Residential Load Flow Modeling and Simulation

Abstract

In recent years, home energy management systems (HEMSs) have emerged as critical components within the concept of smart cities and grids. Within HEMSs, load flow analysis represents one of the fundamental tools for smart grid studies, forming the basis for a wide range of advanced applications, including state estimation, fault diagnosis, and optimal power flow computation. To achieve high levels of load flow accuracy and reliability, HEMSs must incorporate detailed models of all electrical elements found in modern residential units, including appliances, wiring, and energy resources. This paper proposes a load flow solution for smart home networks, featuring detailed models of wiring, appliances, and on-site generation systems. Moreover, a detailed appliance model derived from smart meter data, capable of representing both static-load devices and complex appliances with time-varying consumption profiles, is introduced. Additionally, a measurement-based validation of residential electrical wiring model is presented. The proposed models and calculation procedures have been verified by comparing the simulated results with the literature, yielding a deviation of approximately 0.7%. Analyses of a large-scale network have shown that this approach is up to six times faster compared to state-of-the-art procedures. The developed solution demonstrates practical applicability for use in industry-grade smart power management software.

1. Introduction

1.1. Motivation

Smart buildings and homes have become key components of the modern distribution-level smart grid paradigm. Besides consuming energy, they can also generate it through renewable energy sources such as photovoltaic (PV) systems, wind turbines, and other distributed generation (DG) technologies [1,2]. In that case, they can be called prosumers. Consequently, the traditionally passive nature of low voltage (LV) distribution networks (DNs) is evolving into an active, bidirectional system. In this context, prosumers contribute to the overall energy balance by partially meeting their own consumption demands and, in many cases, by exporting surplus energy back to the utility grid. To effectively manage these interactions, the development of advanced home energy management systems (HEMSs) [1,2] is essential. Such systems must integrate and coordinate a diverse range of residential loads (including smart appliances such as refrigerators, washing machines, televisions, and electric vehicle (EV) charging units) with on-site renewable energy resources (ERs) [3,4,5,6,7,8,9]. Those resources are expected to play an increasingly significant role for two primary reasons: first, the rapid population growth in both developed and developing countries is contributing to a substantial rise in energy demand and second, the cost of electricity supplied by large-scale distribution systems is increasing, largely due to the uneven expansion of infrastructure relative to consumption growth [9]. HEMSs are therefore used to manage on-site generation and consumption, to influence the demands in the energy market on user behalf, and to improve efficiency in both consumption and generation of the local communities. Additionally, there is a growing need for detailed and accurate models of residential ERs, which often employ complex control strategies under varying operating conditions [4,5,6].

Home electrical wiring systems vary globally due to differences in line section configurations. In European countries, single-phase two-wire and three-phase four-wire systems are predominantly utilized [7]. In these systems, the voltage between the neutral and the live (phase) conductor is typically 230 V, while the inter-phase voltage in a three-phase system is approximately 400 V, with a phase angle difference of 120 degrees. On the other hand, in countries such as the United States and Taiwan, single-phase three-wire systems are commonly employed [7]. In this configuration, the voltage between the neutral (grounded) wire and either of the two ungrounded (live) wires is approximately 110 V, with a phase angle difference of 180 degrees. The voltage between the two ungrounded wires is around 220 V. This configuration may originate from a split-phase system (110/220 V) or from a three-phase four-wire delta system. The adoption of such multi-wire configurations offers several advantages, including reduced line losses, voltage drops, and the need for fewer wires to supply equivalent areas within residential or commercial buildings.

Load flow analysis [5,7,8] is one of the most widely utilized functions in power system studies as it serves as the foundation for numerous advanced techniques such as state estimation, fault analysis, and optimal power flow. Consequently, HEMSs depend on accurate modeling of residential grid parameters and a reliable load flow calculation procedure which incorporates more detailed models of smart home wiring systems and appliances.

1.2. Literature Review

In recent years, numerous load flow algorithms have been developed to support distribution-level smart grid applications [4,5,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. These methods address several key challenges inherent to DNs, including the following: (i) low X/R ratios, which are characteristic of distribution line segments and can affect numerical stability in power flow solutions [10,11]; (ii) multiphase line configurations that require more complex modeling techniques [12]; (iii) large-scale distribution network representations aimed at enabling real-time computational performance; (iv) unbalanced operating conditions, which necessitate more detailed and accurate network models to reflect phase asymmetries and load diversity [12]. Furthermore, the topology of DNs is generally more susceptible to structural changes compared to transmission networks. As a result, load flow solutions originally developed for transmission systems are often not directly applicable to DNs due to their distinct operational and structural characteristics [5,8]. In the context of distribution systems, load flow models are typically categorized as either node-oriented or branch-oriented, each offering specific advantages in terms of modeling flexibility and computational efficiency. Node-oriented load flow methods include Newton-based approaches [4,18,19], the Implicit Zbus method [22,23,24], and the Current Injection method [25]. In contrast, branch-oriented methods primarily comprise the backward–forward sweep (BFS) algorithm [12,13,15,16], Loop Impedance Matrix-based Load Flow [7], and the Distribution System Load Flow Solution (DADS) [11,21]. Among these, BFS techniques have demonstrated notable performance advantages in radial and weakly meshed DNs. This is largely due to their independence from matrix-based formulations, including the following: (i) the Bus Admittance Matrix [4]; (ii) the Bus-Injection to Branch-Current (BIBC) and Branch-Current to Bus-Voltage (BCBV) matrices [21]; (iii) the Branch Impedance Matrix [12]. Additionally, the simplicity and computational efficiency of BFS methods make them particularly attractive for industrial applications [5,12,26]. However, a known limitation of the BFS approach is the need for special handling of mesh structures in weakly meshed systems, which can lead to increased iteration counts and longer convergence times. Finally, linear models are used when performing distribution network (DN) planning and operation. These linearized models are approximate and allow the study of realistic (large-scale) networks. When dealing with virtual power plants and microgrids [27], it is important to have a precise simulator of residential units. To address highly uncertain parameters and to simplify the model, the utilization of a linear power flow model along with the adoption of a backward–forward sweep algorithm has been presented in [28].

Accurate and detailed models of residential networks offer numerous benefits, supporting both load flow analyses and grid-interactive efficient building applications [29,30]. Specialized load flow methods have been developed to address the specific requirements of residential networks across different regions of the world [7,29,30]. Paper [31] proposes a method to evaluate building energy consumption using the energy consumption monitoring index which is obtained from different functional sectors. It is based on energy consumption monitoring data, and it uses the linear regression model to predict the overall consumption. The study of HEMS with battery storage systems for peak load shaving was considered in [32]. The model includes PV with battery storage which is externally connected to the load. This model does not include any other types of consumers or electric installations in the home network. The impact of the voltage imbalance on the LV distribution networks is considered in [33]. The authors stated that the voltage imbalances could seriously damage the distribution equipment and speed up the equipment replacement cycle. Among all effects, the following ones are highlighted: extra power loss, safety deficiency, motor failure, and life-cycle attenuation. The higher the voltage imbalance, the greater the power consumption. Again, these imbalances were not considered in smart home wiring systems. The urgent need for smart energy coordination of residential homes is stated in [34]. The smart grid technology must be intelligent to coordinate small- and large-scale energy generation for the consumers since the rapid employment of DG units and Energy Storage Systems (ESSs) cannot be operated by standard system operators. This technology must be based on real-time control implementation which would improve the reliability, security, and efficiency of the distribution network. Additionally, in [34], the residential home is considered as a single consumer without concern for the impact of individual home appliances, DGs, and ESSs on the internal home power grid.

Paper [35] performs the assessment of household devices and appliances in terms of their power usage. The results show that modern appliances are changing the reactive load flow. This is due to the increasing usage of switching-mode power supplies. The measurement has not been performed over a longer period of time and covers the single-phase three-wire system only. Measurement in real time are necessary since some of the appliances can have different power load profiles. Paper [36] gave realistic load profiles of individual appliances based on the power consumption measurement over a longer period of time. This approach is more suitable for load flow modeling of the home network since it gives the opportunity to make more realistic models of the consumers. Both papers [35,36] do not use the load flow analysis based on the consumer profiles and they also did not consider the home wiring as an important smart home component.

The comparison of different load flow methods in distribution and residential networks has been presented in

Table 1. Literature review comparison.

Ref	Method	Residential Network Included	DG Incorporated into the Residential Network	All Types of Residential Electrical Wiring System Included
[4,18,19]	Newton-based approaches	No	No	No
[10]	Newton- and BFS-based approach	Yes	No	No
[22,23,24]	Implicit Zbus method	No	No	No
[25]	Current Injection method	No	No	No
[12,13,15,16]	Backward–Forward Sweep	No	No	No
[7]	Loop Impedance Matrix-based	Yes	Yes	No
[11,21]	Distribution System Load Flow Solution	No	No	No
[29]	Implicit Zbus method	Yes	No	No
[30]	Newton-based approach	Yes	Yes	No

.

