Analysis of Transmission System Stability with Distribution Generation Supplying Induction Motor Loads

Abstract

A distributed-power-generating source (DPGS) is intended to locally supply the increased power demand at a load bus. When applied in small amounts, a DPGS offers many technical and economic benefits. However, with large DPGS penetrations, the stability of the transmission system becomes a significant issue. This paper investigates the stability of a transmission system equipped with a DPGS at load centres supplying power to both a constant power (CP) and induction motor (IM) load. The DPGSs considered in the present study are microturbine and diesel turbine power generators (MTGS and DTGS), both interfaced with synchronous generators. The influence of an IM load supplied by the DPGS on small-signal stability is studied by a critical damping ratio analysis. On the other hand, time-domain indicators of the transient response following a short circuit are employed in the analysis. Further, a variance analysis test (VAT) is performed to determine the contribution of IM and CP loads on the system stability. The study revealed that large penetration levels of IM loads significantly affect the stability and depend on the kind of DPGS technology used.

1. Introduction

Distributed-power-generating sources (DPGSs) have created a paradigm contrasting the conventional power system. The sources have paved the way for a technology suitable for electricity generation in a limited capacity that is located near the point of consumption. The specific definition and implementation of DPGSs can vary significantly from one country to another and even within regions of the same country [1]. This variation is due to several factors, including differences in available renewable and other energy resources, the grid infrastructure, government policies, and local energy demands. The diversity in the definitions and approaches to DPGS reflects the adaptability of this concept to the unique needs and circumstances of different countries and regions, making it a flexible solution for addressing various energy challenges [2]. Generally, DPGS technologies can be categorized based on the devices and fuels utilized, broadly encompassing generators and storage systems. The majority of these systems can produce both active and reactive power, although their capabilities vary significantly from one another. A detailed technical overview and analysis are presented for each DPGS technology in [1]. Many of these technologies harness energy resources, characterized as sources typically resistant to depletion. These encompass solar radiation, wind, biomass, hydropower, marine energy, and geothermal sources [3]. Few other technologies, including small-scale hydroelectric systems, photovoltaic arrays, diesel and wind generators, solar–thermal units, fuel cells, and battery storage technologies, comprise numerous small modules assembled in factories. These technologies’ manufacturing and on-site construction need considerably less time than the process for large centralized power stations [4].

As the popularity and integration of DPGSs continue to grow, it has become essential to develop and implement standards that ensure interoperability, efficiency, and safety across these distributed resources [5]. Several standard organizations [6] have emerged to address this need and have developed various standards related to DPGSs. One of the notable standard organizations in this field is the International Electrotechnical Commission (IEC) [7]. The IEC has published several standards relevant to DPGSs, including IEC 61850, a communication standard for intelligent electronic devices used in power substations. This standard enables the exchange of information between different DPGSs and facilitates their integration into the grid.

Similarly, the IEEE [8] has also developed several standards related to DPGSs. IEEE 1547 is a widely recognized standard that defines the interconnection requirements for DPGSs with the electric power system. It specifies the technical and operational requirements for a reliable, secure grid integration interfaced with DPGSs. Moreover, the National Electrical Manufacturers Association (NEMA) has developed the NEMA SG 4 standard, providing guidelines for evaluating, selecting, and integrating storage energy systems into electrical systems. There are also regional and national standards specific to certain countries or regions. For instance, in the United States, the Smart Grid Interoperability Panel (SGIP) has developed standards and guidelines for energy resource integration, including the Open Field Message Bus (OpenFMB) standard, which enables interoperability between different DPGSs. When comparing these standards, it is imperative to consider factors such as scope, applicability, and adoption. Some standards may have a broader scope, covering multiple aspects of DPGS integration, while others may focus on specific technologies or applications.

The adoption and implementation level of the above standards also varies across different regions and countries. Overall, the analysis and comparison of these standards help stakeholders in the energy sector understand the requirements and best practices for integrating DPGSs into the grid [9]. These standards play a vital role in successfully deploying distributed energy resources worldwide by promoting interoperability, efficiency, and safety. An overview of varied distributed-power-generating source technologies has been presented by authors in [10]. The study gave an insight into the environmental impacts of DPGSs over conventional-power-generating sources. Studies on DPGSs, such as wind turbines and PV panels, have revealed the positive and negative effects on power systems in recent years. While DPGSs contribute to renewable energy integration and reduce greenhouse gas emissions, their increased penetration levels can pose challenges to the power grid’s reliability and stability [11]. Some adverse impacts associated with high levels of distributed generation include frequency and voltage instability, the reduction of the transient stability margin, protection co-ordination challenges, and grid imbalance [12].

In a distribution system, electricity is typically transmitted to substations through high-voltage transmission lines, thereafter lowering the voltage and distributing it to homes, businesses, and other consumers. Distributed-power-generating sources are connected to the transmission system and can help meet the local load demand without requiring extensive transmission and distribution infrastructure.

Integrating DPGSs into the transmission system offers the following technical and operational advantages [13]:

• Decreased line losses and improved voltage profile: DPGS systems located closer to the load reduce the distance electricity travels, minimize transmission losses, and help maintain a stable voltage level in the transmission system. In addition, lower costs for system operation and its maintenance, associated with reduced transmission losses, contribute to cost savings for utilities.
• Enhanced stability and power quality: DPGS systems can provide local stability, ensuring a consistent power supply and improving the overall quality of electricity.
• Increased network efficiency and reliability: DPGS integration optimizes the utilization of existing infrastructure, making the system more efficient. Moreover, by providing backup power during grid outages, DPGS systems can enhance reliability.
• Security enhancement: Distributed generation can enhance the security of the electricity supply by reducing the dependence on centralized power plants, making the system less vulnerable to large-scale failures or attacks.
• Reduced transmission and distribution (T&D) congestion: By generating power closer to the point of consumption, the DPGS mitigates congestion issues by reducing the strain on T&D lines.
• Delayed investments for upgrading facility: Integrating DPGSs can defer the necessity for significant investments in new T&D infrastructure, saving costs for utility companies.
• Suppression of fuel costs for distributed energy resources: Renewable energy sources used in DPGS (wind/solar) systems have no fuel costs, making them economically attractive and environmentally friendly alternatives to traditional fossil-fuel-based generation.

In addition to the above, integrating DPGSs into the transmission system offers economic advantages by reducing costs, creating revenue streams, providing environmental benefits, lowering emissions, and improving air quality. Further, the operational improvements in grid reliability and stability contribute to long-term sustainability by diversifying energy resources and promoting renewable energy adoption [14]. With the advent of several technologies contributing to the advantages mentioned above, however, stability is one of the major concerns when DPGSs are integrated into the transmission system. Researchers and industry professionals are focusing on developing advanced monitoring, control, and mitigation strategies to maintain the stability and resilience of transmission systems in the face of these evolving demands and challenges [15].

The present work investigates the transmission system stability when power demand increases in terms of induction motor load at the load centre interfaced with DPGSs. Hence, the following section presents previous studies focusing on the impact and evaluation of power system stability with DPGSs.

