© The Author(s), published by EDP Sciences, 2024

Nomenclature

s and r : Sending and receiving bus indices

m : Mono frequency system

S : Bus system

T’: Variable tap transformer

n : Iteration counter

N : Total number of buses

Nc: Number of STATCOM locations

NL: Number of security constraint buses

NLsr: Number of low frequency lines

Variables

V_s : Voltage magnitude at sending bus

V_r : Voltage magnitude at receiving bus

Z_o : Characteristics Impedance

$P_{\mathrm{sr}}^g$ : Power generation bus

$P_{\mathrm{sr}}^d$ : Power delivery bus

$Q_{\mathrm{sr}}^g$ : Reactive power of power generation bus

$Q_{\mathrm{sr}}^d$ : Reactive power of power delivery bus

P_vc, Q_vc : Real and reactive power of STATCOM VSC

P_sr, Q_sr : Real and reactive power injected

${\varpi }_{\mathrm{sr}}$ : Angle of VSC of STATCOM

δ_sr : Angle of bus by which STATCOM is connected

$T_s^{\min },T_s^{\max }$ : Transformer maximum and minimum tap setting

$Q_{c,s}^{\min },Q_{c,s}^{\max }$ : STATCOM max. and min. reactive power

$f_{\mathrm{Ls},r}^{\min },f_{\mathrm{Ls},r}^{\max }$ : Minimum and maximum value of operating frequency of LF-HVac subsystem f_L,sr

ΔA_e, ΔV_mn : Change in active DC power value and change in voltage of VSC converter of STATCOM

Abbreviations

OLF: Optimal Load Flow

LF-HVac: Low Frequency High Voltage Alternating current

STATCOM: Static Synchronous Compensator

MJAYA: Modified Jaya

1 Introduction

Optimal Load Flow (OLF) is inherently applied for mono-frequency systems, either 50 Hz or 60 Hz from the beginning. The change in frequency has resulted in a divergence between the aforementioned studies that prioritize voltage and the current study. Multiple converters have been developed to address frequency variations in power systems resulting from advancements in technology. The introduction of HVac brought about significant changes in the transmission of power, involving various constraints and parameters. This marked a change from the previous utilization of HVdc for power transmission. This solution facilitates the management of a multi-frequency system by considering frequency as a key parameter for achieving optimal performance. The successful implementation of the frequency-dependent OLF necessitates the fulfillment of three specific requirements. The requirement for a system that considers frequency arises from the fact that various renewable energy sources, such as wind farms, Pumped Storage Hydro (PSH), and gas turbines, produce power at different frequencies. This model incorporates frequency into existing power system models, wherein the frequency varies based on dependent parameters. Moreover, the incorporation of frequency into renewable interconnected systems can lead to fluctuations in additional parameters, including voltage stability and losses. These factors need to be carefully monitored in order to analyze the implications of frequency alterations within the system. In this study, our focus was on maintaining voltage stability and minimizing losses in the presence of frequency changes. Thirdly, the OLF (Optimal Load Flow) itself involves complex computations due to the presence of constraints, non-constraints, and parameters. In this study, the matrix formulation can be reduced by employing a reduction method, such as Telegrapher’s method, which takes into account the frequency-dependent OLF [1–11].

The utilization of power transmission at frequencies below 50 Hz or 60 Hz offers multiple benefits compared to power transmission at a fundamental frequency. The power flow solution and the system are both affected by the frequency, which is dependent on various parameters of the lines. Low frequency is commonly utilized for the analysis of transient and steady stability concerns. Therefore, it is necessary to incorporate low frequency in load flow analysis in order to accurately assess the depth calculations within the system [12–16]. Low frequency is widely utilized in various industrial applications, particularly in the development of new converters and railway systems. Recent studies have highlighted the significance of determining the optimal frequency for load flow in low-frequency transmission systems [17–19].

Nguyen and Lao in the OLF for minimizing low-frequency losses [19] introduced the predictor–corrector primal-dual interior-point method. The authors established a connection between the fundamental frequency grid and a low-frequency system in order to conduct an analysis of losses. In the thirteenth iteration, the objective function was expanded to accommodate multi-objective solutions that incorporate interfaces to low-frequency systems. BTB converters are commonly employed for interfacing multi-frequency systems. The optimal operating frequency for the BTB converter has been determined by power dispatch. Researchers have not yet determined the optimal frequency for transmission lines and converters. In order to mitigate the impact of frequency changes on system voltage, it is imperative to conduct an analysis of custom power devices. The current study does not incorporate the utilization of compensating devices to ensure voltage stability.

