Dynamic Modeling of Distribution Power Systems with Renewable Generation for Stability Analysis

Abstract

This paper presents a comprehensive study on the dynamic modeling of distribution power systems with a focus on the integration of renewable energy sources (RESs) for stability analysis. Our research delves into the static and dynamic behavior of distribution systems, emphasizing the need for enhanced load modeling to mitigate planning and operational uncertainties. Using MATLAB/Simulink^®, we simulate four distinct study cases characterized by varying load types and levels of distributed generation (DG), particularly solar PV, under both balanced and unbalanced conditions. Our findings highlight the critical role of DG in influencing voltage stability, revealing that deviations in voltage and current during grid imbalances remain within acceptable limits. The study underscores the importance of DG-based inverters in maintaining grid stability through reactive power support and sets the stage for future research on microgrid simulations and battery storage integration to further enhance system stability and performance.

1. Introduction

The European Green Deal and the “FIT for 55” package have established a legal framework that requires EU member states to meet stringent climate goals, primarily focusing on achieving carbon neutrality [1]. Two critical strategies for achieving these goals involve the electrification of heating and transport, as well as increasing the production of electricity from renewable energy sources (RESs). Over the past two decades, Europe has seen significant integration of RES into the power grid, but even more profound changes are anticipated in the coming years [2]. The growing adoption of Electric Vehicles (EVs) and new, highly efficient heating and air conditioning technologies will further reshape energy demand [3,4].

These transformations place significant pressure on distribution power systems, which must evolve to handle the dynamic conditions associated with variable generation and shifting loads [5]. Ensuring grid stability and security in this context will require increased flexibility in power system operations. To prevent congestion and ensure reliable power delivery, distribution systems must accommodate flexible solutions like prosumers (consumers who also produce energy), energy storage systems (ESSs), demand response (DR), and electric vehicles (EVs), which can contribute to balancing supply and demand.

The integration of these flexible energy resources, alongside a growing share of RESs, underscores the importance of developing a deep understanding of the static and dynamic behavior of distribution networks. Addressing the technical, economic, and environmental challenges posed by these new scenarios will be key to ensuring the long-term success of the EU’s climate initiatives.

Load modeling is essential for both the planning and operation of power systems because it allows for a detailed understanding of how different types of loads behave under various system conditions. Accurate load models improve the prediction of power system performance in both steady-state and transient conditions [6,7,8]. This helps system operators and planners to anticipate how the power grid will respond to changes in load demand, which is increasingly important with the integration of renewable energy sources (RESs) and new, flexible power technologies like EVs and ESSs [9]. Accurate load modeling reduces planning and operational uncertainties, contributing to better system reliability and cost-efficiency. In distribution grids, static load models typically capture how loads respond to voltage changes, whereas at the transmission level, dynamic and composite load models are used to represent more complex behaviors, such as time-dependent variations in load response. These models are critical for understanding system performance under both steady-state and disturbed conditions [10]. Historically, transmission system operators (TSOs) modeled distribution grids as passive loads without dynamic responses for stability analysis [11,12]. The integration of RESs of flexible power directly into the distribution grid means that the usual approach of load modeling that was performed in the past by the TSOs has to be adapted in a certain way [10].

The main objective of this paper is to implement a proper dynamic modeling of the distribution power system with different percentages of generation based on renewable energy in order to perform effective stability analysis. Additionally, we conduct electromagnetic transient (EMT) simulations to capture the detailed dynamic behavior of the system under various operational conditions. The modeling includes a comprehensive dynamic load representation to ensure more accurate stability assessments. The paper is organized as follows: Section 2 illustrates the state of the art regarding distribution systems and their modeling, Section 3 describes the proposed modeling of the distribution grid for stability analysis, Section 4 presents four study cases that were simulated, and Section 5 gives the overall conclusion of this paper.

2. State of the Art of Distribution Power Systems

2.1. Latest Issues and Trends

The integration of RESs is driving the transition towards low-carbon energy systems. Current trends indicate that high levels of RES integration into distribution grids introduce numerous operational challenges. The reliability of distribution grids with high levels of RESs becomes critical to the overall operation and stability of the power grid.

Traditionally, synchronous generators (SGs), connected to transmission systems, provide significant system inertia, which determines the rate of change in system frequency after a disturbance and ensures stability in a significant portion of the power system. It is assumed that each SG connected to the grid contributes to maintaining the stability and security of the large power system [13]. However, RESs are integrated into the AC grid through power electronics. Inverter-based RESs are progressively replacing conventional SGs, decreasing the overall system inertia and the fault tolerance capability [14]. As more conventional SGs are decommissioned and DERs are integrated on a large scale, power systems may become increasingly vulnerable to sudden disturbances [15].

The integration of inverter-based RESs primarily at the distribution grid level presents several challenges for distribution system operators (DSOs). One major issue is voltage regulation, as the intermittent and variable nature of solar and wind energy causes fluctuations, making it difficult to maintain voltage levels within operational limits. In addition, ensuring the effectiveness of protection systems becomes more complex. Unlike traditional SGs, inverter-based RESs generate much lower fault currents, which can prevent protection devices from accurately detecting faults, leading to potential malfunctions. Furthermore, directional protection schemes that rely on negative-sequence components often fail due to altered phase angle characteristics in systems dominated by inverter-based RESs [16]. DSOs must ensure operational flexibility and adopt new planning strategies to accommodate RES intermittency, which involves investments in smart grid technologies and real-time monitoring systems.

These challenges underline the need for further analysis at the distribution grid level. The increased variability and dynamic behavior introduced by high levels of RES integration require more sophisticated modeling techniques to fully grasp the complexities involved. By deepening analysis at this level, DSOs will be better equipped to plan and operate their systems effectively, ensuring grid reliability.