2. Literature Review

At various stages of research, researchers have investigated the impact of interfacing DPGSs on the stability aspects by employing different types of DPGS technologies in a power system that includes synchronous generator-based power-generating sources, renewable- and non-renewable-power-generating technologies, and sources interfaced with power converters (like a wind turbine, photovoltaics, fuel-cell, and gas- and microturbine-power-generating sources).

In [16], the authors examined the influence of a synchronous generator interfaced with a small hydropower-generating source of a Brazilian power grid by analyzing the stability aspects of the system. The study unveiled an improved voltage stability margin and transfer capacity due to increased power-generating source levels. A corresponding investigation involving synchronous-generator-based-power-generating sources in the distribution network is carried out in [17]. A technique based on projective integration is applied for rapidly simulating transient stability, particularly in the presence of the high penetration of distributed-power-generating sources and virtual synchronous generators [18]. Few other researchers [19] have emphasized the study of synchronous and asynchronous generator-based distributed-power-generating source technology. The study focuses on the distinct characteristics of these technologies and their impact on the stability of the power grid. The authors in [20] simulated various fault conditions to demonstrate the effect of DPGSs on transmission system transient stability. The study analyzed different fault scenarios occurring in multiple branches of the system, considering N-1 security to assess the system’s robustness.

A detailed survey was carried out to analyze the behavior of voltage variation in a DPGS-interfaced distribution network [21,22]. In [23], the influence of system parameters on voltage oscillations is examined with the aid of participation factors and eigenvalues. Several researchers have suggested a number of techniques to analyze the voltage stability of the system. In [24], an extensive voltage profile study is carried out under the influence of different power-generating sources. The study revealed that a power system under stress undergoes voltage instability, frequently due to the increased load demand at a particular bus. The authors [25] formulated an analytical approach to enhance the reliability and voltage profile by strategically placing and sizing the renewable generating sources on distribution systems. Subsequently, various intelligent methods have also been reported due to the potential benefits of installing DPGSs on the distribution network [26].

In [27], the transient stability was analyzed by studying a hybrid system incorporating various renewable power generation sources. The authors concluded that the influence of power generation sources is contingent on both its location and type. Furthermore, the authors reported that stability declined as the penetration level of power-generating sources increased. In [28], authors have proposed an optimal algorithm to control the frequency in a hybrid system and co-ordinate between various power-generating sources.

A comprehensive mathematical model and several simplified versions of the microturbine power-generating sources that utilize liquid and gas fuel systems were detailed in [29]. Apart from the study in [30] focusing on the system’s transient response, a comprehensive investigation on small-signal stability incorporated a microturbine power-generating source. The analysis indicated that transient stability is enhanced with increasing DPGS penetration levels, whereas the excitation control parameters of the power-generating source influence the small-signal stability. In [31], authors investigated the small-signal and transient stability in a transmission system with rotating types (such as micr-turbine, diesel turbine, and wind turbine generators) of DPGSs interfaced at the load buses. The study revealed that the system’s stability will be degraded when DPGS penetration exceeds a certain optimum level. Moreover, the study in [32] showed that the stability performance under the unequal load increase condition differs considerably compared to the equal load increase condition even for the same DPGS penetration level.

An extensive study on dynamic voltage stability is carried out on a hybrid system comprising conventional- and renewable-power-generating sources [33]. This study presented a comparative analysis involving various types of static VAR compensators, utilizing minimum first swing and damping associated with subsequent oscillations as indicators. Authors in [34] analyzed the small-signal stability of a distribution system interfaced with renewable-power-generating sources. In this study, the authors employed supplementary controllers to improve the damping ratio of critical modes. Furthermore, the study reported substantial involvement of both induction and synchronous generators in the oscillating modes.

Apart from the integration of DPGSs, researchers have been investigating the influence of dynamic loads on the distribution system [35]. These studies aim to understand and address the challenges posed by dynamic loads separately from other types of loads [36]. The studies in [37,38] reported the influence of dynamic loads on short- and long-term voltage stability.

In recent years, DPGS technologies have received significant interest in smart grids, microgrids, and distribution network applications [39]. Although this technology focused on improving efficiency, reliability, and sustainability, many investigations have reported specific technological issues concerning DPGSs and their operation. Researchers in [40] have analyzed the stability aspects considering the interaction between the dynamics of DPGSs with constant power loads. A study based on techniques of synchronization [41] to extract positive–negative sequence components under the influence of inverter-based DPGSs is performed under varied grid conditions.

Some of the significant issues that have been identified based on the literature review are summarized below:

• Most of the earlier findings reported the influence of DPGS penetration on system stability as applied to distribution networks. However, the latest articles have focused on installing DPGSs either at the load centres or distribution substations of transmission networks to meet any growth in load demand in the near future;
• Most of the stability studies reported are in the area of microgrids, focusing on analyzing the influence of constant power loads with interfaced DPGSs;
• In most articles, researchers have focused more on analyzing the effect of increased constant power loads in a transmission/distribution network interfaced with DPGSs. Therefore, it is essential to examine the dynamic load influence in a transmission/distribution network;
• Across a spectrum of research publications, the emphasis is on analyzing the influence of DPGSs on transient or voltage stability. Conversely, the impact of DPGS penetration subjected to small disturbance conditions needs to be explored.

Inspired by the aforementioned requirements, the paper’s key contributions include:

• Analyzing the transmission system stability when the DPGS supply increased power demand, comprising both constant power and dynamic loads;
• Investigating the influence of constant power and dynamic loads on a system’s small-signal stability with the aid of the critical damping ratio and electromechanical mode eigenvalue analysis;
• To study the impact of both dynamic and constant power load changes on the transient stability of the system and to further quantify the transient performance of DPGSs using time-domain indicators;
• To employ the variance analysis test to determine the relative contribution of various control factors on the variance of system response.

3. Transmission System Modelling

3.1. Model of DPGS-Interfaced Transmission System and Induction Motor Penetration Level

Figure 1 illustrates the DPGS-interfaced transmission system model employed in the present study. In this model, the DPGS is interfaced at a load bus so that the increased load demand at this bus is directly supplied by the interfaced DPGS. The central concept of this model is to locally deliver the increased load demand at a load centre by a DPGS so that the power generation of main generating stations and the line power flow in the transmission system remain almost unaffected.

In the model of Figure 1, P_Lk and Q_Lk are the base real and reactive power demand at load bus k. If ΔP_Lk is the increase in the load real power at bus k, then the real power generation of the interfaced DPGS is set as P_DG = ΔP_Lk.

In the present study, the increased power demand ΔP_Lk at load bus k is assumed to have the following two parts:

• A portion, (ΔP_CP)_k, is the increased CP demand;
• The remaining portion, (ΔP_IM)_k = ΔP_Lk − (ΔP_CP)_k is due to the increased power demand in the form of an induction motor.

The above load demand increase model will enable the study of the effect of induction motor loads on the stability of the transmission system interfaced with a DPGS.

