RETRACTED: A New Hybrid Algorithm for Multi-Objective Reactive Power Planning via FACTS Devices and Renewable Wind Resources

Abstract

The power system planning problem considering system loss function, voltage profile function, the cost function of FACTS (flexible alternating current transmission system) devices, and stability function are investigated in this paper. With the growth of electronic technologies, FACTS devices have improved stability and more reliable planning in reactive power (RP) planning. In addition, in modern power systems, renewable resources have an inevitable effect on power system planning. Therefore, wind resources make a complicated problem of planning due to conflicting functions and non-linear constraints. This confliction is the stochastic nature of the cost, loss, and voltage functions that cannot be summarized in function. A multi-objective hybrid algorithm is proposed to solve this problem by considering the linear and non-linear constraints that combine particle swarm optimization (PSO) and the virus colony search (VCS). VCS is a new optimization method based on viruses’ search function to destroy host cells and cause the penetration of the best virus into a cell for reproduction. In the proposed model, the PSO is used to enhance local and global search. In addition, the non-dominated sort of the Pareto criterion is used to sort the data. The optimization results on different scenarios reveal that the combined method of the proposed hybrid algorithm can improve the parameters such as convergence time, index of voltage stability, and absolute magnitude of voltage deviation, and this method can reduce the total transmission line losses. In addition, the presence of wind resources has a positive effect on the mentioned issue.

1. Introduction

Nowadays, the importance of reactive power (RP) supply is more evident. Even though the RP load flow affects the line losses and the bus voltage and the active power load flow and its costs, unfortunately, the costs of producing and transmitting RP have received little attention [1,2,3,4,5,6,7,8,9,10,11,12]. One reason for this lack of attention is the inherent difficulty in understanding this issue, particularly by economists [12,13,14,15,16,17,18,19,20]. Another reason for the lack of serious attention to RP is the low production price compared to active power [21,22,23,24,25,26,27,28,29,30]. However, according to economics and market calculations, RP is not less valuable than active power [31,32,33,34,35,36,37,38,39,40]. The management and control of RP both in traditional and competitive systems have been one of the main concerns of exploiters [41,42,43,44,45,46,47,48,49,50]. RP in systems was optimized via adjusting the voltage of generators, tap transformers capable of changing tap under load, and the size of parallel capacitors. Due to the expansion of power networks, the exploitation of RP resources for increasing voltage profile and reduction has attracted considerable attention. Although the production of RP does not incur any costs itself, it influences the total cost due to its impact on system losses. In general, RP planning involves the two issues of locating and operating [51,52,53,54,55,56,57,58,59,60]. The issue of locating includes determining the type of RP resources, while the issue of operating emphasizes the optimal arrangement of available RP resources. Parameters that need to ordered RP flow problems are the voltage of controlled buses, tap transformers with the capability to change tap under load, and the size of parallel capacitors. While solving RP flow, it is assumed that economic load flow has been completed, and the amount of real power produced by each generator is determined [61,62,63,64,65,66,67,68]. Therefore, RP planning can be expressed in ways that have functions and constraints [69,70,71,72].

For solving this problem, various objective functions can be considered. RP planning is considered minimizing capital expenditure (CAPEX) and operating expense (OPEX) because their constraints are fulfilled in most papers. CAPEX includes the costs of purchasing, setting up, and conserving novel RP resources in the network. OPEX is considered, and the aim is to minimize these costs [73]. An objective function was considered the primary function and other functions as the constraints, and optimization was conducted to solve this problem. It is challenging to implement such a method to solve multi-objective non-differentiable constraint optimization problems where the solution space is non-convex. In [74], the non-convex and discontinuous constraint in the simplex algorithm is eliminated by implementing the Big M method, which models this disjunctive constraint through a mixed-integer formulation.

On the other hand, due to greenhouse gases and ozone layer depletion, renewable energies such as wind power are currently developing rapidly in most countries. The notion of expansion is synonymous with protection concerning the natural environment and is also discussed in GDP and the use of natural and environmental resources [75,76,77]. The imminent end of fossils has urged researchers to discover renewable energies for substituting the standard energy system [78]. RP control has become more complicated under the influence of renewable resources.