Research in this field emphasizes the importance of load modeling to capture the complex behavior of loads and their impact on power system operations, ensuring reliable simulation results. It further highlights the need to incorporate modeling techniques that simulate the dynamic behavior of distribution grids, especially with a high penetration of RESs [10,11,17]. A thorough analysis of distribution grids requires applying appropriate generation and load modeling techniques [18,19]. In past times, when the system stability analysis was being assessed only by TSOs, the distribution networks were modeled as static loads without any dynamic response. However, with the increasing integration of decentralized RESs and new loads, there is a pressing need for new modeling approaches that treat distribution grids as active systems [11].

It is widely acknowledged that load modeling presents inherent complexities. The demand for electricity arises from various devices connected to the grid and is primarily influenced by human behavior. Due to this uncertainty, achieving the perfect modeling of this behavior is challenging. New loads such as EVs, the widespread use of power electronics, and the electrification of economies further complicate the process. Load modeling is never entirely accurate, as certain uncertainties always exist. It should be noted that the advanced modeling and forecasting of energy production from variable renewables has to some extent allowed, e.g., in Germany, a massive integration of RESs. In this way, a more accurate modeling of the load will help to deal with the new challenges on the path to climate neutrality. As the future operation of power systems will be based on conventional SGs, inverted-based resources, as well as on ESs, DR, and EVs, techniques to guarantee the stability and security of the system must be re-examined or even developed.

Numerous research efforts are underway to represent the behavior of renewable energy resources as well as inverter-based sources for stability studies [18,20].

Grid codes are evolving, requiring inverter-based generators to support power systems by equipping them with features such as fault ride through (FRT) and voltage and frequency support during disturbances.

2.2. Simulation of Distribution Power Systems

The increasing complexity of modern distribution power systems, driven by the widespread integration of DGs and renewable energy sources, presents significant challenges for accurate modeling and simulation. The interaction between grid infrastructure, varying generation sources, and loads necessitates the use of more advanced simulation methods that go beyond traditional tools.

To address these challenges, simulation trends are moving towards multi-processor systems optimized for real-time analysis, such as real-time simulators (RTSs). These simulators allow for the real-time progression of modeled systems, providing accurate time responses that closely mimic the behavior of actual distribution networks. By running simulations in real time, system responses to various scenarios can be analyzed in a practical context, making RTS a powerful tool for system validation. This capability is particularly useful when investigating scenarios that would be difficult or risky to replicate physically, as real-time simulations offer a secure and cost-effective alternative [21,22].

In this context, electromagnetic transient (EMT) simulations have become important for capturing the fast dynamics and transient behaviors of distribution systems, particularly those with a high penetration of inverter-based renewable energy technologies. EMT simulations provide finer temporal resolution compared to traditional root mean square (RMS) models, making them ideal for modeling switching events, harmonics, and rapid system responses. However, the computational intensity of EMT simulations, especially in large-scale distribution systems, can limit their feasibility for full-system studies.

To mitigate the limitations of EMT and RMS simulations, hybrid co-simulation models that combine both are being developed. These models integrate EMT simulations with root mean square (RMS) models, balancing the need for high-detail modeling in specific areas of the network, such as those with inverter-based resources, with the computational efficiency of RMS simulations for larger, slower dynamics. This co-simulation approach allows for the detailed modeling of the critical parts of the grid while maintaining simulation feasibility across the broader system.

Additionally, co-simulation techniques extend beyond hybrid EMT/RMS. They integrate multiple simulation environments, enabling the coupling of information and communication technology (ICT) with traditional power systems software, offering a more comprehensive view of system behavior. Multi-agent systems complement co-simulation by enabling decentralized modeling approaches well-suited to distributed energy resources and smart grid technologies, simulating multiple independent actors interacting within a shared grid environment.

2.3. Measurements in Distribution Power Systems

Another significant challenge in modern distribution power systems is the lack of synchronized measurements to perform state estimation and conduct further analysis on stability and security. Traditional distribution systems have few measurement points, and to estimate the true state of the system, it is necessary to calculate a high volume of pseudo-measurements, which have larger errors than real-time telemetered measurements [23]. To address instability concerns in distribution grids and avoid undesirable system states, the inclusion of synchrophasor measurements and new control strategies is necessary. Synchrophasor measurement units, also known as phasor measurement units (PMUs), provide a resilient tool for observing the dynamic behavior and performance of the power system. The technology has evolved significantly in the last decade and is now used on a global scale [14]. PMUs offer synchronized measurements of positive-sequence voltage and current to within a microsecond, as well as negative- and zero-sequence quantities. Additionally, PMUs measure local frequency, its rate of change, and individual phase voltages and currents. PMUs have a wide range of applications in electric power systems, and their use will continue to expand in the years ahead [24].

3. Stability Analysis

Power system stability is defined as the ability of a power system to remain in a state of equilibrium under normal operating conditions and to regain a state of equilibrium after being subjected to a disturbance [25,26]. Historically, stability analysis has predominantly focused on transmission networks, where rotor angle stability—the synchronism of large-scale synchronous generators—was the primary concern. This form of stability is influenced by the dynamics of rotor angles and power–angle relationships in transmission systems. However, the increasing integration of DG, particularly from RESs such as photovoltaic (PV) systems in distribution systems, has shifted the stability landscape. The power dynamics in such systems, which were traditionally considered passive, now play a critical role in overall system stability. As RES-based generators replace conventional synchronous generators, the need to study stability in distribution networks has grown.

