Power system optimization approach to mitigate voltage instability issues: A review

Abstract

Voltage instability is a major challenge facing power system (PS) that has affected some organizations in achieving their desired goals. Therefore, voltage instability is the incapability of the PS to maintain the voltage standard under no disturbance and after subjecting to disruption. This paper describes the voltage instability phenomena; voltage stability indices include Line Stability Index ($L_P$), Line voltage stability index, Fast Voltage Stability Index (FVSI), line stability factor (LQP), Bus voltage collapse prediction index (BVCPI), L index, voltage stability index (VCP-1), and so on. This review focuses on some stability indices that could identify the weak bus in the electrical PS network. The application of particle swarm optimization (PSO) to minimize losses that cause voltage instability is discussed. It started a detailed understanding of the power blackouts and the detrimental effects on the global economy. This was followed by a thorough understanding of the voltage instability/stability phenomenon, classification in power systems, and the corresponding formulations. The study presents an overview of voltage assessment techniques prior to applying PSO in discrete and multi-objective optimization and the corresponding advantages over others. These are followed by the progress and advances in voltage stability using PSO involving single and hybrid optimization methods. Lastly, to bridge the research gaps, the present study highlighted challenges and future prospects to foster further advancement in the field.

Keywords:

• PS
• voltage instability
• maximum load power
• discrete PSO
• multi-objective PSO

1. Introduction

Day-by-day increase in population has caused power system (PS) to undergo more stressed conditions caused by high load demands, changes in the market structure of electricity, incapable transmission lines, generating station projects, and the use of large protection schemes to enlarge the operating limits of the system. Expansion of the system operating limits leads to severe consequences of increasing the probability of instability voltage. PS has become more expansive and dynamic with increased induction motor loads and electronically controlled loads. The increase in induction motor loads, deficient single phase, residential air conditioner (AC) loads, and inertia makes the PS undergo short-term voltage stability (VS). Short-term VS occurs within a short period (Meng & Pian, 2016; Paramasivam et al., 2013).

Voltage instability has been challenging in several areas, leading to blackouts in some countries (Bo et al., 2015). Therefore, a power blackout is the failure of the PS to supply energy to a particular location or region over time (Ahsan et al., 2012; Atputharajah & Saha, 2009; Haes Alhelou et al., 2019; Kiseng et al., 2021). Blackout could be natural and technical. Technical blackout is tripping offline, overload of line, human error stability issues, and faulty equipment. Natural blackouts are the falling of trees on power lines, landslides, floods, and earthquakes (Fukushima Power Plant in Japan; Hatziargyriou et al., 2005; Larik et al., 2019). Table 1 presents the power blackouts in some parts of the world. Figure 1 gives power outages in some countries of the world.

Table 1. Power blackouts in some parts of the world (Larik et al., 2019)

Year	Country	Causes	Hours of occurrence	Millions of people affected	Reference
1990	Egypt	Voltage collapse	6	50	(El-Sadek, 1998)
2011	Brazil	Defect in transmission line	16	53	(Shuai et al., 2018)
2009	Brazil and Paraguay	Short circuit of transmission line	7	87	(Larik et al., 2019)
1999	Brazil	Striking of lightning	5	97	(Larik et al., 2019)
2001	India	An outage of the transmission line	12	226	(Larik et al., 2019)
2012	India	Overloading	15	670	(Larik et al., 2019)
2005	Indonesia	Failure of the transmission line	7	100	(Larik et al., 2019)
2003	Italy	Tripping the transmission line	18	56	(Zhao et al., 2009)
2006	Europe	Congestion	2	15	(Hatziargyriou et al., 2005; Makarov et al., 2005)
2003	Thialand	Strike of lightning	10	8	(Son & Voropai, 2015; Phuangpornpitak & Tia, 2013)
2016	Kenya	Animal shorted the transformer	4	10	(Lee et al., 2016)
2018	Canada	Winds reached speeds of 100 km/h	0.6	4	(Hatziargyriou et al., 2005)

Developing efficient countermeasure techniques to prevent voltage collapses is essential for the power system. Many methods have been put in place to avoid voltage collapse. Load Tap Changer (LTC) transformer taps are an urgent action that has been put in place to prevent voltage instability. Different emergency schemes may be applied to use the LTCs. An alternative way is load reduction using emergency actions LTCs tap blocking (Amroune et al., 2019; Crow & Lesieutre, 1994). When the reference voltage has been reduced to improve voltage, stability is called the voltage set-point reduction (Amroune et al., 2019; Barboza et al., 2005). An uncontrollable drop in voltage may occur, such as a higher load demand, unbalanced load, disturbance, and variation of system conditions, leading to voltage instability. Insufficient reactive power at each bus can lead to voltage instability (Adebayo & Sun, 2017; Ellithy et al., 2008; Shahzad Ashraf et al., 2020; Stanelyte & Radziukynas, 2020). The issue of system voltage instability is one of the biggest issues facing the PS company (Haes Alhelou et al., 2019; Jayasankar et al., 2010) due to the extensive interconnected and complex nature of the modern PS and environmental constraints. Consequently, PS is stressed to operate near their load-ability limit levels (Suganyadevi & Babulal, 2009). The challenges of voltage instability are a significant concern in PS due to the continuous rise in load demands and restrictions in system expansion (Adebayo & Sun, 2018).

Therefore, voltage instability remains a significant problem facing the PS, thereby affecting some organizations in achieving their desired goals due to the incapability of the PS to keep the voltage profile under no-load conditions and after passing through disturbance. Voltage instability and voltage collapse accompany each other; whenever voltage collapse occurs, it may lead to destruction in the system or cause it to malfunction; it could also result in either total or partial blackout.

Figure 1. Power outages in different regions of the world (Laghari et al., 2013).

The issues of voltage collapse can be reduced/minimized, if proper investigation is carried out to identify the weak bus or node where reactive power compensation (RPC) can be placed. In addition, the effective operation of the system is a significant concern for the PS utility company. Flexible Alternating Current Transmission System (FACTS) controllers have existed for many years and have played an essential role in the PS (Kotsampopoulos et al., 2019; Narain & Laszlo, 2000). The authors (Ahmad & Sirjani, 2020; Pérez et al., 2000; Prabha, 1994), the FACTS device applies in load flow control, improving voltage stability margins and transient stability. The use of evolution algorithms such as PSO, artificial intelligence, Genetic Algorithm (GA), Artificial Neural Network (ANN), Bee Colony, etc., have been extensively introduced to minimize the issue of voltage instability in power systems and studied by quite a several researchers (Ashraf, Muhammad et al., 2020; Ashraf, Saleem et al., 2020; El-Zonkoly, 2011; Preetha Roselyn et al., 2014; Skaria et al., 2014; Verma & Mukherjee, 2016). If PSO is correctly applied to the voltage instability problem, the problem will be minimized, and power loss will also be reduced, which is the novelty of this work.

For several years, PSO has been used compared to other swarm intelligence to cover scientific contributions concerning the PS optimization approach to mitigate voltage instability issues. This is due to its efficiency and ease of implementation, among other advantages such as its fast convergence, compelling performance to reduce losses in the transmission and distribution networks, and improving the system’s voltage profile. This has led to many simulations in the field. Thanks to previous literature: Zhang et al. have provided a comprehensive review of PSO for applications in automation control systems, operations research, fuel and energy, communication theory, mechanical engineering, biology, medicine, and chemistry (Zhang et al., 2015). In addition, Wang et al. discussed various PSO algorithm structures, parameter selections, topology structures, and general engineering applications (Gad, 2022; Wang et al., 2018). Michalis et al. present a comprehensive review on the swarm intelligence dynamic optimization (SIDO) which discusses several problems such as continuous, discrete, constrained, classification problems, multi-objective and real-world applications (Mavrovouniotis et al., 2017). Houssein et al. present an up-to-date review of the PSO algorithm and the advancements and trends in the essential implementation introduced recently. It covered the theory, hybridization/combination, complex optimization, parallelization, and different applications of PSO, which make it more accessible to determine which PSO variant is best to use for a given application and optimization problem (Houssein et al., 2021). Also, a new research direction on particle swarm and new applications, the open issues, and comprehensive challenges of the PSO algorithm are discussed. Vipul et al. (Vipul et al., 2011) reported the practical application of PSO to real-world problems. Banks et al. discussed a short review and timely advancement on the original PSO emanation, opportunities, and difficulties/challenges encountered by the PSO algorithm (Banks et al., 2007). However, the reviews are based on the configuration of the algorithm, dynamic environments, and implementation for parallel adaptations. Bank et al. reported some recent works on the complex area of research, such as multi-objective optimization, hybrid optimization problem, and constraint optimization. However, the metric quality of the swarm-based algorithm was not considered (Banks et al., 2008).

Furthermore, simulations have been done on recent optimization to improve the voltage profile by reducing the real loss affecting electrical power systems. For example, Abd-EI Wahab et al. proposed chaotic turbulent flow of water-based optimization (CTFWO) to solve the ORPD problem. ORPD is a mixed-integer nonlinear optimization problem consisting of continuous and discrete variables. CTFWO was employed to minimize real power loss and voltage deviation (Abd-El Wahab et al., 2022). The virus colony search algorithm (VCS) was used for the optimum sizing and placement of DG in power system networks. The effect of DG on the system’s reliability has been investigated, mitigating the losses in the system and improving the reliability of the system network (Hosseini et al., 2018). Fast non-dominated sorting (FNS) and crowding distance (CD) were integrated with load flow (LF) and used to assess the voltage stability assessment (VSM). The method was employed to find the maximum limits of multiple changes in generator load. The critical voltage bus of PV, PQ, QV, and PQV was obtained using stability limit (VSL), voltage stability margin (VSM), and voltage stability curve (Candelo & Caicedo Delgado, 2019).

A marine predator algorithm (MPA) has been proposed to solve optimal power flow (OPF); Islam et al., 2021). OPF was used in power systems to plan, operate, and manage electric power systems. The objective functions that the proposed method solved are real and reactive power loss, fuel cost, voltage deviation, and improvement of voltage stability index, which were considered as single-objective functions (Islam et al., 2021).

In addition, duponchelia fovealis optimization (DFO) and enriched squirrel search optimization (ESSO) algorithms have been used to solve the ORPD problem (Lenin, 2020). The methods solved the multi-objective functions that reduced the power loss, voltage deviation, and improved the voltage profile (Lenin, 2020). Balasubbareddy and Dwivedi used a novel squirrel search algorithm (SSA) to search for OPF problems with the installation of the FACT device, namely Thyristor controlled series capacitor (TCSC). SSA effectively handled the optimization problem, which is capable of optimizing the objective of the electrical power system. The method minimized the single-objective functions such as generation of fuel cost and emission with incorporated TCSC by keeping the equality and inequality constraint and improving the bus voltage (Balasubbareddy & Dwivedi, 2020; Harika & Balasubbareddy, 2020).