The constant power load change and the induction motor load are expressed as a fraction of base real power load demand at bus k, and are defined as:

$${\beta }_{k,1}=\frac{{(\Delta P_{CP})}_k}{P_{Lk}}$$

(1)

$${\beta }_{k,2}=\frac{{(\Delta P_{IM})}_k}{P_{Lk}}$$

(2)

Hence, the net load power change at bus k is given as:

$$\Delta P_{Lk}={(\Delta P_{CP})}_k+{(\Delta P_{IM})}_k=({\beta }_{k,1}+{\beta }_{k,2})P_{Lk}$$

(3)

The degree of the induction motor penetration level (IMPL) is the ratio of the net induction motor load to the net real power demand of the system and is given by:

$$\%\hspace{0.17em}IMPL=\frac{{\displaystyle \sum _k{(\Delta P_{IM})}_k}}{P_{Load}+{\displaystyle \sum _k\Delta P_{Lk}}}\times 100$$

(4)

where P_Load is the net base real power demand of the transmission system.

3.2. Transmission System and Distributed Power Generation Models

The various assumptions made while modeling the entire system are as follows:

• The power system is modeled by employing all synchronous generators by fourth-order model-1.1 (d-q axis model), operated by a static exciter (IEEE type-ST1 single time-constant model, automatic voltage regulator with time-constant T_E = 0.05 s and gain K_E = 50);
• A speed-based, two-stage lag-lead-type power system stabilizer (fixed structures PSS) and a simplified model of a steam turbine for main synchronous generators [42] are used;
• A constant impedance representation of all constant power loads is used during the simulation process.

All the dynamic equations of the synchronous generator are provided in Appendix A.

3.2.1. Induction Generator Model

In the present study, a third order model of IM is employed [43]. This model assumes that the rotor dynamics are slower than stator flux transients. The IM state equations are given as:

$$\frac{d{\omega }_m}{dt}=\frac{1}{2H_m}[T_e-T_m]$$

(5)

$$\frac{d{E}_q^{\prime }}{dt}\hspace{0.17em}=\hspace{0.17em}\frac{-1}{T_0^{\prime }}[E_q^{\prime }\hspace{0.17em}+(X_s-X_s^{\prime })\hspace{0.17em}{i}_d]+S_m{\omega }_s\hspace{0.17em}{E}_d^{\prime }$$

(6)

$$\frac{d{E}_d^{\prime }}{dt}\hspace{0.17em}=\hspace{0.17em}\frac{-1}{T_0^{\prime }}[E_d^{\prime }\hspace{0.17em}-(X_s-X_s^{\prime })\hspace{0.17em}{i}_q]-S_m{\omega }_s\hspace{0.17em}{E}_q^{\prime }$$

(7)

The complex voltage behind transient reactance and the motor terminal voltage relationship is given by:

$$(E_q^{\prime }+j{E}_d^{\prime })=(v_q+j{v}_d)-(i_q+j{i}_d)(R_s+j{X}_s^{\prime })$$

(8)

In Equations (6) and (7),

$T_0^{\prime }=\frac{X_{lr}+X_m}{{\omega }_s{R}_r}$ is the transient open-circuit time constant;

$X_s^{\prime }=\frac{X_m{X}_{lr}}{X_m+X_{lr}}+X_{ls}$ is the transient reactance;

$X_s=X_{ls}+X_m$ is the rotor open-circuit reactance.

3.2.2. Distributed-Power-Generating Sources

The following synchronous-generator-interfaced DPGSs are employed in the investigation:

• Microturbine power generator source (MTGS);
• Diesel turbine power generator source (DTGS).

The microturbine is the prime mover for a synchronous generator for a MTGS and Figure 2 shows the general model. The MTGS is primarily based on the gas turbine model and was used in many past studies [44,45,46].

The MTGS prime-mover consists of a compressor, combustor, and turbine, which drives a synchronous generator; the transfer function model of the MTGS is depicted in Figure 3. The speed controller associated with the microturbine is operated in the isochronous mode. In the isochronous mode, the rate of change of the speed controller output varies proportional to the speed error input of the connected PI controller so that the generator maintains its speed even under load changes. The speed governor transfer function has gain K_W and governor time constants T_x and T_y.

The main dynamic component of the MTGS is the compressor turbine modeled by a first-order transfer function with time constant T_cd. Due to the associated compressor discharge volume, the compressor output cannot respond instantaneously to any changes in its input. The combustor time constant T_ecr is a small value related to the combustion reaction. The fuel system comprises a valve positioner and an actuator; the inertia of both governs the dynamics of the entire fuel flow mechanism. In the MTGS model of Figure 3, W_min is the minimum fuel flow, whereas K_f and K_a are the fuel system feedback and gain of valve positioner, respectively. The mechanical torque output of the turbine is expressed by a function T_mt and is given by:

$$T_{mt}=0.5(1-w_e)+1.3(W_F-0.23)$$

(9)

where $w_e=1+S_m$ is the electrical rotor speed.

A DTGS comprises a synchronous generator driven by a diesel turbine prime mover; the general block diagram of the DTGS model is shown in Figure 4 and consists of an electric control box, actuator, and IC engine.

A diesel turbine uses liquid fuel or natural gas as the primary fuel and operates on air compression and fuel. Initially, the air is blown into the engine until it is compressed and then the fuel is injected to generate the heat, which triggers the fuel inflammation. Figure 5 depicts the simplified transfer function model of the DTGS, representing the essential dynamics and is adapted from [44,47]. The main processes accounted for in this model are the (i) fuelling actuation, (ii) combustion or torque production, and (iii) crankshaft torque balance of the engine. The time delay between the fuel-flow actuation and subsequent power stroke is represented by the time constant T_Dd.

4. Problem Formulation and Solution Methodology

The influence of induction motor loads on the small-signal stability of the transmission system interfaced with a DPGS is studied using eigenvalue analysis and the critical damping ratio (ζ_cr) of oscillating mode eigenvalues. The linearized state-space model of the system at an initial operating point assuming constant inputs can be formulated as:

$$\mathbf{\Delta }\stackrel{•}{\mathit{X}}=\mathit{A}\hspace{0.17em}\mathbf{\Delta }\mathit{X}$$

(10)

In Equation (10), ΔX is the vector of state variables, and A is the state matrix. The state matrix A can be used to determine the system’s eigenvalues for specified system parameters and operating state. Considering a complex conjugate (oscillating mode) eigenvalue ${\lambda }_k={\alpha }_k\pm j{\omega }_k$, the associated damping ratio can be determined as:

$${\zeta }_k=\frac{-{\alpha }_k}{\sqrt{{\alpha }_k^2+{\omega }_k^2}}$$

(11)

The critical damping ratio is the minimum of the damping ratio associated with oscillating mode eigenvalues, given as:

ζ_cr = Minimum {ζ; ζ є oscillating modes}

(12)

The transient performance of the transmission system with a DPGS and supplying IM loads is analyzed by simulating self-clearing faults of duration t_c seconds on the mid-point of a transmission line. The following time-domain indicators are used to quantify the transient performance:

• Maximum slip deviation (MSD): the maximum slip deviation among relative slip response generators. It indicates the worst value of peak responses;
• Settling time (ST): the maximum settling time among the relative slip deviation responses of generators, measured with a 2% tolerance band.

