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Article

Analyzing Small-Signal Stability in a Multi-Source Single-Area Power System with a Load-Frequency Controller Coordinated with a Photovoltaic System

by
Ghazanfar Shahgholian
1 and
Arman Fathollahi
2,*
1
Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad 8543131, Iran
2
Department of Electrical and Computer Engineering, Aarhus University, 8200 Aarhus, Denmark
*
Author to whom correspondence should be addressed.
AppliedMath 2024, 4(2), 452-467; https://doi.org/10.3390/appliedmath4020024
Submission received: 8 February 2024 / Revised: 18 March 2024 / Accepted: 27 March 2024 / Published: 3 April 2024
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Abstract

:
The frequency deviation from the nominal working frequency in power systems is a consequence of the imbalance between total electrical loads and the aggregate power supplied by production units. The sensitivity of energy system frequency to both minor and major load variations underscore the need for effective frequency load control mechanisms. In this paper, frequency load control in single-area power system with multi-source energy is analysed and simulated. Also, the effect of the photovoltaic system on the frequency deviation changes in the energy system is shown. In the single area energy system, the dynamics of thermal turbine with reheat, thermal turbine without reheat and hydro turbine are considered. The simulation results using Simulink/Matlab and model analysis using eigenvalue analysis show the dynamic behaviour of the power system in response to changes in the load.