However, there is currently no review on the application of PSO to voltage stability/instability and key measures for furthering the advance in the field. To escalate the satiation further, the absence of an article that discusses the progress in the field has created difficulty in understanding the research trend/hotspot. Thus, alleviating this challenge necessitates the discussion of progress and advancement of this field to understand the strength and weakness of this technology for voltage stability, thereby drawing insight and directions for future research. This review aims to present a recent overview of the optimization of the PS to mitigate voltage instability issues using PSO as an optimization tool. This paper discussed some of the voltage stability indexes, such as FVSI, Lmn, $L_P$, LQP, and VCP-1 and their advantages and disadvantages. For a stable system, the value of indices must be less than unity. An overview of VS is also discussed, while some necessary steps to prevent voltage collapse are also mentioned; likewise, some factors affecting voltage instability are highlighted. In addition, the application areas of PSO and its advantage are also discussed. The remaining parts of this work are organized as follows: Section 2, presents voltage instability phenomenal, Section 3 gives VS, its classification in the power system, and the associated advantages over other methods. Section 4 presents the progress and advances of VS assessment techniques. In contrast, section 5 shows the application of PSO, discrete and multi-objective PSO, section 6 gives an overview of voltage stability using PSO, and the last section presents the challenges and future prospects to bridge the knowledge gap and foster advancement in the field.

2. Voltage instability phenomenal

2.1. Maximum load power

When more power is transmitted over long distances, the transmission network can cause PS instability. More care is needed to transfer power between the load centers and generation of voltage instability. A simple circuit is considered based on the fundamentals of the power transfer between a generator and a load, as shown in Figure 2. For more simplicity/clarification, a purely reactive transmission impedance, jX, was considered and assumed that the synchronous generator (SG) behaves like a static voltage source of magnitude -E.

Figure 2. Two-bus system (Phenometrâ et al., 2000).

Under three-phase source, steady-state sinusoidal conditions, the system operation is described by the LF equations for a three-phase balanced system (Phenometrâ et al., 2000; Prabha, 1994; Sauer et al., 1993).

(1) $\mathrm{P}=-\left\{\frac{\mathit{EV}}{X}\right\}\sin \mathit{\theta }$(1)

(2) $\mathrm{Q}=\mathit{\hspace{0.17em}}-\frac{V^2}{\mathit{X}}+\frac{\mathit{EV}}{X}\cos \mathit{\theta }$(2)

Substituting equations (1) and (2) with respect to V gives

(3) $\mathrm{V}={⌈\frac{E^2}{2}-\mathit{QX}\pm \left(\frac{E^4}{4}-X^2{P}^2-\mathit{X}{E}^2\mathit{Q}\right)⌉}^2$(3)

Where;

P is the active power consumed by the load

Q is the reactive power consumed by the load.

V is the load bus voltage magnitude, and

$\mathit{\theta }\mathrm{is}$ the phase angle difference between the load and the generator buses.

3. Voltage stability (VS) and its classification in PS

3.1. Definition of VS

Many researchers have worked on VS, and several definitions have been given to it. Their work was important to maintain the PS network to keep the load voltage profile at all points in time (Haocheng Haocheng Yang et al., 2021). According to the joint workgroup of the IEEE committee and CIGRE38 committee, VS is defined as the capacity of each bus to always keep the voltage level after suffering a disturbance for a given operating condition. The balance between supply (generating station) and the load demand (consumer) plays a major task in VS. Immediately, the voltage profile of the system is destroyed, the system bus voltage will reduce/increase, and the transmission line will experience tripping, which will cause the generator to fallout of working conditions. Hence, this will cause power loss to the load that the generator feeds and eventually results in a blackout, and thus a voltage collapse occurs. Therefore, voltage collapse can be further described as a situation in which the system voltage is reduced beyond the operating limit of the system, which is caused by the instability voltage (Hatziargyriou et al., 2020; Meng & Pian, 2016).

3.1.1. Definition of some terms in voltage stability

Unstable PS failure occurs frequently, and voltage instability plays a more significant role. As a result, numerous efforts have been devoted to VS, which has led to some of the voltage stability terms given below (Glavic, 2015; Hongjie et al., 2005; Meng & Pian, 2016).

3.1.1.1. Voltage stability

VS is the system voltage’s capability to keep the system’s operating limit value depending on its characteristics (line parameter, transformers tap, reactive power compensation, etc.) to control effect when the power system is experiencing large or small disturbance.

3.1.1.2. Voltage instability

Voltage instability is defined as the initial stage of the system that does not meet the operating limit of the system. The bus voltage may increase or decrease, making the power system transmission exceed the limit. Voltage instability occurs after experiencing or suffering much disturbance. It is also defined as the system’s incapability to retain the voltage profile under normal conditions and after the disruption.

3.1.1.3. Voltage stability limit

This is used to measure voltage-level state, either stable or fluctuating. It is also defined that when the system gets to a certain level or stage, and additional loads are added, the system voltage may dramatically fall out, leading to voltage collapse.

3.1.1.4. Voltage collapse

This occurs when the network undergoes disturbance; the reactive network of the system may no longer be in operation. Adjustment and control measures cannot recover the voltage, resulting in the generator or power grid voltage breakdown.

Voltage collapse can also be described as the events that follow voltage instability, leading to a low voltage profile in some parts of the operation of the PS. A gradual decrease in system voltage is characterized by voltage collapse (Kumar et al., 2018).

Some of the factors causing voltage collapse are:

1. High active power loading in the system (stressed power system).

2. Unexpected sudden relay operation that occurs during low voltage magnitudes.

3. Insufficient reactive power to the system.

4. Low voltage magnitude in load characteristics.

5. Response of generators, transformer tap changer exciter limiters to decreasing

voltages at load buses (Kumar et al., 2018).

3.1.1.5. Some steps to put in place to prevent voltage collapse

Adopting various steps/measures to avoid voltage collapse (Naik et al., 2012; Vanishree & Ramesh, 2014).

1) Application of the devices to compensate for VAR requirement.

2) Proper control and operation of protective devices.

3) Proper managing of tap changer transformers.

4) Proper control of VAR output of the generators and system voltage control.

5) Dropping of load during under-voltage.

3.2. Classification of voltage stability

There are different standards by which voltage stability could be classified, based on the disturbance’s nature (Meng & Pian, 2016), as shown in Figure 3.

1. Small disturbance voltage stability (SDVS): This is the capability of the system to keep system voltage within an acceptable range/period when experiencing minor/slight disturbance, usually within a short period (seconds). Factors affecting SDVS are:

(a). Load characteristics and (b). Continuous and discontinuous control.

2. The mid‐term voltage stability (MVS) occurs when the short-term period rises/increases to minutes, from two (2) minutes or more. It usually occurs on the activation of under load tap changers before the engagement of excitation limiters (Larik et al., 2019).

3. Large disturbance voltage stability (LDVS): is the ability of the PS to maintain the bus voltage at a given period when experiencing disturbance (line outage, system failure, breakdown of generator, short circuit). LDVS can be determined by load characteristics, system characteristics, and protection and control.

Depending on the time and period, the classification of VS is based on short-term, mid-term, and long-time. Short-term VS occurs within a shorter period and lasts for seconds, usually studying the induction motor and high voltage direct current (HVDC) converter. Long-term voltage stability occurs more often, usually in minutes, and lasts longer than short-term VS (Hatziargyriou et al., 2020).

3.3. Some factors affecting the voltage instability

Some of the few inherent factors that affect voltage stability (Meng & Pian, 2016) are listed below:

(a). Distance between the generating station and load center leads to long or far transmission distance.

(b). Rise in-circuit transmission line causes multiple failures of lines to go higher.

(c). Load characteristics such as increase in the speed of load type, static and dynamic characteristics load.

(d) The increase in load’s distance at the load and power generator increases the transmission line capacity.

(e). As the synchronous reactance rises, the transmission line power limit decreases, which causes voltage instability.

(f). Transformer tap adjustment, reactive power compensation, on-load tap changer, etc., are practical tools for voltage stability.

3.4. Formulation and theory of voltage stability index [VSI)

Many methods have been used to predict VSI in power system networks and operations to locate the weakest bus (Ranjan et al., 2003). Few among the indices are discussed in this section and Table 2 discusses the advantages and disadvantages of line VSIs. A single-line diagram is shown in Figure 4.

Figure 3. Classification of voltage stability.

Table 2. The comparison of the advantage and disadvantages of VSIs

Index name	Equation	Assumption	Advantage	Disadvantage
Line stability index ($L_P$)	$L_P=\frac{4R{P}_2}{{(V_{1}cos(\mathit{\theta }-\mathrm{\partial }))}^2}$	Effect of reactive power neglected, Y $\approx$ 0	It is easy to implement	When the load power factor is minimal, the accuracy is significantly affected. Hence it is only used in a distribution system where the load is purely resistive(Moghavvemi & Faruque, 2001)
Line stability index ($L_{\mathit{mn}}$)	$L_{\mathit{mn}}=\frac{4X{Q}_2}{{[V_{1}sin\left(\mathit{\theta }-\mathit{\delta }\right)]}^2}\le 1$	Effect of active power neglected, Y $\approx$ 0	It is responsive to real power changes due to indirect connection to the real power as a result of differences in voltage angle(Samuel et al., 2017)	Under some operating conditions, the index may not be accurate due to the assumption that real power does not affect voltage instability(Ratra et al., 2018)
Fast voltage stability index (FVSI)	FVSI = $\frac{4Z^2{Q}_2}{V_1^2\mathit{X}}\le 1$	sin δ$\approx$ 0; cos δ $\approx$ 1	It is very fast (Samuel et al., 2021)	Sensitivity of index to the resistance reactance ratio of transmission line(Yari & Khoshkhoo, 2017)
Line Stability Factor (LQP)	LQP $=$ 4 $\left(\frac{X}{V{ }_1}\right)\left(Q_2\frac{XP{ }_1}{\mathit{V}{ }_1}\right)$	$\mathit{R}\approx$ 0, Y $\approx$ 0	Under certain operation conditions, it could be inaccurate due to the assumption(Yari & Khoshkhoo, 2017)	Sensitivity of index to the resistance reactance ratio of transmission line(Yari & Khoshkhoo, 2017)
Voltage stability index (VSI-1)	$\mathrm{VSI}-1=\min \left[\frac{{\mathit{P}}_{\mathit{margin}}}{P_{\mathit{max}}},\frac{{\mathit{Q}}_{\mathit{margin}}}{Q_{\mathit{max}}},\frac{{\mathit{S}}_{\mathit{margin}}}{S_{\mathit{max}}}\right]$	R $\approx$ 0; Y $\approx$ 0	It is used in power system networks for real-life monitoring(Kanimozhi & Selvi, 2013)	Based on the assumption, it affects the resistance reactance ratio of the transmission line(Yari & Khoshkhoo, 2017)

where,

$S_1$ and $S_2$ are apparent powers at the sending and receiving ends

$P_2\mathrm{and}{Q}_2$ are the active and reactive powers at receiving ends.