In modern distribution networks with a high penetration of inverter-based RESs, stability challenges can differ significantly [27,28]. Voltage stability is a primary concern due to the fluctuating nature of renewable energy generation and the increased deployment of inverters. Voltage stability is defined as the ability of a power system to maintain voltages into the acceptable range at all the buses in the power system under normal operating conditions and after the occurrence of a disturbance [25,29,30]. Voltage instability arises when the system fails to sustain stable voltage levels, typically due to reactive power shortages or rapid fluctuations in generation and demand. Unlike traditional rotor angle instability, this type of instability can manifest while the system remains synchronized, highlighting the unique challenges faced by distribution networks with significant renewable energy integration. Therefore, addressing voltage stability in distribution grids is important to ensuring reliable operation, particularly as renewable energy sources continue to proliferate.

Inverter-based DERs such as PV systems do not inherently provide frequency support because they lack the rotating mass that traditional synchronous generators possess. This rotating mass in conventional generators offers inertial support, which helps maintain grid stability by naturally resisting frequency changes following disturbances. In contrast, inverters used in DERs typically operate with power electronics that disconnect them from the physical inertia of the grid. However, through advanced control strategies for inertia emulation, inverters can be programmed to provide frequency support by adjusting their power output in response to changes in grid frequency [31].

The focus of this paper is on voltage and frequency stability in distribution networks, driven by the increasing penetration of RESs. The inverters modeled in this study are grid-following inverters in distribution systems, lacking advanced control mechanisms dedicated to dynamic voltage or frequency support, reflecting the typical inverters currently being integrated into such low-voltage systems.

Reactive Power Demand

Reactive power in the power grid causes additional undesirable losses and can also lead to further system stability problems. Numerous studies suggest that local reactive power compensation has proven to be highly effective. Reactive power compensation has traditionally been performed with the help of synchronous generators, capacitor banks, and, more recently, flexible alternating current transmission system (FACTS) devices. Improving reactive power management is crucial for the reliable and efficient operation of the power grid. Since controlling reactive power via inverters is possible, it has become a favorable method, as no additional investment in new devices is required. As a result, the need for PV inverters to provide reactive power to the grid is becoming increasingly important. Recently, this requirement has been mandated for new photovoltaic inverters, often executed with proportional control of the power factor or droop control for local voltage regulation [32]. Due to constraints within the inverter model employed in the MATLAB/Simulink^® library for renewables (specialized power systems), the provision and control of reactive power are currently beyond the scope of this paper. However, this topic is of particular interest for possible future studies.

4. Proposed Modeling for Dynamic Studies

4.1. Distribution Power System

The distribution power system is the last stage of the electrical grid. It distributes the electric power to the end-users. Small industrial, residential, and commercial customers are supplied by feeders in the distribution power system. It consists of lines, transformers, poles, and protection circuits [6]. Managing the distribution network ensures an acceptable power supply quality for the end-users. The electric power delivered from the distribution network must be within acceptable ranges for voltage, frequency, and current levels to ensure quality service to end-users. Effective planning and operation of the distribution network is essential to maintain these power supply standards. Additionally, to ensure safe operation, the distribution network must be resilient to specific disturbances, minimizing adverse impacts on end-users [20]. Recently, the high-level integration of DGs into the distribution power system has improved local voltage regulation and reduced technical losses in the system. However, it can also introduce negative side effects, such as decreasing grid stability and affecting existing protection schemes. In this context, the dynamic modeling of distribution power systems is needed to evaluate and predict both the positive and negative effects of RES integration into the distribution network.

4.2. Modeling of the Power Grid

The design and development of the equivalent distribution grid in this work were carried out using MATLAB/Simulink^® tools. The representation of the outer grid was visualized by the use of the three-phase source block from the Simscape Electrical Specialized Power Systems library of MATLAB/Simulink^®. It implements a balanced three-phase voltage source, with a base phase-to-phase voltage Vrms of 400 V and a frequency of 50 Hz. The integration of distributed RESs into a weak power system is a challenge with regard to the voltage control and reactive power compensation. For the present study, in order to properly simulate the weakness of the distribution network for the different scenarios, the internal impedance of the grid is adjusted by changing the value of the source internal resistance R. The three voltage sources are wye-connected with a neutral ground.

The whole distribution grid model in the MATLAB/Simulink^® software, version R2023b (MATLAB R2023b), is shown in Figure 1.

In Figure 1, it can be seen that the three-phase source block is on the left, providing power to the loads in the distribution grid, and that the grid topology is radial. In this paper, one of the objectives is to analyze the behavior of a weak distribution system. In order to represent the unbalanced characteristic of the power distribution system, the value of the parameters of the static load, represented in the three-phase RLC load block, have been accordingly adjusted for this purpose. In this block, the active and reactive power for each phase has been specified, loading phase A with more inductive reactive power than the other two phases, thus making the grid unbalanced.

For the connections between the three-phase source block and the loads, the three-phase PI section line block is used. The expected voltage drop across the grid is therefore modeled by the PI section line parameters. This block implements a balanced three-phase line model with the length of the line, which is 200 m each. The model consists of one set of RL elements connected between the input and output terminals and two sets of shunt capacitances lumped at both ends of the line. This is illustrated in Figure 2.

In Figure 2, Rs and Rm are the self and mutual resistances, whereas Ls and Lm are the self and mutual inductances of the coupled inductors. Cp and, correspondingly, Cg are the phase capacitances and the ground capacitances, which are deducted from the positive- and zero-sequence RLC parameters.

4.3. Load Modeling

The influence of load modeling on stability analysis is notably significant. A particular disturbance at a specific operating point may render the system stable under one load model, while causing instability under a different load. Multiple challenges make load modeling a difficult task. In real-world scenarios, the load always varies in terms of quantity and variety. Furthermore, under changing voltages and frequencies, the loads behave in a different way. In addition to this, the number of individual devices is huge and the computational requirements for simulating a whole distribution grid presents a hindrance; thus, most of the time, the load is modeled at the level of the transmission system [6]. In this work, composite load models are used to represent the load in a distribution power system. The objective of this modeling is to denote an aggregation of different types of loads that can be found in the residential, commercial, industrial, and agricultural sector at the distribution level and to perform a proper analysis of the effect that these composite loads have on the grid itself.