The present study performs all the simulations using Simulink/Matlab software (version 7.1, R14) [48]. In the approach, the solution of the differential equations of all system components is modeled in Simulink. Further, a non-iterative solution technique is developed to update the non-state variables such as voltage, current, and power during the simulation process.

The updated bus admittance matrix Y_BUS is obtained by representing all constant power loads as equivalent shunt admittances, and, hence, load buses will have zero current injections. The non-iterative voltage solution employing the reduced network (containing only the machines) can be written as:

$$\mathit{V}\mathbf{=}{\mathbf{[}\mathit{T}\mathbf{-}{\mathit{Z}}_{\mathit{M}}\mathit{T}{\mathit{Y}}_{\mathit{R}}\mathbf{]}}^{\mathbf{-}\mathbf{1}}\hspace{0.17em}\mathit{E}$$

(13)

In Equation (13), T is the transformation matrix; and Y_R consists of real and imaginary part of reduced Y_BUS matrix terms. The dimensions of T and Y_R are (2n × 2n), where n is the total number of machines.

For the jth synchronous generator, $T_j=\left[\begin{array}{rr}\mathit{cos}{\delta }_j& \mathit{sin}{\delta }_j\\ -\mathit{sin}{\delta }_j& \mathit{cos}{\delta }_j\end{array}\right]$ and δ_j = 0 for induction motor. Z_M consists of a block diagonal matrix of machine impedance, where

$z_{m,j}=\hspace{0.17em}\left[\begin{array}{cc}-R_{a,j}\hfill & \hfill x_{d,j}^{\prime }\hspace{0.17em}\\ -x_{q,j}^{\prime }\hfill & \hfill -R_{a,j}\end{array}\right]$ for the jth synchronous generator;

$z_{m,j}=\hspace{0.17em}\left[\begin{array}{cc}-R_{s,j}\hfill & \hfill X_{s,j}^{\prime }\hspace{0.17em}\\ -X_{s,j}^{\prime }\hfill & \hfill -R_{s,j}\end{array}\right]$ for the jth induction motor;

E = [E_q1′ E_d1′, …… E_qn’ E_dn’]^t.

The network solution of Equation (13) is implemented using a Matlab function. The Simulink model is embedded with a Matlab function that updates the system’s bus voltage. The methodology of stability simulation in the Matlab/Simulink environment is depicted in the flow diagram of Figure 6. The Matlab commands ‘linmod’ and ‘eig’ are employed to obtain the linearized state model and eigenvalues, respectively, whereas the ‘sim’ command is used to obtain the system dynamic response.

5. Test System, Simulation Setup, and Variance Analysis

The influence of IM on system stability with an interfaced DPGS is investigated on a 15-bus test system with a DPGS interfaced at three load centres [49]. Power demand at load buses 7, 8, and 9 is (1.5 + j0.5) pu, whereas it is (0.5 + j0.3) pu at load buses 12 and 15. Hence, it has been decided to install DPGSs at heavy power demand centres at buses 7, 8, and 9 to supply increased power demand. Figure 7 depicts the test system with a DPGS interfaced at three load centres at buses 7, 8, and 9.

At the DPGS-interfaced load centres, the maximum increase in load demand is assumed to be 50% of the base power demand. Accordingly, (β_k,1 + β_k,2) ≤ 0.5 for k = 7, 8, and 9. Hence, the maximum load change of CP and IM are set as β_k,1 ≤ 0.25, and β_k,2 ≤ 0.25 so that the net load change does not exceed 50% of the base load demand. All the DPGS and IM parameters are provided in Appendix B.

Variance Analysis Test (VAT)

When several factors are influencing the system’s stability performance, it is important to determine the relative contribution of individual factors. VAT is a tool primarily employed in the Taguchi robust designs [50] and is a useful technique to assess the contribution of each factor on the variance of system response.

The first step in the VAT is to define certain discrete levels of various control factors affecting the system response. In the present investigation, there are two control factors at a load bus k (for k = 7, 8, and 9) interfaced with a DPGS, affecting the stability of the transmission system. These are β_k,1 and β_k,2, the fractions of load changes by CP and IM, respectively. Thus, the total number of control factors (F) is six in the selected transmission system. In the present investigation, five discrete levels (L) are defined for β_k,1 and β_k,2 within the specified range [0, 0.25] as summarized in

Table 1. Control factors and their levels.

Control Factors	Level-1	Level-2	Level-3	Level-4	Level-5
β7,1, β7,2 β8,1, β8,2 β9,1, β9,2	0.0	0.0625	0.125	0.1875	0.25

.

To analyze the variance of system response (R) over the entire operating range of control factors, it is necessary to obtain the response considering all possible combinations of six control factors and their five levels as per the full factorial design (FFD) matrix. Therefore, as per FFD, it requires L^F = 5⁶ = 15,625 system responses to be determined, which is cumbersome and time-intense. However, obtaining the responses as per an orthogonal array (OA) makes it possible to analyze the variance with only a few combinations of control factors, thus avoiding time-consuming simulations. In a system consisting of F number of control factors, each defined with L number of distinct levels, the selected OA must have a minimum number of entries as per Equation (14):

$$N_{\mathit{min}}=(L-1)F+1$$

(14)

In our study, L = 5 and F = 6 and, hence, N_min = 25. Accordingly, L₂₅ OA is selected with 25 entries from the standard OA design matrix available [50,51]. The overall mean response R over the entire operating region is given by:

$$m_R=\frac{1}{25}{\displaystyle \sum _{k=1}^{25}{R}_k}$$

(15)

In Equation (15), R represents the transmission system stability indicators (critical damping ratio ζ_cr for small-signal stability and time-domain indicators, MSD and ST for transient stability).

The average response R of a load-j (either CP or IM) due to level-i is the direct effect of each level of a load [51] and is given by:

$${(m)}_{i,j}=\frac{1}{rl}{\displaystyle \sum _{i=1}^{rl}{(R_i)}_j}$$

(16)

For j = β_k,1, β_k,2; k = 7, 8, and 9.

In Equation (16), rl represents the repetition number of level-i of control factor j, and it signifies the number of times each level of a control factor repeats in a column of L₂₅ OA.

The contribution of a control factor j on the total variance of the response [51] is given as:

$$C_j=\frac{{\displaystyle \sum _{i=1}^{rl}5[{(m)}_{i,j}}-m_R{]}^2}{{\displaystyle \sum _{k=1}^{25}[R_k}-m_R{]}^2}$$

(17)

For j = β_k,1, β_k,2; k = 7, 8, and 9.

VAT serves as a valuable technique for assessing the impact of individual control factors on the variance of responses around the mean response of the system, thus indicating the relative importance of each control factor. In the current study, we utilize the ‘anovan’ routine from the Matlab toolbox to conduct the VAT.

6. Simulation Results

The present investigation considers the following two DPGS interface scenarios:

• Case-I: Load buses 7, 8, and 9 are interfaced with a DTGS (All DTGS);
• Case-II: Load buses 7, 8, and 9 are interfaced with a MTGS (All MTGS).

As explained in Section 4, the influence of IM on small-signal stability of the test system with interfaced DPGS is investigated through critical damping ratio and eigenvalue analysis. On the other hand, the system’s response following a fault and its time-domain indicators are used to study the influence of IM loads on transient performance.