$P_1$ and $Q_1$ are the active and reactive powers at sending ends

|$V_1|\mathrm{is}$ the voltage at the sending end, which is equal to |$V_1\mathit{\hspace{0.17em}}|\mathrm{\angle }{\mathrm{\partial }}_1$

|$V_2|$ is the receiving end voltage equal to |$V_2$|$\mathrm{\angle }{\mathrm{\partial }}_2$

$\mathrm{\partial }$ is the power angle $={\mathrm{\partial }}_1$- ${\mathrm{\partial }}_2$

$\mathit{\theta }$ is the angle of the transmission line

Resistance of the line = R

The reactance of the line = X

Impedance of the line = Z = R + jX.

3.4.1. Line stability index (${\mathit{L}}_{\mathit{P}}$)

The $L_P$ proposed by (Moghavvemi & Faruque, 2001) uses the concept of voltage stability index. For a stable system, the value of $L_P$ must less than one, and if it is greater than one, the system is unstable (Moghavvemi & Faruque, 2001). A single-line diagram in Figure 4 is taken into consideration.

Line stability index $L_P$ of a transmission line is defined as follows:

(4) $L_P=\frac{4R{P}_2}{{(V_{1}cos(\mathit{\theta }-\mathrm{\partial }))}^2}$(4)

The shunt admittance and reactive power effect on VS are neglected; only active power affects the line stability.

3.4.2. Line stability index (${\mathit{L}}_{\mathit{mn}})$

$L_{\mathit{mn}}$ was proposed by (Moghavvemi & Faruque, 1999). Using the concept of power flow from a single line diagram, the discriminant of the quadratic voltage equation was set to be greater than or equal to zero. Therefore, for a stable system, the value transmission line $L_{\mathit{mn}}$ must be less than unity (1). The voltage equation is given below

(5) $L_{\mathit{mn}}=\frac{4X{Q}_2}{{[V_{1}sin\left(\mathit{\theta }-\mathit{\delta }\right)]}^2}\le 1$(5)

3.4.3. Fast voltage stability index (FVSI)

The FVSI is proposed by (Musiri & Abdul Rahman, 2002) to calculate the voltage stability margin of a line/bus under given criteria. From Figure 4, using the quadratic voltage equation of the system. The FVSI is defined as

(6) $\mathit{F}\mathrm{VSI}=\frac{4Z^2{Q}_2}{V_1^2\mathit{X}}\le 1$(6)

The value of the line that gives a unity (1) value shows that one of the buses that connect to the line will experience voltage collapse. If the value of FVSI is below 1, the line connected to the bus is stable.

3.4.4. Line stability factor (LQP)

(Mohamed et al., 1998) developed the LQP; the formula was formulated based on $L_{\mathit{mn}}\mathit{\hspace{0.17em}}$ and FVSI. LQP must be less than 1; anything more significant than 1 result in system instability. This index is assumed that the line is lossless (i.e. (R/X≪1), and shunt admittance of the line is negligible (Mohamed et al., 1998). The LQP equation is stated as follows.

(7) $\mathrm{LQP}=\mathit{\hspace{0.17em}}4\left(\frac{X}{V{ }_1}\right)\left(Q_2\frac{XP{ }_1}{\mathit{V}{ }_1}\right)$(7)

3.4.5. Voltage stability index (VSI-1)

VSI-1 is formulated by (Gong et al., 2006), using the concept of power transfer index (PTSI) and voltage collapse proximity index (VCPI); the formulation is as follows

(8) $\mathrm{VS}{\mathrm{I}}_1=\min \left[\frac{{\mathit{P}}_{\mathit{margin}}}{P_{\mathit{max}}},\frac{{\mathit{Q}}_{\mathit{margin}}}{Q_{\mathit{max}}},\frac{{\mathit{S}}_{\mathit{margin}}}{S_{\mathit{max}}}\right]$(8)

(9) $P_{\mathit{margin}}=P_{\mathit{max}}-P_r$(9)

(10) $Q_{\mathit{margin}}=Q_{\mathit{max}}-Q_r$(10)

(11) $S_{\mathit{margin}}=S_{\mathit{max}}-S_r$(11)

(12) $P_{\mathit{max}}={\left[\frac{{V^2}_1}{4X^2}-Q_2\frac{{V^2}_1}{\mathit{X}}\right]}^2$(12)

(13) $Q_{\mathit{max}}=\frac{V^2{ }_1}{4\mathit{X}}-\frac{X{P^2}_2}{{V^2}_1\mathit{\hspace{0.17em}}}$(13)

(14) $S_{\mathit{max}}=\frac{\left(1-sin\mathrm{\Psi }{V}^2{ }_1\right)}{2\mathit{X}{cos}^2\mathrm{\Psi }}$(14)

3.4.6. Power stability index (PSI)

PSI is proposed by (Aman et al., 2012) based on a two-bus system to identify critical bus/node close to voltage collapse in PS. The author proposed that a value less than unity is a stable system. PSI was proposed to realize the optimal placement of DG within the critically sensitive buses closed to the voltage failure (collapse).

(15) $\mathit{PSI}=\frac{4R\left(P_L-P_G\right)}{(\left|V_1\right|cos\left(\mathit{\theta }-\mathit{\delta }\right){)}^2}\le 1$(15)

3.4.7. Voltage deviation index (VDI)

The VDI is an N-bus system based on the overall N voltage deviations calculated for each bus/node in the system. The absolute value of voltage bus deviation is defined by this index as proposed by (Yang et al., 2012)

(16) $\mathit{VD}{I}_j=|1-V_j|$(16)

(17) $\mathit{VD}{I}_T=\sum _{\mathit{j}=1}^N|1-V_j|$(17)

“Where N is the number of buses and $V_j$ is the target values of the index calculated”

3.4.8. Improved voltage stability index (IVSI)

The IVSI is formulated based on load flow (LF), and the equation is given below. The proposed method is to enhance the PS voltage stability. The value of the index for a stable operation is zero (no-load) and unity for an unstable system. The IVSI aims to optimize VS by the optimum setting of compensation devices of an N-bus system, as reported by (Yang et al., 2012).

(18) $\mathit{IVS}{I}_T=\sum _{\mathit{i}=1}^N\mathit{IVS}{I}_i$(18)

3.4.9. Integral steady-state margin (ISSM)

The ISSM index is a preferable method for applications in online security controls. ISSM is a modified index used to evaluate PS in steady-state conditions (Danish et al., 2019). For system stability, the value is between zero and unity.

(19) $\mathit{ISSM}=\left[\frac{J_C}{J_O}\right]$(19)

Where

$J_C$ and $J_O$ are the steady-state condition and system fictitious state of Jacobian, respectively.

3.4.10. Novel line stability index (NLSI)

The equation for NLSI given below was derived from the two bus systems of the LF equation. As reported by (Yazdanpanah-Goharrizi & Asghari, 2007), it is effective for identifying the weak bus/node, and the most vulnerable line in the systems.

(20) $\mathit{NLSI}=\frac{R{P}_1+\mathit{X}{Q}_2}{0.25{V_1}^2}$(20)

3.4.11. Minimum eigenvalue and right eigenvector method

Gao et al. (Gao et al., 1992a) expressed an analysis model based on the smallest eigenvalue associated with the right eigenvector. From the equation, the system is stable if all the eigenvalues are positive and the real part of the eigenvalue shows that it is unstable/critical.

(21) $\mathrm{\Delta V}=\sum _i\frac{{\xi }_i{\eta }_i}{{\lambda }_i}\mathrm{\Delta Q}$(21)

Where

$\mathrm{\Delta V}\mathit{and}\mathrm{\Delta }\mathit{Q}$ = the deviation in voltage magnitude and reactive power injected.

${\xi }_i$ and ${\eta }_i$ are the $i^{\mathit{th}}$ column right and left row eigenvectors, respectively

${\lambda }_i$ is diagonal eigenvalue matrix of a reduced Jacobian matrix

3.4.12. Impedance ration indicator (IRI)

This index assessed the robustness and effectiveness of indicators over the operating ranges. The Thevenin theorem is then used in equation (22). A VCPI based on the impedance ratios of the two-bus systems is applied (Chebbo et al., 1992).

(22) $\frac{{\mathit{Z}}_{\mathit{ii}}}{Z_i}\le 1$(22)

$Z_{\mathit{ii}}\perp {\beta }_i=i^{\mathit{th}}\mathit{diagonal}\mathit{element}\mathit{of}\left|\mathit{Z}\right|$

(23) $\left|\mathit{Z}\right|={\left|Y\right|}^{-1}$(23)

3.4.13. Bus voltage collapse prediction index (BVCPI)

BVCPI method is proposed to predict the collapse voltage in PS based on the LF equation (Balamourougan et al., 2004). BVCPI is suitable for estimating the voltage collapse considering the load effect at the other buses/nodes. The system is stable if the value of the index is zero and unstable/collapse point at the value of unity.

(24) $\mathit{VCPI}=\left|1-\frac{{\sum }_{i=1i\ne j}^{Nb}{V}_m^{\prime }}{V_k}\right|$(24)

3.4.14. L index

This is formulated based on the LF equation to identify the weak bus/node and predict the voltage instability at different contingency states, thereby providing a quantitative measure of the state of real PS (Kessel and Glavitsch 1986).() The L index finds a complete system’s stability condition and is given in the equation below. For a stable system, the L-index must be zero; however, when it is unity, the system approaches a voltage collapse.

(25) $L_j=\left|1-\sum _{\mathit{i}=1}^g{F}_{\mathit{ji}}\frac{V_i}{V_j}\right|$(25)

(26) $\left[\begin{array}{c}{I}_G\\ I_L\end{array}\right]=\left[\begin{array}{cc}{Y}_{\mathit{GG}}& Y_{\mathit{GL}}\\ Y_{\mathit{LG}}& Y_{\mathit{LL}}\end{array}\right]\left[\begin{array}{c}{V}_G\\ V_L\end{array}\right]$(26)

where $I_L,I_{\mathit{G}}\mathit{and}{V}_G,V_L$ are the currents and voltages at the generator and load buses/nodes.

$Y_{\mathit{GG}},Y_{\mathit{GL},}{Y}_{\mathit{LG}}\mathit{and}{Y}_{\mathit{LL}}$ are the Y-bus matrix.

(27) $\left[\begin{array}{c}{V}_L\\ I_G\end{array}\right]=\left[\begin{array}{cc}{Z}_{\mathit{LL}}& F_{\mathit{LG}}\\ K_{\mathit{GL}}& Y_{\mathit{GG}}\end{array}\right]\left[\begin{array}{c}{I}_L\\ V_G\end{array}\right]$(27)

Where,

$Z_{\mathit{LL}}$ is the total impedance at the node bus

$K_{\mathit{GL}}$ is the negative transpose of the matrix of $F_{\mathit{LG}}$

$F_{\mathit{LG}}=$ $-{\left[{\mathit{Y}}_{\mathit{LL}}\right]}^{-1}{Y}_{\mathit{LG}}$

3.4.15. Simplified voltage stability index (SVSI)

SVSI (Pérez-Londoño et al., 2014) was proposed to improve the indicator of voltage stability margin in electric PS. The method used a correction factor ($\mathit{\beta })$, associated voltage magnitude at bus (n and l), and is defined in the equation (28). The PS will be stable if the index value is below unity and unstable if the value of the index is close to unity if and only the drop in voltage at the Thevenin impedance was the same as the voltage at the load bus

(28) $\mathit{\beta }=1-{\left(max\left(\left|V_n\right|-\left|V_l\right|\right)\right)}^2$(28)

(29) $\mathrm{SVSI}=\frac{\mathrm{\Delta }{V}_i}{\mathit{\beta }\ast V_i}$(29)

3.4.16. Proposed stability index (PSI)

The PSI (Gupta & Kumar, 2018; Ismail et al., 2020) was formulated using a two-node/bus power transmission line. The index equation is given in equation (30).