1. Introduction

An essential aspect of fortifying passive defence mechanisms lies in ensuring the accessibility of energy during critical scenarios. The conventional approaches to power generation are poised to undergo a transformative shift, making room for environmentally sustainable alternatives with the integration of renewable energy sources into the power grid. The escalating demand for energy, propelled by societal growth and technological advancement, is compounded by a growing inclination towards harnessing renewable energy sources. This trend is driven by their recognized efficiency and cost-effectiveness, ushering in a transformative shift in the energy landscape. The expansion of power systems to accommodate the integration of renewable sources and the establishment of interconnections between diverse regions reflects this changing paradigm. However, this evolution introduces fresh challenges, particularly in ensuring the stability, reliability, and uninterrupted functionality of these increasingly complex systems. In tandem with this global shift towards sustainable energy, the power system is encountering new challenges and opportunities. The traditional paradigm of centralized power generation is giving way to a decentralized model, characterized by the widespread adoption of distributed generations and diverse industrial loads [1,2].
This shift not only transforms the physical infrastructure of the power grid, but also necessitates a revaluation of control strategies and grid management techniques. Renewable energy sources, like solar and wind, are intermittent, which adds a degree of unpredictability that conventional power systems were not initially built to handle [3,4]. An electric power system consists of a complex set of electromechanical devices, the main aim of which is to convert energy and transfer it from the place of production, with minimum cost and with minimum environmental impact, to the points where the consumers are located [5,6]. The power system is required to consistently fulfil the ongoing demand for both active and reactive power [7,8]. Additionally, it is imperative for the power supply to maintain frequency stability, voltage stability, and a high reliability. In contemporary times, the energy grid faces heightened complexity, attributable to the integration of distributed generators and diverse industrial loads [9,10]. The escalating integration of renewable energy generation units into electric grids poses a significant challenge for frequency stabilization, primarily attributed to the distinctive technical characteristics associated with their implementation [11,12].
Load frequency control constitutes a pivotal control loop within the power system, designed to swiftly mitigate frequency deviations to zero following a disturbance event. Also, the load should be distributed among the generators, based on their capacity. Maintaining the power exchange between the connection lines between the areas within the programmed limits is another task of this control loop [13,14]. The use of different distributed generator sources, such as wind turbine generators [15,16], solar panels [17,18], diesel engine generators [19,20], and energy storage systems [21,22], has significant advantages in improving the quality of the energy network. Nevertheless, the growing adoption of distributed generation sources presents challenges, including the intricacy of the frequency control mechanism, increased voltage instability, and heightened system sensitivity to faults within the energy network.
To date, numerous studies have been put forth concerning frequency control in energy networks, with a subset of these investigations explicitly addressing the implications arising from the incorporation of renewable power resources [23,24]. A superconducting magnetic energy storage system integration is one suggested method to improve the dynamic performance of automatic generation management after sudden load disruptions, as described in [25]. This method is primarily used in a multi-source power generation system with two zones that are connected. The quadratic performance index is adjusted using the integral square error technique in order to ascertain the integral gain of the controller.
Research into load frequency regulation within energy grids, encompassing a diverse range of energy sources such as thermal, hydro, and gas power facilities, is documented in [26]. The study delves into the determination of controller parameters, aiming to achieve optimal transient performance for the system. This is accomplished through the application of the meta-heuristic harmony search algorithm, which facilitates the exploration and optimization of the controller’s settings.
The examination of load frequency regulation within a hybrid energy production framework is the focal point of scrutiny in [27]. This investigation employs first-order PID (proportional-integral-derivative) regulators to regulate and stabilize the system. The hybrid configuration of the power generation system incorporates a combination of thermal, wind, and solar energy sources. Through the utilization of these controllers, this work aims to optimize the performance and balance the load-frequency dynamics in the context of the diverse and interconnected energy generation components.
In [28], a proposal is put forth for determining the optimal settings of the PI regulator specifically tailored for load frequency regulation within a hybrid power system. This hybrid configuration encompasses both a thermal generator and a photovoltaic system. The procedure of determining the tuned controller coefficients is based on solving a time-dependent objective function, which is made easier by employing the firefly algorithm.
Research into the application of the artificial bee colony optimization framework for virtual inertia regulation within wind energy conversion systems interfaced with the grid is documented in [29]. This study delves into the intricacies of the control strategy, illustrating its efficacy through a load frequency investigation conducted across various distinct real operation situations.
In [30], a comprehensive control methodology is introduced to regulate and enhance the performance of three pivotal models within power networks. A thermal power plant’s load frequency management system, an integrated photovoltaic thermal power plant’s frequency variation, and an automatic voltage regulation system’s output variations are all included in these models. The study explores diverse objective functions, aligned with the specific demands of the energy industry, to demonstrate the advancements in system performance parameters, including, but not limited to, maximum overshoot, steady state error, and settling time.
The analysis of small signal stability within an integrated power system is detailed in [31], employing a mathematical model that accounts for the influence of both wind and photovoltaic energy on the electrical grid. Simulation results reveal that in such an integrated energy grid incorporating wind and photovoltaic energy, the system’s frequency can swiftly return to the nominal frequency, simultaneously bolstering the strength of each individual unit. This underscores the efficiency and effectiveness of the integrated system in responding to and recovering from perturbations.
In [32], an adaptive controller featuring auto-tuning parameters is described to facilitate the general performance of the power system in response to load variations. This controller exhibits a capability to effectively respond to disturbances arising from the operation of renewable production units, including, but not limited to, wave power conversion systems and solar energy systems. The adaptive nature of the controller, coupled with its auto-tuning functionality, underscores its potential to dynamically optimize system behaviour amid changing load conditions and the inherent variability associated with renewable energy integration.
Synchronous machines serve as prevalent power generators in traditional power systems. However, the integration of renewable energies, facilitated by power converters lacking inertia, notably diminishes the overall inertia of the network. This study explores the utilization of a solar photovoltaic system to augment the inertial capability of the power system. Focusing on a single-area system with diverse energy production sources, the power system equations are formulated in state space. Through the application of system mode analysis and the development of an energy network model in the Simulink/MATLAB environment, the dynamic behaviour of the energy network is simulated to assess its response to load changes.
Given the outlined problem statement and the significance of the subject matter, the paper is structured as follows: Section 2 provides an overview of the frequency load control system, elucidating the control loops within the power system; Section 3 briefly addresses the Photovoltaic system and its applications; moving on to Section 4, the paper presents the energy network model based on first-order transfer functions; the simulation of the single-area power system’s performance under step changes in consumption load is conducted in Section 5 using MATLAB/Simulink software, and this section also includes a comparative analysis of the impact of various power sources on the frequency deviation of the system; and Section 6 concludes the paper by summarizing key findings and insights.