6.1. Influence of IM on Small-Signal Stability

6.1.1. Effect of IMPL on Critical Damping Ratio and Small-Signal Response

In this analysis, the critical damping ratio of the linearized system is determined by varying the fraction of the CP load (β_k,1) from 0.0 to 0.25 by equal amounts at all load buses with the fraction of IM load (β_k,2) held constant at a pre-specified value. The following settings at the DPGS-interfaced load buses are employed for these simulations:

β_7,1 = β_8,1 = β_9,1 = β₁ (CP load fraction)

β_7,2 = β_8,2 = β_9,2 = β₂ (IM load fraction)

Figure 8 compares the critical damping ratio variation for the two cases (all DTGS and all MTGS) as a function β₁ for different hold values of β₂.

In Case-I, when DTGS are interfaced at all load buses, the following observations are made from Figure 8a:

• For a specified IM load fraction β₂ value, ζ_cr values initially show an increasing tendency with an increase in β₁, indicating improvement in the small-signal stability. However, with a further increase of β₁, ζ_cr values rapidly decrease. This decrease in ζ_cr is more prominent when IM shares a larger portion of the net load demand increase (higher β₂ values). Hence, beyond particular loading, the local supply of increased power demand by a DPGS degrades the system’s small-signal stability.
• It is seen that, when the CP shares a larger portion of the load increase (high β₁), an increase in IM load share reduces ζ_cr values. On the other hand, when the CP shares a smaller portion of the load increase (small β₁), the small-signal stability improves with an increase in β₂ and indicates the mutual dependency between β₂ and β₁.
• Figure 8a shows that the ζ_cr peak value point shifts towards the left with increased β₂ values. This shift indicates that, when IM shares a larger portion of the increased load demand, the small-signal stability will be significantly affected at high load increase values.

The following observations are made from Figure 8b when MTGSs are interfaced at all load buses (Case-II):

• For a specified value of IM share with β₂ ≤ 0.15, an increase in β₁ reduces ζ_cr. However, when β₂ = 0.25, the resulting ζ_cr value is much smaller, although an increase in β₁ marginally improves the small-signal stability.
• For any specified value of β₁, an increase in β₂ values reduces ζ_cr. Therefore, with MTGSs, the small-signal stability will be affected whenever the IM shares a larger portion of the increased load demand.

6.1.2. Small-Signal Response

The small-signal response of the system is obtained for both Case-I and Case-II for a step increase in speed reference (w_ref) of 0.05pu at t = 0 under two different IMPL values, as summarized in

Table 2. Control factors values for small–signal response.

Case	β1	β2	%IMPL
Case-I (All DTGS)	0.25	0.05	3.28
	0.25	0.25	14.56
Case-II (All MTGS)	0.05	0.05	3.78
	0.05	0.25	16.42

.

The β₁ and β₂ values for the two simulations in Case-I and Case-II are chosen to create a large IMPL change and, hence, represent the worst-case scenario. The slip deviation responses of main synchronous generators (MSGs), and synchronous generators associated with DPGSs and IMs are illustrated in Figure 9 and Figure 10, respectively, for Case-I and Case-II.

• Case-I (All DTGS)

Figure 9 shows that all the synchronous generators exhibit growing slip deviation oscillation responses with an increase in IMPL. The peak overshoots of MSG₂ and MSG₃ are slightly higher than those of MSG₁, as seen from Figure 9a. Moreover, the MSG₃ oscillates with a much higher frequency than MSG₁ or MSG₂. The oscillating frequency of all synchronous generators interfaced with a DTGS is significantly higher than the oscillating frequencies of MSGs. In addition, the slip deviation magnitude of DTGS₁ is relatively larger than that of all other synchronous generators. As expected, an increase in IMPL results in increased oscillations of the IM with a higher peak and settling time. In summary, it can be concluded that the effect of an increase in IMPL is quite different for MSGs and synchronous generators interfaced with a DTGS.

2.

Case-II (All MTGS)

Figure 10 shows that the peak overshoot of slip deviation response of all synchronous generators and induction motors increases with an increase in IMPL. The slip deviation magnitude of MTGS₁ is much larger than MTGS₂ and MTGS₃, as seen from Figure 10b. However, MTGS₂ and MTGS₃ oscillate more than MTGS₁, settling slowly to the new steady state. Figure 10c shows that, although the slip deviation magnitude of IM₁ is higher than that of IM₂ or IM₃, it will settle quickly to a new steady state. As in Case-I, the influence of IMPL on MSGs is different from MTGSs.

6.1.3. Effect of IMPL on Electromechanical Modes of Oscillations

This analysis determines the electromechanical modes of oscillations (eigenvalues corresponding to states ΔS_m and Δδ) of MSG and synchronous generators interfaced to a DPGS for different IMPL values using the participation matrix [52].

• Case-I (All DTGS)

In this case, β₁ is kept constant at 0.25 and β₂ is varied as 0.05, 0.15, and 0.25. The electromechanical modes of the MSG and DTGS determined are summarized in

Table 3. Electromechanical eigenvalues of the system—Case-I (All DTGS).

β2	IMPL (%)	Electromechanical Modes (ζ, f in Hz)
		Main Synchronous Generator	Synchronous Generator Associated with DTGS
0.05	3.28	−0.489 ± j 5.381 (0.0904, 0.8564)	−1.501 ± j 16.458 (0.0908, 2.6194)
		−0.928 ± j 8.934 (0.1033, 1.4219)	−1.623 ± j 18.541 (0.0872, 2.9508)
		−1.219 ± j 13.051 (0.0930, 2.0772)	−1.685 ± j 19.274 (0.0871, 3.0675)
0.15	9.25	−0.499 ± j 5.374 (0.0924, 0.8553)	−1.548 ± j 16.633 (0.0926, 2.6472)
		−0.964 ± j 8.942 (0.1072, 1.4232)	−1.626 ± j 18.875 (0.0858, 3.0040)
		−1.223 ± j 13.099 (0.0930, 2.0847)	−1.687 ± j 19.645 (0.0856, 3.1265)
0.25	14.52	−0.510 ± j 5.367 (0.0946, 0.8542)	−1.597 ± j 16.838 (0.0944, 2.6798)
		−1.002 ± j 8.952 (0.1112, 1.4247)	−1.618 ± j 19.253 (0.0838, 3.0642)
		−1.222 ± j 13.144 (0.0926, 2.0919)	−1.677 ± j 20.076 (0.0832, 3.1951)

for three different IMPL values. The damping ratio and oscillating frequency (in Hz) related to these modes are shown within the brackets.

It is seen from Table 3 that, with an increase in IMPL, the damping of MSG₁ and MSG₂ increases while it decreases for MSG₃. Similarly, the damping ratio of oscillating modes associated with DTGS₂ and DTGS₃ shows a decreasing tendency, while that of DTGS₁ exhibits an increasing tendency. Further, it is observed that the mode-oscillating frequencies of the All DTGS case (in the range of 2.6–3.2 Hz) are much higher than the mode oscillating frequencies of the All MSG case (in the range of 0.85–2.1 Hz). Notably, the critical electromechanical mode is mode-3, associated with DTGS₃ with the lowest damping and highest oscillating frequency.