The frequency-dependent Optimal Power Flow (OLF) was calculated in this study by incorporating a multi-objective function. This function aimed to minimize losses and ensure voltage stability across the entire system. The utilization of STATCOM in power systems has proven effective in enhancing voltage stability and mitigating voltage fluctuations [17, 20–24].

The optimal load flow analysis is a method used to determine the operating state of a power system at the optimal point [25]. The control variables are responsible for generating voltages at precise angles and power levels. The control variables are solely dependent on the line parameters. In the past, there was no existing method that took into consideration the possibility of the frequency being dependent on line parameters. The individual in question is Roy MC. The lumped circuit model is commonly used to represent frequency-dependent line parameters. This model allows for the relationship between the variables and constraints of the load flow to be expressed as a function of frequency [26]. The components were constructed using the Telegrapher’s method. However, the current approach does not prioritize optimal load flow solutions that consider constraints and multi-objective functions.

The implementation of two-port network modeling of Telegrapher’s solution was conducted in this study, with the aim of incorporating frequency into the proposed model. The Telegrapher’s method solutions were utilized in the modeling of the objective functions for the optimal load flow solution.

The OLF, or optimal load flow, is classified as a mixed-integer non-concurring problem in conventional multi-frequency systems operating at 50/60 Hz. It involves nonlinear programming techniques. Various methodologies have been employed by researchers in order to address the OLF (Optimal Load Flow) predicament. These methodologies encompass linearization techniques as well as nonlinear programming methods. Nonlinear programming is a reliable and accurate approach for solving the OLF problem, in contrast to the limited accuracy and reliability of linearization. In a previous study [27], the authors demonstrated that the conventional Levenberg approach of Marquardt’s method may encounter convergence difficulties. Metaheuristic search methods are known for their robustness and flexibility, making them a valuable tool in tackling a wide range of complex problems. The primary objective of both metaheuristic and heuristic algorithms is to generate improved outcomes within a specified timeframe.

In 2016, a population-based approach known as the JAYA algorithm was introduced [28]. One advantage of JAYA is its ability to provide global or nearly global solutions to the optimization problem without the need for any parameter adjustments. As per JAYA, it is recommended that each individual in the population make efforts to enhance their own abilities by emulating the exemplary members and refraining from following the unproductive ones [28]. The JAYA algorithm possesses notable characteristics, such as its user-friendly interface and efficient convergence capabilities. The JAYA algorithm, like other heuristic algorithms, exhibits susceptibility to computational complexity and the potential to get stuck in local optima [29]. Researchers have proposed different versions of the original JAYA algorithm, such as the modified JAYA, in order to mitigate these drawbacks. In this study, we have employed a modified Jaya algorithm to tackle these concerns. Our approach involves the utilization of a frequency-dependent objective function. The MJAYA algorithm is designed to address the OLF problem by minimizing losses while simultaneously ensuring the preservation of both the voltage stability index and the voltage deviation index [30–34].

The remainder of this paper is organized as follows: Section 1 describes the literature related to LF-HVac, OLF, and Telegrapher’s equation. Section 2 describes the system methodology. Section 3 discusses the various aspects of the proposed formulation. In Section 4, we discuss the solution approach. Finally, different case studies are discussed in Section 5, and conclusions are presented in Section 6.

2. Formulation

In this section, the frequency dependence of the line parameters proposed using Telegrapher’s equation in multi-objective formation is explained. STATCOM was introduced for voltage stability in transmission systems. In addition, a two-port network with an impedance analysis is described in this section.

2.1 Frequency-dependent model and STATCOM

The linear combination of frequency as a function of the parameters of multi-frequency system is expressed in this section. f_1, f₂,…_. f_s are the different frequencies of the different systems m₁, m₂,…m_s.

