends on the flux induced in the machine’s rotor, and therefore on the excitation current. It can be assumed that the internal voltage ~ Eof the generator remains constant during the duration of the fault. The generator could be modelled during the short-circuit using the classic Thévenin equivalent, that is, as an ideal voltage source in series with the generator’s impedance, as shown in the electrical circuit of Figure 18. ~ Ua ~ Ub ~ Uc ~ Ea ~ Eb ~ Ec ~ Za ~ Zb ~ Zc Figure 18: Thévenin equivalent circuit scheme This circuit allows us to obtain the value of the induced electromotive force, given the voltage ~ Upf and power ~ Spf before the fault at the generator’s output bus: ~ E=~ Upf +~ Zgen ·~ Ipf =~ Upf + ~ S∗ pf ~ Ygen ·~ U∗ pf (31) However, this would require adding an additional bus to the original system between the generator’s impedance and the ideal voltage source. Therefore, the generator is modelled using its Norton equivalent, that is, an ideal current source in parallel with the generator’s impedance, as shown in the schematic of Figure 19. The Norton current source will take the value of the internal voltage multiplied by its admittance: ~ IN=~ Ygen ·~ E(32) Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 35 of 63 FLEXDIS project ~ INa ~ Za ~ INb ~ Zb ~ INc ~ Zc ~ Ua ~ Ub ~ Uc Figure 19: Norton equivalent circuit scheme 3.5 Voltage Source Converters Voltage Source Converters (VSCs) have emerged as prevalent devices in modern power systems, playing a pivotal role in the integration of renewable energy sources, energy storage systems, and HVDC links, to name a few. VSCs are power electronics-based devices built with a set of semiconductors that when properly controlled allow the adjustment of electrical magnitudes. VSCs interface DC systems with AC systems as depicted in Figure 20. More complicated configurations can exist, some involving the neutral, yet they are discarded for this project and assumed this is a three-wire VSC on the AC as common with transmission systems. Pt,b +jQt,b Pt,a +jQt,a Pt,c +jQt,c Pf,p Pf,n Ut,a∠δt,a Ut,b∠δt,b Ut,c∠δt,c Uf,p Uf,n Figure 20: Representation of a VSC VSCs offer two degrees of controllability when seen from a steady-state perspective. Internally, they synthetize voltages so that these reference magnitudes are met. This typically involves controlling active and reactive powers, voltage magnitudes in the DC or AC side, or even angle/frequency on the AC side in the case of grid-forming converters [26]. Under normal conditions VSCs can regulate a given set of magnitudes, as captured in Table 1 (excluding droops and abnormal cross-combinations of controls). Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 36 of 63 FLEXDIS project Table 1: Non-exhaustive list of control combinations for a VSC under balanced conditions Type Control 1 Control 2 PtQtPt,a +Pt,b +Pt,c Qt,a +Qt,b +Qt,c PtUtPt,a +Pt,b +Pt,c Ut,a, Ut,b, Ut,c PtUfPt,a +Pt,b +Pt,c Uf,p −Uf,n PfQtPf,p +Pf,n Qt,a +Qt,b +Qt,c PfUtPf,p +Pf,n Ut,a, Ut,b, Ut,c PfUfPf,p +Pf,n Uf,p −Uf,n UtδtUt,a, Ut,b, Ut,c δt,a, δt,b, δt,c Table 1 showcases some of the anticipated controls for systems under balanced conditions. In these cases, the powers on the three phases would be the same, voltages magnitudes take the same values, and angles maintain a perfect 120 degree shift between them. However, under unbalanced conditions, the situation changes considerably. There is little point in controlling the sum of powers, as large unbalances can take place, potentially forcing VSCs to saturate some of phases so as to not surpass its current limits. Hence, it is conventionally proposed to control powers and voltages in the positive sequence as described in [27]. During short-circuit conditions, and depending on the grid codes and priorities of the grid operator, it is proposed to inject current in the negative sequence to minimize the negative sequence voltages, while keeping some current margin in the positive sequence to boost the positive sequence voltage [28]. The grid-support converters can provide is dependent upon the maximum current its internal semiconductors can stand. Compared