The reasons mentioned above led power industry engineers to revise the existing power systems’ design and present the design of flexible AC transmission systems (FACTS). This design, which involves the use of power electronics devices to increase the transmission capacity and controllability of power systems, has established a new approach to current dynamic problems and control of power systems and has introduced new solutions in this field along with the advancements in power electronics technology and the introduction of the concept of flexible AC transmission systems called FACTS [79]. Planners have found it interesting that concerning RP, FACTS technology provides new chances to control and improve the operational capacity of systems and the new and upgraded lines.

The possibility of controlling the flow within a transmission line at a reasonable cost makes it possible to increase the capacity of the existing lines in the form of lines with more prominent conductors and to increase power flow through such lines under normal and unpredictable conditions by using one of the FACTS controllers [78]. In RP planning, various methods have been proposed that can be divided into mathematics and optimization intelligence.

The first category is based on mathematical models such as non-linear, linear, and quadratic programming [80,81,82]. Unfortunately, the methods described above lose their effectiveness despite the success achieved when the system has many variables in the non-linear structure.

The second category is based on optimization intelligence, which is generally faster and more reliable than the mathematical methods. The volume of computations in this method is less than that of the classical methods. In this category, the following references can be referred to. Shaheen et al. [81] used a modified multi-agent evolutionary model of the RP planning problem. The weakest buses are selected for the installation of RP. Then they are solved using the evolutionary algorithm and target functions based on loss costs and losses. The proposed method in this reference is applied in various standard systems, and the results obtained are compared using genetic and harmonic search algorithms. Bin et al. [82] proposed a multi-objective group search method to accommodate capacitors with different target functions optimally. The voltage drop in lines and the cost of capacitor installation are selected as the target functions. In this paper, the capacitor is used ideally and continuously, and, therefore, the result obtained cannot be applied directly and without changes.

Bhattacharyy et al. [83] addressed the teacher–student learning method for solving RP planning problems by considering linear and non-linear constraints. The proposed method in this reference is applied in a variety of standard systems. Shaheen et al. [84] provide a comprehensive overview of all available methods in the first and the second categories. In this reference, the weaknesses and strengths of the available methods are described meticulously. In [85,86], the gravitational search method is employed to RP problems with FACTS devices presence. Xian et al. [87] modeled the effect of energy on economic distribution for power and pollution. In this model, probabilistic methods are employed for wind energy modeling. El-sobky et al.’s [88] method based on a reliable search area was used for solving the dynamic distribution problem of pollution. Arul et al. [89] used the self-adaptive harmony search algorithm for solving the problem. Control coefficients are focused in a self-regulated manner with program repetitions. The chaos theory based on logistics has also been utilized to improve local search. Despite the methods mentioned earlier, due to the non-linear, discrete, and continuous nature of the constraints in the RP programming problem, a hybrid model that can cover this issue more accurately is required. For overcoming this weakness, the virus colony search optimization algorithm is presented [90]. Because voltage limits and variations are introduced as a constraint, the bus voltage can decrease to a maximum, which increases the probability of voltage instability in those buses. This issue is fundamental in power systems under stress that has been brought on in recent years due to economic issues. Therefore, in this paper, two other target functions, namely, increasing the system security and minimizing the voltage bias of the bus, are considered, and the problem is taken into account as multi-objective programming. The proposed method, coded as a multi-objective method, is one of the highlights that has never been presented in studies thus far.

The model efficiency was investigated on sample systems with FACTS resources formed on proper criteria, and the obtained data were compared to another model. According to the explanations mentioned earlier, the following points can be made about the innovation and highlights of this paper:

• (A).Improving local and general search via algorithm form on virus search, particle swarm optimization. This hybrid algorithm tries to employ its powerful searching operators in the optimization problem. In addition, it develops the standard particle swarm optimization with the best guiding during the search. Furthermore, it presents a fuzzy mechanism to select the best solution from several solutions.
• (B).Modeling the practical constraints in RP programming with continuous and discrete variables in an optimization problem and considering the wind units and FACTS devices to make an accurate model of power system
• (C).Consider several analyses and scenarios to evaluate the proposed model and optimization algorithm. In addition, present some comparisons with another optimization algorithm.
• (D).The second section deals with modeling RP distribution by considering renewable sources and FACTS devices. The third section is devoted to modeling the proposed multi-objective hybrid algorithm. The fourth section describes how to implement the algorithm on the RP programming, and the fifth section presents simulation data. The sixth section was dedicated to a conclusion.