Let us consider m₁ as a subsystem with one frequency f₁ and so on, as shown in Figure 1. Then, the power combination of the multi-frequency system with a fundamental frequency system can be represented by$${\left(P_{m_r}\right)}_{f_{m_r}}={\left(P_{m_1}\right)}_{f_{m_1}}+{\left(P_{m_2}\right)}_{f_{m_2}}+.......{\left(P_{m_s}\right)}_{f_{m_s}}$$(1) f can be less than or equal to 60 Hz.

Figure 1 Basic model of LF-HVac.

In Figure 1, a back-to-back converter was used as a frequency converter for each renewable system. Therefore, to integrate the multi-frequency system, the fundamental ratio coefficient k was introduced to correlate the two systems. The load flow line parameters were modified to make them frequency-dependent. K can be represented by equation (2), where f_r is the fundamental frequency, and f₁ and f₂ represent the lower frequencies of the power system.$$\frac{{ f}_{1,2\dots ,s}}{f_r}=k,$$(2)

Therefore, to modeling the branch and line parameters, the line resistance R, inductance L, capacitance C, and shunt conductance G in p. u. are taken as it is but frequency-dependent X as series reactance and B as susceptance of transmission lines are modified in terms of k as follows:$$X_{s,r}\left(f\right)=k·L_{\mathrm{sr}}\hspace{0.5em}\forall s,r\in S,$$(3) $$B_{s,r}\left(f\right)=k·C_{\mathrm{sr}}\hspace{1em}\forall s,r\in S,$$(4)

where s and r denote the sending and receiving buses, respectively. As we know, three-phase winding transformers are independent of frequency, so they depend only on tap values. In the proposed model, we generalized the STATCOM model to a real tap-changing transformer, where the tap ratio ‘T’ of the transformer changes with a change in the voltage of the respective bus.$$V_{\mathrm{sr},\mathrm{vc}}=T^{\text{'}}{e}^{j{\varpi }_{\mathrm{sr},\mathrm{vc}}}{E}_{\mathrm{dc}}.$$(5)

A simplified model of the STATCOM can be obtained using equation (5). Where V_sr,vc is the phase voltage of Voltage Source Converter (VSC) and ${\varpi }_{\mathrm{sr},\mathrm{vc}}$ is the phase angle, T ^′ represents the variable tap of the transformer and E_dc shows the voltage across the DC source of the converter).

For a three-phase winding transformer, the tap magnitude can be expressed as [20]$$T^{\text{'}}=\sqrt{3}/2 T .$$(6)

In the above equation, the tap ratio will be a continuously changing variable, and T ^′ will be rounded off to an integer with the nearest value.

2.2 Two port equivalent model of transmission system based on Telegrapher’s method

A simplified formulation between the two buses is shown in Figure 2 where Rdx, Ldx, Gdx, and Cdx are considered as transmission lines of small lengths.

Figure 2 Simplified model of transmission line.

The frequency G, L, and C are dependent quantities that change with transmission. Therefore, the transmission losses also depend on the G value.

Admittance of line can be expressed as$$Y=G+\mathrm{jB},$$(7) $$G^n\left(f\right)=G_1·{\left(\frac{f_s}{f_r}\right)}^{g_e},$$(8)where shunt conductance G is dependent on frequency and g_e is Terman’s statement constant, which is considered to be 1 for a uniform transmission line [7].

Voltage and current solution in case of sinusoidal steady state can be expressed as$$V\left(x,t\right)=\mathrm{Re}\left\{V\left(x\right)·e^{\mathrm{j\phi }t}\right\},$$(9) $$I\left(x,t\right)=\mathrm{Re}\left\{I\left(x\right)·e^{\mathrm{j\phi }t}\right\},$$(10)

where φ denotes angular frequency$$\frac{\mathrm{d}V}{\mathrm{d}x}=-L\frac{\mathrm{d}I}{\mathrm{d}t}=-\mathrm{j\phi }L·I,$$(11) $$\frac{\mathrm{d}I}{\mathrm{d}x}=-C\frac{\mathrm{d}V}{\mathrm{d}t}=-\mathrm{j\phi }C·V .$$(12)

Differentiating equations (11) and (12),$$\frac{{\mathrm{d}}^{2}V}{\mathrm{d}{x}^2}=-p^{2}V,$$(13) $$\frac{{\mathrm{d}}^{2}I}{\mathrm{d}{x}^2}=-p^{2}I,$$(14)

where $p=\phi \sqrt{L·C}$ is a frequency-dependent equation in the solution of the Telegrapher’s method. So,$$Y^n= G^n+\mathrm{j\phi }C .$$(15)

For a lossy transmission line of fixed length, the ports are excited by voltage sources at the sending end with an angular frequency φ.