2.

Case-II (All MTGS)

In Case-II, β₁ is kept constant at 0.05, and β₂ is varied as 0.05, 0.15, and, 0.25. The electromechanical modes of the MSG and MTGS obtained are shown in

Table 4. Electromechanical eigenvalues of the system—Case-II (All MTGS).

β2	IMPL (%)	Electromechanical Modes (ζ, f in Hz)
		Main Synchronous Generator	Synchronous Generator Associated with MTGS
0.05	3.78	−0.435 ± j 5.257 (0.0824, 0.8366)	−1.086 ± j 12.231 (0.0885, 1.9466)
		−1.289 ± j 11.714 (0.1094, 1.8643)	−1.289 ± j 11.714 (0.1094, 1.8643)
		−1.049 ± j 15.179 (0.0690, 2.4160)	−1.216 ± j 13.369 (0.0906, 2.1277)
0.15	10.55	−0.444 ± j 5.248 (0.0842, 0.8352)	−1.138 ± j 12.303 (0.0921, 1.9581)
		−1.325 ± j 11.774 (0.1118, 1.8739)	−1.325 ± j 11.774 (0.1118, 1.8739)
		−1.032 ± j 15.2039 (0.0677, 2.4197)	−1.243 ± j 13.471 (0.0919, 2.1439)
0.25	16.42	−0.454 ± j 5.238 (0.0862, 0.8337)	−1.205 ± j 12.41 (0.0966, 1.9766)
		−1.344 ± j 11.854 (0.1126,1.8867)	−1.344 ± j 11.854 (0.1126, 1.8867)
		−1.025 ± j 15.258 (0.0670, 2.4284)	−1.274 ± j 13.632 (0.0930, 2.1696)

for three different IMPL values. The damping ratio and oscillating frequency (in Hz) associated with these modes are also shown.

As seen from Table 4, the damping of the electromechanical modes of MSG₁ and MSG₂ increases, while it decreases for MSG₃ for any increase in IMPL. On the other hand, the damping of electromechanical modes associated with the All MTGS case shows an increasing tendency with IMPL increase. The oscillating frequency of mode-1 of the MSG is much lower (approximately 0.83 Hz) than that of other modes of main synchronous generators. In contrast, the electromechanical modes of the All MTGS case lie in the range of 1.86–2.16 Hz. In this case, the critical mode is mode-3, associated with MSG₃ with the lowest damping ratio (around 0.067–0.069) and highest oscillation frequency (around 2.41 Hz).

6.1.4. Variance Analysis Test of Small-Signal Stability

In this analysis, the VAT is performed on the critical damping ratio (ζ_cr) measured as per L₂₅ OA using five levels of control factors (Table 1).

Table 5. L₂₅ Orthogonal Aaray and measured values of critical damping ratio.

Sl. No	Control Factors						%IMPL	ζcr
	β7,1	β7,2	β8,1	β8,2	β9,1	β9,2		Case-I (All DTGS)	Case-II (All MTGS)
1	0	0	0	0	0	0	0	0.0833	0.0705
2	0	0.0625	0.0625	0.0625	0.0625	0.0625	4.7120	0.0850	0.0686
3	0	0.125	0.125	0.125	0.125	0.125	8.7379	0.0869	0.0676
4	0	0.1875	0.1875	0.1875	0.1875	0.1875	12.2172	0.0863	0.0674
5	0	0.25	0.25	0.25	0.25	0.25	15.2542	0.0843	0.0681
6	0.0625	0	0.0625	0.125	0.1875	0.25	8.6124	0.0870	0.0681
7	0.0625	0.0625	0.125	0.1875	0.25	0	5.7416	0.0868	0.0674
8	0.0625	0.125	0.1875	0.25	0	0.0625	10.0478	0.0868	0.0673
9	0.0625	0.1875	0.25	0	0.0625	0.125	7.1770	0.0878	0.0680
10	0.0625	0.25	0	0.0625	0.125	0.1875	11.4833	0.0869	0.0696
11	0.125	0	0.125	0.25	0.0625	0.1875	9.9057	0.0863	0.0671
12	0.125	0.0625	0.1875	0	0.125	0.25	7.0755	0.0872	0.0685
13	0.125	0.125	0.25	0.0625	0.1875	0	4.2453	0.0869	0.0678
14	0.125	0.1875	0	0.125	0.25	0.0625	8.4906	0.0865	0.0687
15	0.125	0.25	0.0625	0.1875	0	0.125	12.7358	0.0881	0.0673
16	0.1875	0	0.1875	0.0625	0.25	0.125	4.1860	0.0869	0.0680
17	0.1875	0.0625	0.25	0.125	0	0.1875	8.3721	0.0862	0.0675
18	0.1875	0.125	0	0.1875	0.0625	0.25	12.5581	0.0873	0.0678
19	0.1875	0.1875	0.0625	0.25	0.125	0	9.7674	0.0881	0.0671
20	0.1875	0.25	0.125	0	0.1875	0.0625	6.9767	0.0857	0.0692
21	0.25	0	0.25	0.1875	0.125	0.0625	5.5046	0.0854	0.0677
22	0.25	0.0625	0	0.25	0.1875	0.125	9.6330	0.0871	0.0674
23	0.25	0.125	0.0625	0	0.25	0.1875	6.8807	0.0840	0.0701
24	0.25	0.1875	0.125	0.0625	0	0.25	11.0092	0.0864	0.0682
25	0.25	0.25	0.1875	0.125	0.0625	0	8.2569	0.0873	0.0674

summarizes the L₂₅ OA and the ζ_cr measured for each combination of control factors for Case-I (All DTGS) and Case-II (All MTGS).

The balancing property of the columns of L₂₅ OA is quite evident from Table 5; hence the columns of OA are mutually orthogonal. In a mutually orthogonal pair of columns, all combinations of control factor levels appear an equal number of times. Hence, OA represents an experimental region of factors under study.

The VAT provides the contribution of the CP fraction (β_k,1) and IM fraction (β_k,2) of increased load demand at load bus–k (k = 7, 8, and 9) interfaced with a DPGS causing variations of mean critical damping ratio values around the overall mean. The VAT results for the two cases are illustrated in

Table 6. Variance test results of critical damping ratio.

Load Bus No	Load Fraction	% Contribution
		Case-I (All DTGS)	Case-II (All MTGS)
7	β7,1	37.328	5.076
	β7,2	10.609	4.451
8	β8,1	5.158	19.017
	β8,2	13.115	57.777
9	β9,1	13.852	6.005
	β9,2	19.939	7.674

and the corresponding mean response plot is depicted in Figure 11.