2. Materials and Methods

Due to the increase in electric power consumption over the past decades, electric power supply systems have been expanded to such an extent that today the optimal distribution of RP for optimal programming and utilization of power systems between energy production units with the lowest cost has become widespread and complex in power system issues. The formulation of the problem of RP programming by considering linear and non-linear constraints is defined as follows.

2.1. RP with FACTS without Wind Unit

In this paper, to achieve more comprehensive RP programming, different target functions have been applied according to the following descriptions.

2.1.1. Loss Function

It is an essential function in RP programming and aims to reduce network losses, increasing network efficiency. To calculate network losses, the Newton–Raphson method has been employed [48]:

$$P_L={\displaystyle {\sum }_{k=1}^{N_L}{g}_k[V_i^2+V_j^2-2V_i{V}_j\cos ({\theta }_i-{\theta }_j)}$$ (1)

V_i and V_j represent voltage for the beginning and last of lines i and j, and the θ_i and θ_j show the angles at the beginning and end of the line, respectively.

2.1.2. Voltage Equalization

Common issues in RP programming are network voltage stability. As the nature of the network voltage is different from that of the loss function, a new function called network voltage equalization, which indicates difference for desired per bus and obtained after each run of the load distribution program, is as follows:

$$V_{Index}=\frac{1}{N}{\displaystyle \sum _{i=1}^N|V_{act}-V_{des}|}$$ (2)

In the above relation, N is the candidate bus number for installation in RP sources, V_act is the value obtained from the load distribution problem with RP sources, and V_des is the expected value according to the constraints on the power system network.

2.1.3. The Cost of FACTS Devices

The cost of equipment performance consists of two parts: the cost of the active power and the cost including shunt equipment. The target is obtained via:

$$C_{Total}=C_E+C_{Sh}$$ (3)

C_E is the cost of losses ($), and C_Sh is the cost of shunt capacitors ($). C_E generally increases by considering system losses. In this paper, installing shunt equipment cost, installing equipment cost, and the cost of energy losses have been considered to be equal to 3 $/kVar, 1000 $, and 0.06 $/KWh, respectively. Now, the target function for RP programming is expressed by considering the FACTS equipment as follows:

$$C_{FACTs}=C_{TCSC}+C_{SVC}$$ (4)

where C_TCSC and C_SVC show the cost of equipment performance of TCSC and SVC in RP programming, respectively, as follows [17]:

$$C_{TCSC}=0.0015{(TCS{C}_{value})}^2-0.7130(TCS{C}_{value})+153.75(US\$/kvar)$$ (5)

$$C_{SVC}=0.0003{(SV{C}_{value})}^2-0.3051(SV{C}_{value})+127.38(US\$/kvar)$$ (6)

2.1.4. The Objective Function

As seen above, the final target function is presented separately for each section. The target should be considered as consisting of all of the above points. The final target function proposed in this paper is based on the value of losses, voltage fluctuations, and the assignment of FACTS devices and the related constraints as follows:

$$\begin{array}{l}Minimize\alpha {\displaystyle \sum _{k=1,\dots ,N_L}{P}_{loss}^{(k)}}+\beta {\displaystyle \sum _{i=1,\dots ,N}{Q}_{ci}{y}_i}+\gamma {\displaystyle \sum _{i=1}^{Nd}|V_i-V_i^0|}\\ subject.to.\\ {\displaystyle \sum _{i=1}^N{y}_i=N_c}\\ P_{G_i}^{(k)}-P_{L_i}^{(k)}-V_i^{(k)}{\displaystyle \sum _{j{w}_i}{V}_j^{(k)}}\times (G_{ij}\cos {\theta }_{ij}^{(k)}+B_{ij}\sin {\theta }_{ij}^{(k)})=0\\ Q_{G_i}^{(k)}+Q_{C_i}^{(k)}(y_i)-Q_{L_i}^{(k)}-V_i^{(k)}{\displaystyle \sum _{j{w}_i}{V}_j^{(k)}}\times (G_{ij}\sin {\theta }_{ij}^{(k)}-B_{ij}\cos {\theta }_{ij}^{(k)})=0\\ V_{i\min }^{(k)}\le V_i^{(k)}\le V_{i\max }^{(k)}\\ P_{G_i\min }^{(k)}\le P_{G_i}^{(k)}\le P_{G_i\max }^{(k)},Q_{C_i\min }^{(k)}\le Q_{C_i}^{(k)}\le Q_{C_i\max }^{(k)},i\in NG\\ -S_{ij\max }^{(k)}\le S_{ij}^{(k)}\le -S_{ij\max }^{(k)},P_{tieline}^{(k)}\le TTC(Q_C^{(k)})\end{array}$$ (7)