4.3.1. Static Load Model

Since the dynamics of the load is not of particular interest when modeling a static load, there is no need for the usage of differential equations to the describe the dynamics. Simple algebraic equations are sufficient to represent static load models. There are two main methods for describing how static loads respond to voltage variations: the polynomial static load model and the exponential static load model. The polynomial static load model, known as the ZIP model, represents the load as a mixture of constant impedance (Z), constant current (I), and constant power (P) [8]. It is illustrated with the help of Equation (1):

$$P=P_0\left[p_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+p_2{\left({\displaystyle \frac{V}{V_0}}\right)}^1+p_3{\left({\displaystyle \frac{V}{V_0}}\right)}^0\right]=P_0\left[p_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+p_2{\displaystyle \frac{V}{V_0}}+p_3\right]Q=Q_0\left[q_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+q_2{\left({\displaystyle \frac{V}{V_0}}\right)}^1+q_3{\left({\displaystyle \frac{V}{V_0}}\right)}^0\right]=Q_0\left[q_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2+q_2{\displaystyle \frac{V}{V_0}}+q_3\right]$$

(1)

where $V_0$ is the per-unit (pu) bus voltage for the initial condition from the power flow solution, $P_0$ and $Q_0$ are the real and reactive pu loads, $p_1,p_2,p_3$ are the percentages of $P_0$ assigned as constant impedance, constant current, and constant power, in that order, and $q_1,q_2,q_3$ are the percentages for $Q_0$. The conditions in Equation (2) have to be respected:

$$p_1+p_2+p_3=1.0 q_1+q_2+q_3=1.0$$

(2)

The exponential static model expresses the real and reactive components of the load, as shown in Equation (3), where the parameters are the coefficients $p$ and $q$ and the exponents α and β.

$$P=P_0\left[p{\left({\displaystyle \frac{V}{V_0}}\right)}^{\alpha }\right]Q=Q_0\left[q{\left({\displaystyle \frac{V}{V_0}}\right)}^{\beta }\right]$$

(3)

In this paper, the ZIP model is used as the static load model. Specifically, the three-phase series RLC load block from the Simscape Electrical Specialized Power Systems library in MATLAB/Simulink^® is employed, connected to the VI static load measurement block, as shown in Figure 1, under the area labeled “Static (Z) Load Block”. The load block is wye-connected with a neutral ground. It represents a Z-type load, which means that at the given frequency the load has a constant impedance, and the active and reactive power consumed by the load is proportional to the square of the applied voltage. This means that the values of $p_2, p_3$ and, respectively, $q_2, q_3$ from Equation (2) are zero. Therefore, Equation (1) can be re-written in the following form:

$$P=P_0\left[p_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2\right]Q=Q_0\left[q_1{\left({\displaystyle \frac{V}{V_0}}\right)}^2\right]$$

(4)

4.3.2. Dynamic Load Model

To represent the aggregation of various load types and their dynamic behavior in the distribution grid, a dynamic load model from the Simscape library of MATLAB/Simulink^® was adopted. In Figure 3, the modeling of the dynamic load is illustrated.

In the regions labeled “Dynamic Load” in Figure 1, the three-phase dynamic load block from the Simscape Electrical Specialized Systems library is used. It implements a three-phase, three-wire dynamic load, connected to the load flow bus, as shown in Figure 3. The voltage on the dynamic load block is positive-sequence, which means that only the positive sequence is considered for the output calculation of the block. However, the output calculation of the block has an impact on the negative- and zero-sequence quantities of the whole simulated system, although they are not available for analysis. This document primarily focuses on voltage analysis, with specific reference to the insights provided by source [6]. According to the authors, the study of negative- and zero-sequence components of voltages and currents is considered less relevant in the context of stability investigations. Consequently, any potential limitations related to the negative- and zero-sequence components within the MATLAB/Simulink^® model are of lesser concern for the purpose of this research. In Figure 4, the block subsystem that is connected to the PQ signal input of the dynamic load block is shown.

The active power P and the reactive power Q are controlled externally with the desired load profile with the help of this system, shown in Figure 4. There are two possible options to create signal 1 that serves this purpose. As shown in Figure 4, it can be seen that this signal can either be generated through a MATLAB function that contains the code for the desired load profile or using the signal builder, where the signal is generated and then amplified using the gain blocks. The desired option can be selected by changing the position of the manual switch.

Another dynamic block is also used in the modeled network. In this block, the active power P and the reactive power Q are not controlled externally. Instead, when the terminal voltage exceeds the minimum value $V_{min}$, the active power P and reactive power Q of the load vary according to Equation (5):

$$P\left(s\right)=P_0{\left({\displaystyle \frac{V}{V_0}}\right)}^{n_p}{\displaystyle \frac{1+T_{p1}s}{1+T_{p2}s}}$$

(5)

where $V_0$ is the initial positive-sequence voltage, $P_0$ and $Q_0$ are the initial active and reactive powers at the initial voltage $V_0$, and $V$ is the positive-sequence voltage. $T_{p1}$ and $T_{p2}$ are time constants controlling the dynamics of the active power $P$, and $T_{q1}$ and $T_{q2}$ are time constants controlling the dynamics of the reactive power $Q$. The active and reactive components $n_p$ and $n_q$ are exponents controlling the nature of the load, meaning that adjusting these values enables the modeling of different load behaviors. This is illustrated in section composite load models, and the values of the exponents $n_p$ and $n_q$ can be found in

Table 1. The values of $n_P$ and $n_q$ for different load models [33].

Constant Impedance	Constant Current	Constant Power	Industrial	Commercial	Residential	Load Type
2	1	0	0.18	1.51	0.92	$n_P$
2	1	0	6	3.4	4.04	$n_q$

below.