(30) $\mathit{PSI}=\frac{4\mathit{R}}{V^2{ }_i}\left(\frac{Q^2{ }_j}{P_j}+P_j\right)$(30)

3.4.17. Voltage sensitivity index (VSI)

This index used the same concept of a power transmission line at a node’s sending and receiving end (Murty & Kumar, 2015a). The equation for the index is given below.

(31) $\mathit{VSEI}=\frac{4{\mathit{X}}_{\mathit{ij}}}{V^2{ }_i}\left(\frac{P^2{ }_j}{Q_j}+Q_j\right)$(31)

3.4.18. Voltage stability index (VSI)

VSI is proposed by (De & De, 2010) by considering the resistance of a line in the distribution network due to the higher resistance ratio R/X at the sending and receiving end of a node.

(32) $\mathit{VSI}=4\left(Q_j{X}_{\mathit{ij}}+\frac{R^2{Q}_j}{X_{\mathit{ij}}}\right)\frac{\left(1-cos2\mathrm{\varnothing }\right)}{2V^2{ }_i\mathit{\hspace{0.17em}}{sin}^2({\mathrm{\partial }}_i-{\mathrm{\partial }}_j-\mathrm{\varnothing })}$(32)

3.4.19. Line collapse proximity index (LCPI)

LCPI is based on a transmission line proximity index model using the ABCD parameter of a two-port network (Tiwari et al.,).

(33) $\mathit{LCPI}=\frac{4Acos\propto (P_j\mathit{Bcos\beta }+Q_j\mathit{Bsin\beta }}{(V_i\mathit{cos\delta }{)}^2}$(33)

Where $\propto \mathit{and}\mathit{\beta }$ are phase angles for A and B parameters. A, B, C, and D are the parameters of the transmission line, which can be expressed as:

(34) $\left[\begin{array}{c}{V}_i\\ I_i\end{array}\right]\mathit{\hspace{0.17em}}=\left[\begin{array}{cc}\mathit{A}& \mathit{B}\\ \mathit{C}& \mathit{D}\end{array}\right]\left[\begin{array}{c}{V}_j\\ I_j\end{array}\right]$(34)

3.4.20. Stability index ($\mathit{S}{\mathit{I}}_{\mathit{VGE}})$

SivaSankar and Anjaneyulu proposed $\mathit{S}{I}_{\mathit{VGE}}$ focusing on the bus deviation in a system that obtained the difference between the slack bus voltage and voltage at a specific bus (SivaSankar & Anjaneyulu, 2013). Th equation of a stability index of bus i is given in equation (35)

(35) $\mathit{S}{I}_{\mathit{VGE}}=\frac{1}{\surd \mathit{l}}\sqrt{\sum _{\mathit{i}=1}^l{\left({\mathit{V}}_{\mathit{slack}}-\sum _{\mathit{i}=1}^N{Z}_i\left(\frac{P_i-\mathit{j}{Q}_i}{V_i}\right)\right)}^2}$(35)

3.4.21. Improved voltage stability index (IVSI)

Yang et al., 2012 proposed the IVSI using the LF equation formula. The index is measured at each bus/node in a system and is capable of radial and mesh system networks. The system remains stable when the value of IVSI is below unity; once it is closed to unity, the system is unstable, leading to voltage collapse. The equation is given as:

(36) $\mathit{IVS}{I}_T=\sum _{\mathit{i}=1}^N\mathit{\hspace{0.17em}}\frac{-4{\sum }_{j=0}^n\left(G_{\mathit{ij}}-B_{\mathit{ij}}\right)\left(P_i+Q_i\right)}{{\sum }_{j=1}^n\left|V_j\right|{\left[{\mathit{G}}_{\mathit{ij}}\left(\mathit{cos}{\mathrm{\partial }}_{\mathit{ij}}+\mathit{sin}{\mathrm{\partial }}_{\mathit{ij}}\right)-B_{\mathit{ij}}\left(\mathit{cos}{\mathrm{\partial }}_{\mathit{ij}}-\mathit{sin}{\mathrm{\partial }}_{\mathit{ij}}\right)\right]}^2}$(36)

3.4.22. Voltage stability proximity index (VCPI)

Using the concept of maximum power transferred in a single transmission line. The maximum real/actual and reactive powers transmitting from the receiving end are denoted as P and Q, while $P_j$ and $Q_j$ were calculated from the LF equation (Moghavvemi & Faruque, 1998).

(38) $\mathit{VCP}{I}_P=\frac{P_j}{\frac{V^2{ }_j}{Z_{\mathit{ij}}}\frac{\mathit{cos\phi }}{4co{s}^2\frac{\mathit{\theta }-\mathit{\phi }}{2}}}$(38)

(39) $\mathit{VCP}{I}_Q=\frac{Q_j}{\frac{V^2{ }_j}{Z_{\mathit{ij}}}\frac{\mathit{sin\phi }}{4co{s}^2\frac{\mathit{\theta }-\mathit{\phi }}{2}}}$(39)

3.4.23. Integrated transmission line transfer index (ITLTI)

The ITLTI uses the concept of power transfer under leading, lagging, and unity power factors (Chuang et al., 2016). ITLTI was based on the radial topology introduced in equation (40).

(40) $P_R=\frac{A{V}^2{ }_R}{\mathit{B}}cos\left(\mathit{\hspace{0.17em}}\mathit{\beta }-\mathit{\alpha }\right)+\frac{V_R{V}_S}{\mathit{B}}cos\left(\mathit{\hspace{0.17em}}\mathit{\beta }-\mathit{\alpha }\right)$(40)

$V_S$ is the sending end constant voltage, A and B are line parameters, $P_R\mathit{and}{V}_R$ real/active power and voltage at the sending end bus.

3.4.24. Line voltage stability index (LVSI)

The LVSI is proposed by Ratra et al. (2018) to assess the stressfulness of a line. LVSI is used to estimate the stability of PS. The value of the index varies between 2 and 1, which is a stable point to the voltage collapse point. From this, the unstable line will be identified. The equations are given in (41–42):

(41) $\mathit{LVS}{I}_j=\frac{2{\mathit{V}}_{\mathit{Rj}}{A}_{j}cos\left({\beta }_j-{\alpha }_j\right)}{V_{\mathit{Sj}}cos\left({\beta }_{\mathit{j}}-{\delta }_{\mathit{SRj}}\right)}\mathrm{\forall }\mathit{j}=1,2,3\dots \dots ..\mathrm{l}$(41)

(42) $\mathit{LVSI}>1$(42)

Critical boundary index (CBI)

In CBI, the real/active and reactive power was used with some novelties (Furukakoi et al., 2018). The transmission line is stable when the value of CBI is above zero and unstable when it is close to zero. The equations are given in (43–45):

(43) $\mathit{CB}{I}_{\mathit{ij}}=\sqrt{\mathrm{\Delta }{P}_{ij}^2+\mathrm{\Delta }{Q}_{ij}^2}$(43)

(44) $\mathrm{\Delta }{P}_{\mathit{ij}}=\mathit{X}-P_O$(44)

(45) $\mathrm{\Delta }{Q}_{\mathit{ij}}=\mathit{Y}-Q_O$(45)

3.4.26. Second-order index

The second-order index is also called the i index. The index considers the Jacobian matrix inverse’s total system load and maximum singular value. For a stable system, the value must be unity and unstable when it reaches zero, as reported by (Berizzi et al., 1998). The equation is given as:

(46) $\mathit{i}=\frac{1}{i_o}\frac{{\mathit{\sigma }}_{\mathit{max}}}{\mathit{d}{\sigma }_{\mathit{max}}/\mathit{d}{\lambda }_{\mathit{total}}}$(46)

where $i_o$ is the value of $\frac{{\mathit{\sigma }}_{\mathit{max}}}{\mathit{d}{\sigma }_{\mathit{max}}/\mathit{d}{\lambda }_{\mathit{total}}}$, ${\sigma }_{\mathit{max}}$ is the maximum value of the Jacobian matrix inverse,and ${\lambda }_{\mathit{total}}$is the overall load in the system.

3.4.27. Tangent vector index $\mathit{\hspace{0.17em}}(\mathit{TV}{\mathit{I}}_{\mathit{i}}$)

$\mathit{TV}{I}_i$ measured the effect of the load change on the vector element (e.g., magnitude of voltage and angle; De Souza et al., 1997). The index is calculated using system load and vector tangent. However, assessing how a system will operate from unstable/voltage collapse is better. The equation for the index is given as:

(47) $\mathit{TV}{I}_i={\left|\frac{d{V}_i}{\mathit{d\lambda }}\right|}^{-1}$(47)

3.4.28. Predicting the voltage collapse index ($\frac{\mathit{V}}{{\mathit{V}}_{\mathit{o}}}$ Danish et al., 2019)

Danish et al. reported $\frac{V}{V_o}$ for the system’s operating point, the voltage magnitude (V) can be obtained from the LF (Danish et al., 2019). The new value of the no-load system condition is $V_o$ (no load voltage). The smallest value of the index is the weak bus in the system. The benefit of this index is that it can be used for both offline and online applications. The index shows a nonlinear profile when changing the loading parameter. $\frac{V}{V_o}$ is effective in nonlinear performance, prediction of collapse point, and not effective in computational cost as reported by (Cardet, 2010; Cupelli et al., 2012).

3.4.29. Power transfer stability (PTSI)

PTSI was proposed (Muhammad et al., 2006) and used the Thevenin equivalent of a two-bus system. The PTSI value is between unity (weak point) and zero (stable point). The equation of PTSI is given in equation (48).

(48) $\mathit{PTSI}=\frac{2S_L{Z}_{\mathit{Thev}}(1+cos\left(\mathit{\beta }-\mathit{\alpha }\right)}{E^2{ }_{\mathit{Thev}}}$(48)

$\mathit{\beta }\mathit{and}\mathit{\alpha }$ are the phase angle for load impedance and phase angle for Thevenin impedance, respectively.

3.4.30. Voltage stability factor (VSF)

VSF was formulated from a two (2)-bus system. At the unstable/collapse point, the VSF is zero, and the high value indicates the stable operation. At the receiving end of the bus, the voltage magnitude is half of the sending end voltage magnitude (Sultana et al., 2016). The equation is given in (49):

(49) $\mathit{VS}{F}_T={\sum }_{n=1}^{Nb}\left(2V_{\mathit{n}+1}-V_n\right)$(49)

where $V_n$ and $\mathit{Nb}$ are voltage magnitude at the substation and the overall number of buses/nodes in the system.

