Optimal Reactive Power Flow of AC-DC Power System with Shunt Capacitors Using Backtracking Search Algorithm

Abstract

In this paper, it is proposed that a two-terminal high voltage direct current (HVDC) be integrated into the power system. Line-commutated converter (LCC)-HVDC is used because of its ability to reduce line losses, which improves overall system efficiency. Shunt capacitors also aid in voltage maintenance by compensating for the reactive power demand. In essence, limiting voltage drops in electrical networks promotes a more efficient power transmission and distribution by lowering resistive losses. In power system investigations, it was discovered that the HVDC link and SCB exist separately. So, for the first time, the backtracking search algorithm (BSA) is used to solve the optimal reactive power flow (ORPF) of a power system with a HVDC link and shunt capacitor banks (SCB). Although BSA simulations on a modified IEEE 30 bus yielded successful results, ABC was also utilized for comparing the outcomes of different methods. Overall, three separate cases of the modified IEEE 30 bus system were examined. When the acquired results are compared to other methods, the suggested algorithm is found to be better at concerning effectiveness as well as performance.

1. Introduction

As the world’s population continues to rise, more homes, companies, and industries use electricity for a variety of functions, such as lighting, cooling, heating, other equipment, and industrial activities [1]. Increasing the load causes a higher current flow, more resistance heating in power transmission lines, and an increasing voltage drop, all of which reduce the efficiency as well as the reliability of electric power transmission lines [2]. When the load exceeds the system limit, the chance of a system outage increases. To solve these issues, shunt capacitor banks (SCB) and HVDC are proposed.

When SCBs are used in power systems, they offer a variety of effects and benefits. These advantages include reactive power compensation, loss reduction, increased transmission capacity, and improved power quality [3]. SCBs aid in boosting voltage by eliminating voltage dips and increasing voltage stability. Voltage levels can be kept within acceptable limits by compensating for reactive power, especially during periods of heavy load demand. This prevents voltage drops and offers a stable and dependable power source. Shunt capacitors help to reduce power losses in power systems [3]. The system’s apparent power requirement is decreased by correcting for the reactive power. As a result, the I2R losses associated with transmission and distribution lines, transformers, and other components are reduced [4].

When compared to traditional optimization techniques, heuristic optimization algorithms can generate the best solution for the location and sizing of SCBs in distribution networks [5]. To optimize the position of the SCBs in power systems, many methods and methodologies have been utilized. For example, ref. [5] used a hybrid of the harmony search algorithm (the) and Particle Artificial Bee Colony algorithm (PABC), ref. [6] implemented Integrated Deferred-Merge Embedding (IDME), ref. [7] utilized multi-objective particle swarm optimization (MOPSO), ref. [8] employed the Bacterial Foraging Optimization Algorithm (BFOA), ref. [9] implemented the Fuzzy Genetic Algorithm (FGA), ref. [10] employed PSO, ref. [11] proposed an analytical approach, ref. [12] used the Memetic algorithm (MA), ref. [13] utilized a hybrid of the imperialist competitive algorithm (ICA) and GA, ref. [14] used A hybrid of ABC with evolution algorithm (EA), and ref. [15] applied a backtracking search algorithm (BSA). Ref. [16] used System-based testing methodologies for protection systems which were implemented using Relaysimtest software, which is compatible with OMICRON injection test sets. Ref. [17] applied a genetic algorithm developed for the optimum planning of shunt capacitor banks. Ref. [18] used GA, PSO, Bat Optimization Algorithm (BOA), and the Whale Optimization Algorithm (WOA) and the Sperm-Whale Algorithm (SWA).

Many countries are beginning to employ HVDC transmission in their systems thanks to its many positive effects. The advantages of the HVDC link include cost-effective long-distance power transmission, improved grid stability, and the opportunity to integrate renewable energy sources [19]. Although HVDC has advantages in some instances, it should be noted that HVDC installations are often more expensive than HVAC systems [19]. As a result, the decision to adopt HVDC in a power system is influenced by several factors, including transmission distance, system needs, and economic considerations.

LCC (Line Commutated Converter) and VSC (Voltage Source Converter) represent two forms of HVDC (High Voltage Direct Current) systems employed for long-distance power transmission or connecting asynchronous AC (Alternating Current) systems.