2. Load Frequency Control

When it comes to power system dynamics, conventional generators with inertial response are essential because they automatically modify rotational speed in response to variations in the system frequency. This inherent feature makes use of the kinetic energy held in rotating masses to dynamically match patterns of demand with active power supply, guaranteeing grid stability. But when renewable energy generators—which do not have these rotating masses—are integrated into the landscape, the intrinsic inertia of contemporary power systems is challenged. Transient disturbances within a power system induce instability, resulting in an imbalance between load and generation. These instabilities can significantly impact the normal operation of the power system [33,34]. Notably, the flow of active power and reactive power in the transmission network is nearly independent of each other. Active power control hinges on frequency control, while reactive power control is contingent upon voltage control. In the realm of modern energy management systems, automatic generation control emerges as a crucial function. However, various load frequency control methods grapple with challenges stemming from uncertainties, parameter variations, diverse system characteristics, and varying operating point load scenarios across a wide spectrum [35,36].
Load frequency regulation holds a pivotal role in the design and operation of modern electrical systems, ensuring high reliability [37,38]. The hierarchical execution of load-frequency regulation is determined according to the magnitude of frequency deviation [39,40]. Maintaining constant frequency and voltage is crucial for the proper functioning of the power system. The performance and reliability of the power system are intricately linked to the extent of frequency deviation [41]. Figure 1 illustrates how frequency controls are categorized according to the length and magnitude of frequency deviations. The four sections that make up frequency control levels are as [42,43]: primary control (droop control), secondary control (acting based on measurement of frequency change and power flow between areas), tertiary control (the system operator’s use of various protection methods to reduce the magnitude of the frequency if the secondary control loop is not capable), and emergency control (continuous control to avoid blackouts, massive outages, or cascading failures in case of large disturbances) [44,45].
The shift from traditional synchronous generators to generators controlled by electronics for the integration of renewable energy sources results in a notable decrease in system inertia, which may present challenges to the stability and dependability of the system over an extended period of time. In addition to stressing the idea of Fast Frequency response (FFR) and illuminating the changing dynamics and difficulties related to frequency control in the face of evolving power production technologies, Figure 2 graphically depicts the dynamic reaction of system frequency across various time scales. As discussed, the power system uses Primary Reserves (PRs), Secondary Reserves (SRs), and Tertiary Reserves (TRs) in a tiered manner to preserve stability in the face of disruptions. When a disturbance occurs, PRs react quickly to stabilize the frequency within a ±20 MHz range. To maintain system stability, primary control must be applied before the frequency deviation rises above ±200 MHz. SRs activate within a response window of 30 to 15 minutes’ post-disturbance once PRs have stabilized the frequency. Their job is to replenish previously used PRs in order to maintain continuous stability, while restoring and maintaining the frequency at its nominal value. On the other hand, TRs use a manual method to neutralize and absorb the impacts of activated SRs by using tertiary control over a period of minutes to hours. Figure 3 shows a single-area energy grid with four frequency control loops along with an inertial control loop. Leveraging virtual inertia control technology boosts fortification of the total inertia of the power system and augments its frequency performance [46].

3. Photovoltaic System

Concerns regarding environmental issues have arisen as a result of rising energy use. Solar energy is one of the major renewable energy sources and one of the sustainable solutions to this issue. Photovoltaic solar energy, which transforms solar heat into electrical energy, and solar thermal energy, which converts solar heat into thermal energy, are the two uses of solar energy. The ever-increasing need for electrical energy has spurred the rapid development of renewable energy sources, which has accelerated the photovoltaic production. A photovoltaic system is one of the practical tools for converting solar energy into electricity [47,48]. This system represents one category within the spectrum of solar energy facilities, further classified into two types: photovoltaic units connected to the grid and those disconnected from the grid [49,50]. The general components of a photovoltaic system are: solar panels, storage batteries, inverter, control device, metal structure, and communication cables. The two main types of exploiting solar thermal energy are heat collection and electricity generation [51,52].
The transfer function of the composite circuit representing the photovoltaic system, composed of the photovoltaic plane, highest power point monitoring, filter, and inverter, is analysed with the following relationship [53,54]:
G P V ( s ) = K P s + p 1 s + z 1 s + p 2 .
in which KP is the gain of the photovoltaic system, p1 are p2 are poles, and z1 is zero in the transfer function. Poles and zero are considered real and negative.