3.4.31. Bus participation factor (BPF)

BPF is one of the indices for identifying weak bus/node in a system. It was reported that BPF used the concept of voltage collapse (Gao et al., 1992b; Ismail et al., 2014; Song et al., 2019; Vassilyev et al., 2017).

3.4.32. Voltage collapse index (VCI)

Taylor’s theorem was used to formulate VCI based on system apparent power as reported by (Haque, 2007). In order to linearize the trend of the VCI at the weak point, the value of $\mathit{VC}{I}_k\ge 0$. The equation is given as:

(50) $\mathit{VC}{I}_k={\left[1+\left(\frac{I_k\mathrm{\Delta }{V}_k}{V_k\mathrm{\Delta }{I}_k}\right)\right]}^{\propto }$(50)

3.4.33. Stability index (SI)

The quadratic voltage equation is commonly used to calculate the line VSIs, based on the SI (Eminoglu & Hocaoglu, 2009). The line at which the value of SI is at its lowest is the weakest node for voltage collapse. Therefore, when SI reaches zero, voltage collapse occurs.

(51) $\mathit{SI}=2V_s^2{V}_r^2-V_r^4-2V_r^2\left(P_r\mathit{R}+Q_r\mathit{X}\right)-Z^2({P^2}_{\mathit{r}}+{Q^2}_{\mathit{r}}$(51)

3.4.34. Voltage collapse critical bus index (VCCBI)

This was proposed by (Adebayo & Sun, 2019). The formulation is based on the concept of the LF solution of the load bus. The equation is given in (52):

(52) $(\mathrm{VCCBI}{)}_{\mathit{I}=\mathrm{1..}\mathit{T}}=\frac{{({\mathrm{\Delta }}_i)}_{\mathit{I}=\mathrm{1..}\mathit{T}}}{\mathit{T}{\mathrm{\Delta }}_q}$(52)

The most critical bus is the load bus with the smallest value of T or smallest reactive power load-ability and maximum value of VCCBI.

3.4.35. Reactive power loss index (RPLI)

Reactive power loss index (RPLI) involves the addition of weighted normalized values of reactive power loss at the various load nodes under system intact conditions. Several contingencies are computed as illustrated in the following equation (Moger & Dhadbanjan, 2019).

(53) $\mathrm{RPL}{\mathrm{I}}_i=\mathit{Qlos}{s}_{i,o}^n+{\sum }_{k=1}^{Nc}\mathit{Qlos}{s}_{i,k}^n\ast \mathit{NCOS}{I}_k$(53)

Here

(54) $\mathit{Qlos}{s}_{i,o}^n=\frac{\mathrm{Qlos}{s}_{i,o}^n}{\mathit{max}\left(\mathrm{Qlos}{\mathrm{s}}_o\right)}$(54)

(55) $\mathit{Qlos}{s}_{i,k}^n=\frac{\mathrm{Qlos}{s}_{i,k}^n}{\mathit{max}\left(\mathrm{Qlos}{\mathrm{s}}_k\right)}$(55)

(56) $\mathrm{NCOSI}=\mathit{OS}{I}_{\mathit{LL}}+\mathit{OS}{I}_{\mathit{VP}}+\mathit{OS}{I}_{\mathit{VSI}}$(56)

(57) $\mathit{OS}{I}_{\mathit{VSI}}={\sum }_{\mathit{pq}}\mathit{S}{I}_{VSI}^n$(57)

(58) $\mathit{OS}{I}_{\mathit{VP}}={\sum }_{\mathit{pq}}\mathit{S}{I}_{VP}^n$(58)

(59) $\mathit{OS}{I}_{\mathit{LL}}={\sum }_{\mathit{nl}}\mathit{S}{I}_{LL}^n$(59)

where, $\mathit{pq}$ is the load buses/nodes, $\mathit{OS}{I}_{\mathit{VP}}$ is the overall severity index of the bus voltage profile, $\mathit{S}{I}_{VP}^n$ is the normalized severity index of post contingent bus voltage profile, $\mathit{OS}{I}_{\mathit{VSI}}$ is the overall severity index of the voltage stability index (VSI), $\mathit{S}{I}_{VP}^n$ is the normalized severity index of post contingent, VSI, $\mathit{nl}$ represents the line number, $\mathit{OS}{I}_{\mathit{LL}}$ is the overall severity index of the line loading, and $\mathit{S}{I}_{LL}^n$ is the normalized severity index of post contingent line loading.

3.4.36. Nodal equation of real power loss

The formulation of real power load into a system contributed by a specific generator starts with LF equations (Khalid et al., 2009):

(60) ${\mathrm{P}}_{\mathit{Gj}}={\sum }_{i=1}^{NL}{P}_{\mathit{Gji}}^{\mathrm{\Delta }{l}_G}+{\sum }_{i=1}^{NL}{P}_{\mathit{Gji}}^{\mathrm{\Delta }{V}_G}$(60)

The loss caused by various load i is given as:

(61) $P_{\mathit{Loss}\left(\mathit{i}\right)}^{n_L}={\mathrm{P}}_{\mathit{Li}}-{\mathrm{P}}_{\mathit{Gj}}$(61)

where $P_{\mathit{Gji}}^{\mathrm{\Delta }{l}_G}$ is the current dependent of load $\mathit{i}$ to ${\mathrm{P}}_{\mathit{Gj}}$, $P_{\mathit{Gji}}^{\mathrm{\Delta }{V}_G}$ is the voltage dependence of load $\mathit{i}$ to ${\mathrm{P}}_{\mathit{Gj}}$, ${\mathrm{P}}_{\mathit{Li}}$ is the real load vector

3.4.37. Power loss sensitivity (PLS)

Power loss sensitivity (PLS) was formulated based on real power loss. PLS helps to reduce the search space and obtain faster results for the optimization process (Abdelaziz et al., 2016; Ismail et al., 2020). The node with the normalized voltage is less than 1.01, and the highest ranking is the bus that needs reactive support.

Real power loss:

(62) $\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}=\frac{2P_j{R}_{\mathit{ij}}}{V^2{ }_j}$(62)

(63) $\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}=\frac{2Q_j{R}_{\mathit{ij}}}{{V^2}_{\mathit{j}}}$(63)

Reactive power loss:

(64) $\frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}=\frac{2P_j{X}_{\mathit{ij}}}{V^2{ }_j}$(64)

(65) $\frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}=\frac{2Q_j{X}_{\mathit{ij}}}{V^2{ }_j}$(65)

3.4.38. Combined power loss sensitivity (CLPS)

CLPS was proposed to consider the real and reactive power losses. The real and reactive power losses sensitivity index in equations (62–65) were used (Murthy & Kumar, 2013). The formulation is given below:

(66) $\frac{\mathrm{\partial }{\mathit{S}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}=\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}+\mathit{j}\frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}$(66)

(67) $\frac{\mathrm{\partial }{\mathit{S}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}=\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}+\mathit{j}\frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}$(67)

The loss sensitivity matrix (LSM) is given below:

(68) $\mathit{LSM}=\left[\begin{array}{c}\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}\frac{\mathrm{\partial }{\mathit{P}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}\\ \frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{P}_j}\frac{\mathrm{\partial }{\mathit{Q}}_{\mathit{loss}}}{\mathrm{\partial }{Q}_j}\end{array}\right]$(68)

The node with the highest index value is considered unstable and needs reactive support.

3.4.39. Power loss index (PLI)

PLI is an efficient approach sensitive to the real power loss reduction. In PLI, the overall power loss is normalized between zero and one for low and high reduction, respectively (Ismail et al., 2020). The equation for PLI is given below:

(69) $\mathit{PLI}\left(\mathit{i}\right)=\frac{\mathit{M}\left(\mathit{i}\right)-{\mathit{M}}_{\mathit{min}}}{M_{\mathit{max}}-M_{\mathit{min}}}$(69)

where M is power loss reduction, $M_{\mathit{min}}$ is minimum power loss reduction, $M_{\mathit{max}}$ is maximum power loss reduction.

4. Overview of voltage assessment techniques

According to Shuaibu Hassan et al., 2020; Sultana et al. (2016), the optimum sitting and sizing of distributing generating (DG) is the proper way to increase the performance of the PS during the network reconfiguration and capacitor placement. Many researchers, power system engineers, and operators are looking forward to solving the problem of distributed systems such as VS, power loss, and voltage profile based on the optimal allocation of DG. Moreover, the optimal allocation of DG guides the distributing system from unnecessary events and makes the PS operator run the system in isolate/islanding mode. Also, Iqbal et al. (2018); Isha & Jagatheeswari (2021); Zhang & Yu (2020) used 33 and 30 bus systems that provided a guide on minimizing losses and improving the bus voltage in the distribution system, which is the placement of renewable energy sources (RES) such as biomass, solar photovoltaic (PV), and wind. The study gave Distributed Energy Resources (DER) that the system network will be stable when it is put near the load. It can still work on islanding if there is a fault outside the network. It further discussed that the placement of DG results in under-voltage in some buses, which does not have total effect on reactive power needed in the system, and optimal placement of Distribution Static Compensation (DSTATCOM) was used to solve the under-voltage.

Shaik et al. present the benefit of inserting DG unit into the distribution system (DS) to enhance the voltage profile of the system and reduce the system power loss (Ali Shaik et al., 2022). The equilibrium optimizer is used as an optimization method to determine where the best DG can be positioned and how to reconfigure the distribution network. The methods were carried out in MATLAB software, and IEEE 33 bus system was used. The author reported that there was an improvement in voltage profile as well as the power loss was reduced. Also, Amroune et al. presented the use of event-driven emergency demand response (EEDR) using the whale optimization algorithm (WOA) to keep the voltage stability margin (VSM) at an acceptable limit during emergency occurrence by the amount of load reduction (Amroune et al., 2019). The method is tested on IEEE 14 and a real Algerian 14 bus power system. The method was proved as an efficient tool in maintaining voltage stability.

Ranjan et al. gave a new VSI for selecting the bus that is liable for voltage collapse using load modeling to analyze the stability of the voltage in a radial distributing network (Ranjan et al., 2003). The author reported that the method was an effective tool to be used. (Hongjie et al., 2005) discussed the usefulness of the L index that can be used in region operating centers for interconnection of PS; it also gives an accurate indication of voltage instability. Munkhchuluun et al. (Munkhchuluun et al., 2020) reported the solar PV and reactive power of the SG and investigated the long-term VS combined with large solar PV. It was shown that the large-scale solar PV has a good impact on the PS and the type of loading level applied to give the kind of impact over the excitation limiter (OEL). The effect of ambient temperature on the longer time VS and solar irradiance were also investigated

Another study (Ingole & Gohokar, 2017) on the FACTS showed that stability keeps the PS stable. The type of FACTS used was the Static Synchronous Series Compensator (SSSC) connected in series to a transmission line. The simulation was done using MATLAB, and the VS and reactive power compensations were further investigated using IEEE 4 and IEEE 9 bus systems. Also, some sets of devices have been presented for placing the thyristor controlled series compensator (TCSCs), shunt VARs compensator (SVCs), and unified power flows controllers (UPFCs) in the weak bus in the networks (Nadeem et al., 2020). The line was determined using the P-V curve of load buses/nodes through the line stability index. Whale optimization (WOA) was used to determine the optimum coordination of TCSC, SVC, and UPFC with a reactive power source, and the rating of each device was found. The objective was to minimize the system’s operation cost, which contains the FACTS device and real power losses in the system. The result was compared with GA and PSO, and the WOA was observed to give a better reduction in operating costs.