to synchronous generators, VSCs have a much more limited current provision, often in the range 1.0 to 2.0 p.u., which is several orders of magnitude smaller in comparison to traditional generators. The inclusion of VSCs into the power flow and short-circuit formulation must ensure the following limits are met at all times: It,a ≤Imax,(33) It,b ≤Imax,(34) It,c ≤Imax,(35) where Imax is the upper limit current magnitude a converter can admit, as specified by the manufacturer, and currents of the form It,x, where xwould symbolize one of the three phases, can be extracted from the powers and voltages depicted in Figure 20 through: It,x =qP2 t,x +Q2 t,x Ut,x .(36) Following grid codes, as the specification of currents is given in the sequences, the Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 37 of 63 FLEXDIS project Fortescue transformation has to be applied:      ~ It,a ~ It,b ~ It,c      =     1 1 1 1~a2~a 1~a ~a2           ~ It,0 ~ It,1 ~ It,2      ,(37) where the current ~ It,0becomes zero in a three-wire AC system and the currents ~ It,1and ~ It,2are commonly a function of the voltage drop. Traditional power system analysis for faults and power flows relies on symmetrical components. This method is effective because the sequences are decoupled in conventional networks, allowing the system’s state to be solved by superimposing the results from each sequence. However, this paradigm breaks down with power electronic converters, as their operational limits, particularly current magnitude, exist in the physical abc domain. One must first synthesize the total current in each phase from all three sequences to determine if a limit is violated. Mapping the dynamic behavior of grid-following or grid-forming converters into a steadystate model is also fraught with difficulty. The analysis must account not only for current limits but also for control priorities between active and reactive power, compliance with intricate grid codes, and discrete control actions, such as fault-ride-through logic, which alter the converter’s operating state during a disturbance. Consequently, accurately modeling converter behavior for system studies remains a significant and largely unsolved challenge, marking it as a critical area for ongoing research. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 38 of 63 FLEXDIS project 4 Validation of unbalanced models In this chapter, the steady-state results are presented and validated for both the power flow studies conducted on IEEE test feeders and the subsequent short-circuit simulations for different fault types. 4.1 IEEE 13 Node Test Feeder To validate the implementation of the three-phase power flow algorithm developed by Alex Blanco [24], the IEEE 13 Node Test Feeder has been selected as a reference benchmark [29]. This distribution system model, published by the IEEE Distribution System Analysis Subcommittee, represents a typical North American urban radial distribution network [30]. 680 652 611 684 671 692 675 646 645 632 633 634 650 Grid abc abc abc abc abc abc bcbc abc abc acc a abc −Yb −Ybc −D a−Y abc −D abc −Yabc −Yca −Dc −Y c −Y abc −Y abc −Y Figure 21: IEEE 13 Node Test Feeder representation The network, depicted in Figure 21, includes a variety of distribution elements such as: • Unbalanced overhead power lines with different phasing configurations (three-phase, two-phase, and single-phase). Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 39 of 63 FLEXDIS project • A substation transformer between the external grid and the slack bus 650, and a distribution transformer between buses 633 and 634. • Multiple balanced and unbalanced loads modelled as constant impedance (Z), constant power (P), and constant current (I), with three-phase connections in star (Y) and delta (D), connections between two phases, and also between one phase and the ground. • Shunt capacitor banks for reactive power compensation, also with different phasing configurations (three-phase and single-phase). The base voltage levels of the system are 115 kV on the transmission side and 4,16 kV on the distribution side, with a downstream low-voltage level of 0,48 kV