Using this setup, however, means that it is not possible to set a desired load profile. Although the name of the block is the dynamic load block, it may not constantly behave as a dynamic load. If the terminal voltage $V$ of the load is lower than a specified value $V_{min}$, then the load impedance is kept constant; therefore, in the time interval when this specific condition is active, the load behaves like a static Z load.

4.3.3. Composite Load Model

In practical application and as commonly observed in the literature, the modeling of distribution grids often entails the representation of loads as constant current, constant power, or constant impedance, primarily due to data limitations. Because of the varying behavior of loads nowadays, it is evident that outcomes derived from these conventional models may not faithfully mirror the actual picture. More accurate results can be delivered when modeling with the help of the composite load model [33,34]. It combines the various kinds of loads that can be found in the power grid. It can be a combination of static loads and different types of dynamic loads such as three-phase induction motors, single-phase induction motors, power electronic loads, and even distributed generation [25]. In this work, by varying the exponents $n_P$ and $n_q$ from Equation (1) that are implemented in dynamic load block 2 used in MATLAB/Simulink^®, composite load models are created. In Table 1, various kinds of loads that can be found in a distribution grid are shown: the industrial, commercial, residential, constant power, constant current and constant impedance load types in correlation with the values of $n_P$ and $n_q$.

4.4. Modeling of the Distributed Generation

The high-level integration of DERs in the existing distribution grid may lead to voltage fluctuations and overloads of different elements within the grid. The power factor may also be seriously affected, and additional unbalanced voltage conditions may occur as well [35]. The results in this work also point to the importance of the power factor problem. Consequently, to comprehensively assess the influence of DGs on the distribution grid, it is imperative to undertake a proper and accurate modeling of DG systems. In this work, photovoltaics (PVs) are integrated into the modeled distribution network. This is achieved by using the subsystem grid-connected PV array. In Figure 5, an illustrated view of the subsystem is given.

In our implementation, we use the inverter model employed for the low-voltage PV farm simulation in the MATLAB/Simulink^® library for renewables (specialized power systems). This inverter operates as a grid-following device, where the control strategy is based on measuring the AC-side voltages and currents, which are transformed into the dq0 reference frame. This allows for the regulation of both the DC bus voltage and the output currents. A phase-locked loop (PLL) is employed to synchronize the inverter to the grid frequency, while a maximum power point tracking (MPPT) algorithm, specifically the perturb and observe (P&O) method, is utilized to optimize power extraction from the PV panels [36,37]. The MPPT adjusts the output voltage periodically and compares the power obtained during each cycle, optimizing the system’s performance under varying climatic conditions. However, although the inverter synchronizes with the grid and follows its frequency, it lacks control functionalities that support grid stability, such as transient voltage regulation through reactive power control (e.g., Q–V droop) or frequency stabilization through active power control (e.g., frequency–Watt droop). As a result, it is not expected to contribute to dynamic voltage or frequency regulation in the grid. Despite this, inverters of this type are commonly used at the low-voltage level in many countries. This presents a challenge, as future grid codes—particularly with the growing penetration of renewable energy sources—may require inverters to not only maximize power generation but also actively participate in grid support functionalities.

For study cases 3 and 4 in this paper, the number of PV modules is changed, so different scenarios—with low, average, and high integration of DGs—can be tested, adjusting the load ratio with respect to the nominal load to 50%, 75%, and 100%. This can be adjusted in the block PV array in the subsystem, as shown in Figure 6.

The x-axis represents the time duration of 30 s. The y-axis corresponds to the sun irradiance that is given in W/m². In real-world applications, the created signal would represent a sudden change in the sun irradiance from 1000 W/m² to 750 W/m², from the 5th to the 10th s, further decreasing from 750 W/m² to 500 W/m², from the 15th to the 20th s. These three values of sun irradiance correspond to 100 kW, 75 kW, and 50 kW of generation from the PV farm. The transformer in the subsystem was also adjusted accordingly with respect to the nominal conditions of the modeled grid. The PV farm subsystem offers a multitude of possibilities and encompasses several variables that can be changed, including the choice of the module manufacturer and its specifications, sun irradiance levels, ambient temperature conditions, and converter specifications. This flexibility renders the PV subsystem a highly valuable resource, presenting ample opportunities for in-depth investigations within this domain.

4.5. Study Cases

In this work, four distinct study cases have been formulated, each further subdivided into two main subcases: the balanced and the unbalanced subcase. The introduction of the imbalance in the grid is achieved by increasing the capacitive reactive power of the static RLC load in phase A by 50% with respect to its nominal power in the balanced case.

4.5.1. Study Case 1

In the first study case, the system is modeled using only a static Z load without dynamic load modeling or renewable energy integration. The simulated grid, as shown in Figure 1, includes only the “Main Grid” and the “Static Z load”, with all other elements deactivated. The power factor of the grid is 0.9, with a capacitive (leading) nature. The results, illustrated in Figure 7, compare two subcases: balanced and unbalanced scenarios.

In the unbalanced subcase, the voltage of the static load bus for phase A decreases to 0.925 pu, while the voltage for phase B slightly increases to 0.98 pu, and phase C remains constant at 0.96 pu. In contrast, in the balanced case, the voltage for all three phases is 0.96 pu.

In the unbalanced subcase, phase A of the static load bus experiences a voltage drop to 0.925 pu, while phase B shows a slight increase to 0.98 pu, and phase C remains constant at 0.96 pu. For the balanced scenario, all phases maintain a voltage of 0.96 pu. A similar behavior is observed at the voltage receiving bus. In the balanced subcase, the voltage across all phases is 0.98 pu. In contrast, in the unbalanced subcase, phase A decreases to 0.97 pu, phase B rises to 0.99 pu, and phase C remains unchanged at 0.98 pu.