1.3. Research Gaps

To the best of our knowledge, references [7,10,29,30] represent the only studies that examine the electrical networks of residential and commercial buildings where appliances are modeled with time-invariant active and reactive power load profiles. Reference [7] investigates systems in both the United States and Taiwan, focusing exclusively on single-phase three-wire configurations. In contrast, references [10,29,30] are restricted to the modeling of European household electrical networks. The load flow algorithms presented in the rest of the referenced literature generally disregard the internal electrical networks of residential and commercial buildings, treating them instead as aggregated consumer nodes characterized by specified active and reactive power values. However, for a HEMS to operate with high accuracy, it is essential to develop detailed models that account for individual electrical appliances and the typical wiring infrastructure found in modern residential dwellings, which are largely absent from the power flow methodologies reviewed. Additionally, DGs are often modeled as standalone units connected directly to the distribution network, rather than as integrated components of a prosumer’s local network. When implementing HEMS optimization strategies [37,38,39], comprehensive and precise simulations of the dwelling unit are imperative. A proposed load flow approach that incorporates residential wiring layouts, appliance-level consumption profiles, and local generation sources is therefore a critical component of such simulations.

1.4. Contributions

To address the identified research gaps, this paper proposes a novel power flow calculation procedure tailored to a detailed electrical network model of residential buildings. The key contributions of this work are as follows:

• Development of line models for single-phase two-wire, single-phase three-wire, and three-phase four-wire installation systems suitable for load flow calculations in various types of installations, such as smart homes across Taiwan, America, and Europe. Measurements used to determine the series and shunt parameters of residential wiring sections validate the modeling of home electrical networks.
• Prosumer-based models for single-phase and three-phase DGs, as well as Electronically Coupled Distributed Generators (ECDGs), have been developed and integrated into the load flow framework. These models incorporate advanced control strategies to represent the behavior of ECDGs more accurately within residential energy systems.
• A comprehensive load flow methodology and simulator tailored for smart home networks has been developed using the proposed network models. To achieve more realistic test scenarios, measurements of various household electrical appliances have been incorporated into the load flow simulator. The developed simulator constitutes a versatile platform for the evaluation of additional models and scenarios in the context of future experimental studies.

The remainder of this paper is organized in alignment with its primary contributions. Section 2 presents the modeling of single-phase and multiphase residential wiring systems, along with typical household consumers. This section also details the measurement methodology used to determine the series and shunt parameters of home network line sections. Section 3 introduces the proposed load flow model and the associated computational procedure tailored for residential networks. Section 4 provides model validation results, includes comparative analyses between simulated and measured data, and demonstrates the load flow simulations for several representative home circuit configurations. Additionally, this section addresses the integration of prosumer-based generation units into the residential network. Finally, Section 5 concludes the paper with a summary of key findings and potential directions for future research.

2. Home Network Models

Home network models of wiring and consumers are important parts of the load flow calculation. This section presents single- and multiphase home network wiring models. Those models cover the most frequent types of home wiring, including the USA and Europe ones. This section also presents two types of consumer models: simple consumer that has a constant active and reactive power consumption, and a complex one which varies its active and reactive powers in real time.

2.1. Single-Phase Three-Wire System and Consumer Model

The simple home network shown in Figure 1a is considered in [7]. Home installation from this figure represents a single-phase three wire system (1φ3w). The application of the 1φ3w system is more convenient than the single-phase two-wire system (1φ2w). The reason is explained as follows. The two sources connected between grounded and ungrounded conductor have the same magnitude. However, they have the opposite phase angle. In this manner, the voltage between two ungrounded conductors is twice the magnitude over the voltage between grounded and ungrounded conductor. The other advantages are moderate voltage drops, reduced line loss, and savings in materials during home installation construction.

The electrical circuit of the home installation from Figure 1a is represented in Figure 1b. The current sources represent loads from Figure 1b. The consumers are modeled as active and reactive powers known as ZIP consumer model [40]:

$$P_k\left(U_k\right)=z_{Pk}{\left({\displaystyle \frac{U_k}{U_n}}\right)}^2{P}_k^{\mathit{spec}}+i_{Pk}\left({\displaystyle \frac{U_k}{U_n}}\right)P_k^{\mathit{spec}}+p_{Pk}{P}_k^{\mathit{spec}},$$

(1a)

$$Q_k\left(U_k\right)=z_{Qk}{\left({\displaystyle \frac{U_k}{U_n}}\right)}^2{Q}_k^{\mathit{spec}}+i_{Qk}\left({\displaystyle \frac{U_k}{U_n}}\right)Q_k^{\mathit{spec}}+p_{Qk}{Q}_k^{\mathit{spec}},$$

(1b)

where $U_k$ and $U_n$ are actual and nominal voltage magnitudes at load terminal. $P_k^{\mathit{spec}}$ and $Q_k^{\mathit{spec}}$ are specified active and reactive powers at nominal voltage. The coefficients $z_{Pk}$, $i_{Pk}$, $p_{Pk}$ and $z_{Qk}$, $i_{Qk}$, $p_{Qk}$ are constant impedance and current and power coefficients, and the following relations are applied:

$$z_{Pk}+i_{Pk}+p_{Pk}=1,$$

(2a)

$$z_{Qk}+i_{Qk}+p_{Qk}=1.$$

(2b)

If the load power factor is specified at the nominal voltage, then the specified reactive power is calculated in the following manner:

$$Q_k^{\mathit{spec}}=P_k^{\mathit{spec}}\sqrt{{\displaystyle \frac{1}{{\left(cos\left({\varphi }_k^{\mathit{spec}}\right)\right)}^2}}-1}.$$

(3)

If the consumer does not change active power and power factor in real time, it is sufficient to model it with just those two parameters. However, if the consumer varies those two parameters in real time (as described in [36]), then the model is based on the measured consumption in real time using the smart meter connected to the cloud [41], as described in Section 4.3 and [36].

Since the backward–forward sweep is practical for the industrial application [5,12,26], the neutral (grounded) line of the home network 1φ3w line must be integrated into the grounded wires. In this manner, the model of home line sections will be presented as standardized distribution line models from [5,8]. The circuit from Figure 1b with inserted artificial nodes is presented in Figure 1c. With artificial node insertion, the number of nodes is the same in all three conductors of 1φ3w lines. In addition, during the insertion of artificial nodes, the sections are divided into as many parts as there are nodes in ungrounded conductors (e.g., ${\widehat{Z}}_3$ from Figure 1b is divided into ${\widehat{Z}}_3^I$, ${\widehat{Z}}_3^{\mathit{II}}$, and ${\widehat{Z}}_3^{\mathit{III}}$ in Figure 1c). The IDs of new generated nodes are shown in

Table 2. Generated nodes.

IDs of new generated nodes	A0’, A0”, A9’, A10’, A11’, A12’, A13’, A14’, A15’
Node pairs	A0’, A0”	A1, A9’	A2, A10’	A11’, A3	A4, A12’	A13’, A5	A14’, A6	A15’, A7	A8
Layer-type numeration	0a, 0b	1a, 1b	2a, 2b	3a, 3b	4a, 4b	5a, 5b	6a, 6b	7a, 7b	8a

with the corresponding node pairs. From Figure 1c, the grounded conductor nodes which correspond to the voltage source are named as root nodes (A0’ and A0”). Since the layer-type numeration must be implemented in order to apply the backward–forward sweep load flow procedure, the new node IDs are also shown in Table 2 (layer-type numeration row) and presented in the electrical circuit shown in Figure 1d.

From Figure 1d, the branch impedances of each 1φ3w line section are numerated in the following manner: each line section has three conductors (one grounded and two ungrounded). Each section has two sets consisting of three nodes (one node set closer to the voltage source node set and one further from the source node set). The IDs of branches from branch sets are the same with further section node sets.

The model of each 1φ3w line section from Figure 1d is presented in Figure 2a. The 1φ3w line section k from this figure has two node sets: one node set closer to the root (in, ia, ib) and one further from the root (kn, ka, kb). Using Kirchhoff’s laws, the model of 1φ3w line section k is derived:

$${\widehat{U}}_{ka}={\widehat{U}}_{ia}-{\widehat{Z}}_{ka}{\widehat{I}}_{ka}-{\widehat{U}}_{kn},$$

(4a)

$${\widehat{U}}_{kb}={\widehat{U}}_{ib}-{\widehat{Z}}_{kb}{\widehat{I}}_{kb}-{\widehat{U}}_{kn},$$

(4b)

$${\widehat{U}}_{kn}={\widehat{Z}}_{kn}{\widehat{I}}_{kn}={\widehat{Z}}_{kn}\left({\widehat{I}}_{ka}+{\widehat{I}}_{kb}\right).$$

(4c)

This model is further optimized by dropping out the grounded conductor voltage drop ${\widehat{U}}_{kn}$ from first two equations of the previous model (Figure 2b):

$${\widehat{U}}_{ka}={\widehat{U}}_{ia}-\left({\widehat{Z}}_{ka}+{\widehat{Z}}_{kn}\right){\widehat{I}}_{ka}-{\widehat{Z}}_{kn}{\widehat{I}}_{kb},$$