at bus 634. The total nominal load is around 3,8 MVA, distributed across residential, commercial, and industrial consumers. Due to the network’s unbalanced and mixed-phase nature, this test feeder is widely recognised as a challenging yet representative case for validating unbalanced power flow solvers. The parameters used for the modelling and calculation of the transformers are presented in Table 2, for the loads in Table 3, for the lines in Table 4, and finally for the capacitor banks in Table 5. The impedance and admittance matrices for each line configuration are shown in the main author’s master thesis [24]. Table 2: Transformer data of the IEEE 13-bus Test Case From Bus To Bus UHV [kV] ULV [kV] Conn. Sr[kVA] R [%] X [%] Grid 650 115 4,16 Dy5 5 000 1 8 633 634 4,16 0,48 Yy0 500 1,1 2 Table 3: Load data of the IEEE 13-bus Test Case Bus Mod. Pa[kW] Qa[kVAr] Pb[kW] Qb[kVAr] Pc[kW] Qc[kVAr] 634 P - Y 160 110 120 90 120 90 645 P - Y 0 0 170 125 0 0 646 Z - D 0 0 230 132 0 0 652 Z - Y 128 86 0 0 0 0 671 P - D 385 220 385 220 385 220 675 P - Y 485 190 68 60 290 212 692 I - D 0 0 0 0 170 151 611 I - Y 0 0 0 0 170 80 632/671 P - Y 17 10 66 38 117 68 Total 1175 616 1039 665 1252 821 Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 40 of 63 FLEXDIS project Table 4: Line data of the IEEE 13-bus Test Case From Bus To Bus Length [ft.] Configuration Phasing 632 645 500 3 B C 632 633 500 2 A B C 645 646 300 3 B C 650 632 2000 1 A B C 684 652 800 7 A 632 671 2000 1 A B C 671 684 300 4 A C 671 680 1000 1 A B C 671 692 - Switch A B C 684 611 300 5 C 692 675 500 6 A B C Table 5: Capacitor bank data of the IEEE 13-bus Test Case Bus Qa[kVAr] Qb[kVAr] Qc[kVAr] 675 200 200 200 611 - - 100 Total 200 200 300 The following comparison contrasts the magnitude and angle of voltages obtained using the VeraGrid’s developed simulation tool with those provided as reference data by the IEEE [31]. Voltage magnitude and angle results are shown for phase a in Table 6 and plotted in Figures 22 and 23, respectively. The values are perfectly matched. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 41 of 63 FLEXDIS project 835 840 845 850 855 860 865 870 875 880 885 890 895 900 1.04 1.045 1.05 Time [min] Uc[p.u.] IEEE VeraGrid Figure 29: Phase cvoltage profile comparison for hour 14 295 300 305 310 315 320 325 330 335 340 345 350 355 360 1.046 1.048 1.05 Time [min] Ub[p.u.] IEEE VeraGrid Figure 30: Phase bvoltage profile comparison for hour 5 4.3 Single Line-to-Ground Fault (SLG) The short-circuit calculation method will be tested using the 13-bus test network, which was already constructed for the power flow and whose results are exactly the same as the reference, thus fully validated. As shown in the schematic of Figure 31, a fault will be simulated on phase ato earth at bus 634 with two fault impedance values. A generator has been added to bus 632, providing an equivalent power to the network to which the system was connected. The sequence impedance values for the generator are Z1= 0.004 + 0.5jp.u. for the positive sequence, Z2= 0.02 + 0.5jp.u. for the negative sequence, and Z0= 0.01 + 0.08jp.u. for the zero sequence. The fault admittance is defined by the user in VeraGrid, while its matrix depends on the fault type and the phases involved. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 48 of 63 FLEXDIS project 680 652 611 684 671 692 675 646 645 632 633 634 Ia SC G abc abc abc abc abc bcbc abc abc acc a abc −Yb −Ybc −D a−Y abc −D abc −Yabc −Yca −Dc −Y c −Y abc −Y abc −Y Figure 31: Phase-to-ground short-circuit at the bus 634 of the IEEE 13 Node Test Feeder A single line-to-ground fault (SLG) occurs when one phase conductor accidentally makes contact with the ground. It is illustrated in Figure 32 for phase a, and the corresponding fault admittance matrix is given by: ~ Yf=     ~ Ya f0 0 0 0 0 0 0 0      (38) Bus a Bus b Bus c ~ Ya f~ Ia SC Figure 32: Single Line-to-Ground Fault (SLG) Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 49 of 63 FLEXDIS project 632 633 634 671 684 675 680 652 0 0.5 1 Bus Number Ua[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 33: Voltage magnitude per