• Case-I (All DTGS)

From the VAT results shown in Table 6, it is seen that the CP load fraction at load bus 7 (β_7,1) and IM load fraction at load bus 9 (β_9,2) have significant contributions (37.328% and 19.939%, respectively), while the rest of the factors have low to medium contributions. This is also evident from the mean response plot of Figure 11a. In Figure 11a, the overall mean (m_R) value of ζ_cr is 0.0864 and is shown in a dotted line. It can be seen that the average response of β_7,1 undergoes a large variation around the overall mean value when the levels of β_7,1 is varied from level 1 to level 5. A similar tendency is also seen from the control factor β_9,2. Therefore, it can be concluded that the CP fraction of load at bus 7 and IM fraction of load at bus 9 are significant factors affecting the variations of the critical damping ratio and, hence, the small–signal stability of the transmission system.

2.

Case-II (All MTGS)

It is interesting to note from Table 6 that, in the case of All MTGS, load bus 8 is critical to the small-signal stability of the system. At load bus 8, the IM load fraction (β_8,2) contributes around 57.7%, while the CP load fraction (β_8,2) contributes moderately, around 19% on the variance of ζ_cr around the overall mean (0.0686). The CP load and IM load fractions at load buses 7 and 9 are negligible. These results can also be observed from the mean response plot of Figure 11b. Figure 11b shows that the variations of average ζ_cr around the overall mean due to level variations of both β_7,1 and β_7,2 are minimal. A similar observation can also be made for β_9,1 and β_9,2.

6.2. Influence of IMPL on Transient Stability

In this investigation, the influence of IMPL on the transient stability of the transmission system is analyzed by simulating the system’s response for a self-cleared fault of duration t_c seconds at different IMPL values. Time-domain indicators (MSD and ST) were measured and employed to quantify the system’s transient performance to perform the variance analysis test.

6.2.1. Case-I (All DTGS)

A three-phase self-clearing fault of 200 ms is simulated at the middle of line 9–15, and the transient response of MSG, DTGS, and IM is obtained for three IMPL values of 3.78%, 9.85%, and 14.51%. Figure 12 illustrates the comparison of transient responses for increasing values of IMPL.

Figure 12a shows that the peak overshoot of MSG₁ and MSG₂ increases with an increase in IMPL while it decreases for MSG₃. Moreover, the slip deviation response of MSG₁ is more oscillatory than that of MSG₂ or MSG₃. However, the slip deviation magnitude of MSG₁ is much smaller than other main synchronous generators. It can be observed from Figure 12b that the effect of IMPL on the transient response of DTGS₁ and DTGS₃ is more prominent as compared to DTGS₂ although the slip deviation magnitude of DTGS₂ is much larger than DTGS₁ and DTGS₃. In general, it can be concluded that a more oscillatory transient response results in the case of All DTGS. The peak overshoots of all induction motors increase with an increase in IMPL, as observed in Figure 12c. Since IM₃ is located nearer to the fault location, it can be seen that the slip deviation magnitude and peak overshoot of IM₃ are much larger than those of other induction motors.

To quantify the effect of IMPL on the transient response of the transmission system, time–domain indicators of MSG and DTGS are measured with 100 ms and 200 ms fault durations (t_c) as per L₂₅ OA, as illustrated in Figure 13.

It can be observed from Figure 13 that, for a specified IMPL value, both MSD and ST increase with an increase in fault duration. Moreover, the ST values of DTGSs are much smaller than that of MSGs for all clearing times and IMPL values. VAT is carried out using the above time-domain indicator values to determine the relative importance of various control factors.

Table 7. Variance test results of MSG (Case-I).

Control Factor	%Contribution
	tc = 100 ms		tc = 200 ms
	MSD	ST	MSD	ST
β7,1	4.587	20.403	5.752	8.269
β7,2	9.232	15.977	9.178	18.742
β8,1	1.773	14.484	1.764	18.056
β8,2	4.166	14.728	3.786	18.307
β9,1	32.733	18.917	24.343	18.238
β9,2	47.509	15.491	55.179	18.387

and

Table 8. Variance test results of DTGS (Case-I).

Control Factor	%Contribution
	tc = 100 ms		tc = 200 ms
	MSD	ST	MSD	ST
β7,1	6.865	3.491	12.626	18.186
β7,2	2.733	7.613	2.990	6.351
β8,1	6.508	14.139	8.004	21.841
β8,2	3.868	10.607	5.519	32.297
β9,1	39.116	38.252	37.049	3.943
β9,2	40.911	25.899	33.811	17.383

summarize the VAT results of MSGs and DTGSs, respectively. Here, for comparison purposes, VAT is performed on both time-domain indicators, MSD and ST, for 100 ms and 200 ms fault durations.

The following observations are made from the VAT results of Table 7 and Table 8:

• The IM fraction at load bus 9 (β_9,2) is a significant control factor on MSD of MSG for both fault clearing duration times (Table 7). β_9,2 contributes around 47% and 55%, respectively, for t_c of 100 ms and 200 ms, respectively. On the other hand, the CP fraction at bus 9 (β_9,2) contributes moderately (32.7% and 24.3%). Thus, transient stability is affected whenever the IM₃ shares a larger load demand increase at bus 9, and it may be because both CP and IM loads at bus 9 are nearer to the fault location. It is worth noting here that the % contribution of β_9,2 on MSD of main synchronous generators increases from 47.5% to 55.17% as fault duration increases;
• As seen from Table 7, the contributions of all control factors (β_k,1 and β_k,2 for k = 7, 8, and 9) on the settling time of MSGs is more or less the same for both 100 ms and 200 ms fault durations;
• Table 8 shows that, when t_c = 100 ms, both CP and IM fractions at load bus 9 (β_9,1 and β_9,2) contribute significantly to MSD and ST variations of the DTGS. On the other hand, when t_c = 200 ms, β_9,1 and β_9,2 contribute significantly only on MSD of the DTGS. For a 200 ms fault duration, the contributions of β_7,2 and β_9,1 are much lower than other control factors.

The VAT gives the relative significance of each control factor on the variance of a response (MSD or ST) around the overall mean response.

6.2.2. Case-II (All MTGS)

Figure 14 illustrates the comparison of transient response values of IMPL for a three-phase 200 ms self-clearing fault at the mid-point of line 9–15. As in Case-I (All DTGS), increased IMPL increases the oscillation peaks of all synchronous generators except the MSG₃. Moreover, as expected, all induction motor transient responses become more oscillatory with increased IMPL. The slip deviation magnitude of MSG₁ and MTGS₁ is less than that of other synchronous generators. As the IM₃ location is nearer to the fault point, IM₃ exhibits higher peak overshoots, particularly at high IMPL values.

Figure 15 depicts the time-domain indicators of MSGs and MTGSs, measured for fault durations (t_c) of 100 ms and 200 ms as per L₂₅ OA. It is clear from Figure 15a,b that the MSD of MTGSs is much smaller than MSGs. Moreover, for a specified IMPL value, MSD and ST of MSGs and MTGSs increase with an increase in clearing time.

The VAT results performed on time-domain indicators are summarized in

Table 9. Variance test results of MSG (Case-II).

Control Factor	%Contribution
	100 ms		200 ms
	MSD	ST	MSD	ST
β7,1	1.579	11.530	1.537	35.161
β7,2	6.316	17.066	12.701	13.077
β8,1	1.053	16.409	1.720	12.768
β8,2	8.947	14.432	9.590	11.513
β9,1	23.684	23.738	12.701	14.316
β9,2	58.421	16.824	61.749	13.165

and

Table 10. Variance test results of MTGS (Case-II).