According to Alzaareer et al., a new approach was chosen to control variables to prevent voltage instability in the PS; Thevenin-based VS margins were used to perform the analysis derived from the concept of a single-port circuit. In order to verify the accuracy of the method, it was then tested on IEEE 39 and 118 bus systems (Alzaareer et al., 2020).

El-Sadek et al. reported using load shedding as an emergency action to prevent voltage instability in transmission PS. Optimum required quantities are found together with optimum locations of loads to be shed. L-indicator index was used for this purpose with a modified new technique. The application of the test system was reported to test the capability of the new technique for any size of the system (El-Sadek et al., 1999). Also, the ABC algorithm was used to determine the optimum load to be shed with high penetration of RES. Overload and generation contingency was considered on IEEE 30 bus system. The result was compared with traditional methods, and it was very accurate, as reported by (Mogaka et al., 2020)

The use of the phasor measurement units (PMUs) to predict Short Time Voltage Instability (STVI) has been proposed (Lambrou et al., 2021; Yang et al., 2018). An online contingency analysis was used to obtain the three elements look-up tables comprising the presumed contingency, the corresponding post-fault equilibrium point (SEP), and unstable equilibrium point (UEP) for each induction motor IM load. When there is a fault, the time-series of IM slip is computed using the Euler algorithm using local PMU measurements, and a new—time series method was proposed for rolling prediction of IM. When the IM stability mechanism has been viewed, the status of STVI is detected by monitoring the predicted slip trajectory to get to the IM’s UEP in the look-up table. The method was tested on the New England 39-bus system.

According to previous studies by (Ettehadi et al., 2013; Prabha, 1994), the VS problem is a dynamic character and can be solved using static analysis techniques. Increasing voltage rate changes with high demands in some nodes, called sensitive nodes, leading to voltage collapse (Canizares et al., 1996; Mithulananthan & Oo, 2006; Roy et al., 2012). Identifying the critical bus/node that is liable for voltage collapse and controlling the bus voltage using DG placement (Roy et al., 2012; Sreedharan et al., 2020) is important in the power system. (Samuel et al., 2017) came up with a new method and a mathematical formulation, for predicting voltage collapse. Several indices methods have been used to predict voltage collapse points. For example, Musirin and Abdul Rahman (Musirin & Abdul Rahman, 2002) discussed an FVSI method to find the maximum load-ability in a PS. Moghavvemi et al. (Moghavvemi & Faruque, 1999) also came out with a method of predicting voltage collapse based on finding the most stressful line liable to voltage collapse (Mohamed et al., 1998;Mohamed & Jasmon, 1989). However, the authors use the concept of a single line diagram for power transmission.

Another category of voltage assessment indices uses the eigenvalues vector and eigenvalues of the Jacobian matrix, which is used to detect voltage collapse points by studying the lowest eigenvalues of the Jacobian matrix, the value of zero at the voltage collapse point. The Eigen vector’s study or analysis is used to find the unstable bus in the system with the correct vector and the direction of change of injected power with the left vector. These indices are nonlinear and insensitive to parameter variations; when the system is extensive. However, the process is expensive in computation (Canizares et al., 1996; Gao et al., 1992a; Löf et al., 1993).

Having identified the sensitive nodes in the voltage stability analysis, we study the changes in the unstable points caused by adjusting the PS parameters (Pai & Stankovic, 2007). To study the load power at saddle-node bifurcation (SNB), two methods have been proposed, the direct method (DM) and the continuation method (CM). The CM method is based on computational power flow (CPF), which many researchers widely use to study the load flow solution, such as P-V, P-Q-V, and V-Q curves (Chiang et al., 1995) which others have previously described (Hedayati et al., 2008; Pai & Stankovic, 2007; Prabha, 1994). P-V curves have been used to investigate the VS of PS. The voltage stability margin (VSM) information shows how far the system is close to the voltage collapse point or SNB. However, it was reported that there were few changes in the active power consumption after the critical point, which made the voltage disappear while observing the P-V nose curve, and the load power factor (PF load) maintained constant. Therefore, the Q-V curve has been posited to be useful in studying the voltage instability issues in power systems (Mogaka et al., 2021; Van Cutsem, 2000; Vournas, 2020). Reactive power reserve data can be determined through reactive power margin (MVAR) since VSM and MVAR alert the system operator about load-ability and critical conditions. Therefore, proper planning, protection, and control make the system far away from any contingency that may want to occur (Al-Shaalan, 2020; Hedayati et al., 2008; Pai & Stankovic, 2007; Prabha, 1994).

In addition, the CPF has been observed to be sufficient to find VSM and voltage collapse points or SNB. On the other hand, modal analysis has been used to find the best place to place reactive power compensation, load shedding scheme, and generator re-dispatch (Ettehadi et al., 2013). Many VSI can be used to determine the weakest bus in the system (Abdel-Akher et al., 2011; Aman et al., 2012; Juanuwattanakul & Masoum, 2012; Kaya & Chanda, 2013; Kayal & Chanda, 2013; Murty & Kumar, 2015b; Raja et al., 2013; Sedighizadeh et al., 2010; Vinoth Kumar & Selvan, 2009).

5. Application of PSO in discrete and multiobjective optimization (MOO)

Kennedy and Eberhart develop PSO in 1995. PSO is an intelligent algorithm that came to overtake genetic algorithm (Zhu, 2008). PSO is based on the population search algorithm method, which searches the social behavior of birds/animals. PSO intends to emulate a bird flock’s innovative and unpredictable choreography (Eberhart & Kennedy, 1999), planning to get fly bird ability and change direction by regrouping in optimal formation (Pedrycz et al., 2016). It was realized that the principle of operation is simple and efficient optimization algorithms (Eberhart & Kennedy, 1999).

A population (swarm) of particles starts flying in hyper-dimension search space from the initial position. It is memory influences the charge of each particle’s position within search space, and that of neighbors is called social knowledge or cooperation. Each particle position represents a solution. The individual in a particle swarm follows a simple attitude that emulates the neighboring individual’s success and success in discovering the optimal region in the search space (Andries_P._Engelbrecht, 2007).

PSO algorithms exist in two forms: global PSO (GPSO) and local PSO (LPSO; Fukuda et al., 1992; Imran et al., 2013). The GPSO only searches for the best global solution. The LPSO algorithms are practical optimization problems that search for more than one global solution needed (Slowik & Kwasnicka, 2018). At the beginning of PSO, the randomly selected neighbor learning strategy is introduced, which is an improved version (Sun et al., 2017).

The prior version of PSO is useful for continuous optimization problems without constraints, but now there are many changes/modifications (Del Valle et al., 2008; Imran et al., 2013). PSO algorithms can now be used for a wide group of optimization problems with constraints (Sun et al., 2011; Zahara & Kao, 2009), multi-objective optimization (Preetha Roselyn et al., 2014), and combination optimization (Jarboui et al., 2007; Shayeghi et al., 2010; Steenkamp, 2021). There is an improvement in the convergence of PSO due to the update that has taken place. Among them are inertia weight, velocity clamping, and construction coefficient. Inertia weight is given for better control of exploration and exploitation of particle swarm. Velocity clamping ensures that the new position of particles is located at the acceptable space of a given search space. The primary function of the constriction coefficient is to balance PSO algorithm properties between local and global searched solution space (Slowik & Kwasnicka, 2018). The flow chart of the PSO algorithm is given in Figure 5.

Figure 4. A one-line diagram.

Figure 5. Flowchart of PSO algorithm

The position and particle velocity equation is given below

(70) $\mathrm{V}\left(\mathrm{t}\right)=\mathrm{V}\left(\mathrm{t}-1\right)+C_1\ast r_1\ast (P_{\mathit{best}}-\mathrm{X}\left(\mathrm{t}-\mathit{\hspace{0.17em}}1\right)+C_2\ast r_2\ast (G_{\mathit{best}}-\mathrm{X}\left(\mathrm{t}-\mathit{\hspace{0.17em}}1\right)$(70)

(71) $\mathrm{X}\left(\mathrm{t}\right)=\mathrm{X}\left(\mathrm{t}-1\right)+\mathrm{V}\left(\mathrm{t}\right)$(71)

Where,

X(t) is the position of the particle.

V(t) is the velocity of the particle.

$G_{\mathit{best}}$ is the global best.

$P_{\mathit{best}}$ is the personal best.

$r_1$ and $r_2$ are two random numbers in the interval of (0,1) with uniform distribution.

$C_1$ and $C_2$ are the coefficients of accelerated particles.

V(t—1) is the inertial weight of the particle.

5.1. Discrete PSO

The optimization problem comprises binary or discrete; practical examples include routing and scheduling problems. Initially, the updated formula of the PSO algorithm and the corresponding procedures were planned for a continuous space, limiting its applications to the discrete optimization domain. These thereby necessitate some modifications for adapting to a discrete space (Wang et al., 2018).

In a continued PSO, a trajectory is the change in the position of the dimension numbers. On the other hand, a binary PSO (BPSO) involves the possibility that a coordinate takes one and zero values. For example, Jian et al. (Jian & Xue, 2004; Xu et al., 2021) used discrete BPSO for structural optimization of a neural network where a sigmoid function was used. The velocity was reported to take a value of 0, 1, interpreted as the change in probability When re-defined the velocity and position, continued PSO changed to discrete to give a solution to an optimization problem in discrete form. The method was also extended to solve quantum space by Afshinmanesh et al. (2005); Ratnaweera et al. (2004); Tang et al. (2011), who gave more presentations about discrete PSO.

Furthermore, angle modulation PSO (AMPSO) is another modified form of binary PSO, used to produce a string bit to solve original higher dimensional problems. Using direct mapping back to the binary space by an angle of modulation, the higher dimensional problem is minimized to the four-dimensional problems in continuous spaces (Mohais et al., 2005). The genetic BPSO model was reported by Peer et al. (2003) without fixing the sizes of the swarms. An algorithm comprising two forms, including death and birth, was used to modulate the swarm dynamically. It was observed that death and birth rate changed with time; the BPSO model allowed oscillations in the swarm sizes. Naturally, it is social behaviors were ascribed to the intelligent animal (Peer et al., 2003). The binary PSO enhancement was reported by (Kadirkamanathan et al., 2006), who used the phenotype-genotype methods and gave a mutation operation of GA to BPSO (Kadirkamanathan et al., 2006). Beheshti and Shamsuddin (2015) reported a memetic BPSO with a hybrid global and local search in BPSO. The topology of binary hybrid PSO has been used to solve the problem of optimization in a binary search space (Beheshti & Shamsuddin, 2015).