Unbalanced single-conductor transmission lines were assumed in the proposed system, as shown in Figure 3.

Figure 3 Two port model of unbalanced transmission line.

Because x is constant for our proposed model, the voltage and current of the two-port network of the solution of the Telegrapher’s equations as an ABCD type can be expressed as$$\left[\begin{array}{c}{V}_s\\ I_s\end{array}\right]=\left[\begin{array}{cc}\cosh \left(\gamma x\right)& \sinh \left(\gamma x\right)·Z_o\\ \frac{\sinh \left(\gamma x\right)}{Z_o }& \cosh \left(\gamma x\right)\end{array}\right] \left[\begin{array}{c}{V}_r\\ I_r\end{array}\right],$$(16)where A = cosh(γx), B=sinh(γx)·Z_o, $C=\frac{\sinh \left(\gamma x\right)}{Z_o }$ and D = cosh(γx).

Equations (9)–(15) show that the solutions are frequency-dependent and are considered for optimal load flow solutions. From equation (16), the receiving and sending end voltages can be derived and expressed as$$\left\{ \begin{array}{c} V_s=V_r\cosh \left(\gamma x\right)+I_r\sinh \left(\gamma x\right)·Z_{o }\\ I_s=\frac{V_r\sinh \left(\gamma x\right)}{Z_o }+I_r\cosh \left(\gamma x\right) \end{array}\right\}.$$(17)

From (17) $$V_r=\frac{Z_o \left(I_s-I_r\cosh \left(\gamma x\right)\right)}{\sinh \left(\gamma x\right)}.$$(18)

3 Proposed formulation

3.1 Objective functions

3.1.1 Loss minimization

3.1.1.1 Losses in LF-HVac system

Losses can be expressed as$$F=\sum _{s=1}^N\sum _{\begin{array}{c}r=1\\ s\ne r\end{array}}^N\frac{G_{\mathrm{sr}}^n}{2}\left[{\left|V_s^n\right|}^2+{\left|V_r^n\right|}^2-2\left|V_s^n\right|\left|V_r^n\right|\cos \left({\delta }_s-{\delta }_r\right)\right].$$(19)

For a perfectly grounded system, Z_o = 0$.$Therefore, the transmission losses become localized and cosh becomes 1 and sinh ~ 0; hence, (18) can be reduced to$$V_s=V_r,$$(20)

Substituting (20) into (19), losses are modified to locally estimated transmission losses by one of the ends [11]. This loss also depends on the angle difference between the two ends. Hence, the new losses that can be designated as node losses F_n are as follows:$$F_n=\sum _{s=1}^N\sum _{\begin{array}{c}r=1\\ s\ne r\end{array}}^N{G}_{\mathrm{sr}}^n\left[{\left|V_r^n\right|}^2(1-\cos \left({\delta }_s-{\delta }_r\right))\right].$$(21)

The plot of the node loss F_n as a function of V_r in (21) is shown in Figure 4.

Figure 4 New loss function F1 and Fc plot.

3.1.1.2 Losses in converters

The VSCs loss F_c in the low-frequency system, as shown in Figure 5 is calculated using the generalized formula of converter loss, which is dependent on the injected current shown in (22) [8].$$F_c=\alpha +\beta ·I_c+\gamma I_c^2,$$(22)where α, β, and γ are the loss coefficients of the VSC connected to LF lines. The injected current I_c can be expressed as$$I_c= \frac{V_{L,s}-V_r}{1/Y}.$$(23)

Figure 5 Simplified model of low frequency converters connected to AC grid.