4. Small-Signal Model

To guarantee the seamless and efficient operation of the power system, the development of highly effective controllers is imperative. The optimization of these controllers relies heavily on an intricate process of modelling. The complexity of power systems, characterized by their nonlinear nature and dynamic parameters dependent on operating points, demands a meticulous approach to modelling for the design of optimal controllers. Power systems are inherently intricate, with their parameters varying as a function of the system’s operating point. Moreover, the load within a power system undergoes continual fluctuations rather than remaining constant.
Appliedmath 04 00024 g003
Figure 3. Control frequency in single-area energy system.
Figure 3. Control frequency in single-area energy system.
Appliedmath 04 00024 g003
Consequently, the significance of modelling within the realm of power systems cannot be overstated. It serves as a crucial tool for comprehending system behaviour and crafting controllers capable of adapting to the ever-changing dynamics. In the unfortunate event of a power outage in the national grid, it becomes paramount to ensure the uninterrupted functionality of subsystems. The challenges arising from the frequency load control system predominantly revolve around addressing minor disturbances in demand load. The analysis of power system stability, specifically concerning minor disturbances, constitutes the focal point of small signal stability analysis. This intricate examination takes place during the operation of the power system, aiming to safeguard its stability in the face of subtle disruptions.
Small signal stability analysis hinges on a multitude of factors, including the initial operating conditions of the system, the efficacy of electrical connections interlinking various components, and the unique characteristics of control devices. By scrutinizing these elements, experts gain insights into the system’s robustness and its ability to weather minor disturbances, ensuring a resilient and dependable power infrastructure. The single-area LFC is based on the block representation of the transfer function for the multi-source energy network shown in Figure 4. In this linearized structure designed for load frequency control, for the sake of simplification, a first-order delay function is employed to characterize the transfer functions of both the steam power facility and the hydro energy facility [55,56].

4.1. Participation Coefficients of Generation Units

The rate of participation in load frequency control may not be the same for all manufacturers. The contribution factors of thermal units without re-heater, thermal with re-heater, and water are KWR, KRH, and KHY, respectively. The contribution of the units in the nominal load is determined by these factors. Within a particular control area, the aggregate of membership factors is equivalent to 1 [57,58].

4.2. Rotating Mass Model

The rotor and load inertia equivalent circuit are considered with the first order differential equation [59,60]:
Δ F Δ P M − Δ P L = 1 J M s + K D .
where KD is the load damping constant and JM is combined inertial constant. System frequency changes, mechanical power changes, and load changes are indicated by ΔF, ΔPM, and ΔPL, respectively [61,62].
The alterations in the mechanical power of the entire system are equivalent to the summation of the changes in the mechanical power of each unit, accounting for the unit’s participation factor in the overall production.
Δ P M = K H Y Δ P M H + K R H Δ P M R + K W R Δ P M W .
where ΔPMH, ΔPMR, and ΔPMW are the changes in mechanical power in generative units, water, thermal with reheater and thermal without reheater. Furthermore, the participation coefficients for hydro power plants, thermal power plants with reheaters, and thermal power plants without reheaters are denoted as KHY, KRH, and KWR, respectively.
By selecting the state variable x1 for the system frequency changes and according to the state variables representing the mechanical power changes of the production units (x2, x5, and x8), the differential equation representing the rotor and load inertia equivalent model is expressed as follows:
d d t x 1 = − K D J M x 1 + K H Y J M x 2 + K R H J M x 5 + K W R J M x 8 − 1 J M u 1 .