(5a)

$${\widehat{U}}_{kb}={\widehat{U}}_{ib}-\left({\widehat{Z}}_{kb}+{\widehat{Z}}_{kn}\right){\widehat{I}}_{kb}-{\widehat{Z}}_{kn}{\widehat{I}}_{ka}.$$

(5b)

The final model suited for the BFS load flow procedure application is as follows:

$${\widehat{U}}_{ka}={\widehat{U}}_{ia}-\left({\widehat{Z}}_{\mathit{kaa}}\right){\widehat{I}}_{ka}-{\widehat{E}}_{ka},$$

(6a)

$${\widehat{U}}_{kb}={\widehat{U}}_{ib}-\left({\widehat{Z}}_{\mathit{kbb}}\right){\widehat{I}}_{kb}-{\widehat{E}}_{kb}.$$

(6b)

The impedances ${\widehat{Z}}_{\mathit{kaa}}$ and ${\widehat{Z}}_{\mathit{kbb}}$ are self-impedances while the electromotive forces ${\widehat{E}}_{ka}$ and ${\widehat{E}}_{kb}$ represent inductive couplings between two single-phase ungrounded conductors. The model described above corresponds with the final decoupled section circuits (Γ_ks, $s=\mathrm{a},\mathrm{b}$) shown on Figure 2c. The Γ_ks circuits have one predecessor and, in general cases, more than one successor Γ_js, $j\in {\alpha }_{ks}$, $s=\mathrm{a},\mathrm{b}$.

The final home network circuit is presented in Figure 3. It can be observed from this figure that the ungrounded conductor wires are decoupled using the model described above. Furthermore, the required layer-type numeration has been used: the Γ segments directly supplied from the root with second nodes are put in the first layer. The successor Γ segments which are supplied from its predecessors from the first layer are put in the second layer. Finally, the last layer consists of Γ segments that have no successors (e.g., segments 8a and 7b from Figure 3). The network from Figure 3 has eight layers in phase a and seven layers in phase b. The BFS procedure adjusted for the smart home application is described as follows:

Backward sweep

$${\left({\widehat{I}}_{ks}\right)}^{h+1}={\left({\widehat{I}}_{\mathit{conks}}\right)}^{h+1}+{{\sum }_{i\in {\alpha }_{ks}}\left({\widehat{I}}_{is}\right)}^{h+1},$$

(7a)

$${\left({\widehat{I}}_{\mathit{conks}}\right)}^{h+1}={\displaystyle \frac{{\widehat{S}}_{ks}\left(U_{ks}\right)}{{({\widehat{U}}_{ks}^h)}^{*}}},s=\mathrm{a},\mathrm{b};$$

(7b)

Forward sweep

$${\left({\widehat{U}}_{ks}\right)}^{h+1}={\left({\widehat{U}}_{is}\right)}^{h+1}-{\widehat{Z}}_{kss}{\left({\widehat{I}}_{ks}\right)}^{h+1}-{\left({\widehat{E}}_{ks}\right)}^{h+1},s=\mathrm{a},\mathrm{b},$$

(7c)

$${\left({\widehat{E}}_{ks}\right)}^{h+1}={\widehat{Z}}_{kn}{\left({\widehat{I}}_{kt}\right)}^{h+1},s,t=a,b\wedge t\ne s.$$

(7d)

The load flow procedure starts with the backward sweep, where the branch currents are calculated starting from branches from the last layer. Afterwards, in the forward sweep the node voltages are calculated starting from the first layer. The procedure is finished when the convergence criteria $\left|U_{ks}^{h+1}-U_{ks}^h\right|<{\epsilon }_U$, $\left|{\theta }_{ks}^{h+1}-{\theta }_{ks}^h\right|<{\theta }_U$ is achieved, where $U_{ks}$ and ${\theta }_{ks}^h$ are node k voltage magnitude and phase angle. The flow-chart of the procedure is presented in detail in Section 3.

2.2. Three-Phase and Multiphase Systems

The European-style home installation mainly consists of four- and five-wire three-phase (3φ4w, 3φ5w) and single-phase three-wire line sections (1φ3w). In this system, an additional earth wire is added for safety reasons with zero potential and will not be considered since there is no regime in normal operating conditions for it. Additionally, using Carson’s equation with the Kron reduction the ground impedance is accounted into the model. The general multiphase Π_k section is presented in Figure 4. The ${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$ from this figure is a matrix of series impedances representing self and mutual inductive couplings of conductors. The ${{\widehat{\mathit{Y}}}^{\prime }}_k^{\mathit{abc}}$ and ${{\widehat{\mathit{Y}}}^{″}}_k^{\mathit{abc}}$ represent shunt matrices of self and mutual capacitive phase couplings and are usually considered as the same. The series and shunt matrices are of dimensions $s\times s$, where s is the number of phase conductors. The model of the multiphase home installation line is as follows:

$${\widehat{\mathit{U}}}_k^{\mathit{abc}}={\widehat{\mathit{U}}}_i^{\mathit{abc}}-{\widehat{\mathit{Z}}}_k^{\mathit{abc}}{\widehat{I}}_k^{\mathit{abc}},$$

(8a)

$${{\widehat{\mathit{I}}}^{\prime }}_{k0}^{\mathit{abc}}={{\widehat{\mathit{Y}}}^{\prime }}_k^{\mathit{abc}}{\widehat{\mathit{U}}}_i^{\mathit{abc}}, {{\widehat{\mathit{I}}}^{″}}_{k0}^{\mathit{abc}}={{\widehat{\mathit{Y}}}^{″}}_k^{\mathit{abc}}{\widehat{\mathit{U}}}_k^{\mathit{abc}},$$

(8b)

$${{\widehat{\mathit{I}}}^{\prime }}_k^{\mathit{abc}}={\widehat{\mathit{I}}}_k^{\mathit{abc}}+{{\widehat{\mathit{I}}}^{\prime }}_{k0}^{\mathit{abc}}, {\widehat{\mathit{I}}}_k^{\mathit{abc}}={{\widehat{\mathit{I}}}^{″}}_{k0}^{\mathit{abc}}+{\widehat{\mathit{I}}}_k^{\mathit{abc}}.$$

(8c)

The meaning of voltage vectors ${\widehat{\mathit{U}}}_k^{\mathit{abc}}$, ${\widehat{\mathit{U}}}_i^{\mathit{abc}}$ and current vectors ${\widehat{\mathit{I}}}_k^{\mathit{abc}}$, ${{\widehat{\mathit{I}}}^{\prime }}_k^{\mathit{abc}}$, and ${{\widehat{\mathit{I}}}^{″}}_{k0}^{\mathit{abc}}$ are obvious from Figure 4 and have the dimensions $s\times 1$. For the three-phase home line section, the series and shunt matrices are as follows:

$${\widehat{\mathit{Z}}}_k^{\mathit{abc}}=\left[\begin{array}{ccc}{\widehat{Z}}_{\mathit{kaa}}& {\widehat{Z}}_{\mathit{kab}}& {\widehat{Z}}_{\mathit{kac}}\\ {\widehat{Z}}_{\mathit{kab}}& {\widehat{Z}}_{\mathit{kbb}}& {\widehat{Z}}_{\mathit{kbc}}\\ {\widehat{Z}}_{\mathit{kca}}& {\widehat{Z}}_{\mathit{kcb}}& {\widehat{Z}}_{\mathit{kcc}}\end{array}\right], {{\widehat{\mathit{Y}}}^{\prime }}_k^{\mathit{abc}}={{\widehat{\mathit{Y}}}^{″}}_k^{\mathit{abc}}=\left[\begin{array}{ccc}{\widehat{Y}}_{\mathit{kaa}}& {\widehat{Y}}_{\mathit{kab}}& {\widehat{Y}}_{\mathit{kac}}\\ {\widehat{Y}}_{\mathit{kab}}& {\widehat{Y}}_{\mathit{kbb}}& {\widehat{Y}}_{\mathit{kbc}}\\ {\widehat{Y}}_{\mathit{kca}}& {\widehat{Y}}_{\mathit{kcb}}& {\widehat{Y}}_{\mathit{kcc}}\end{array}\right].$$

(9)

The rows and columns which correspond to the missing phases of multiphase home line section are excluded from these matrices. Since the electric line parameters stored in these matrices are important for the load flow calculation and the specifications are rarely given in the factory specifications, the new method of series and shunt matrices measurement of the three-phase home line section has been presented in the next subsection.

Measurement of Series and Shunt Parameter Matrices of Home Line Sections

The series parameter matrix contains the series resistances and internal inductances, as well as the mutual inductances. The matrix of shunt parameters contains shunt conductances, self capacitances, and mutual capacitance between phases. In standard models, the resistance, inductance, and capacitance of the phases with the grounded neutral wire are integrated into the matrices ${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$, ${{\widehat{\mathit{Y}}}^{\prime }}_k^{\mathit{abc}},$ and ${{\widehat{\mathit{Y}}}^{″}}_k^{\mathit{abc}}$. For the stated reasons, the matrices of the three-phase home line sections are size 3 × 3. Although there are models whose matrices can be analytically calculated, it is very complex to apply such relations in practice, especially for the purpose of the low-voltage home network modeling. For that reason, matrices of series and shunt parameters need to be obtained empirically, i.e., by short- and open-circuit methods, respectively. Finally, it is important to emphasize that this measurement is used to verify the line models used in the BFS procedure presented in Section 4 (Results and Discussion).