According to (Afshinmanesh et al., 2005), who proposed the techniques of estimation and fitness inheritance, involving achieve four (4) estimations and fifteen (15) effects of inheritance techniques need to be applied to the MOPSO algorithm.

There are two (2) approaches to keep the MOPSO diversity, ε-dominance and the sigma techniques (Criticality & Løvbjerg, 2002; Juang et al., 2011; Robinson & Rahmat-Samii, 2004). The multi-swarm PSO algorithm can be divided into three equal sub-swarm parts. Every equal sub-swarm part needed to be applied with a diverse mutation coefficient to improve the search particle’s capacity.

5.2. Multi-objective optimization (MOO) PSO

In the MOO problem, each individual target function is optimized separately and then determines the optical values for each, but conflicts/issues usually arise among the objects, thus making it difficult to get the exact solutions for all the objectives. Based on this, only Pareto optimum solution has been proven.

In PSO, the exchange of information is significantly diverse from another optimization tool based on a swarm. In GA, the exchange of information chromosomes is through crossover operation, thus making it a two-way (called bi-directional) information exchange mechanism. The PSO algorithm uses nBest or gBest to provide information to other individuals. Due to attractive point features, the conventional PSO algorithms could not locate many optimal points containing the Pareto frontiers (Wang et al., 2018). Many optimal solutions can be obtained by giving various weights to all objective functions, running, and combining them many times. Therefore, the need to find a method that can be used to obtain a group of Pareto optimal solutions simultaneously is needed. Ghodratnama et al. reported a wide-ranging PSO algorithm with Pareto dominant to solve the MOO problem (Ghodratnama et al., 2015). Also, the elitist MOPSO, which combines the elitist mutation coefficients to improve individual capacity exploitation (Ozcan & Mohan, 1997), has been reported. The proposed iterative MOPSO vector-based controlled parameterizations for coping with the dynamic optimizations of the states constrained chemical and biochemical engineering challenges (Wang et al., 2011) have been developed. Some other research groups have developed the corresponding MOPSO algorithms (Bornapour et al., 2020; Chen et al., 2014; Clerc & Kennedy, 2002; Fan & Yan, 2014). Li (Li, 2004) reported a new cultural MOQPSO algorithm where the cultural evolution mechanism has been incorporated into quantum-behaved PSO for dealing with multi-objective problems. In the MOQPSO algorithm, every individual particle’s position was obtained based on belief space containing a diverse knowledge type. To increase diversity in the population and attain a well (of evenly) distributed Pareto front and continuous, a combined-based operator was used to update the external population (Li, 2004).

5.3. Advantages of PSO

The following are the advantages of PSO.

1) Easy implementation and lesser parameters.

2) Good memory capability.

3) More efficient in maintaining the diversity of the swarm since all the particles use the information related to the most successful particle to improve themselves (Clerc, 2010; Del Valle et al., 2008).

5.4. Application area of PSO to PS

There are many areas in which PSO can be applied to PS engineering

5.4.1. Load flow (LF) and optimal power flow (OPF)

The essential tools for designing and analyzing power systems are the load flow used for planning, operation, economic schedule, voltage stability, transient stability, and contingency studies. LF is an optimization problem with the objective function of finding the magnitude of voltage and reducing/minimizing the input and output difference. The LF equation finds the active power (P) and reactive power (Q) at each bus/node. Solving LF problem either by using a conventional method like Gauss-Seidel (GS), fast decouple, and Newton-Raphson (NR); this conventional method may fail at heavy load (El-dib et al., 2004). For this reason, the use of PSO algorithms comes into place. The mathematical formulation is given below:

(72) $\mathrm{Minimized}\mathit{f}\left(\mathit{v},\mathit{q}\right)={\sum }_i(f_{pi}^2+f_{qi}^2)$(72)

Where,

$f^2{ }_{\mathit{pi}}$ and $f^2{ }_{\mathit{qi}}\mathit{\hspace{0.17em}}$ is nonlinear LF equations.

Subjected to the following constraint

1. Scheduled the value of power and PV bus voltage

2. Scheduled the value of slack bus voltage

For the particle, it is given as

$\mathrm{X}=(V_i,\dots \dots .V_n,q_i,\dots \dots q_n)$

$q_i$ $\mathrm{and}{V}_i$ are the phase voltage and magnitude voltage of the bus.

To improve the performance of PSO, decrease constriction factors, and establish mutation in particles as in GA, PSO can solve LF problems with minimum tolerance and give a solution to more loading case where NR fails to provide the solution. For OPF, the main goal is to determine the best optimal solution to the power system objective function like voltage magnitude difference at various buses, total loss, and real and reactive power (adjusting system control variable) (Abido, 2002). Conventionally, the traditional methods are optimization tools that are used to give solutions to the nonlinear problem, linear and nonlinear optimization (LP and NLP), quadratic programming (QP), interior point method, etc. However, those methods have their associated advantages and disadvantages. Due to the weaknesses, the PSO method has solved OPF problems (Abido, 2002).

(73) $\mathrm{Minimized}\mathrm{f}\left(\mathrm{x},\mathrm{u}\right)$(73)

Where,

F (x, u) is the objective function of the power system

$\mathit{x}$ is the load bus voltages or apparent power flow.

$\mathit{u}$ is the vector of the control variables such as active and reactive powers and voltage magnitudes at generator buses.

The constraints that needed to be met include.

1. Load flow equations.

2. Upper and lower limits for each generator (Generation constraints).

3. Minimum and maximum tap setting (Transformer constraints).

4. Shunt VAR constraints.

5. The upper and lower boundaries of control variables.

6. Security constraints (Abido, 2002; Del Valle et al., 2008; El-dib et al., 2004).

5.4.2. Reactive power and voltage control

As both line and load are subjected to change, either in operation or not, the electric power network will vary; therefore, keeping voltage within an acceptable range for the consumer/end-user is essential for an electric utility company. To achieve this, the power utility company will control the transformer tap settings, FACTS devices, and synchronous generator to give them an exact value of reactive power needed to maintain the bus/node voltage at required values. For this to be achieved, an online control strategy must be used.

Essentially, voltage control and reactive power strategy must ensure that the voltage does not migrate toward the voltage collapse (Van Cutsem, Thierry, 1998). The equation is formulated as

(74) $\mathrm{Minimize}\sum \mathit{loss}$(74)

Therefore, the following constraints need to satisfy.

1. The voltage at all buses must be at its exact range.

2. The power flow at all branches must be smaller than the maximum allowable.

3. Transformer tap positions should be at the same range of steps.

4. Reactive power of the generator must be at an allowable range.

The constraints can be formulated to maintain the security of the voltage in the PS (Van Cutsem, Thierry, 1998; Yoshida et al., 2001).

5.4.3. Power system reliability and security

The reliability of a PS is to supply enough facilities to the system to meet the load demands for a given system condition. Security of PS is the capability of the system to overcome disturbances that occur within (Meeuwsen, 1998). Many researchers have analyzed the power system reliability indices using contingency analysis, considering multiple failures and an extensive, more complicated, and time-consuming network. Binary PSO is applied to identify disrupted network elements, possibly leading to a series of cascades resulting in extensive network damage.

Reconfiguration of feeder is a method used by some authors to improve the quality of the service provider for the customers, while the network reliability is maintained. The NLP problem is subjected to the security constraints of the network distribution system, so it does not exceed busbar capacity, while radial network structure is maintained. Many researchers have reported the applications of binary PSO as a successful tool for reconfiguring feeders (Chang & Lu, 2002; Jin & Zhao, 2005; Robinson, 2005; Shen & Lu, 2002).

Applied PSO is another tool to increase the reliability of the PS to find a good place for decriminalized devices in distribution lines. The objective function is to reduce the annual feeder interruption cost, and the particle is located at the network switch. It directly affects the outage time of the feeder, but if it is correctly done, it improves the reliability of the system network (Kurutach & Tuppadung, 2004a, 2004b).

6. overview of voltage stability using PSO

Voltage instability using PSO is discussed based on the single and combined optimization techniques. Single optimization is when only one technique is used without combining it with other methods, while combine optimization is when one or more processes are employed to solve a given problem. Combined optimization techniques are also called hybrid methods of optimization. One of the significant advantages of PSO over other methods in PS is that it accommodates the use of combine/hybrid algorithm methods that compensate for the liability gap of one algorithm, such that one algorithm improves the weakness of the other, giving better results than a single algorithm. PSO is valuable and relevant in PS due to its fast convergence and effective performance in reducing losses in the transmission and distribution networks. It helps to improve the voltage profile of the system more than other algorithms. Based on this, the next subsections discuss the applicability of single and hybrid PSO in voltage stability.

6.1. Voltage stability using single optimization method (PSO only)

According to Ansari & Joshi (2015), the usefulness of PSO for optimum placement and size of static VAR compensator in a transmission system was studied. By using the objective function to minimize the power loss and deviation in voltage or a combination of the two, its effectiveness was tested in IEEE 14 and 5 bus by applying the MATLAB software.

Joshi and Pandya used the PSO algorithm to minimize the deviation of rescheduled values of the generator’s active and reactive power by considering voltage profile improvement and VS enhancement. Generator active and reactive power sensitivity factors to the congestion line were used to find the total number of generators involved in congestion management. It was concluded that the PSO minimized the cost of rescheduling active and reactive power. The algorithms were tested in IEEE 30 bus system, and the quality of the result was compared with other literature (Joshi & Pandya, 2011).

Rajalakshmy and Paul used the generator rescheduling for voltage stability enhancement, the active and reactive power rescheduling was perfectly done due to system disturbance. PSO was used to obtain the optimum values of rescheduling, and it was simulated to reduce the total reactive power loss. The system’s constraints of load balance bus voltage and reactive and active generator outputs were reported to be within the minimum and maximum loads (Rajalakshmy & Paul, 2015).

The discussion about the use of the optimum location of Static Var Compensator (SVC) and Thyristor Control Series Capacitor (TCSC) for reactive power planning using PSO-based technique to locate the optimum placement of TCSC and SVC has been documented by (Auchariyamet & Sirisumrannukul, 2010; Jamnani & Pandya, 2020). The objective function was considered, such as reduction in the installation cost of SVC or TCSC, energy loss cost, and the total cost. These were subjected to the LF equation, the generators’ reactive power output at voltage-controlled buses, bus voltage limits, and limits of setting values for SVC and TCSC. A modified IEEE 14 bus system was used to test its performance, giving the optimal solution, while the constraint and objective function were met.

Sonwane and Kushare used PSO for optimum placement of capacitors and sizing. It helps reduce the power losses, improves the power factor quality, and helps keep the voltage profile. It is reported that the method improves the reliability of the system (Sonwane & Kushare, 2014).