4.3. Governor-Turbine Dynamic Model of Hydro Energy Facility

At elevated altitudes, water courses through the blades of a hydro turbine within a hydroelectric power plant, undergoing a conversion of potential energy into kinetic energy that ultimately generates electricity. A notable advantage of water turbine power plants lies in the elimination of fuel supply costs. Moreover, these installations boast a longer average lifespan compared to their thermal power plant counterparts. The hydro turbine model is considered with a transfer function having one zero (1/TW) and one pole (−2/TW) [63]. By selecting state variables x2, x3, and x4, the first order differential equations of the generating unit with hydraulic turbine in the state space are:
d d t x 2 = − 2 T W x 2 + 2 T W x 3 − 2   d d t x 3 = 2   K G 1 α   R P 1 T G 1 x 1 − 2 T W x 2 + ( 2 T W + 2 α   T R )   x 3 − ( 2 α   T R − 2 α   T G 1 )   x 4 − 2   K G 1 α   T G 1 u 2 .
where TW represents the nominal starting time of water in the penstock, RP1 stands for the droop constant, TG1 denotes the governor time constant, and TR represents the long reset time. Furthermore, KG1 is a positive coefficient associated with the governor.
d d t x 3 = − 1 α   T R x 3 + 1 α   T R x 4 + 1 α d x 4 d t = − K G 1 α   R P 1 T G 1 x 1 − 1 α   T R x 3 + ( 1 α   T R − 1 α   T G 1 )   x 4 + K G 1 α   T G 1 u 2 .
d d t x 4 = − K G 1 R P 1 T G 1 x 1 − 1 T G 1 x 4 + K G 1 T G 1 u 2 .
For further details on the extraction of the mathematical differential equation of the nonlinear dynamics, see [64,65].

4.4. Governor-Turbine Dynamic Model of Thermal Power Plant

A steam turbine could be conceptualized as a multi-dimensional regulation system with the main steam valve lifter and the reheat valve serving as its inputs [66]. By selecting state variables x5, x6, and x7, the first order differential equations of the generating unit with reheat steam turbine in the state space are:
d d t x 5 = − 1 T H x 5 + 1 T H x 6 + F H d d t x 6   = − 1 T H x 5 + ( 1 T H − F H T T ) x 6 + F H T T x 7 .
d d t x 6 = − 1 T T x 6 + 1 T T x 7 .
d d t x 7 = − K G 2 T G 2 R P 2 x 1 − 1 T G 2 x 7 + K G 2 T G 2 u 3   .
where TH denotes the reheater time constant, FH indicates the Strong pressure coefficient, TG1 represents the governor time constant, and TT is the turbine time constant. In addition, KG2 is the governor positive coefficient, RP2 stands for the droop constant, and TG2 represents the governor time constant.
By selecting state variables x8 and x9, the first order differential equations of the generating unit without reheat steam turbine in the state space are:
d d t x 8 = − 1 T C x 8 + 1 T C x 9 .
d d t x 9 = − K G 3 T G 3 R P 3 x 1 − 1 T G 3 x 9 + K G 3 T G 3 u 4   .
where TC indicates the turbine time constant and RP3 represents the droop constant. Furthermore, KG3 and TG3 denote the governor’s positive coefficient and time constant, respectively.

4.5. Governor-Turbine Dynamic Model of Thermal Power Plant

Radiation and temperature are pivotal factors influencing the energy output of solar panels [67]. Given the instantaneous variability in the output power of photovoltaic systems, their impact on the primary frequency response holds significant importance [68].
By choosing two state variables, the photovoltaic system structure is expressed in the state-space model as follows, which includes the photovoltaic panel, inverter, highest power point monitoring, and filter.
d d t x 10 = − p 2   x 10 + z 1 x 11 + d d t x 11 − p 2   x 10 + ( z 1 + p 1 ) x 11 + K P V u 2 .
d d t x 11 = − p 1 x 11 + K P V u 2  
Note that all the parameters and constants in the denominator of Equations (1)–(14) are positive.