A short-circuit method implies that the far end of the three-phase cable is short-circuited (Figure 5a). In this manner the capacitive effects are annulled. In the first stage of the method (Figure 5a), a voltage source is connected to phase a, which dictates the voltage ${\widehat{U}}_{ia}^{\mathit{SC}}$ of node ka, which causes the nominal branch current of the phase a. The voltage magnitudes and angles at the single-phase nodes of the other two phases (nodes kb and kc) can be measured. The following relations for the described method can be written:

$$\left[\begin{array}{c}{\widehat{U}}_{ia}^{\mathit{SC}}\\ {\widehat{U}}_{ib}^{\mathit{SC}}\\ {\widehat{U}}_{ic}^{\mathit{SC}}\end{array}\right]=\left[\begin{array}{c}{\widehat{Z}}_{\mathit{kaa}}\\ {\widehat{Z}}_{\mathit{kba}}\\ {\widehat{Z}}_{\mathit{kca}}\end{array}\right]I_{\mathit{nomka}}{e}^{j{\psi }_{ka}}.$$

(10)

The current ${\widehat{I}}_{ka}^{\mathit{SC}}$ from Figure 5a has nominal magnitude value ($I_{\mathit{nomka}}$) which is specified by the installation manufacturers. The current phase angle is denoted as ${\psi }_{ka}$. Since there are no branch currents of phases b and c, the corresponding columns of matrix ${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$ from (9) are nullified in this stage of the method. The inductive self and mutual couplings with the other two phases are manifested through the current flow in phase a. The columns corresponding to phase a are calculated as follows:

$$\left[\begin{array}{c}{\widehat{Z}}_{\mathit{kaa}}\\ {\widehat{Z}}_{\mathit{kba}}\\ {\widehat{Z}}_{\mathit{kca}}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{{\widehat{U}}_{ia}^{\mathit{SC}}}{\left(I_{\mathit{nomka}}{e}^{j{\psi }_{ka}}\right)}}\\ {\displaystyle \frac{{\widehat{U}}_{ib}^{\mathit{SC}}}{\left(I_{\mathit{nomka}}{e}^{j{\psi }_{ka}}\right)}}\\ {\displaystyle \frac{{\widehat{U}}_{ic}^{\mathit{SC}}}{\left(I_{\mathit{nomka}}{e}^{j{\psi }_{ka}}\right)}}\end{array}\right].$$

(11)

In the same manner, the columns corresponding to phases b and c are obtained: first, the voltage source is connected to the phase b of the home line causing the nominal current magnitude value of the same phase. In this manner, the column corresponding to phase b is obtained. Second, the voltage source is connected to the phase c causing also the nominal current magnitude value of the same phase. Finally, the column corresponding to phase c is also obtained.

The open-circuit method implies that the line at the far end is left open (node k in Figure 5b). In this manner, the inductive couplings are annulled. In the first stage of this method, a voltage source of the nominal magnitude ${\widehat{U}}_{ia}^{\mathit{OC}}$ is connected to phase a (Figure 5b). The nominal magnitude $U_{\mathit{nomia}}$ is specified by the installation manufacturers. The currents that are measured are the capacitive-caused currents. Since the node voltages of phases b and c are nullified, the corresponding columns of matrices ${{\widehat{\mathit{Y}}}^{\prime }}_k^{\mathit{abc}}$ and ${{\widehat{\mathit{Y}}}^{″}}_k^{\mathit{abc}}$ from (9) are nullified as well. The columns corresponding to phase a are calculated as follows:

$$\left[\begin{array}{c}{\widehat{I}}_{ja}^{\mathit{OC}}\\ {\widehat{I}}_{jb}^{\mathit{OC}}\\ {\widehat{I}}_{jc}^{\mathit{OC}}\end{array}\right]=2\times \left[\begin{array}{c}{\widehat{Y}}_{0\mathit{kaa}}\\ {\widehat{Y}}_{0\mathit{kba}}\\ {\widehat{Y}}_{0\mathit{kca}}\end{array}\right]U_{\mathit{nomia}}{e}^{j{\theta }_{ia}}.$$

(12)

The phase voltage angle is denoted as ${\theta }_{ia}$. The current magnitudes and angles at the single-phase nodes of the other two phases (phase kb and kc from Figure 5b) can be measured. The capacitive self and mutual couplings with the other two phases are therefore measured using this method. The column corresponding to phase a is calculated:

$$\left[\begin{array}{c}{\widehat{Y}}_{0\mathit{kaa}}\\ {\widehat{Y}}_{0\mathit{kba}}\\ {\widehat{Y}}_{0\mathit{kca}}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{{\widehat{I}}_{ka}^{\mathit{OC}}}{\left(2\times U_{\mathit{nomia}}{e}^{j{\theta }_{ia}}\right)}}\\ {\displaystyle \frac{{\widehat{I}}_{kb}^{\mathit{OC}}}{\left(2\times U_{\mathit{nomia}}{e}^{j{\theta }_{ia}}\right)}}\\ {\displaystyle \frac{{\widehat{I}}_{kc}^{\mathit{OC}}}{\left(2\times U_{\mathit{nomia}}{e}^{j{\theta }_{ia}}\right)}}\end{array}\right].$$

(13)

Relations (12) and (13) contain factor 2 as a consequence of the elimination of the series impedance matrix (${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$) from Figure 4 (Section 2.2). During the open-circuit method, the inductive couplings are annulled. The assumption of equal voltages on both line ends was made (because of the reduced inductive effects) and this situation corresponds to Figure 6, where series impedance matrix (${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$) is annulled. Since the equal shunt admittance matrices in Π circuits are assumed in most of the literature, the parallel shunt matrices are substituted with the double value of the admittance matrix elements.

3. Home Network Load Flow Model and Calculation

Considering the possibility of 1φ3w, 1φ2w, or 3φ4w systems, the decoupled Γ_ks, $s\in {\varphi }_k$ section circuit is derived (Figure 7), where ${\varphi }_k$ is the set of phases of Π_k section from Figure 4. The Γ_ks, in general cases, could have multiple successor segments (Γ_js from Figure 7). The successor indices are stored in the successor set ${\alpha }_{ks}$.

The impedance ${\widehat{Z}}_{kss}$ from Figure 7 corresponds to the self inductance of section k and they are the same as the diagonal element in the matrix ${\widehat{\mathit{Z}}}_k^{\mathit{abc}}$ from (9) which corresponds to the same phase. The admittance ${\widehat{Y}}_{\Sigma ks}$ represents total capacitance of the Π_k section and Π_j successor segments. This admittance is obtained by summing the admittances of the predecessor Π_k section and the successor Π_j segments. The capacitive couplings with the other phases are represented with ${\widehat{I}}_{\mathit{CCks}}$. The inductive couplings with the other phases are represented with the ${\widehat{E}}_{ks}$. The load flow procedure for the home installations is as follows:

Backward sweep

$${\left({\widehat{I}}_{\mathit{conks}}\right)}^{h+1}={\displaystyle \frac{\left[{\widehat{S}}_{ks}\left(U_{ks}^h\right)\right]}{{\left({\widehat{U}}_{ks}^h\right)}^{*}}},$$

(14a)

$${\left({\widehat{I}}_{\mathit{CCks}}\right)}^{h+1}={\sum }_{t\in {\varphi }_k\wedge t\ne s}{\widehat{Y}}_{kst}{\left({\widehat{U}}_{kt}\right)}^{h+1},$$

(14b)

$${\left({\widehat{I}}_{ks}\right)}^{h+1}={\widehat{Y}}_{\Sigma ks}{\left({\widehat{U}}_{ks}\right)}^{h+1}+{\left({\widehat{I}}_{\mathit{CCks}}\right)}^{h+1}+{\left({\widehat{I}}_{\mathit{conks}}\right)}^{h+1}+{\sum }_{t\in {\alpha }_{ks}}{\left({\widehat{I}}_{ts}\right)}^{h+1}, s=\mathrm{a},\mathrm{b},\mathrm{c};$$

(14c)

Forwardsweep

$${\left({\widehat{E}}_{ks}\right)}^{h+1}={\sum }_{t\in {\varphi }_k\wedge t\ne s}{\widehat{Z}}_{kst}{\left({\widehat{I}}_{kt}\right)}^{h+1},$$

(15a)

$${\left({\widehat{U}}_{is}\right)}^{h+1}={\left({\widehat{U}}_{ks}\right)}^{h+1}-{\widehat{Z}}_{kss}{\left({\widehat{I}}_{ks}\right)}^{h+1}-{\left({\widehat{E}}_{ks}\right)}^{h+1}, s=\mathrm{a},\mathrm{b},\mathrm{c}.$$

(15b)

The BFS procedure is presented in Figure 8. The procedure is finished when the convergence criteria is achieved. If the 1φ3w system is observed, the number of phases is two and the capacitive coupling calculation (14b) is not executed. As stated earlier in this paper, before the BFS execution, the layer-type node numeration of the home grid must be performed first. In this case, the BFS load flow procedure does not handle loops since the home electric installations have a strictly radial structure. In the weakly meshed DNs, the loops are managed with the Thevenin theorem [5].