LCC-HVDC is appropriate for the long-distance transmission of power but has limited capacity for reactive power and is less expensive, whereas VSC-HVDC has better controllability but is more expensive.

Table 1. Algorithms and objective functions of the HVDC field.

	Method	Objective Function	Test System	Year
[20]	Fitness-distance Balance-based Stochastic Fractal Search (SFS), (FDB-SFS)	Power Loss Voltage Deviation, Cost	IEEE 30 bus system	2023
[21]	ETAP	Voltage Profile	36 bus (real system)	2022
[22]	Genetic Algorithm	Power Loss	IEEE 14 and 30-bus system The modified New England 39	2014
[23]	Differential evolution algorithm	Cost	IEEE 5, 9, 118-bus The modified New England 39	2018
[24]	Differential evolution with neighborhood mutation (DENM)	Power Loss	The modified New England 39, 114 bus (real system)	2018
[25]	Power Flow (PF) algorithm	Power Loss	IEEE 5 bus and Modified England IEEE 39 bus system	2019
[26]	Artificia Bee Colony (ABC)	Power Loss Voltage Deviation Cost	IEEE 30 bus system	2016
[27]	Harmonic Power Flow	Harmonic	IEEE 30 bus system	2022
[28]	Fast Differential Equation Power Flow (DEPF)	relative speed of computations (RSC)	IEEE 39, 118, 145, 300-bus system, 1153 bus (real system), and 4438 bus(real system)	2023
[29]	Semidefinite programming (SDP)	Power Loss	Modified CIGRE, IEEE 33 bus system	2022
[30]	OPF model	Cost	IEEE 39 bus system	2022
[31]	BSA	Power Loss	IEEE 30 bus system	2021
[32]	Teaching Learning-based Optimization (TLBO)	Cost	IEEE14, 30 and 57	2021

demonstrates some of recent findings that have considered the HVDC in the last few years [20,21,22,23,24,25,26,27,28,29,30,31,32].

Optimal reactive power flow research has concentrated on developing algorithms and approaches for optimizing reactive power flow in AC-DC hybrid grids. These insights help to manage reactive electricity more efficiently, reduce losses, and improve overall grid efficiency. The optimal reactive power flow in a power system is the process of determining the most efficient and cost-effective distribution of the reactive power sources in order to keep system voltages within acceptable limits.

Scientists have used a variety of approaches to solve the optimal reactive power flow problem in hybrid AC-DC systems [20,22,24,25,29,31]. The reference [20] employed FDB-SFS in a power system with HVDC links to reduce power loss, improve voltage variation, and save costs. In ref. [22], GA was used to reduce power loss in an alternating current–direct current system. In ref. [24], DENM was utilized in power systems with hybrid AC-DC systems to reduce power loss [25]. The PF algorithm was used to minimize losses in an AC-DC system. In ref. [29], SDP is used in AC-DC systems to reduce power loss. In ref. [31], BSA was utilized in a hybrid AC-DC system to reduce power loss and costs. In ref. [33], power flow was used as a systematic way to compare the HVDC and HVDC systems based on power loss. In [34], the Songo–Apollo link was used to network model with DIgSILENT Power Factory software (made by DIgSILENT GmbH in Gomaringen, Germany) in a system with HVDC and flexible alternating current technology systems (FACTS). In [35], the Teaching Learning-Based Algorithm (TLBA) was used to reduce both system power losses and fuel costs.

During system disturbances, capacitors can aid by offering voltage support. Capacitors are able to release their stored energy to help maintain the DC voltage level in the case of a voltage dip or sag in the AC system. This increases system stability and helps to reduce voltage swings.

In this article, the backtracking search algorithm will be used to configure three shunt capacitors and an LCC-HVDC link in an IEEE 30 bus system. This work is completed in three different scenarios to demonstrate the effects of SCB and LCC-HVDC. Another approach was used for comparison with the proposed method, and results from both methods were compared and analyzed.

The paper is organized into five sections. Section 1 starts with an introduction, which included a literature review on the shunt capacitor bank and HVDC link in the power system. Section 2 discusses materials and methods which are AC and DC system modeling, objective function, and fitness function. Section 3 includes a backtracking search algorithm construction, pseudo, and flow chart. Section 4 presents the paper’s results and discussions, and Section 5 presents the paper’s conclusions.