5. Simulation Results

Ensuring the quality and reliability of power supply while simultaneously maintaining stability in the power system hinges significantly on the effective control of frequency. This pivotal parameter plays a crucial role in orchestrating the seamless operation of diverse energy facilities within the overall power infrastructure.
In this section, our focus narrows down to the examination of a synergistic system. This composite configuration is comprised of a thermal energy facility equipped with a reheater, a hydro energy facility, and a thermal power plant that operates without a reheater. By scrutinizing the collective dynamics and interactions of these components, we aim to gain insights into the intricate interplay of various energy sources and their impact on frequency control within the broader context of the power system. This analysis allows us to discern the nuanced relationships and dependencies among different types of power generation facilities, shedding light on their collective role in ensuring the robustness and reliability of the power supply.
The above system is described using 13 state variables and 5 independent inputs in the state space. Information on energy production sources for simulation is given in Table 1, and information on the power system and resource participation coefficients in energy production is summarized in Table 2. The assessment of the energy grid’s performance under varying loads is conducted through computer simulations utilizing MATLAB software. The study’s process, focusing on simulating a power system with diverse energy sources and assessing the impact of the photovoltaic system on the load frequency control system, is illustrated in Figure 5.
Three distinct simulation scenarios are examined to illustrate the frequency fluctuations within the energy system. The first case involves load changes exclusively in the broader area of the electrical energy system, encompassing three diverse energy sources. In the second case, load changes are confined to the area of the power system incorporating the photovoltaic unit. The third case explores load changes occurring simultaneously in both areas.

5.1. Demand Step Increase without PV System

In this case, it is assumed that the power sub-system including three energy sources is subject to load demand change. Figure 6 illustrates the frequency variations within the power system, revealing a notable frequency drop exceeding 0.25 Hz. In Figure 7, alterations in the mechanical power of the entire power system are presented in response to step variation in load demand. This power is derived from the output power of three power plants, considering the participation coefficients of the individual units. Figure 8 depicts the changes in the mechanical power of each power plant unit within the energy system, specifically responding to variations in demand side. As can be seen, the steam power plant with a preheater has a lower speed compared to the other two power plants.

5.2. Effect of Photovoltaic System

In this section, it is assumed that the input of the photovoltaic system is 5%. Figure 9 illustrates the frequency fluctuations in the energy grid resulting from variations in load, while Figure 10 specifically presents the frequency variations within the photovoltaic system area. Notably, the frequency changes approach nearly zero in the steady-state.

6. Conclusions

The proliferation of distributed energy resources, coupled with a decline in tradition-al generators, has ushered in a transformation in the power system, marked by a discernible reduction in rotational inertia. This shift in the energy landscape has, in turn, contributed to a noteworthy decrease in overall frequency stability. Within this evolving paradigm, this paper delves into the critical challenge of load frequency control within a single-area energy grid characterized by a diverse mix of energy sources. The constituents of this singular system encapsulate a thermal energy facility featuring a reheat turbine, an-other equipped with a turbine without reheat, and a hydro power plant. The examination extends to include the incorporation of a photovoltaic unit, investigating its impact on the frequency deviation within the energy grid. The paper’s findings highlight that effectively integrating renewable photovoltaic systems not only improves the environmental sustainability of overall power generation, but also alleviates frequency fluctuations within the energy system. Consequently, this integration contributes significantly to enhancing the stability and reliability of the next generation of green energy systems. Through exploration and simulation, this study aims to unravel the intricate dynamics of the energy grid, shedding light on the dynamic behaviour induced by a multi-source energy production environment. This analysis not only serves to elucidate the complex interactions among diverse power sources, but also provides valuable insights into the efficacy of load frequency regulation strategies in navigating the evolving landscape of energy resources. By addressing these pertinent issues, the paper contributes to the ongoing discourse on enhancing the resilience and adaptability of power systems in the face of emerging challenges associated with the integration of varied and decentralized energy resources.