Three-Phase Distributed Generator Model Integration in Load Flow Procedure

The modern home premises are categorized as prosumers, meaning that the electric power can be generated through distributed generators (DGs) such as solar panels, wind turbines, and small-scale hydro turbines. The load flow model must integrate the various types of DG models in order to have robust and easy maintainable software [5,6,26].

The DGs are presented in Figure 9. There are three main categories of the three-phase DGs: (i) The traditional type of DGs based on the diesel generators or the older type of the wind turbines with the traditional synchronous machines (Figure 9a). (ii) The new type of the wind turbines based on the Double-Fed Induction Generator (DFIG), where the stator is directly connected to the grid and the rotor is connected through the power electronic devices. These DGs are categorized as DGs partially connected via power electronics devices (Figure 9b). (iii) PV systems and some type of wind turbines where all the resources are connected to the grid through the power electronic devices (Figure 9c). The configuration of the power electronics could be three-wire or four-wire. The three-phase node k from Figure 9 is considered as the Point of Common Coupling (PCC). In this paper, the described control strategies are implemented at the PCC.

The control strategies of the traditional DGs (based on the synchronous machines) are implemented in the following manner: the three-phase active ($P_{3\varphi k}$) and the three-phase reactive power ($Q_{3\varphi k}$) are controlled at the PCC. Thus, the three-phase active power is $P_{3\varphi k}>0$ with the three-phase reactive power possible values being as follows: $Q_{3\varphi k}>0$$\vee$$Q_{3\varphi k}=0$$\vee$$Q_{3\varphi k}<0$. The values of the reactive power depend on the status of the synchronous machine (over-excited, normal operating condition, under-excited).

If the induction machine is considered, then the active power is $P_{3\varphi k}<0$, and the reactive power depends on the machine parameters and is strictly $Q_{3\varphi k}<0$. In this paper they are denoted as DG type 1.

The control strategies of DFIG-based DGs and DGs that are connected to the grid through the power electronic devices are more complex since they can employ more strategies than the traditional ones. Additionally, they can dictate or block the inverse sequence current ($I_{\mathit{DGk}}^i$) and they can block the zero-sequence current ($I_{\mathit{DGk}}^0$). Thus, if both sequence currents are blocked, the symmetrical three-phase current is injected into the network. In this paper, they are denoted as DG type 2 and the injected three-phase balanced current strategy is considered.

Since the home network is unbalanced, the more detailed DG model must be considered. This is due to the phase voltage and current disbalances which are significantly higher in the modern active multiphase DNs compared to the traditional ones. Thus, the impact on the machine rotor or power electronics is more expressed.

The DG_k sequence circuits are presented on Figure 10a, and they are represented as the Thevenin equivalent circuits. In this figure, the positive sequence impedance is denoted as ${\widehat{Z}}_{\mathit{DGk}}^d$. The reactive part of this impedance is the subtransient reactance $X_{\mathit{DGk}}^d$. The DG_k internal voltage (electro motive force) is denoted with ${\widehat{E}}_{\mathit{EXCk}}^d$. The negative and zero sequence impedances are denoted as ${\widehat{Z}}_{\mathit{DGk}}^i$ and ${\widehat{Z}}_{\mathit{DGk}}^0$, respectively. The impedance between the neutral and the ground is denoted as ${\widehat{Z}}_{Nk}$.

In this paper, the Thevenin equivalent is transformed into the Norton equivalent since it is more suited for the BFS procedure. The Norton equivalent sequence circuits are presented in Figure 10b. The model in matrix form is as follows:

$$\left[\begin{array}{c}{\widehat{I}}_{\mathit{DGk}}^{\mathit{d}}\\ {\widehat{I}}_{\mathit{DGk}}^i\\ {\widehat{I}}_{\mathit{DGk}}^0\end{array}\right]=\left[\begin{array}{c}-{\widehat{I}}_{\mathit{EXCk}}^d\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}{\widehat{Y}}_{\mathit{DGk}}^d& 0& 0\\ 0& {\widehat{Y}}_{\mathit{DGk}}^i& 0\\ 0& 0& {\widehat{Y}}_{\mathit{DGk}}^0\end{array}\right]\left[\begin{array}{c}{\widehat{U}}_k^d\\ {\widehat{U}}_k^i\\ {\widehat{U}}_k^0\end{array}\right],{\widehat{I}}_{\mathit{EXCk}}^d={\widehat{E}}_{\mathit{EXCk}}^d/{\widehat{Z}}_{\mathit{DGk}}^d.$$

(16)

Considering the control strategies represented using DG types from

Table 3. Specified and unknown state variables for particular generator type.

Generator Type	Specified Variables	Unknown State Variables
DG type 1	$P_{3\varphi k}$, $Q_{3\varphi k}$	${\widehat{U}}_{ka}$, ${\widehat{U}}_{kb}$, ${\widehat{U}}_{kc}$
DG type 2	$P_{3\varphi k}$, $Q_{3\varphi k}$, $I_{\mathit{DGk}}^i$, $I_{\mathit{DGks}}^0\left(=0\right)$	${\widehat{U}}_{ka}$, ${\widehat{U}}_{kb}$, ${\widehat{U}}_{kc}$

, the following equations apply:

$${\widehat{Y}}_{\mathit{DGk}}^i\left\{\begin{array}{c}\ne 0,\mathit{if} {\mathit{DG}}_k \mathit{is} \mathit{type} 1,\\ =0,\mathit{if} {\mathit{DG}}_k \mathit{is} \mathit{type} 2; \end{array}\right.$$

(17a)

$${\widehat{Y}}_{\mathrm{DG}k}^0\left\{\begin{array}{l}\ne 0, \mathrm{if} {\mathrm{DG}}_k \mathrm{is} \mathrm{grounded} \mathrm{or} \\ { \mathrm{DG}}_k \mathrm{is} \mathrm{implemented} \mathrm{as} \mathrm{four} \\ \mathrm{wire} \mathrm{configuration};\\ =0, {\mathrm{if} \mathrm{DG}}_k \mathrm{is} \mathrm{ungrounded} \mathrm{or} \\ { \mathrm{DG}}_k \mathrm{is} \mathrm{implemented} \mathrm{as} \\ \mathrm{a} \mathrm{three} \mathrm{wire} \mathrm{configuration}. \end{array}\right.$$

(17b)

Since the phase domain of the home network circuit model is considered, the DG model in this domain is considered as well. Using the transformation matrices

$$\widehat{\mathit{T}}={\displaystyle \frac{1}{3}}\left[\begin{array}{ccc}1& \widehat{a}& {\widehat{a}}^2\\ 1& {\widehat{a}}^2& \widehat{a}\\ 1& 1& 1\end{array}\right],{\widehat{\mathit{T}}}^{-1}=\left[\begin{array}{ccc}{\widehat{a}}^2& \widehat{a}& 1\\ \widehat{a}& {\widehat{a}}^2& 1\\ 1& 1& 1\end{array}\right],\widehat{a}=e^{j{\displaystyle \frac{2\pi }{3}}},$$

(18)

the DG_k model from (16) is transformed from the sequence to the phase domain:

$$\left[\begin{array}{c}{\widehat{I}}_{\mathit{DGka}}\\ {\widehat{I}}_{\mathit{DGkb}}\\ {\widehat{I}}_{\mathit{DGkc}}\end{array}\right]=\left[\begin{array}{c}-{\widehat{I}}_{\mathit{EXCka}}\\ -{\widehat{I}}_{\mathit{EXCkb}}\\ -{\widehat{I}}_{\mathit{EXCkc}}\end{array}\right]+\left[\begin{array}{ccc}{\widehat{Y}}_{\mathit{DGkaa}}& {\widehat{Y}}_{\mathit{DGkab}}& {\widehat{Y}}_{\mathit{DGkac}}\\ {\widehat{Y}}_{\mathit{DGkba}}& {\widehat{Y}}_{\mathit{DGkbb}}& {\widehat{Y}}_{\mathit{DGkbc}}\\ {\widehat{Y}}_{\mathit{DGkca}}& {\widehat{Y}}_{\mathit{DGkcb}}& {\widehat{Y}}_{\mathit{DGkcc}}\end{array}\right]\left[\begin{array}{c}{\widehat{U}}_{ka}\\ {\widehat{U}}_{kb}\\ {\widehat{U}}_{kc}\end{array}\right].$$

(19)

The DG_k internal current sources ${\widehat{I}}_{\mathit{EXCks}},s=\mathrm{a},\mathrm{b},\mathrm{c}$ from (19) are balanced; therefore, the following rule applies: ${\widehat{I}}_{\mathit{EXCkb}}={\widehat{a}}^2{\widehat{I}}_{\mathit{EXCka}}$, ${\widehat{I}}_{\mathit{EXCkb}}=\widehat{a}{\widehat{I}}_{\mathit{EXCka}}$. The presented phase domain model in matrix form (19) is further developed in the following decoupled form:

$${\widehat{I}}_{\mathit{DGks}}={\widehat{Y}}_{\mathit{DGkss}}{\widehat{U}}_{ks}-{\widehat{I}}_{\mathit{PCks}}-{\widehat{I}}_{\mathit{EXCks}},$$

(20a)

$${\widehat{I}}_{\mathit{PCks}}=-{\sum }_{t\in \left\{a,b,c\right\}\wedge t\ne s}\left({\widehat{Y}}_{\mathit{DGkst}}{\widehat{U}}_t\right), s=\mathrm{a},\mathrm{b},\mathrm{c}.$$

(20b)

The phase couplings (PC) with the remaining phases $t\in \left\{a,b,c\right\}\wedge t\ne s$ of the phase s are modeled with the current source ${\widehat{I}}_{\mathit{PCks}}$. Considering the model (20), the phase domain decoupled Norton equivalent circuit of the DG_k is presented in Figure 11.