2. Materials and Methods

The mathematical representation of the single objective optimization problems under consideration is as follows:

$$\begin{array}{cc}\mathrm{Minimize}& f(x,u)\hfill \\ \mathrm{Subject} \mathrm{to}& g(x,u)=0\hfill \\ & h(x,u)\le 0\hfill \end{array}$$

(1)

where g represents the equality constraints, h depicts the inequality constraints, x and u are state and control parameters, respectively, and f denotes the objective functions to be minimized. The state and control variables of an optimization problem for a two-terminal HVDC system are taken into account as in [28].

Figure 1 depicts the basic configuration of a two-terminal HVDC scheme, which consists of two converters connected by a DC link [22]. The rectifier (with abbreviation r) and inverter (with abbreviation i) are linked to AC system buses.

Figure 2 depicts the equivalent circuit for the two terminal HVDC link.

The following are the equations linked to a converter’s rectifier operation [36,37]:

$${\mathrm{V}}_{\mathrm{dor}}=\left(\raisebox{1ex}{$3\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$\Pi $}\right.\right){\mathrm{V}}_{\mathrm{r}}{\mathrm{t}}_{\mathrm{r}}$$

(2)

$${\mathrm{V}}_{\mathrm{dr}}={\mathrm{V}}_{\mathrm{dor}}\cos \mathsf{\alpha }-\frac{3{\mathrm{X}}_{\mathrm{cr}}}{\Pi }{\mathrm{I}}_{\mathrm{d}}$$

(3)

where ${\mathrm{V}}_{\mathrm{dor}}$ represents the optimum no-load direct voltage and $\mathsf{\alpha }$ is the ignition delay angle. The equivalent commutating reactance ${\mathrm{X}}_{\mathrm{cr}}$ accounts for the voltage drop caused by a commutation overlap. The rectifier side’s active power is determined by:

$${\mathrm{P}}_{\mathrm{dr}}={\mathrm{V}}_{\mathrm{dr}}{\mathrm{I}}_{\mathrm{d}}$$

(4)

On the rectifier side, the reactive power is calculated as follows:

$${\mathrm{Q}}_{\mathrm{dr}}=\left|{\mathrm{P}}_{\mathrm{dr}}\tan {\Phi }_{\mathrm{r}}\right|$$

(5)

where ${\Phi }_{\mathrm{r}}$ is the phase angle separating the alternating current and voltage, and is determined as follows:

$${\Phi }_{\mathrm{r}}={\cos }^{-1}\left(\frac{{\mathrm{V}}_{\mathrm{dr}}}{{\mathrm{V}}_{\mathrm{dor}}}\right)$$

(6)

The formulas relating to a converter’s inverter operating are as follows:

$${\mathrm{V}}_{\mathrm{doi}}=\left(\raisebox{1ex}{$3\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$\Pi $}\right.\right){\mathrm{V}}_{\mathrm{i}}{\mathrm{t}}_{\mathrm{I}}$$

(7)

$${\mathrm{V}}_{\mathrm{di}}={\mathrm{V}}_{\mathrm{doi}}\cos \mathsf{\gamma }-\frac{3{\mathrm{X}}_{\mathrm{ci}}}{\Pi }{\mathrm{I}}_{\mathrm{d}}$$

(8)

$${\mathrm{P}}_{\mathrm{di}}={\mathrm{V}}_{\mathrm{di}}{\mathrm{I}}_{\mathrm{d}}$$

(9)

$${\mathrm{Q}}_{\mathrm{di}}=\left|{\mathrm{P}}_{\mathrm{di}}\tan {\Phi }_{\mathrm{i}}\right|$$

(10)

$${\Phi }_{\mathrm{i}}={\cos }^{-1}\left(\frac{{\mathrm{V}}_{\mathrm{di}}}{{\mathrm{V}}_{\mathrm{doi}}}\right)$$

(11)

The relationship between the two DC voltages can be represented as:

$${\mathrm{V}}_{\mathrm{dr}}={\mathrm{V}}_{\mathrm{di}}+{\mathrm{R}}_{\mathrm{dc}}{\mathrm{I}}_{\mathrm{d}}$$

(12)