bus in the faulted phase a during a SLG fault 632 645 646 633 634 671 675 680 1 1.02 1.04 1.06 Bus Number Ub[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 34: Voltage magnitude per bus in the health phase b during a SLG fault 632 645 646 633 634 671 684 611 675 680 0.94 0.96 0.98 1 1.02 Bus Number Uc[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 35: Voltage magnitude per bus in the health phase c during a SLG fault Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 50 of 63 FLEXDIS project Figures 33–35 show the VeraGrid results for the voltage magnitudes across all buses during a single line-to-ground (SLG) fault. In phase a, the voltage collapses nearly to zero at the faulted bus (634) and drops significantly at the adjacent bus (633), with the severity increasing as the fault resistance decreases. In contrast, phases b and c experience a slight rise at the faulted location due to the local reference shift, while farther from the fault the dominant effect is the voltage drop caused by fault current flowing through the network impedances, leading to reduced voltages compared to the pre-fault condition [36]. 4.4 Line-to-Line Fault (LL) A line-to-line fault (LL) occurs when two phase conductors come into contact with each other. Figure 36 shows the fault between phases c and a, and the corresponding fault admittance matrix is given by: ~ Yf=     ~ Yca f0−~ Yca f 0 0 0 −~ Yca f0~ Yca f      (39) Bus a Bus b Bus c ~ Yca f ~ Ia SC ~ Ic SC Figure 36: Line-to-Line Fault (LL) A line-to-line (LL) fault is simulated in VeraGrid between phases cand aat bus 634 of the IEEE 13-Node Test Feeder. Figures 37 and 39 show that both faulted phases undergo a strong voltage drop, with the relative severity depending on the fault resistance: phase a is more affected for Rf= 1 p.u., while phase cshows a deeper dip for Rf= 0.1p.u. This difference reflects how the two phases share the fault current path and how the drop distributes with network impedance and Rf. In the surrounding buses the impact is still visible, but voltages gradually recover with distance from the fault. Figure 38 presents the results for the healthy phase b, which remains nearly uniform but slightly reduced across the network. Unlike the SLG case, no voltage rise is observed in the healthy phase, since two phases are simultaneously pulled down and the neutral shift effect is suppressed. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 51 of 63 FLEXDIS project 632 633 634 671 684 675 680 652 0.4 0.6 0.8 1 Bus Number Ua[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 37: Voltage magnitude per bus in the faulted phase a during a LL fault 632 645 646 633 634 671 675 680 1.02 1.04 1.06 Bus Number Ub[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 38: Voltage magnitude per bus in the health phase b during a LL fault 632 645 646 633 634 671 684 611 675 680 0.4 0.6 0.8 1 Bus Number Uc[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 39: Voltage magnitude per bus in the faulted phase c during a LL fault Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 52 of 63 FLEXDIS project 4.5 Double Line-to-Ground Fault (DLG) A double line-to-ground fault (DLG) occurs when two phase conductors simultaneously make contact with the ground. Figure 40 illustrates the fault involving phases c and a, and the corresponding fault admittance matrix is given by: ~ Yf=     ~ Ya f0 0 0 0 0 0 0 ~ Yc f      (40) Bus a Bus b Bus c ~ Ya f ~ Yc f ~ Ia SC ~ Ic SC Figure 40: Double Line-to-Ground Fault (DLG) A double line-to-ground (DLG) fault is simulated in VeraGrid at bus 634 of the IEEE 13-Node Test Feeder, involving phases aand cconnected to ground. Figures 41 and 43 show that both affected phases undergo a severe voltage collapse, almost reaching zero at the faulted bus, which represents a deeper drop than in the LL case where the voltages remained above 0.4 p.u. In the neighbouring buses the impact is still evident, while farther away the voltages gradually recover. The behaviour of the healthy phase b, shown in Figure 42, is markedly different: a clear voltage rise appears at the buses closest to the fault (633 and 634). This occurs because the simultaneous grounding of two phases strongly distorts the local reference, displacing the neutral point and forcing the remaining phase upward. This contrasts with the LL fault, where no rise was observed and the healthy phase experienced a more uniform slight depression across the system [37]. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 53 of 63 FLEXDIS project 632 633 634 671 684 675 680 652 0 0.5 1 Bus Number Ua[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 41: Voltage magnitude per bus in the faulted phase a during a DLG fault 632 645 646 633 634 671 675 680 1 1.02 1.04 1.06 Bus Number Ub[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 42: Voltage magnitude per bus in the health phase b during a DLG fault 632 645 646 633 634 671 684 611 675 680 0 0.5 1 Bus Number Uc[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 43: Voltage magnitude per bus in the faulted phase c during a DLG fault Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 54 of 63 FLEXDIS project 4.6 Three-Phase Fault (LLL) A three-phase (LLL) fault occurs when all three phase conductors come into contact with each other. Figure 44 shows the fault between phases a, b, and c, and the corresponding fault admittance matrix is given by: ~ Yf=     ~ Yab f+~ Yca f−~ Yab f−~ Yca f −~ Yab f~ Yab f+~ Ybc f−~ Ybc f −~ Yca f−~ Ybc f~ Ybc f+~ Yca f      (41) Bus a Bus b Bus c ~ Yca f ~ Ybc f ~ Yab f ~ Ia SC ~ Ib SC ~ Ic SC Figure 44: Three-Phase Fault (LLL) A three-phase (LLL) fault is simulated in VeraGrid at bus 634 of the IEEE 13-Node Test Feeder. Figures 45–47 show the voltage profiles of phases a,b, and cacross the system during the disturbance. In this case, all three phases behave symmetrically, with the voltages collapsing almost to zero at the faulted bus and showing a marked drop at the upstream bus 633. The influence of the fault resistance is negligible, since the short circuit equally involves all phases. At the remaining buses, the voltages settle to nearly uniform values slightly below the pre-fault condition, reflecting the balanced nature of this type of fault. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 55 of 63 FLEXDIS project 632 633 634 671 684 675 680 652 0 0.5 1 Bus Number Ua[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 45: Voltage magnitude per bus in the faulted phase a during a LLL fault 632 645 646 633 634 671 675 680 0 0.5 1 Bus Number Ub[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 46: Voltage magnitude per bus in the faulted phase b during a LLL fault 632 645 646 633 634 671 684 611 675 680 0 0.5 1 Bus Number Uc[p.u.] Power Flow Rf= 1 p.u. Rf= 0.1p.u. Figure 47: Voltage magnitude per bus in the faulted phase c during a LLL fault Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 56 of 63 FLEXDIS project 4.7 Three-Phase-to-Ground Fault (LLLG) A three-phase-to-ground (LLLG) fault occurs when all three phase conductors come into simultaneous contact with the ground. Figure 48 illustrates the fault involving phases a, b, and c, and the corresponding fault admittance matrix is given by: ~ Yf=     ~ Ya f0 0 0~ Yb f0 0 0 ~ Yc f      (42) Bus a Bus b Bus c ~ Ya f ~ Ya f ~ Yc f ~ Ia SC ~ Ib SC ~ Ic SC Figure 48: Three-Phase-to-Ground Fault (LLLG) Finally, a three-phase-to-ground (LLLG) fault is simulated in VeraGrid at bus 634 of the IEEE 13-Node Test Feeder. Figures 49–51 show that all three phases decrease uniformly, collapsing close to zero at the faulted bus and exhibiting a marked drop at the upstream bus 633. The remaining buses present nearly uniform voltages slightly below the pre-fault condition, similar to the LLL case. The main difference is that the fault resistance plays a more significant role: for Rf= 1 p.u. the voltages at the faulted bus do not fall completely to zero, highlighting the stronger influence of grounding impedance in this type of fault. A detailed explanation of the three-phase power flow and short-circuit analysis implemented in VeraGrid can be found in the related master thesis [24]. Deliverable D1.1 – Steady-state Models of Distribution Grids for Voltage Stability Evaluation 57 of 63