From (23), I_c is also dependent on Y, which in turn is dependent on the frequency, as shown in (7) and (8). Thus, the overall losses comprise converter losses and node losses of the LF-HVac system, which can be expressed as$$F_1=F_n+F_c.$$(24)

3.1.2 Voltage stability index

VSI can be expressed as$$\mathrm{VSI}=F_2={-V}_r^4-2\left(P_S{R}_{\mathrm{sr}}+Q_s{X}_{\mathrm{sr}}\right)V_r^2-\left|Z^2\right|\left(P^2+Q^2\right)+2V_s^2{V}_r^2.$$(25)

For a perfectly grounded system, as per (20), (25) can be expressed as$$\mathrm{VSI}=V_r^2\left(V_r^2-2\left(P_S{R}_{\mathrm{sr}}+Q_s{X}_{\mathrm{sr}}\right)\right).$$(26)

3.1.3 Voltage deviation Index

The Voltage Deviation Index (VDI) of the network voltage profile can be expressed as:$$F_3=\sum _{m=1}^{S_{\mathrm{PQ}}}\left|V_m-V_{\mathrm{ref}}\right|,$$(27)where S_PQ is load bus, V_ref is taken as 1 p.u.

3.1.4 Quadratic weighted sum objective function

The multi-objective function was reduced to a single objective function by allocating a weight to each objective function and then optimizing the weighted sum quadratic function. However, there are no restrictions on choosing a form for this problem. This method provides good solutions for the Pareto front, as shown in Figure 6. Thus, the objective function can be expressed as$$\begin{array}{c}{F}_q=\min \sum _{i=1}^k{W}_i{F}_i^2,\\ ={\min \{W}_1{F}_1^2+W_2{F}_2^2+W_3{F}_3^2\},\end{array}$$(28)where the W_i weighing coefficients can be determined by the linear weighted sum method, which can be determined by equation (30).

Figure 6 Pareto front of objective functions.

Because the objective is to minimize the inconsistent objective function F_q to implement without hampering generality, it can be regarded as$$\min \left\{F_i\right\}=-\max \left\{-F_i\right\},$$(29)where F_i is a quadratic function in the proposed case, W_i € R and is the weight of the ith objective function.$$\genfrac{}{}{0pt}{}{W_i\le 1,\hspace{1em}\hspace{1em}\forall i=1,\dots , n,}{{\sum }_{i=1}^n{W}_i=1, }$$(30)

From Figure 7. The best solution using a quadratic weighted sum can be observed with linear weighted coefficients where n is taken as 3.

Figure 7 Quadratic weighted objective function.

3.2 Constraints

3.2.1 Equality constraints

The mathematics of power systems are connected to the OPF equality criteria. Moreover, inequality limitations reflect the restrictions placed on the physical components of power systems, as well as the restrictions put in to ensure network security. The equality constraints for the power flow equations are given by$$P_{\mathrm{cal}}=P_{\mathrm{sr}}^g-P_{\mathrm{sr}}^d,$$(31) $${ Q}_{\mathrm{cal}}=Q_{\mathrm{sr}}^g-Q_{\mathrm{sr}}^d .$$(32)

3.2.2 Inequality constraints

Inequality constraints for different units of the system are$$P_{g,s}^{\min }\le P_{g,s}\le P_{g,s}^{\max },\hspace{1em}\hspace{1em}s=1, 2\dots N,$$(33) $$Q_{g,s}^{\min }\le Q_{g,s}\le Q_{g,s}^{\max },\hspace{1em}\hspace{1em}s=1, 2\dots N,$$(34) $$V_{g,s}^{\min }\le V_{g,s}\le V_{g,s}^{\max },\hspace{1em}\hspace{1em}s=1, 2\dots N,$$(35) $$T_s^{\min }\le T_s\le T_s^{\max },\hspace{0.5em}s=1, 2\dots N,$$(36) $$Q_{c,s}^{\min }\le Q_{c,s}\le Q_{c,s}^{\max },\hspace{1em}s=1,2\dots \mathrm{Nc},$$(37) $$V_{L,s}^{\min }\le V_{L,s}\le V_{L,s}^{\max },\hspace{0.5em}s=1, 2\dots \mathrm{NL},$$(38) $$S_{\mathrm{sr}}\le S_{\mathrm{sr}}^{\max },\hspace{0.5em}s,r=1, 2\dots N,$$(39) $$f_{\mathrm{Ls},r}^{\min }\le f_{\mathrm{Ls},r}\le f_{\mathrm{Ls},r}^{\max }\hspace{1em}\hspace{1em}s=1, 2\dots N{L}_{\mathrm{sr}} .$$(40)