2.1. The State and Control Variables of DC and AC System

The following AC and DC system state (x) and control variables (u) have been determined as follows:

$$x=\left[p_{\mathit{gslack}}, q_{g1},\dots ,q_{gNg}, v_{L1},\dots ,v_{L{N}_l}{t}_r, t_i, {\alpha }_r, {\alpha }_i,v_{\mathit{dr}},v_{\mathit{di}}\right]$$

(13)

The AC and DC system control variables per unit are also selected as follows:

$$\begin{array}{l}{u}_{AC}=[p_{g2},\dots ,p_{gNg}, v_{g1},\dots ,v_{gNg},t_1,\dots ,t_{N_T},\\ q_{c1},\dots ,q_{c{N}_c},q_{cb1},\dots ,q_{cb3},lo{c}_1,\dots ,lo{c}_3{p}_{\mathit{dr}}, p_{\mathit{di}}, q_{\mathit{dr}}, q_{\mathit{di}},i_d]\end{array}$$

(14)

2.2. Power Loss

The objective of minimizing power line losses is mathematically represented by the following:

$$\mathit{Ploss}={\displaystyle \sum _{i=1}^{Ngn}{p}_{gngi}}-{\displaystyle \sum _{j=1}^N{p}_{ldj}}$$

(15)

where N is the total number of system buses and $p_{ldj}$ denotes the active load of jth bus, respectively.

System restrictions on equality and inequality.

Each bus’s equality restrictions include active and reactive power balancing [22]. The active–reactive power of the generators, voltage magnitude, transformer tap, and reactive compensator limits are the inequality constraints, as shown in (16)–(21) [28].

The HVDC line parameters’ limitations of inequality (minimum and maximum limits) can be seen below [22]:

$$p_{gm}^{\min }\le p_{gm}\le p_{gm}^{\max } m=1,\dots ,N_g$$

(16)

$$q_{cbm}^{\min }\le q_{cbm}\le q_{cbm}^{\max } m=1,\dots ,N_b$$

(17)

$$v_m^{\min }\le v_m\le v_m^{\max } m=1,\dots ,N$$

(18)

$$q_{gm}^{\min }\le q_{gm}\le q_{gm}^{\max } m=1,\dots ,N_g$$

(19)

$$q_{cm}^{\min }\le q_{cm}\le q_{cm}^{\max } m=1,\dots ,N_c$$

(20)

$$t_m^{\min }\le t_m\le t_m^{\max } m=1,\dots ,N_T$$

(21)

The HVDC line parameters’ limitations of inequality (minimum and maximum limits) can be seen below [22,31]:

$$p_{dk}^{\min }\le p_{dk}\le p_{dk}^{\max } k=r,i$$

(22)

$$v_{dk}^{\min }\le v_{dk}\le v_{dk}^{\max } k=r,i$$

(23)

$$q_{dk}^{\min }\le q_{dk}\le q_{dk}^{\max } k=r,i$$

(24)

$$t_{dk}^{\min }\le t_{dk}\le t_{dk}^{\max } k=r,i$$

(25)

$${\alpha }_k^{\min }\le {\alpha }_k\le {\alpha }_k^{\max } k=r,i$$

(26)

$$i_d^{\min }\le i_d\le i_d^{\max }$$

(27)

2.3. Fitness Function

To deal with the inequality constraints associated with state variables, they can be integrated as the quadratic penalty factors in the objective function. The following enhanced objective function then describes the fitness function:

$$\mathit{Fitness}=\mathit{objective}+{\displaystyle \sum _i^{ke}{k}_i}\left|\mathsf{\Delta }{x}_i\right|$$

(28)

where $\mathsf{\Delta }x$ is the difference between the state variable and its limit, k_i is the penalty factor, and k_e is the number of state variables.

3. Backtracking Search Algorithm

The backtracking search algorithm (BSA) is a generic algorithmic strategy for solving combinatorial issues. When compared to other algorithms, BSA has a much simpler structure. BSA is capable of solving nonlinear, nonconvex, and difficult optimization problems. This algorithm’s most essential quality is that it is not very sensitive to the parameter’s starting value [38]. The proposed approach has been used successfully to address issues in a variety of technical domains [38,39,40,41]. Ref. [42] provides a detailed description of the BSA pseudo code.

BSA employs a random mutation technique, selecting individuals from the previous generation at random, and employing a non-uniform crossover mechanism [42].