The DGk internal current source is calculated using the following equations:

$${\widehat{I}}_{\mathit{EXCka}}=-{\displaystyle \frac{{\widehat{S}}_{3\varphi k}-{\sum }_{s=a,b,c}\left[{\widehat{U}}_{ks}^{*}\left({\widehat{Y}}_{\mathit{DGkss}}{\widehat{U}}_{ks}-{\widehat{I}}_{\mathit{PCks}}\right)\right]}{{\widehat{U}}_{ka}^{*}+{\widehat{a}}^2{\widehat{U}}_{kb}^{*}+\widehat{a}{\widehat{U}}_{kc}^{*}}},$$

(21a)

$${\widehat{I}}_{\mathit{EXCkb}}={\widehat{a}}^2{\widehat{I}}_{\mathit{EXCka}},$$

(21b)

$${\widehat{I}}_{\mathit{EXCkc}}=\widehat{a}{\widehat{I}}_{\mathit{EXCka}}.$$

(21c)

It should be noted that the single-phase DG is modeled as the consumer with the negative active and reactive powers from Section 2.1.

The DG model is integrated into the load flow procedure in the following manner: following the backward and forward procedure execution, the DG internal currents are calculated. The flow-chart of the DG model integration into the BFS procedure is presented in Figure 12.

4. Results and Discussion

This section presents the validation of the proposed load flow model, supported by empirical measurement results and simulation outcomes for several representative residential circuits. These include the following: (i) A home wiring installation network adopted from [7], used to verify the accuracy of the developed model and load flow procedure (Section 4.1). (ii) Single-phase and three-phase residential line segments, used to validate the proposed home network wiring model (Section 4.2). (iii) A comprehensive 24 h load flow simulation of a representative European dwelling, providing insight into temporal load dynamics (Section 4.3). The simulation was based on the load measurement performed by the Smart Meter which provides both active power and power factor. This feature provides model with the degree of lagging or leading of the current with respect to the voltage. Furthermore, this subsection discusses the integration of DG units within the home network, highlighting their relevance in the evolving role of residential prosumers. (iv) The performance of the proposed procedure on the large-scale DNs was compared to the state-of-the-art procedures presented in [7,29,30] (Section 4.4).

The load flow algorithm has been implemented in C# using Visual Studio 2019 and executed on a personal computer equipped with an Intel Core i7 processor (2.3 GHz, four cores, eight threads) and 16 GB of RAM. A flat-start initialization was employed for all simulations, and the convergence criterion for voltage magnitude and phase angle corrections was set to 10⁻⁶ per unit.

4.1. Model Verification on the Single-Phase Three-Wire Configuration

The model introduced in Section 3 is initially validated using the home line installation network described in [7]. This single-phase three-wire (1φ3w) configuration, depicted in Figure 1a (from Section 2.1), serves as a widely accepted benchmark for low-voltage residential premises wiring systems. The electrical parameters of the conductors employed in this network are listed in

Table 4. Conductor parameters.

Branch	Phase/Neutral	R [Ω/km]	X [Ω/km]	Length [m]
A0–A1	a	3.62	0.11	2.5
A1–A2	a	3.62	0.11	2.7
A2–A4	a	3.62	0.11	3.95
A4–A8	a	3.62	0.11	4.45
A0–A3	b	3.62	0.11	8
A3–A5	b	3.62	0.11	1.8
A5–A6	b	3.62	0.11	1.9
A6–A7	b	3.62	0.11	1.5
A0–A9	n	10.1	0.119	2.5
A9–A10	n	10.1	0.119	2.7
A10–A11	n	10.1	0.119	2.8
A11–A12	n	10.1	0.119	1.15
A12–A13	n	10.1	0.119	0.65
A13–A14	n	10.1	0.119	1.9
A14–A15	n	10.1	0.119	1.5
A15–A16	n	10.1	0.119	0.4

. To enable detailed load flow analysis, the original home network has been restructured by incorporating artificial nodes, as illustrated in Figure 13a. In this modified configuration, the conductor lengths have been subdivided to correspond with the positions of the artificial nodes. Node 0 is designated as the slack node, with phase voltage magnitudes of 109.1 V for phase a and 108.7 V for phase b. The corresponding phase angles are 0° and 180°, respectively. The simplified one-line diagram corresponding to Figure 13a is shown in Figure 13b. Load data for the network under study is provided in

Table 5. Load data for sample 1φ3w home network.

Consumer	Original Numeration/(New Numeration)	Phase	Case 1		Case 2
			Active Power [W]	Power Factor	Active Power [W]	Power Factor
Fluorescent light	A1–A9/L1	a	30	0.95	30	0.95
TV	A2–A10/L2	a	80	0.95	160	0.9
Other load	A4–A12/L4	a	80	0.95	240	0.7
Fluorescent light	A8–A16/L8	a	80	0.95	560	0.55
Fluorescent light	A3–A11/L3	b	40	0.95	40	0.95
Laptop	A5–A13/L5	b	60	0.75	60	0.75
Incandescent lights	A6–A14/L6	b	25	0.8	25	0.9
Fan	A7–A15/L7	b	30	0.95	30	0.8

. Two distinct load scenarios have been analyzed to evaluate the performance and accuracy of the proposed load flow procedure.

Since all phase and neutral conductors of the home line installation network have the same parameters (Table 4), the line section parameter matrix is derived for all home network lines using the following relation:

$${\widehat{\mathit{Z}}}_k^{\mathit{ab}}=\left[\begin{array}{cc}{\widehat{Z}}_{ka}+{\widehat{Z}}_{kn}& {\widehat{Z}}_{kn}\\ {\widehat{Z}}_{kn}& {\widehat{Z}}_{kb}+{\widehat{Z}}_{kn}\end{array}\right]=\left[\begin{array}{cc}13.72+\mathrm{j}0.229& 10.1+\mathrm{j}0.119\\ 10.1+\mathrm{j}0.119& 13.72+\mathrm{j}0.229\end{array}\right]\left[{\displaystyle \frac{\mathsf{\Omega }}{\mathrm{km}}}\right], k=1,2,\dots ,7,$$

(22a)

$${\widehat{Z}}_{8a}=\left(13.72+\mathrm{j}0.229\right)\left[{\displaystyle \frac{\mathsf{\Omega }}{\mathrm{km}}}\right].$$

(22b)

The finalized circuit representation of the home line installation is provided in Figure 3 (refer to Section 2.1). This configuration is derived using the section lengths from Figure 13b in combination with the corresponding line section parameter matrix ${\widehat{Z}}_k^{\mathrm{ab}}$. Based on this data, the complete equivalent model of the circuit in Figure 3 has been constructed.

Voltage magnitudes at all network nodes, corresponding to home consumer connection points, are summarized in Figure 14 and Figure 15 for the two evaluated load scenarios. As the line network configuration is not supported by [29,30], the simulation results are benchmarked against only those reported in [7]. The observed discrepancies are minimal, with voltage magnitude mismatches remaining within approximately 0.3%, thereby confirming the accuracy and reliability of the proposed load flow model.

4.2. Three-Phase and Single-Phase Sections

This subsection presents the modeling approach for typical European residential wiring installations, specifically focusing on four-wire three-phase and two-wire single-phase cable configurations. The physical and electrical characteristics of the three-phase cables are provided in

Table 6. Three-phase cable data.