3.3 STATCOM power constraints

The power constraints of the STATCOM equivalent model of STATCOM as shown in Figure 8 can be represented by$$P_{\mathrm{cal}}=\sum _{s=1}^r{V}_s{V}_r\left(G_{\mathrm{sr}}\cos {\delta }_{\mathrm{sr}}-B_{\mathrm{sr}}\sin {\delta }_{\mathrm{sr}}\right),$$(41) $$Q_{\mathrm{cal}}=\sum _{s=1}^r{V}_s{V}_r\left(G_{\mathrm{sr}}\cos {\delta }_{\mathrm{sr}}+B_{\mathrm{sr}}\sin {\delta }_{\mathrm{sr}}\right),$$(42) $$\begin{array}{c}{P}_{\mathrm{vc},\mathrm{sr}}=V_{\mathrm{sr}}^2{G}_{\mathrm{vc},\mathrm{sr}}+V_{\mathrm{sr}}{V}_{\mathrm{sr},\mathrm{vc}}\\ \left(\left[\begin{array}{c}{G}_{\mathrm{sr},\mathrm{vc}}\cos \left({\varpi }_{\mathrm{sr},\mathrm{vc}},{\delta }_{\mathrm{sr}}-{\delta }_{\mathrm{sr}},{\varpi }_{\mathrm{sr},\mathrm{vc}}\right)+\\ B_{\mathrm{sr},\mathrm{vc}} \sin \left({\varpi }_{\mathrm{sr},\mathrm{vc}},{\delta }_{\mathrm{sr}}-{\delta }_{\mathrm{sr}},{\varpi }_{\mathrm{sr},\mathrm{vc}}\right)\end{array}\right]\right), \\ \end{array}$$(43) $$\begin{array}{c}{Q}_{\mathrm{vc},\mathrm{sr}}=-V_{\mathrm{mn}}^2{G}_{\mathrm{vc},\mathrm{sr}}+V_{\mathrm{sr}}{V}_{\mathrm{sr},\mathrm{vc}}\\ \left(\left[\begin{array}{c}{G}_{\mathrm{sr},\mathrm{vc}}\cos \left({\varpi }_{\mathrm{sr},\mathrm{vc}},{\delta }_{\mathrm{sr}}-{\delta }_{\mathrm{sr}},{\varpi }_{\mathrm{sr},\mathrm{vc}}\right)-\\ B_{\mathrm{sr},\mathrm{vc}}\sin \left({\varpi }_{\mathrm{sr},\mathrm{vc}},{\delta }_{\mathrm{sr}}-{\delta }_{\mathrm{sr}},{\varpi }_{\mathrm{sr},\mathrm{vc}}\right)\end{array}\right]\right) . \\ \end{array}$$(44)

Figure 8 Equivalent diagram of STATCOM.

4 Solution method based on MJAYA

The Jaya Algorithm is a population-based algorithm inspired by the social behavior of individuals in a society, particularly in terms of cooperation and mutual improvement and is used to solve OLF. This algorithm was developed by Rao in 2009. Suppose d_th is a member of the t_th variable; then, the position is updated by$$X_{d,t}=X_t^{\min }+\mathrm{rand}\left(·\right)\left[X_t^{\max }-X_t^{\min }\right],$$(45) $$X_{t,d,s}^{\mathrm{new}}=X_{t,d,s}+r_{1,t,m}\left[\left(X_{t,\mathrm{best},s}\right)-\left(\left|X_{t,d,s}\right|\right)\right]-r_{2,t,s}\left[\left(X_{t,\mathrm{worst},s}\right)-\left(\left|X_{t,d,s}\right|\right)\right],$$(46)

where $X_{t,d,s}^{\mathrm{new}}$ are the new solutions of the t_th variable and r_1,t,s and r_2,t,s are randomly generated values between 1 and 0 of the t_th variable. X_t,best,s and X_t,worst,s are the best and worst values, respectively, of the t_th variable.

Therefore, to eliminate the disadvantages of the Jaya algorithm, the updating solution equation can be modified as$$X_{t,d,s}^{\mathrm{new}}=X_t^{d,s}+r_{1,t,s}\left[X_{t,\mathrm{worst},s}-\left|X_{t,d,s}\right|\right]-{L\times r}_{2,t,s}\left[{X_{t,d,s}}^2-{X_{t,\mathrm{best},s}}^2\right],$$(47)where L is the coefficient determined in every iteration, and L = 1 if rand > 0.5; otherwise, it is −1. The flowchart demonstrates the proposed algorithm in more detail, as shown in Figure 9.