The proposed algorithm’s functions can be divided into five steps, including initialization, selection I, mutation, crossover, and selection II.

Figure 3 depicts the BSA’s flow chart.

The basic steps of the BSA are outlined as follows [42]:

Initialization

At first, two distinct populations ($\mathit{Pop}$ and $\mathit{oldPop}$) develop as follows:

$${\mathit{Pop}}_{i,j}~\mathit{Rand}\left({\mathit{low}}_j,{\mathit{up}}_j\right) i=1,\dots ,S\mathrm{N} j=1,\dots ,D$$

(29)

$${\mathit{oldPop}}_{i,j}~\mathit{Rand}\left({\mathit{low}}_j,{\mathit{up}}_j\right) i=1,\dots ,S\mathrm{N} j=1,\dots ,D$$

(30)

Selection-1

The old population ($\mathit{oldPop}$) at the initial step is formed in this section employing:

$$if a<b, \mathit{oldPop}:=\mathit{Pop} \mathit{end}| a,b~\mathit{Rand}\left(0,1\right)$$

(31)

The update operator in Equation (31) is :=. This operator transfers variables from $\mathit{Pop}$ persons to $\mathit{oldPop}$ individuals at random. Then, Equation (32) is used to alter individuals in the $\mathit{oldPop}$ as random as follows:

$$\mathit{oldPop}:=\mathit{Randshuff}(\mathit{oldPop})$$

(32)

where $\mathit{Randshuff}$ represents a random mixing tool.

Mutation

As a result of the mutation process,

$$\mathit{mutant}\mathit{Pop}=\mathit{Pop}+W\left(\mathit{oldPop}-\mathit{Pop}\right)$$

(33)

The amplitude of the search line matrix is controlled by $W$ in Equation (33). The goal here is to partially process the preceding generation’s experience. The function “$\mathit{randn}$” creates random numbers between 0 and 1 using ordinary normal distribution.

Crossover

The trial population ($\mathit{Tpop}$) is generated in this step using Equation (34). The crossover procedure is divided into two steps. The binary number system is completely valued in the first stage, yielding the $SN\ast D$ size of a matrix (map). This matrix is used to decide whether we need to change $\mathit{Tpop}$ one row at a time (“individually by individual” in heuristic expressions). After forming the matrix, the second stage yields the following equation:

$$\mathit{if} {\mathit{map}}_{i,j}=1 \mathit{then} {\mathit{Tpop}}_{i,j}={\mathit{Pop}}_{i,j} \mathit{else} {\mathit{Tpop}}_{i,j}=\mathit{mutant}{\mathit{Pop}}_{i,j} \mathit{end}$$

(34)

The mutation technique utilizes a single individual from a previous population, but the BSA’s crossover mechanism is more complicated than that of the differential evolution algorithm and developed versions [42].

Selection-II

This part computes all of the “individual” fitness values generated. Individuals are ranked the from best to worst in terms of fitness. The SN of them is then transmitted to the following iteration, $\mathit{Pop}$. The rest are omitted. The best “individuals” from the entire population are thus passed down to the next generation.

4. Results and Discussion

The proposed algorithm was implemented in the IEEE 30 bus system [43]. This study takes into account three separate cases by BSA (Method 1) and ABC (Method 2). The cases include:

Case 1: The standard system.

Case 2: Three SCBs configured in the default system.

Case 3: Configuration of three SCBs with HVDC at lines (2–14).

Case 1: The power flow is applied in the passive test system, and the results are given in

Table 2. Method 1 and Method 2 produced comparable results for Case 1 and 2.