Control Factor	%Contribution
	100 ms		200 ms
	MSD	ST	MSD	ST
β7,1	2.786	5.221	14.051	25.454
β7,2	2.179	17.681	9.682	12.601
β8,1	1.875	40.922	3.942	30.033
β8,2	2.217	4.745	5.719	1.516
β9,1	48.034	15.167	52.352	13.297
β9,2	42.909	16.263	14.254	17.099

for MSG and MTGS, respectively. The following observations are made from VAT results:

• The IM fraction at load bus 9 (β_9,2) contributes significantly to MSDs of the MSG at both fault duration times (58.4% and 61.7%, respectively). Thus, it can be said that the peak overshoot of the MSG is mainly governed by the amount of load shared by the IM at bus 9 since, at this bus, the contributions from the CP fraction (β_9,1) are 23.6% and 12.75, or 100 ms and 200 ms fault duration, respectively.
• When the fault duration is 100 ms, all control factors contribute to the variance of the settling time of the MSG, although β_9,1 contributes slightly higher (23.7%) than other factors. However, at 200 ms, the CP fraction (β_7,1) is the dominant factor with 35% contribution.
• The MSD of the MTGS is mainly controlled by the CP and IM fractions (β_9,1 and β_9,2) at load bus 9. As seen from Table 10, the contribution from the IM fraction at bus 9 (β_9,2) on MSD of the MTGS is 42.9% and 14.2%, respectively, for 100 ms and 200 ms, while the CP fraction (β_9,1) contributes 48% and 52.35%.
• Table 10 shows that, for the 100 ms fault, the CP fraction at bus 8 (β_8,1) alone contributes around 40.9% towards ST variations of the MTGS and, hence, is a dominant factor. On the other hand, for the 200 ms fault, β_8,1, β_7,1, and β_9,2 contribute moderately.

7. Discussion

It is seen from the simulation results that the DPGS interfacing reduces the system damping considerably, particularly at high IMPL. Even though the high gain and fast-acting excitation system of synchronous generators improves the transient stability, it will affect the small-signal stability by reducing the damping torque. One way to enhance the damping of low-frequency oscillations is to add optimally tuned PSS in the excitation control of synchronous generators interfaced with DTGSs and MTGSs. The authors in [22] reported that small-signal stability would be poorer when the IM is located nearer a DPGS, resulting in voltage oscillatory modes. The VAT results in our study also indicated significant contributions of IM loads on variations of the critical damping ratio. The low damping ratio of the system at higher IMPL might be due to IMs of low inertia (H_m = 0.5), as it was reported in [53] that low-inertia systems are more susceptible to small-signal instability.

The fault was simulated at the mid-point of line 9–15 in the transmission system. Thus, the fault location is nearer to MSG₂, and DPGS₃ and IM₃ are connected to bus 9. Figure 16 shows the transient response during the fault duration of 200 ms in Case-II (All MTGS).

It is seen from Figure 16a that, soon after the fault application at t = 0, both MSG₂ and MTGS₃ accelerate, increasing the stator frequency of IM₃. Since the terminal voltage of IM₃ (bus 9) decreases significantly during the fault period, as observed from Figure 16c, the real power drawn by IM₃ and the real power generated by the synchronous generator interfaced to MTGS₃ decrease accordingly. This is shown in Figure 16d. This results in increased synchronizing power and the action helps to slow down the accelerating MTGS₃-interfaced synchronous generator. It can be observed from Figure 16a that, at around 0.1 s, the slip deviation of MTGS₃ starts reducing. Hence, if the location of the IM is nearer to the synchronous generators, it helps slow down the accelerating synchronous generators due to the neutralizing effect. This observation agrees with the findings by Davood Khani et al. [49]. However, at higher IMPL values, the induction motors affect the performance of other synchronous generators as the IM draws heavy reactive power from the system, causing voltage problems [21]. Therefore, it is necessary to employ reactive power compensation at IM terminals. It is worth noting here that, in [21,22], the DPGS does not directly supply the IM loads, whereas, in our study, the influence of the IM on stability is investigated when the DPGS directly supplies power to both the IM and CP loads.

The development of state matrix A by an analytical approach is quite complex. In addition, separate simulation programs to solve all state equations must be developed to obtain the system’s time responses. On the other hand, the key feature of the proposed approach is that a single Simulink model developed gives both the state matrix and time-domain simulation results.

The system’s transient response is a function of several factors, such as fault location, fault clearing time, system parameters, and the DPGS technology. Employing OA to study the factor effects is advantageous when there are many affecting factors. OA permits us to perform the stability analysis with a few selected combinations of factor levels, thus reducing the computational time. Therefore, the balancing property of OA helps arrive at valid conclusions over the entire experimental region defined by control factors and their level settings. Moreover, with many control factors affecting stability, it is challenging to attribute the dominant factors that affect the system’s response and stability. In such cases, VAT becomes very useful as it provides the relative importance of various factors. Even though VAT results on time-domain indicators could not establish a general tendency, they highlighted the influence of IMPL on transient stability.

8. Conclusions

This paper investigated the stability of the transmission system comprising distributed-power-generating sources and induction motor loads. The main objective was to analyze the stability when the DPGS supply increased power demand comprising both constant-power- and induction-motor-type loads. The following are the observations made from the present investigation:

• The influence of IMPL on both small-signal and transient stability of the transmission system is also governed by the DPGS technology employed to locally supply the increased load demand at a load bus;
• With a DTGS in the system, the critical damping ratio reduces when both the CP and IM share large portions of increased load demand. The critical damping ratio sharply reduces with increased load demand when the IM shares a larger fraction of the increased load demand. On the other hand, in the case of MTGSs, the critical damping ratio shows a decreasing tendency with increasing load demand when the IM shares a larger portion of increased load demand;
• Eigenvalue analysis revealed that, with DTGS, the critical electromechanical mode is associated with a synchronous generator interfaced with a DTGS with the lowest damping and high oscillating frequency. On the other hand, with the MTGS, the electromechanical mode of the main synchronous generator is critical, affecting the small-signal stability. Variance test results indicated that the CP fraction is the dominant contributor causing the variance of the critical damping ratio when the DTGS is set to supply the increased load demand, whereas it was the IM fraction in the case of the MTGS;
• The system’s transient response following a fault is affected by the IMPL and fault duration. Although the IM located near a synchronous generator helps to slow down the accelerating synchronous generator during the fault duration, its action will adversely affect the performance of remote generators.
• A variance test was employed to identify the relative contribution of various control factors on the variance of time-domain indicators. The VAT results revealed the importance of the IM fraction on both maximum slip deviation and settling time of transient response.

Further research is necessary to investigate the effects of IM parameters on stability in a large power grid. In addition, the influence of IM loads, when supplied by different DPGS technologies such as PV, wind, and fuel cells need to be explored. The effect of adding PSS with DTGS and MTGS excitation controllers on damping needs further research. The conclusions are drawn from a small system which represents a realistic power grid and, hence, can be generalized to a large interconnected system. Moreover, the methodology presented is general and applies to all systems.