Figure 9 MJAYA algorithm for optimal frequency selection.

Each optimization algorithm had unique parameters. In PSO, c1 and c2 were set to 1.5, the population size was set to 310, and the number of iterations was set to 400. The default weights of w1 and w2 are taken as 0.5 and 0.9, respectively. For the BRO algorithm, the number of iterations and population size remained the same, and the weight factor W1 varies from [0–1]. Because Jaya is a gradient-free algorithm, it requires no parameters other than random values. Figure 10 depicts the convergence characteristics of the various algorithms with and with frequency constraints on the transmission lines. According to figure, the proposed MJAYA OLF provided the best results for all LF-HVac lines. Twenty separate runs were performed for each algorithm to gauge how frequently the optimal or nearly optimal solution was reached while preserving the same stopping conditions, thereby demonstrating the resilience and consistency of the proposed method (a maximum of 400 iterations).

Figure 10 Convergence comparison with different optimization algorithms with LF-HVac line.

5 Case studies

In this section, the benefits of the proposed formulation in Section 3 and solution method in Section 4 are presented. The proposed OLF algorithm can reduce losses, increase the stability index, and maintain voltage stability in LF-HVAC transmission lines. The proposed methodology not only reduces the matrix calculation significantly but also reduces the parameters in the OLF. The two-port network solution of Telegrapher’s method reduces the loss equation to a single-parameter variation.

The proposed OLF was tuned and performed best with a 310 person population and 400 iterations, as shown in Table 1.

5.1 System design

The test system comprised 10 generating units connected by 39 transmission network branches, as shown in Figure 11. The following topology diagram represents the IEEE 39-bus system, where bus 39 is a slack bus: With the possible exception of the slack bus, the buses (30–38) connected to each generator had voltage levels of 20–38 kV, 11–38 kV, and 500 kV for all other buses. In this study, a MATLAB-based simulation tool was used to perform OLF on a two-port system. A metaheuristic algorithm was presented with a quadratic-weighted single-objective function. This method reduces the number of calculations and matrix formations.

Figure 11 Line diagram of modified IEEE 39 bus test system.

To solve the complexity of voltage and frequency changes in low-frequency HVAC systems, the ratio coefficient k is introduced in Section 2, where V_base and f_base are taken as 500 kV and 60 Hz, respectively. Table 2 represnts the voltage levels of RES. For any transmission line, the variation in reactance and resistance is very small, that is, 0.10% and 0.20%, respectively, as shown in [15]. However, in our proposed model, low frequency affects the X_n, Y_n, and G_n of the transmission lines as per the k coefficient between the s and r buses. With each iteration of the optimal load flow, the values of conductance G_n and susceptance B_n are updated with the change in frequency of the system, as per [15].

Subsequently, the Modified JAYA with STATCOM inclusion is updated with new variables, constraints, and parameters while achieving the optimal load flow. Using two-port network analysis and Telegrapher’s method, the objective functions are modified to form new functions. Frequency consideration in a simplified two-port network provides a better solution than in a conventional one.

5.2 Assessment of voltage stability

The STATCOMs were placed at six buses (5, 16, 21, 25, 27, and 29) in the proposed system to maintain voltage stability. Initially, the OLF was run without STATCOM on a 39 IEEE bus system and was estimated in terms of a modified objective function (F_q). To visualize the impact of the frequency change, two separate runs of the OPF were formulated and compared. In the first run, the optimal load flow with the function F_q and frequency change parameters were included. STATCOM includes the same parameters, functions, and Z for the next run.

Figure 12 shows that the buses with the STATCOM have a significant improvement in voltage stability over the first case. The voltage level changes by up to 0.4% on bus 5 with LF-HVac lines and 0.1%–0.4% on other buses, demonstrating that the bus with STATCOM and LF-HVac lines provides better voltage stability than buses without LF-HVac lines. This indicates that the placement of the STATCOM is favorable for the proposed system, which provides optimistic results.

Figure 12 Voltage profile for LF-HVAC lines with STATCOM.