Author Contributions

Conceptualization, G.S. and A.F.; methodology, G.S. and A.F.; software, G.S.; validation, G.S. and A.F.; investigation, G.S.; writing—original draft preparation, G.S. and A.F.; writing—review and editing, G.S. and A.F.; visualization, G.S. and A.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data is available upon reasonable request for the academic purposes from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

ΔFFrequency deviation
ΔPMMechanical power deviation
ΔPDDemand load deviation
JMCombined inertial constant
KDLoad-damping constant
ΔPMHHydro turbine mechanical power deviation
ΔPMRThermal with reheater turbine mechanical power deviation
ΔPMWThermal without reheater turbine mechanical power deviation
TWNominal starting time of water in penstock
TTSteam turbine time constant
TRLong reset time

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Figure 1. Classification and time range of frequency control.
Figure 1. Classification and time range of frequency control.
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Figure 2. The dynamic reaction of system frequency across various time scales.
Figure 2. The dynamic reaction of system frequency across various time scales.
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Figure 4. Block diagram of an interconnected energy system consisting of photovoltaics and various energy sources with only primary speed control.
Figure 4. Block diagram of an interconnected energy system consisting of photovoltaics and various energy sources with only primary speed control.
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Figure 5. The process of simulating the presented power system and investigating the impact of the photovoltaic system.
Figure 5. The process of simulating the presented power system and investigating the impact of the photovoltaic system.
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Figure 6. Power system frequency changes for step changes in load demand (without PV system).
Figure 6. Power system frequency changes for step changes in load demand (without PV system).
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Figure 7. The variations in the mechanical power of the energy network in response to step changes in demand side (without PV system).
Figure 7. The variations in the mechanical power of the energy network in response to step changes in demand side (without PV system).
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Figure 8. The variations in the mechanical power of each power unit within the power system in response to variations in demand side (without PV system).
Figure 8. The variations in the mechanical power of each power unit within the power system in response to variations in demand side (without PV system).
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Figure 9. The frequency changes within the power grid in response to variations in demand side (with PV system).
Figure 9. The frequency changes within the power grid in response to variations in demand side (with PV system).
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Figure 10. The frequency variations specific to the area associated with the photovoltaic system (with PV system).
Figure 10. The frequency variations specific to the area associated with the photovoltaic system (with PV system).
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Table 1. Parameters of energy production resources in the hybrid system.
Table 1. Parameters of energy production resources in the hybrid system.
ParameterSymbolHydro Energy FacilityThermal Energy Facility
With Reheat-TurbineWithout Reheat-Turbine
Governor time constantTG1, TG2, TG30.20.20.2
Turbine time constantTW, TT, TC10.30.3
Droop constantRP1, RP2, RP30.050.050.05
Reheater time constantTH-7-
Strong pressure coefficientFH-0.3-
Slope ratioα0.76--
Long reset timeTR5--
Table 2. Rotor and load parameters, along with the contribution coefficients of energy production sources.
Table 2. Rotor and load parameters, along with the contribution coefficients of energy production sources.
ParameterSymbolValue
Damping coefficientKD1
System inertia constantJM6
Hydro power plant participation coefficientKHY0.3
Coefficient of participation of thermal power plant with reheaterKRH0.5
Coefficient of participation of thermal power plant without reheaterKWR0.2
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MDPI and ACS Style

Shahgholian, G.; Fathollahi, A. Analyzing Small-Signal Stability in a Multi-Source Single-Area Power System with a Load-Frequency Controller Coordinated with a Photovoltaic System. AppliedMath 2024, 4, 452-467. https://doi.org/10.3390/appliedmath4020024

AMA Style

Shahgholian G, Fathollahi A. Analyzing Small-Signal Stability in a Multi-Source Single-Area Power System with a Load-Frequency Controller Coordinated with a Photovoltaic System. AppliedMath. 2024; 4(2):452-467. https://doi.org/10.3390/appliedmath4020024

Chicago/Turabian Style

Shahgholian, Ghazanfar, and Arman Fathollahi. 2024. "Analyzing Small-Signal Stability in a Multi-Source Single-Area Power System with a Load-Frequency Controller Coordinated with a Photovoltaic System" AppliedMath 4, no. 2: 452-467. https://doi.org/10.3390/appliedmath4020024

APA Style

Shahgholian, G., & Fathollahi, A. (2024). Analyzing Small-Signal Stability in a Multi-Source Single-Area Power System with a Load-Frequency Controller Coordinated with a Photovoltaic System. AppliedMath, 4(2), 452-467. https://doi.org/10.3390/appliedmath4020024

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