Further analysis of the current measured at the receiving bus shows an increase across all three phases, with phase A exhibiting the largest rise to 1.4 pu. This increase is primarily due to the 50% rise in capacitive reactive power in phase A with respect to the nominal power. All the voltages in the three phases A, B, and C differ from each other, which is proof of the unbalanced behavior of the system.

In Figure 8, the active and reactive power measured from the receiving current bus, which flows from/to the grid in both balanced and unbalanced cases, is shown.

Figure 8 presents the active and reactive power flow measured at the receiving bus. In the balanced case, the active power flowing from the grid to the load is around 1.32 pu. In contrast, in the unbalanced case, the active power demand increases slightly to 1.53 pu. The negative sign in the reactive power indicates that the capacitive load is feeding reactive power back to the grid. In the balanced scenario, this reactive power feedback is around 0.58 pu, while in the unbalanced case, due to the increase of capacitive reactive power in phase A, the reactive power flowing back to the grid increases to 0.65 pu.

This first case study highlights the impact of unbalanced conditions on system voltages and currents, as well as the increased demand on active and reactive power in the unbalanced scenario, providing critical insights into the system’s response to unbalanced load conditions.

4.5.2. Study Case 2

In study case 2, the grid model incorporates both static Z load and dynamic load modeling, but without integrating renewable energy sources. The components of the grid used in this case, as shown in Figure 1, include the “Main Grid”, “Static Z load”, and both “Dynamic (profile) load” and “Dynamic (composite) load”. All other components are deactivated. This case also explores both balanced and unbalanced conditions in the network, with three further subdivisions to represent different types of composite loads: industrial, residential, and commercial. The total power supplied by the grid is evenly divided, with 50% consumed by the static load and 50% allocated to the dynamic composite load model.

Study Case 2.1

In this particular study case, along with the simulated static Z load, there are also simulated industrial loads that can be commonly encountered in real-world scenarios, by setting the values of $n_P$ to 0.18 and $n_q$ to 6 into the “dynamic (composite) load” part. The results are shown in Figure 9.

As in the first study case, a clear distinction between the balanced and unbalanced scenarios is observed. In the unbalanced subcase, the capacitive reactive power in phase A is increased by 50% relative to the nominal value. This leads to an increase in the receiving current, most significantly in phase A, where the impact of the reactive power increase is most pronounced. Analyzing the voltages at the load side, the unbalanced case results in a voltage drop in phase A, while phase B experiences a slight voltage rise, and phase C remains constant.

With the addition of dynamic and composite load models, the voltages and currents at these buses show behavior similar to the static load buses but with more noticeable variations due to the dynamic nature of the loads. Figure 9 illustrates that the voltages at the buses connected to the dynamic and composite loads exhibit a more pronounced variation in waveforms compared to the static load scenario. However, these voltage fluctuations remain within acceptable limits, indicating that the system remains stable despite the unbalanced conditions.

Interestingly, the currents at the dynamic and composite loads remain symmetrical even though the system is unbalanced. In practical situations, some deviation from this ideal behavior are expected. This situation is due to the limitations of modeling. The inherent characteristics of the “Three-Phase Dynamic Load” block from the Simscape library, as detailed in the Dynamic Load Model chapter, explain this phenomenon. The block does not account for negative- and zero-sequence currents, ensuring that the load currents remain balanced even when the voltage is unbalanced.

This study case underscores the importance of dynamic load modeling in analyzing real-world industrial scenarios, where the load behavior can significantly affect the overall system’s voltage and current characteristics under unbalanced conditions.

Study Case 2.2

In this particular study case, the simulated loads can be typically encountered in commercial areas. This is achieved by configuring the specific parameter value of $n_P$ to 1.51 and $n_q$ to 3.4 into the “dynamic (composite) load” part. The results are shown in Figure 10.

The key distinction between study case 2.1 and study case 2.2 lies in the parameter values of $n_P$ and $n_q$ in the three-phase dynamic block 2, which are changed in order to simulate a commercial load type. As the measurements from that block are depicted in Figure 10 under the labels “Voltage DYN npnq Load” and “Current DYN npnq Load”, respectively, for the balanced and the unbalanced scenarios, a small difference is barely noticeable. In this case, the voltage and current levels have insignificantly decreased.

Study Case 2.3

In this study case, the loads in residential areas are simulated by setting the values of $n_P$ to 0.92 and $n_q$ to 4.04 into the “dynamic (composite) load” part to reflect the behavior of such loads. The results of this simulation are shown in Figure 11.

In Figure 11, the measured voltage and current, represented as “Voltage of DYN profile Load” and “Current of DYN profile Load”, exhibit slight differences compared to the previous study cases. The variations in voltage and current between the balanced and unbalanced cases are relatively small, indicating that while residential loads introduce more fluctuations compared to industrial loads, these differences remain moderate.

Comparison of Study Cases “Static Z Load”

The key distinction among study case 2.1 (industrial load), study case 2.2 (commercial load), and study case 2.3 (residential load) lies in the adjustment of the parameters $n_P$ and $n_q$ in the three-phase dynamic block. In study case 2.1, $n_P$ = 0.18 and $n_q=$ 6, simulating industrial loads that typically exhibit stable and predictable power demand with minimal sensitivity to voltage variations. In contrast, study case 2.2 models commercial loads, with $n_P=$1.51 and $n_q=$3.4, which reflect more variable usage patterns due to lighting, HVAC systems, and office equipment. For study case 2.3, residential loads are simulated using $n_P=$0.92 and $n_q=$ 4.04, capturing the fluctuating power consumption caused by varying household activities, such as appliance usage and heating/cooling systems.