Type of Cable	Size of Phase Wires	Size of Neutral Wire	Length
PPY	2.5 m2	2.5 m2	35 m

. By applying the measurement verification outlined in Section 2.2, the series impedance matrix Z and the shunt admittance matrix Y of the installation cables have been obtained:

$$\begin{gathered} {\widehat{Z}}^{\mathrm{abc}}=\left[\begin{array}{ccc}26.743\angle {0.34}^{°}& 11.686\angle {0.31}^{°}& 11.742\angle {0.48}^{°}\\ 11.686\angle {0.31}^{°}& 23.086\angle {0.33}^{°}& 11.714\angle {0.2}^{°}\\ 11.742\angle {0.48}^{°}& 11.714\angle {0.2}^{°}& 23.257\angle {0.34}^{°}\end{array}\right]\left[{\displaystyle \frac{\mathsf{\Omega }}{\mathrm{km}}}\right]=\left\{\left[\begin{array}{ccc}26.742& 11.686& 11.742\\ 11.686& 23.085& 11.714\\ 11.742& 11.714& 23.257\end{array}\right]+\mathrm{j}\left[\begin{array}{ccc}0.159& 0.063& 0.098\\ 0.063& 4.654& 0.041\\ 0.041& 1.431& 4.830\end{array}\right]\right\}\left[{\displaystyle \frac{\mathsf{\Omega }}{\mathrm{km}}}\right], \\ {\widehat{Y}}^{\mathrm{abc}}=\left[\begin{array}{ccc}48.544\angle {86.7}^{°}& 27.955\angle {87.3}^{°}& 5.618\angle {86.9}^{°}\\ 27.955\angle {87.3}^{°}& 48.209\angle {86.6}^{°}& 15.244\angle {87.2}^{°}\\ 5.618\angle {86.9}^{°}& 15.244\angle {87.2}^{°}& 47.684\angle {86.7}^{°}\end{array}\right]\left[{\displaystyle \frac{\mathrm{nS}}{\mathrm{km}}}\right]=\left\{\left[\begin{array}{ccc}5.587& 2.634& 0.608\\ 2.634& 5.718& 1.489\\ 0.608& 1.489& 5.490\end{array}\right]+\mathrm{j}\left[\begin{array}{ccc}96.926& 55.848& 11.220\\ 55.848& 96.249& 30.451\\ 11.220& 30.451& 95.210\end{array}\right]\right\}\left[{\displaystyle \frac{\mathrm{nS}}{\mathrm{km}}}\right]. \end{gathered}$$

The accuracy of the series and shunt parameter matrices derived in Section Measurement of Series and Shunt Parameter Matrices of Home Line Sections has been experimentally validated using the setup shown in Figure 16. In this experiment, a three-phase voltage source is connected at node 1, with the following voltage magnitudes and phase angles: 393.759 V at 1.52° (phase a), 391.736 V at −118.73° (phase b), and 395.446 V at −238.311° (phase c). At node 2, two single-phase resistive loads heater 1 and heater 2 are connected to phases a and b, respectively. Heater 1 has an active power of 1.936 kW and reactive power of 0.019 kVAr, while heater 2 has an active power of 1.766 kW and reactive power of 0.014 kVAr. In addition, a vacuum cleaner is connected to phase c, with an active power of 1.770 kW and reactive power of 0.171 kVAr. Given the known three-phase voltages at node 1 and the active and reactive power consumption of the connected loads, the three-phase voltage magnitudes and angles at node 2 were computed using the proposed load flow procedure described in Section 3. The resulting voltage magnitudes and angles are presented in

Table 7. Calculated voltage magnitudes comparison at node 2 with measured values.

Ua			Ub			Uc
Calculated [V]	Measured [V]	Mismatch [%]	Calculated [V]	Measured [V]	Mismatch [%]	Calculated [V]	Measured [V]	Mismatch [%]
222.828	223.891	0.475	223.744	223.177	0.254	225.519	224.994	0.233

and

Table 8. Calculated voltage angles comparison at node 2 with measured values.

θa			θb			θc
Calculated [o]	Measured [o]	Mismatch [%]	Calculated [o]	Measured [o]	Mismatch [%]	Calculated [o]	Measured [o]	Mismatch [%]
1.488	1.52	2.105	−119.102	−118.96	−0.119	121.106	121.2	0.078

, respectively. Comparison with measured values at node 2 shows that the discrepancies in voltage magnitudes and angles do not exceed 0.5% and 2%, respectively. These results confirm the accuracy of the model introduced in Section Measurement of Series and Shunt Parameter Matrices of Home Line Sections and the load flow model and computational procedure proposed in Section 3.

For the single-phase cable sections, the series impedance and shunt admittance parameter matrices reduce to scalar values due to the single-conductor configuration, resulting in 1 × 1 matrices. By applying the same measurement verification described in Section Measurement of Series and Shunt Parameter Matrices of Home Line Sections, adapted for a single-phase conductor, the corresponding series impedance Z and shunt admittance Y parameters have been accurately determined as follows:

$${\widehat{Z}}^s=\left(13.428+\mathrm{j}0.162\right)\left[{\displaystyle \frac{\mathsf{\Omega }}{\mathrm{km}}}\right], {\widehat{Y}}^s=\left(7.2+\mathrm{j}49.335\right)\left[{\displaystyle \frac{\mathrm{nS}}{\mathrm{km}}}\right], s=\mathrm{a}\vee \mathrm{b}\vee \mathrm{c}$$

The single-phase cable parameters were validated through the experimental setup illustrated in Figure 17. In this experiment, a single-phase voltage source was connected at node 1, supplying a voltage magnitude of 398.891 V with a phase angle of −3.92°. At node 2, a single-phase heater was connected, characterized by an active power consumption of 3.547 kW and a reactive power consumption of 0.031 kVAr. Using the proposed load flow procedure described in Section 3 and the previously obtained cable parameters, the voltage magnitude and angle at node 2 were calculated. The results are presented in

Table 9. Calculated voltage magnitude and angle comparison at node 2 with measured values.

θa			Ua
Calculated [°]	Measured [°]	Mismatch [%]	Calculated [V]	Measured [V]	Mismatch [%]
−3.926	−3.996	1.752	222.818	222.521	0.133

. Comparison with the measured values at node 2 shows that the mismatch in voltage magnitude does not exceed 0.15%, while the phase angle deviation remains within 1.8%. These results further validate the precision of the installation parameter measurement presented in Section Measurement of Series and Shunt Parameter Matrices of Home Line Sections, as well as the reliability of the overall load flow model and computational approach presented in Section 3.

4.3. Three-Phase European Home Network

In this study, a real-life European residential dwelling unit is utilized to implement and validate the proposed load flow methodology, as depicted in Figure 18. While standard medium-voltage benchmark networks, such as the IEEE 13, 34, 37, and 123 bus test feeders are commonly used for evaluating load flow algorithms, they are not suitable for modeling detailed low-voltage home electrical networks due to their scale and abstraction. In contrast, the residential network modeled here captures the physical and operational characteristics of a typical household electrical installation. In Figure 18, the main installation cable corridors are illustrated using black lines, indicating the spatial layout of wiring across the dwelling. Consumer connection node points include the following: (i) single- and double-gang socket outlets; (ii) ceiling and wall-mounted lighting fixtures; (iii) fixed-position loads such as the bathroom water-heater and air-heater. The complete one-line diagram of the home installation network is provided in Figure 19. This network consists of 14 primary circuit feeders connected to a Three-Phase and Neutral Distribution Board (TPN). The physical lengths of all feeder branches are annotated alongside the corresponding line sections in Figure 19 and expressed in meters. Each branch is uniquely identified with a feeder ID, numbered from 1 to 14. Additionally, each consumer connection point (e.g., socket outlets, lighting fixtures) is associated with a unique node identification pair in the format (Feeder ID/Connection Node Point ID). All nodes within the residential electrical system, including wall junction boxes, are distinctly numbered and represented with circled numerals in Figure 19.

The list of loads distributed throughout the entire residential network is presented in

Table 10. One load variant scattered through entire home network with electrical data.

Connection Node Point	Load	Active Power [W]	Power Factor	Phase
1.1	Electric Water Heater E52–50R-045DV	4500	1	c
2.1	LG clothes washer F2DV5S7N0E	Figure 20a	Figure 20a	a
2.2	LED bulb Osram—470 Lm	5.5	−0.57	a
2.3	Hair dryer	598	0.99	a
2.4	Hair dryer	598	0.99	a
3.1	Bathroom heater	1200	1	b
4.1	LED bulb—Phillips 806 Lm	16.7	−0.55	c
4.2	LED bulb Osram—470 Lm	5.5	−0.57	c
4.3	LED bulb—Phillips 806 Lm	16.7	−0.55	c
4.4	LED bulb Osram—470 Lm	5.5	−0.57	c
5.1	Electric range/oven Kenmore 790.9131 2013	5100	0.98	abc
6.1	Water Kettle—Anko	2129.3	0.99	a
7.1	Dish washer	Figure 20b	Figure 20b	b
8.1	Moulinex UNO Bread Maker	Figure 20c	Figure 20c	c
8.2	Microwave—Samsung	1716.3	0.92	c
8.3	Refrigerator Maytag, 25.6-cuft side-by-side	Figure 21a	Figure 21a	c
9.1	Air conditioner LG ES-H123LLA0	Figure 21b	Figure 21b	a
10.1	Vacuum—Dyson	1335.0	0.98	b
10.2	Mobile phone charger	10	−0.47	b
10.3	LED—Dick Smith—40”	64.5	−0.62	b
10.4	Charger Power Tool—Ryobi	38.2	−0.62	b
11.1	Mobile phone charger	10	−0.47	c
11.2	Mobile phone charger	10	−0.47	c
11.3	Mobile phone charger	10	−0.47	c
11.4	Iron—Kambrook	1863.7	0.99	c
12.1	Air conditioner LG ES-H123LLA0	Figure 21b	Figure 21b	a
12.2	Laptop Charger—Apple	58.1	−0.47	a
12.3	Charger Vacuum—Anko	13.5	−0.58	a
12.4	LED—Dick Smith—40”	64.5	−0.62	a
13.1	LED bulb—Phillips 806 Lm	16.7	−0.55	b
13.2	LED—Dick Smith—40”	64.5	−0.62	b
13.3	Mobile phone charger	10	−0.47	b
13.4	Laptop Charger—Apple	58.1	−0.47	b
13.5	USB Adapter Powerbank—Cosmos	33.2	−0.54	b
14.1	LED bulb Osram—470 Lm	5.5	−0.57	c
14.2	LED bulb Osram—470 Lm	5.5	−0.57	c
14.3	LED bulb—Phillips 806 Lm	16.7	−0.55	c
14.4	LED bulb Osram—470 Lm	5.5	−0.57	c
14.5	LED bulb Osram—470 Lm	5.5	−0.57	c
14.6	LED bulb—Phillips 806 Lm	16.7	−0.55	c
14.7	LED bulb—Phillips 806 Lm	16.7	−0.55	c
14.8	LED bulb—Phillips 806 Lm	16.7	−0.55	c
14.9	LED bulb—Phillips 806 Lm	16.7	−0.55	c