Mandal and Tudu implemented an improved PSO technique for designing a hybrid PV, wind turbine, battery bank, and diesel generator as a backup system. The results obtained by the improved method are compared with an iterative method, giving better results. The use of an Interline Power flow Controller (IPFC) to maintain the stability of the PS has also been presented (Praveen & Srinivasa Rao, 2016). The PSO technique was used to identify the best position for placement of IPFC. The objective function is the generation cost, L-index, and transmission losses (Mandal et al., 2016).

Haider et al. presented the issue of voltage instability and real power in PS, which has reduced the system’s performance. Therefore, the optimum location and sizing of DGs and multi-objective PSO were used in radial distribution networks before and after reconfiguration. It was reported that with the optimum network configuration of DG, the voltage profile was improved, power loss was reduced, and improved the system’s reliability and efficiency. The technique was evaluated on IEEE 33 distribution system (Haider et al., 2021).

Sensitivity analysis was introduced to find the optimum placement and sizing of the DG unit. Autonomous group PSO (AGPSO) is the optimization method used to reduce the active power loss in the system. The IEEE 33 test system was used to test the method’s performance, and it was concluded that the voltage profile was improved (Kiran & Chandana, 2017).

6.2. Voltage stability using a combination of algorithms (PSO plus any other)

Having realized the effectiveness of a combined system in PSO for examining the voltage stability mentioned above, numerous studies have been conducted to establish the performance of the hybrid PSO method over the single method. For example, Ibrahim and El-Amary (Ibrahim & El-Amary, 2018) presented the recurrent neural network (RNN) in voltage instability. The RNN is used together with PSO. The method is tested on 14 and 30-bus IEEE standard system and simulated using MATLAB/power system toolbox program. The PSO and Backpropagation (BP) algorithms were also compared.

The use of hybrid techniques has been reported by (Bhattacharyya et al., 2009), who discussed reducing actual power loss and improving the system’s voltage profile, PSO, and fuzzy logic for optimal reactive power planning. Transmission loss was expressed as the increments in voltage by relating the tap position of transformers, reactive power injection by shunt capacitors, and the reactive VAR generator. PSO was used for the optimal reactive generation and transformer tap position setting. Therefore, the solution was compared with differential evolution (DE) and GA.

The discovery of modifying PSO and Artificial Bee Colony (ABC) with hybrid-GA (H-GA) for appropriate sizing of FACTS devices such as the unified power flow series compensator (UPFC) controller to improve the bus voltage and voltage stability was reported (Harish Kiran et al., 2016). The FVSI method of VSI was used to identify the bus where the FACTS device was placed. Itis effectiveness was then tested on the standard IEEE 30 bus system. However, it has been reported that the capacitor bank (CB) and DG allocated in the distribution system (DS) have the potential to improve the performance of the radial distributing system (RDS; Venkatesan et al., 2021). They used hybrid grey wolf optimization and PSO (EGWO-PSO) for optimum sizing and placement of DGs and CBs. EGWO optimization algorithms were simulated by a grey wolf. Through particle movement, PSO was used to find an optimum solution to a given problem. According to their investigation, the hybrid methods have high converged speeds and do not trap in local minima. The advantage was enhanced using a multi-objective function (MOF) such as minimized voltage deviation index (VDI), real power losses, the total cost of electrical energy, etc. At the same time, the method’s effectiveness was tested on IEEE 30 and 69 bus systems. The results were compared with other optimization algorithms, and EGWO-PSO was observed to give better performance (Venkatesan et al., 2021).

(Al-Ismail & Abido, 2011) presented STATCOM-based stabilizers to stabilize power systems. The PSO, DE, Tabu Search, GA, Simulated Annealing, and Evolutionary Programming (EP) optimized the stabilizer tuning parameters. PSO and DE showed a significant improvement regarding settling time and PS stability.

The use of a modified PSO (MPSO) has been reported by (Lin et al., 2012), which gave a technique for the proper location of a unified power flow controller (UPFC) with an ECI model to increase VS. The method’s effectiveness was carried out in the IEEE 30 bus system. It was reported that the MPSO gave the best performance to optimize the UPFC and improve the VS.

Singh presented multi-objective optimal reactive power control to minimize the power loss and improve the VS using the hybrid Multi-Swarm PSO (HMPSO) algorithm. The results were tested on IEEE 30 bus system and compared with the other three (classical PSO, classical DE, and modified DE algorithms). HMPSO was observed to give good performance and efficiency (Singh, 2016).

Another study conducted by Rekha & Kannan (2013) proposed a mathematical model for optimizing reactive power using PSO and GA. The GS method was used with PSO to get the optimal power value. The result showed that the PSO has a better performance than GA.

Jumaat et al. proposed a new approach to minimize transmission loss, improve the voltage, and monitor installation costs. The meta-heuristic technique (Evolutionary PSO) was reported to be feasible with PSO, and EP gave a better result when tested on the IEEE 30 bus system (Jumaat et al., 2014).

Balakumar et al. reported using Moth Flame Optimization (MOF) and PSO to locate and size SVC in DN to overcome the problem of voltage instability and line losses. The authors minimized the voltage deviation and power loss in the system. The superiority of the methods was tested on MATLAB software using Wolaita Sodo radial distribution of 34 bus systems. It was reported that the voltage profile was improved even during peak hours (Balakumar et al., 2021).

A hybrid firefly and PSO (HFPSO) algorithm is used to find the best sizing of DSTACOM and DG. Multi-objective functions were considered to improve the voltage profile of the system. VSI is used to locate the optimum place of DSTACOM, and DG and HFPSO were used to find the optimum sizing of both DSTACOM and DG. The method was demonstrated on IEEE 33 and 65 radial distribution systems and improved the voltage stability effectively (Al-Wazni & Al-Kubragyi, 2022).

Firefly algorithm and adaptive particularly tunable fuzzy PSO (APT-FPSO) was reported to minimize the power loss, VDI, and VSI. The method’s accuracy was tested on many benchmarks optimization functions and IEEE 30, 57, and 118 test systems. The result showed that the method could improve the system’s profile and handle complex optimization problems (Nasouri Gilvaei et al., 2020).

7. Challenges and future prospects

1. PSO has been proved effective in PS design at the distribution level. PSO has been presented to balance loads between the feeders or by finding the location for sectionalized devices. Also, it has been effective in distribution system reconfigurations for aggregating the loads. Future research can be expanded to distribution layout design, overhead line configuration, transformer design, substation location, etc. Expansion in the future and financial computation can be incorporated into fitness functions.

2. There is a need for mathematical investigation of PSO characteristics and search behavior for the optimal solution. Further work is needed to improve the overall performance characteristics to avoid being stocked into local optimal minima in the search space.

3. PSO, techniques have been used to solve nonlinear optimization problems for capacitor placement in the PS network. The work could be further extended to the placement of the FACTS device and on-load tap changing transformers to optimize the result compared with and without optimization. Also, in an analytical-based technique that may undergo the curse of dimensionality, PSO will still manage to give optimal firing angle and switch position.

4. PSO application should not be limited to a single objective, deterministic optimization problem, and continuous. There is a need to focus on multiobjective, discrete, constraint, and dynamic optimization. Therefore, there is a need to further expand PSO applications in the areas.

5. Also, the application areas in which PSO has been successfully applied to the PS were discussed. Yet, some areas are not looked into, like restoration, protection, electric machinery, etc. The previous works have shown the excellent performance of using combination algorithms (PSO with many others) compared to only one algorithm. The combination algorithms perform fantastically because one improves the weakness of the other. Also, it provides a good balance by making the algorithms reach all the search space and search for an optimum solution within the search space. In the research survey, most of the work done is based on PSO to minimize loss without using indices to identify the weak bus in the system. Therefore, more work is needed to use PSO and indices to identify weak buses and optimize loss in the system. Also, there is a need to use more hybridization algorithms than single ones because hybridization algorithms give excellent results than a single algorithm.

6. The proposed algorithm’s performance and robustness have been compared to other metaheuristic algorithms. None of the studies has covered the hybrid’s performance and effectiveness based on a conventional metaheuristic approach. Therefore, this is recommended for future researchers.

7. The problem facing power system operation is a challenging factor that needs proper attention due to the enormous population growth, which increases the load daily. New industries are being established daily, raising the load and making power systems operate beyond their capacity limit, leading to voltage instability. If proper attention is not given to voltage instability, the world will continue to experience voltage collapse, and both accompany each other. Due to issues of voltage instability, many researchers have been given various methods of VSI, such as the line stability index ($L_P$), line stability index ($L_{\mathit{mn}}$), fast voltage stability (FVSI), line stability factor (LQP),L-index, and voltage stability index (VCI-1) and so on. This review emphasized that the value of the index must be below unity (i.e., 1) for a stable system; anything greater than 1 leads to an unstable system in one of the lines connected to the bus. This study has presented various measures to be adopted to prevent voltage collapse. The use of optimization algorithms such as PSO comes in place to overcome the challenge, which also reduces transmission losses, the computation time, and offers an accurate result which makes it effective when compared to other optimization techniques, thereby providing it with numerous opportunities that are applicable in many areas to solve real-life problems. Therefore, more work is needed to combine two or more indices to form a single one to be more robust and efficient in identifying the weakest bus/node in PS networks.

8. Concluding remarks

This review has discussed some stability indices that could identify the weak bus in the electrical PS network. The application of PSO to reduce/minimize losses that cause voltage instability, leading to voltage collapse, was discussed. Insight into the detrimental effect of power blackouts on the global economy resulting in voltage instability was provided. This study gave an understanding of the voltage instability/stability phenomenal, voltage stability and its classification in PS, and the corresponding formulations. Distinctly, it presents an overview of voltage assessment techniques, the application of PSO in discrete and multiobjective optimization, and the corresponding advantages over others. The progress and advancements in voltage stability using PSO involving single and hybrid optimization methods have been discussed. At the same time, the current research gaps and some challenges/prospects to foster further advancement in the field were also highlighted. PSO has been used to reduce/minimize losses and to identify a proper location where FACTS devices are to be placed to improve the voltage profile and reduce voltage instability that can lead to voltage collapse. This work will be helpful for future researchers because it will make identification of the bus that is liable to voltage collapse easier, and PSO will provide insight into better losses reduction in transmission networks compared to other algorithms.

Authors Contributions

Samson Ademola Adegoke: Conceptualization, data curation, formal analysis and investigation, methodology, validation, visualization, writing-original draft preparation, and writing- review, and editing. Yanxia Sun: Conceptualization, validation, visualization, review, editing, and supervision. All authors read and approved the manuscript.

Availability of data and materials

All data and materials used in this study are available within this article.

Acknowledgements

This work was supported in part by grant of Global Excellence and Stature (GES) University of Johannesburg, South Africa, National Research Foundation under Grant 120106 and Grant 132797, and in part by the South African National Research Foundation Incentive under Grant 132159.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by grant of Global Excellence and Stature (GES) University of Johannesburg, South Africa, National Research Foundation under Grant 120106 and Grant 132797, and in part by the South African National Research Foundation Incentive under Grant 132159.