The measurements from each case, named the voltage DYN npnq load and current DYN npnq load, respectively, for the balanced and the unbalanced cases, show small differences across all load types. Figure 12 provides a detailed comparison of the total power (both active and reactive) for each load type, highlighting how these small variations affect overall system performance.

The main distinction between study case 2.3 (residential load) and the previous cases—study case 2.1 (industrial load) and study case 2.2 (commercial load)—is the adjustment of $n_P$ and $n_q$ in the three-phase dynamic load block. These values simulate the characteristics of residential loads, which tend to fluctuate based on varying household activities such as appliance usage, lighting, and heating/cooling systems.

In study case 2.1, which represents the industrial load type, the value of $n_P$ is 0.18, which is close to zero. This explains why it can be seen that the blue curve bares more of a resemblance to a straight line than a curve (from Table 1 it can be seen that for the value of 0 for $n_P$, it represents a constant power load type, a straight line). In the commercial load case, where $n_P=1.51$, the active power exhibits noticeable oscillations over time, indicating a higher sensitivity to voltage variations compared to the industrial load case (with $n_P$ = 0.18), where the active power remains nearly constant. This is consistent with the operational behavior of commercial loads, which typically experience greater fluctuations due to more variable usage patterns. The residential load case (where $n_P=$0.92) also shows oscillations in active power, but to a lesser extent than the commercial load. However, these oscillations remain within operational limits, ensuring grid stability is maintained despite the load variability.

Similarly, the reactive power for the commercial load shows slight variation compared to the industrial load. The higher value of $n_q$ reflects the reactive power demand typical of commercial environments with a lot of inductive loads. In the residential load case, slight fluctuations in reactive power are observed but remain within acceptable limits.

The comparison of these curves highlights how reactive power demands differ between the load types, with industrial loads being the most stable and predictable, and commercial and residential loads showing more variability. While the reactive power does fluctuate, the magnitude of these changes is small and does not significantly affect the system’s voltage stability.

4.5.3. Study Case 3

In study case 3, the static Z load model is implemented without dynamic load modeling, but this case includes the integration of renewable energy into the distribution network. As shown in Figure 1, the modeled grid incorporates the “Main Grid”, “Static Z Load”, and a PV farm, while other elements are deactivated in the simulation. This case also divides the network into balanced and unbalanced scenarios. The electric power supplied by the PV farm follows a generation profile, varying over time and reflecting the integration of 50%, 75%, and 100% of the power consumed by the loads, as illustrated in Figure 13.

In the first 5 s of the simulation time, the power consumed by the loads in the network comes 100% from the PV farm. Then, in the next 5 s, it gradually decreases to 75% and stays constant. It again starts to decrease in the 15th s of the simulation until it reaches 50% in the 20th s of the simulation, and it stays at this level for 10 s more. Once again, the imbalance of the grid is achieved by increasing the capacitive reactive power in phase A by 50% with respect to the nominal power, and just like in the previous study cases, the values of the voltage for phase A in the static load bus expectedly decrease, whereas the values of the voltage for phase B slightly increase, and the voltage of phase C maintains the same value. A similar situation can be observed in the receiving voltage bus.

Of particular interest is the behavior of the receiving current bus under unbalanced conditions. The integration of the PV farm affects the outer grid, as seen in the behavior of the current in phase A during the initial 5 s. The current value in phase A is about 0.7 pu, while phase C remains around 0.5 pu, and phase B hovers at approximately 0.2 pu. During this time, the PV farm generates 100% of the power consumed by the loads, which would typically suggest that the current flowing from the grid would be near 0 pu. However, the current remains notably higher. The direction of the current, which flows from the static load back to the grid, is noteworthy. This phenomenon occurs due to the increased capacitive reactive power in phase A, which is raised by 50% relative to the nominal power.

Figure 14 further illustrates the active and reactive power observed at the receiving bus. The most significant observation is the inability of the PV farm’s inverter to provide reactive power to support the grid. The active power closely follows the sun’s irradiance profile, while the reactive power remains constant at approximately 0.6 pu in the balanced case and 0.65 pu in the unbalanced case. This constancy indicates that the reactive power surplus or demand is independent of the PV farm, and the inverter used cannot provide reactive power support to the grid.

4.5.4. Study Case 4

In study case 4, the static Z load model is implemented along with dynamic load modeling and the integration of renewable energy into the network. All components shown in Figure 1 are active in this simulation, including the main grid, static Z load, dynamic (composite and profile) load, and the PV farm. The network is divided between balanced and unbalanced conditions, as in previous study cases. The composite load model is configured to represent industrial, residential, and commercial loads, as detailed in Table 1. Half of the total power is consumed by the static load, while the other half is consumed by the dynamic load model. The PV farm provides power according to a generation profile that varies over time, supplying 50%, 75%, and 100% of the total load, as seen in study case 3.

Study Case 4.1

In this scenario, the composite load is modeled to represent industrial loads, combining dynamic and composite load characteristics. The key difference between study case 4 and study case 3 is the inclusion of dynamic load behavior.

Figure 15 illustrates the current and voltage behavior, with the dynamic load undergoing rapid fluctuations every 2 s, ranging between approximately 0.24 pu and 0.26 pu. These fluctuations impact the entire grid, influencing the voltage on the static load as well as the sending and receiving buses. Compared to study case 3, the graphs show more pronounced fluctuations, due to the dynamic loads’ behavior. Nevertheless, the overall effect of this load on the grid in the voltage remains within acceptable limits. The ”Receiving current” measurements are minimal during nominal power injection from the PV system, indicating that the system is predominantly supplying the loads. Only a small amount of power is being drawn from the external grid. Consequently, the curves can be interpreted as primarily driven by the power injected by the PV system. At the 5th s, under balanced conditions, the load is fully supplied (100%) by the PV system.