[35,36]. These loads are connected at physical node points corresponding to various branch feeders, as detailed in Table 10 and shown in Figure 19. For certain complex consumers, whose load profiles vary over time, individual load data (such as constant power and phase angle) was insufficient to accurately model their behavior. Instead, these loads have been modeled as time-varying data, reflecting their dynamic consumption patterns: (i) Consumer 2.1, a combined clothes washer and dryer (Figure 20a); (ii) Consumer 7.1, a dishwasher (Figure 20b); (iii) Consumer 8.1, a bread maker (Figure 20c); (iv) Consumer 8.3, a refrigerator (Figure 21a); (v) Consumers 9.1 and 12.1, air conditioners (Figure 21b). This approach ensures that the more complex consumption patterns of these appliances are accurately represented in the introduced load flow model (Section 3).

As part of the simulation procedure, load consumption data was gathered using a smart meter, as shown in Figure 22. The system includes a communication module (1), a smart kWh meter (2), and a socket (3) for connecting the device under test. The communication module utilizes the MODBUS protocol to retrieve data from the smart meter, which is then transmitted to the cloud via Wi-Fi. The smart meter is classified with an accuracy of 0.2% for current and voltage measurements, 0.5% for active power, and 2% for reactive power. The measurement data is available for download in CSV (Comma-Separated Values) or JSON (JavaScript Object Notation) formats.

As the load flow procedures presented in [29,30] are not compatible with the methodology described above and employed in this study, the load flow for the entire home network was calculated at time t = 2:00 PM, with the results summarized in Figure 23. Given the extensive volume of results, only those derived from the procedures developed in this work are presented. The deviations from the procedures described in [29,30] do not exceed 0.7%.

In the subsequent phase of testing, a 24 h period was considered for the load flow simulation. The results, including voltage and current magnitudes, were obtained for the nodes corresponding to consumers with more complex load consumption patterns (as illustrated in Figure 20 and Figure 21). The calculated voltage at each node and the corresponding consumer current magnitudes over the 24 h period are presented in Figure 24 and Figure 25, respectively.

The final results are presented in Figure 26, which illustrates the total active and reactive power consumption over the 24 h period. From this figure, a significant power imbalance is evident throughout the day. Notably, during specific time intervals, such as from 9:10 to 10:05 AM, negative reactive power is observed in phase B, as shown in Figure 26b.

Integration of DG in Home Network

As smart homes evolve into prosumers, the home network depicted in Figure 18 is modified to incorporate distributed generation. Specifically, a three-phase DG is connected at connection point 5.2 in the main circuit feeder 5 (Figure 19). In Case 1, DG type 1, as detailed in Table 4 (Section Three-Phase Distributed Generator Model Integration in Load Flow Procedure), is employed. This generator is a traditional synchronous machine, with specified direct, inverse, and zero-sequence admittances: Y_d = 110%, Y_i = 20% and Y₀ = 10%, respectively. The nominal apparent power is 3 kVA. The corresponding results for Case 1 are presented in

Table 11. Node voltages and currents in node 5.2.

Case	Voltages [V]			Injected Currents [A]
	Ua	Ub	Uc	Ia	Ib	Ic
1	227.502	226.337	228.480	2.163	2.441	2.362
2	227.547	226.315	228.460	2.317	2.317	2.317

. In Case 2, DG type 2 is connected, which is a Double-Fed Induction Generator (DFIG) with an additional control strategy that ensures balanced current injection into the system. The total active and reactive power consumption over the 24 h period for Case 2 is shown in Figure 27. Comparing the total consumption results from Figure 26 and Figure 27, it is evident that the integration of the generator significantly reduces the total home consumption, highlighting the role of DG in enhancing energy efficiency in prosumer-based smart homes. In both cases, the reverse reactive power flow is detected in phase B.

4.4. Performance of the Developed Procedure on the Large-Scale Distribution Network

The performance of the proposed procedure is evaluated using a large-scale IEEE European DN from [42] and compared with ones from [7,29,30]. Two test cases are considered: (i) Since the model in [7] assumes a single-phase three-wire (1φ3w) configuration, the European distribution network from [42] is adapted accordingly by converting it to the same configuration, using the line parameters specified in Section 4.1. The line lengths remain unchanged and the original loads (PQ nodes) are replaced with household electric networks described in Section 4.1. (ii) In this case, only the loads from [42] are replaced with the household electric networks defined in Section 4.2, while the original European DN configuration is retained. This configuration is used due to the incompatibility of the models in [29,30] with the single-phase three-wire (1φ3w). In both cases, the load flow for European DN is solved using the procedure from [5], whereas the proposed procedure, along with the procedures from [7,29,30], are applied to the household electrical networks. The results are summarized in

Table 12. Procedure performance comparisons with [7].

Total CPU Time [ms]
Proposed load flow	Load flow procedure from [7]
159	668

and

Table 13. Procedure performance comparisons with [29,30].

Total CPU Time [ms]
Proposed load flow	Load flow procedure from [29]	Load flow procedure from [30]
432	1505	2881

. Based on the results, it is evident that the proposed load flow calculation method for residential networks demonstrates the highest efficiency.

4.5. Limitations of the Proposed Methodology

The limitations of the proposed solution are as follows:

(i)

The distribution energy resource voltage control in situation where part of the residential network forms the microgrid, operating in islanded mode.

(ii)

When the distribution energy resource is intermittent, the output power could be estimated using the weather, solar irradiation, ambient temperature, and wind speed model. Once these complex models are tailored for power flow model, they can be easily incorporated into the developed power flow calculation procedure since the efficiency is demonstrated in the previous subsection.

(iii)

Since EVs represent complex load, it is mandatory to measure their consumption at each stage of charging (grid to vehicle) or discharging (vehicle to grid) and then form the consumption diagram for this complex type of load. Based on this diagram it is possible to integrate the EV model into the load flow model and develop the load flow calculation procedure.

(iv)

The non-linearity loads results in a non-sinusoidal current which is periodic and needs Fourier expansion. It is possible to extend power flow models by increasing the number of relations by using the first Kirchhoff’s law for current and power balance.

5. Conclusions

Load flow analysis and simulations are crucial components of the smart grid concept. A wide range of advanced applications within HEMS are dependent on the results of the load flow calculation procedure. Within load flow models, traditionally, residential buildings have been modeled as aggregated loads with specified values of active and reactive powers. Although this approach has proven adequate for traditional distribution network analyses, the evolving complexity of residential building networks and smart grid concepts require more detailed modeling framework that explicitly incorporates the characteristics of residential buildings.

The proposed load flow procedure is based on the accurate models of all home network elements, which include appliances (with complex and traditional load profiles), measurement validation that is used to check the accuracy of the home wiring models, and on-site generator models for single-phase and three-phase distributed generators.

The wiring models have been validated by comparing the measured values to the simulated ones having the voltage magnitude mismatch of maximum 0.475% for the three-phase cables, and up to 0.133% for the single-phase cables. The single home model was also verified by comparing the simulation results of the introduced model to the model from the literature, and the voltage magnitude mismatch was less than 0.7%. Since some of the appliances have quite complex power profiles, which change both active and reactive power in real time, this paper introduced a realistic test case that includes such appliances by using their load patterns in real time. The load flow model includes not only the wiring and appliances models, but also the prosumer-based generator models such as photovoltaics or wind turbines.

The performance of the proposed load flow procedure is compared against those from the literature and shown to have the highest efficiency. The overall solution presented in this paper can be incorporated in both home energy management systems and more complex distribution energy management systems.

Future work will include the following: (i) microgrid operation in islanded mode; (ii) intermittent nature of DGs; (iii) EV model support including vehicle to grid and grid to vehicle modes; (iv) the support for the non-linear loads; (v) electromagnetic interference impact on the power flow.