Control Variables	Limit		Method 1 (Case 2)	Method 2 (Case 2)	Default (Case 1)
	Low	High
pg1	0	3.602	1.22745	1.27188	2.57549
pg2	0	1.4	1.40000	1.36250	0.51450
qg1	−1	1	−0.11026	0.03135	−0.18687
qg2	−0.4	0.5	0.27830	0.08594	2.17192
qc5	−0.4	0.4	0.400000	0.366644	--
qc8	−0.1	0.4	0.400000	0.400000	--
qc11	−0.06	0.24	0.240000	0.194316	--
qc13	−0.06	0.24	0.090000	0.240000	--
v1	1	1.15	1.078	1.071	1.000
v2	1	1.15	1.073	1.058	1.000
v5	1	1.15	1.046	1.029	0.875
v8	1	1.15	1.038	1.027	0.857
v11	1	1.15	1.089	1.078	0.877
v13	1	1.15	1.022	1.054	0.872
t(6–9)	0.9	1.1	0.9896	0.9878	0.9500
t(6–10)	0.9	1.1	0.9716	0.9315	0.9500
t(4–12)	0.9	1.1	1.0180	1.0236	0.9500
t(28–27)	0.9	1.1	0.9753	0.9864	0.9500
CB1(MVAR)	0	10	10.0(5)	9.5(5)	--
CB2(MVAR)	0	10	10.0(24)	10.0(30)	--
CB3(MVAR)	0	10	10.0(30)	10.0(19)	--
Power loss(p.u)			9.3450	9.5373	25.599

.

Case 2:

A total of three shunt capacitors are integrated into the test system in this scenario, as shown in Figure 4. The total reactive power injected by three SCBs in the buses 5, 24, and 30 using Method 1 is 10.0 MVAR for each unit, whereas the capacitors used in Method 2 are placed in the buses (5, 19, and 30) with capacities of 9.5, 10, and 10 MVAR, respectively. The results of Method 1 and Method 2 for Case 2 of the test system are compared.

Methods 1 and 2 reduced power loss by 63.49% and 62.74%, respectively, based on the default setup. Table 2 shows the outcome of Case 2.

Case 3:

In this case, three SCBs have been installed in the test system, with an HVDC link connecting bus 2 and 14. The total reactive power injected by three SCBs in buses 7, 23, and 30 using Method 1 is 10.0, 10.0, and 9.0 MVAR, respectively, whereas the capacitors used in Method 2 are located in buses 5, 21, and 30, with capacities of 8.75, 9.5, and 8.5 MVAR. Method 1 and Method 2 outcomes for Case 3 of the test system are compared.

Based on the default configuration, Methods 1 and 2 reduced power loss by 64.21% and 63.78%, respectively. Case 3’s outcome is shown in Figure 5 and

Table 3. Method 1 and Method 2 produced comparable results for Case 3.

Control Variables	Limit		Method 1	Method 2
	Low	High
pg1	0	3.602	1.23817	1.29303
pg2	0	1.4	1.40000	1.37267
qg1	−1	1	−0.02148	0.09383
qg2	−0.4	0.5	0.25030	0.33611
qc5	−0.4	0.4	0.40	0.39
qc8	−0.1	0.4	0.40	0.40
qc11	−0.06	0.24	0.24	0.13
qc13	−0.06	0.24	0.23	0.24
v1	1	1.15	1.066	1.079
v2	1	1.15	1.055	1.064
v5	1	1.15	1.028	1.036
v8	1	1.15	1.027	1.030
v11	1	1.15	1.083	1.070
v13	1	1.15	1.036	1.081
t(6–9)	0.9	1.1	0.97	0.99
t(6–10)	0.9	1.1	1.00	0.90
t(4–12)	0.9	1.1	1.02	0.96
t(28–27)	0.9	1.1	1.01	0.98
CB1(MVAR)	0	10	10.0 (7)	8.75(5)
CB2(MVAR)	0	10	10.0 (23)	8.50(30)
CB3(MVAR)	0	10	9.0(30)	9.50(21)
	Ploss(p.u)		9.1607	9.272

.

The DC parameters and their limit values are shown in

Table 4. Control variable of DC system given by Methods 1 and 2 for Case 3.

Control Variables	pdr	pdi	qdr	qdi	tr	ti	alfa(o)	theta	vdr	vdi	id
Minimum limit	0.1	0.1	0.05	0.05	0.9	0.9	9.74	8.59	1	1	0.1
Maximum limit	1.5	1.5	0.75	0.75	1.1	1.1	26	30	1.5	1.5	1
Method 1	0.2271	0.2246	0.1136	0.1123	0.96	1.01	25.6539	26.1374	1.2278	1.2139	0.1850
Method 2	0.3400	0.3336	0.1700	0.1668	0.90	0.90	25.8333	25.2593	1.1564	1.1343	0.2941

. All DC parameter values are within their ranges.