5.3 Numerical analysis of proposed system for fixed frequency range

The purpose of the presented results was to demonstrate the agility of the proposed algorithm with a defined solver. To visualize this, the system was tested with the proposed OLF, which consists of six LF-HVAC transmission lines, that is, (31–29), (34–20), (23–24), (21–22), (2–25), and (26–29) connected to wind farms, pumped storage hydropower, and gas turbines. All transmission lines have BTB converters with a working range of 0.9–1.0 p.u. with discrete step sizes of 0.01. Table 2 shows that the generator buses maintained the voltage limits with a maximum change of 0.089% in low-frequency transmission lines. MJAYA also outperforms the others in resolving the OLF issue while achieving the optimal frequency of each low-frequency line for a predefined range [15–20 Hz], as shown in Table 3.

5.4 Loss analysis for different frequency range

In this scenario, losses were incurred as a function of the frequency range. The losses were calculated with and without the optimal load flow. Figure 13 without OLF, the proposed system’s losses remain constant at 0.4491 MW, whereas after OLF, the losses decrease in a different frequency range. In the 31–40 Hz range, the losses are minimal with OLF, so an optimal frequency range can be depicted. From Figure 14 loss was found for the proposed system, and a frequency range of 31–40 Hz was depicted as the optimal range.

Figure 13 Losses with OLF and without OLF.

Figure 14 Convergence characteristics of losses with OLF.

5.5 Loss analysis for different frequency range

In this scenario, losses were incurred as a function of the frequency range. The losses were calculated with and without the optimal load flow. Figure 13 without OLF, the proposed system’s losses remain constant at 0.4491 MW, whereas after OLF, the losses decrease in a different frequency range. In the 31–40 Hz range, the losses are minimal with OLF, so an optimal frequency range can be depicted. From, Figure 14 Loss was found for the proposed system, and a frequency range of 31–40 Hz was depicted as the optimal range.

5.6 Statistical analysis for each low frequency lines

The optimal frequency with the proposed algorithm and solver is shown in Figure 15 shows the frequency deviation of low-frequency transmission lines (WT, PSH, and GT), which can reach a maximum of ±5 Hz. Also Figure 16 shows the standard error of the frequency for each low-frequency line.

Figure 15 Frequency deviation of low-frequency lines.

Figure 16 Standard deviation of low-frequency lines.

6 Conclusion

The optimal frequency selection for the LF-HVac transmission system was studied in this work. We investigated OLF by utilizing a reduced quadratic weighted objective function. A two-port equivalent transmission system was implemented to modify the objective function and reduce the variables based on Telegrapher’s method. We proposed the MJAYA algorithm with optimal coefficient selectivity to validate the correctness of our proposed algorithm. Conducting load flow analysis using STATCOM effectively enhanced the voltage stability under various frequency constraints. The results are compared with those of alternative optimization methods, which show that the application of Telegrapher’s equation consistently outperforms the existing approaches. The proposed quadratic weighted objective function provided better results for the stability index and loss minimization function with frequency attributes. We determined the optimal frequency for each low-frequency line. In addition, the proposed formulation for the optimal frequency selection for transmission lines can incorporate new custom power devices.

References

All Tables

All Figures

	Figure 1 Basic model of LF-HVac.
In the text

	Figure 2 Simplified model of transmission line.
In the text

	Figure 3 Two port model of unbalanced transmission line.
In the text

	Figure 4 New loss function F1 and Fc plot.
In the text

	Figure 5 Simplified model of low frequency converters connected to AC grid.
In the text

	Figure 6 Pareto front of objective functions.
In the text

	Figure 7 Quadratic weighted objective function.
In the text

	Figure 8 Equivalent diagram of STATCOM.
In the text

	Figure 9 MJAYA algorithm for optimal frequency selection.
In the text

	Figure 10 Convergence comparison with different optimization algorithms with LF-HVac line.
In the text

	Figure 11 Line diagram of modified IEEE 39 bus test system.
In the text

	Figure 12 Voltage profile for LF-HVAC lines with STATCOM.
In the text

	Figure 13 Losses with OLF and without OLF.
In the text

	Figure 14 Convergence characteristics of losses with OLF.
In the text

	Figure 15 Frequency deviation of low-frequency lines.
In the text

	Figure 16 Standard deviation of low-frequency lines.
In the text