As in previous cases, the grid imbalance is introduced by increasing the capacitive reactive power in phase A by 50% relative to nominal power. This is evident in the current receiving block, where phase A shows a higher current magnitude compared to phases B and C.

Study Case 4.2

In this study case, the composite load is modeled to represent loads in the commercial sector. Upon comparing the graphs in study case 4.1 and 4.2, it can be seen relatively minor differences. As the composite load model consists of only 25% of the total load, it can be concluded that it makes no significant difference whether the composite load model represents loads in the industrial or commercial sector. The results are represented in Figure 16.

Study Case 4.3

In this study case, the composite load is modeled to represent loads in the residential sector. The results from this study case are shown in Figure 17.

Comparison of Study Cases “Static Z Load, Dynamic Load and RES”

When comparing the graphs from previous three study cases, relatively minor differences in behavior can be observed. However, to facilitate a clearer comparison, Figure 18 illustrates the total power consumed by the composite load in each case. The blue curve represents the active power consumption, while the orange curve represents the reactive power consumption across the industrial, commercial, and residential load scenarios.

Figure 18 shows that in the industrial and residential load types, the active power consumption is higher than the reactive power. In contrast, for the commercial load type, reactive power consumption is more significant than active power. This behavior highlights the distinct power consumption characteristics of each load type. Additionally, the power consumption of the composite load is influenced by the generation profile of the PV farm. As the PV generation decreases, it leads to a voltage drop in the grid, which causes a corresponding reduction in power consumption from the composite load.

Due to the composite load being a mixture of static ZIP loads and various dynamic loads, the decrease in power consumption does not occur at the same rate as the voltage drop caused by the reduced PV generation. The power consumption of the composite load adjusts, but not in a direct proportion to the generation decrease. As the PV farm generates less power, the composite load compensates by drawing more power from the main grid. This can be seen in Figure 15, Figure 16 and Figure 17, where the measurements from the receiving bus show an increase in the current drawn from the main grid as the PV generation decreases.

The total power consumed by the composite load is influenced by both load type and the generation profile of the PV farm. While industrial and residential loads consume more active power than reactive power, commercial loads show a higher proportion of reactive power consumption. Furthermore, the decrease in PV generation leads to increased reliance on the main grid to meet the power demand of the composite load.

4.6. Comparison of All Study Cases

The four study cases collectively highlight the impact of different load types, dynamic modeling, and renewable integration on grid stability under balanced and unbalanced conditions and are summarized in

Table 2. Simulated Study Cases.

Study Case	Static Load Used	Dynamic Load Used	RES Used
1	Yes	No	No
2	Yes	Yes	No
3	Yes	No	Yes
4	Yes	Yes	Yes

.

Study case 1, which only includes static Z loads, serves as a baseline and shows that even modest imbalances in the grid can result in voltage and current asymmetries. Study case 2 demonstrates that adding dynamic loads introduces more variation in voltage behavior, especially under unbalanced conditions, but the system remains largely stable due to the inherent properties of the dynamic load models.

Study case 3 introduces renewable energy into the mix, showing that the grid’s response becomes more complex when renewable generation fluctuates. A notable limitation is the inability of the PV farm’s inverter to support the grid with reactive power, a critical factor in maintaining voltage stability under unbalanced conditions. Finally, study case 4 reveals that the combination of dynamic load modeling and renewable integration leads to more significant current and voltage fluctuations, although these remain within acceptable operational limits.

In conclusion, the study cases underscore the importance of considering both dynamic loads and renewable integration when analyzing grid stability, particularly in systems where unbalanced conditions may arise. The findings suggest that while renewable integration enhances grid resilience, it also requires advanced inverter technologies capable of providing both active and reactive power support to ensure stable operation.

5. Conclusions

This study offers a thorough analysis of the dynamic modeling of distribution power systems with integrated RESs, focusing on stability assessment. Using EMT modeling and detailed simulations conducted in MATLAB/Simulink^®, we examined distribution system behavior across various levels of distributed generation (DG) penetration and diverse load models. Both balanced and unbalanced network conditions were analyzed to capture how distributed photovoltaic (PV) systems influence voltage stability and overall grid performance.

The results underscore the critical role of inverter control techniques in maintaining voltage stability, particularly through reactive power control. Significant limitations were observed in the reactive power compensation capabilities in the implemented inverters. This shortfall poses a substantial risk to stability in grids with high DG penetration. Without proper voltage and frequency control techniques implemented in inverters, voltage and frequency stability could be further compromised, especially as DG levels increase.

A contribution of this study lies in the modeling of the external grid’s behavior. By adjusting the internal impedance of the equivalent upstream grid, we simulate its inherent weaknesses, particularly in upstream systems with high DG penetration and numerous dynamic loads (such as motors). The study demonstrates that grid stability could be at risk without proactive reinforcements. This finding emphasizes the urgency for strengthening distribution grids to handle increasing operational demands as DG integration continues to grow in both low and high voltage.

Dynamic load modeling proved crucial for understanding interactions between RESs and distribution networks. The study reveals that different load types—industrial, commercial, and residential—affect grid performance in distinct ways. The diverse behaviors of these loads, when combined with renewable integration, highlight the need for meticulous planning and operational management to ensure reliable and stable grid operation.

Future research will focus on modeling advanced voltage control techniques in inverters, integrating battery storage systems, and exploring islanding methods to further enhance grid stability. Addressing the limitations in reactive power control in current inverter technologies will be pivotal, especially as distribution grids evolve to accommodate higher levels of renewable energy.

This work establishes a robust foundation for ongoing advancements in dynamic modeling, offering critical insights into the challenges and opportunities presented by the transition to low-carbon energy systems. Proactive grid reinforcements, especially in the context of high DG integration, and the continued innovation of inverter technologies will be essential for ensuring future grid stability and reliability.