Figure 6 depicts variations in reactive power at the rectifier and inverter sides versus iteration number for Case 3. The reactive powers are clearly within the limits as shown in Figure 6.

Figure 7 shows variations in the transformer tap rate at the rectifier and inverter sides versus iteration number for Case 3. The transformer tap ratios are clearly within the limits as shown in Figure 7.

Figure 8 illustrates variations in the DC current in HVDC link versus iteration number for Case 3. The DC current is clearly within the limits as shown in Figure 8.

The analysis in

Table 5. The line loss of the test system in all cases.

Line	Case1	Case2	Case3
1–2	6.1530	0.6980	0.8340
1–3	3.7630	1.3470	1.1520
2–4	2.4390	1.0670	0.7420
2–5	4.5240	2.4080	2.4550
2–6	4.0250	1.7700	1.3600
3–4	1.0440	0.3610	0.3070
4–6	0.8780	0.3550	0.3650
4–12	0	0	0
5–7	0.2240	0.1460	0.2850
6–7	0.4920	0.2820	0.2990
6–8	0.2540	0.0990	0.1000
6–9	0	0	0
6–10	0	0	0
2–28	0.1200	0.0270	0.0220
8–28	0.0040	0.0070	0.0060
9–11	0	0	0
9–10	0	0	0
10–20	0.1280	0.1190	0.0710
10–17	0.0230	0.0390	0.0220
10–21	0.1610	0.0840	0.1070
10–22	0.0750	0.0360	0.0490
12–13	0	0	0
12–14	0.1070	0.0540	0.1220
12–15	0.3030	0.1150	0.1040
12–16	0.0710	0.0210	0.0800
14–15	0.0080	0	0.2660
15–18	0.0540	0.0270	0.0930
15–23	0.0390	0.0020	0.0020
16–17	0.0090	0.0030	0.0180
18–19	0.0060	0.0030	0.0260
19–20	0.0270	0.0280	0.0160
20–22	0.0010	0.0050	0.0030
22–24	0.0550	0.0340	0.0620
23–24	0.0060	0.0170	0.0660
24–25	0.0350	0.0400	0.0060
25–26	0.0660	0.0450	0.0480
25–27	0.0960	0.0840	0.0380
27–28	0	0	0
27–29	0.1240	0.0120	0.0180
27–30	0.2340	0.0090	0.0150
29–30	0.0480	0.0010	0.0010

revealed that four lines had losses greater than 3.75 kW: 1–2, 1–3, 2–5, and 2–6. This demonstrates that adding the three SCBs, as in Case 2, reduced the number of lines with losses of more than 5 kW in some lines. The improved system had a 16.254 kW difference in real power for Case 2, and a 16.4383 kW difference in real power for Case 3.

It is possible to see a considerable improvement in the bus voltage profile in Cases 2 and 3 when compared to the default. The bus voltage profile is enhanced with integrated SCBs (Case 2) and integrated SCBs with a hybrid system (Case 3), as shown in Figure 9, and all voltages are within their limitations.

Figure 10 depicts the fitness value variations in the Hybrid test system against iteration number for Cases 2 and 3.

5. Conclusions

This paper presents a thorough investigation of the LCC-HVDC link and SCBs that have been integrated with conventional power systems. The LCC-HVDC systems reduce power losses during long-distance transmissions as a result of their great transmission efficiency. Shunt capacitors supply reactive power locally, which reduces the reactive power flow through transmission and distribution lines. This leads to lower line currents and consequently reduces resistive losses.

When reviewing the power system literature, it was discovered that the HVDC link and SCBs can be found individually in the power systems. For the first time, the power system is modeled as closely as possible to reality in this study, and the power loss of test systems is minimized by BSA, while SCBs and HVDC links coexist in the power systems.

In this study, BSA and ABC solved the ORPF problem of the modified IEEE 30 bus test system for three separate scenarios. The obtained results show that, while the power loss does not change greatly depending on the location of SCBs, the integration of SCBs in the presence of LCC-HVDC significantly minimizes the power loss.

It should be noted that the planning and installation of capacitors with LCC-HVDC systems must be properly engineered, taking into account aspects like voltage and current ratings, energy storage ability, and control techniques. Furthermore, suitable protection and coordination mechanisms should be in place to guarantee the integrated system’s safe and reliable operation.