where NG, N, N_L, and N_c represent generators, buses, load levels, and Var resource numbers. k shows load levels, and k, T^(k), α, β, and y_i represent kth load level, cost of energy, coefficient of the maintenance cost, cost of allocating the equipment, and the binary variable, respectively. If y_i = 1, the Var equipment in that busbar is installed; otherwise, y_i = 0. PG_i, QG_i, PL_i, and QL_i represent the output of the active power of generators, the output of the RP, the active load, and load in request in ith, respectively. S_ij^k is the current flowing from the i-j line. When S_ij^k is not equal to zero, then Q_ci will be a specific capacity of Var for the bus i, and Q_ci^k is the value of Var required at k. P_tieline is power flowing with communication lines between them. TTC(Q_c^(k)) is a one-way interpolation following Formula (4) for voltage stability. Q_c is the total set of Q_ci candidates.

$$p(x,y)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^{n}f(x_i,y_i)}}{\displaystyle \prod _{l\ne i}(\frac{x-x_l}{x_i-x_l})}{\displaystyle \prod _{k\ne j}(\frac{y-y_k}{y_j-y_k})}$$ (8)

The y and x were Var capacities, p(x,y), TTC, and Var function capacity. Equation (7) addresses the minimization of cost, voltage stability, and system losses. As stated previously, one of the goals of the present paper is to ensure the system voltage stability during optimal load distribution. The most crucial issue is voltage stability for both standard and critical (unwanted) conditions. For an extensive system with a probability of various errors, these two modes are a challenge. In order to address this problem, the following formulation are able to guarantee system stability under different working conditions:

$$\begin{array}{l}\underset{k}{\min }\{\max {\displaystyle \sum _{\begin{array}{l}i\in Source Area\\ j\in Sink Area\end{array}}(P_{ij}^{(k)}-P_{ij0})}\}\\ subject.to.\\ {\displaystyle \sum _{i=1}^N{y}_i=N_c}\\ P_{G_i}^{(k)}-P_{L_i}^{(k)}-U_i^{(k)}{\displaystyle \sum _{j{w}_i}{U}_j^{(k)}}\times (G_{ij}\cos {\theta }_{ij}^{(k)}+B_{ij}\sin {\theta }_{ij}^{(k)})=0\\ Q_{G_i}^{(k)}+Q_{C_i}^{(k)}(y_i)-Q_{L_i}^{(k)}-U_i^{(k)}{\displaystyle \sum _{j{w}_i}{U}_j^{(k)}}\times (G_{ij}\sin {\theta }_{ij}^{(k)}-B_{ij}\cos {\theta }_{ij}^{(k)})=0\\ U_{i\min }^{(k)}\le U_i^{(k)}\le U_{i\max }^{(k)}\\ P_{G_i\min }^{(k)}\le P_{G_i}^{(k)}\le P_{G_i\max }^{(k)},Q_{C_i\min }^{(k)}\le Q_{C_i}^{(k)}\le Q_{C_i\max }^{(k)},i\in Source Area\\ -S_{ij\max }^{(k)}\le S_{ij}^{(k)}\le -S_{ij\max }^{(k)},\frac{P_{Li}^{(k)}}{P_{Li}^0}=\frac{Q_{Li}^{(k)}}{Q_{Li}^0},i\in Sink Area\end{array}$$ (9)

where Q⁰_Li and P⁰_Li are the initial demand for active RP in ith busbar, respectively. P_ij0, P_ij^(k) were the initial power and the current power flowing in the line between the i-j busbars. K = 1, …, N_cntg shows the different working conditions, including normal conditions and those containing the error.

2.1.5. Fuzzy Classification

One of the essential points in RP was investigated optimal location for FACTS devices. In other words, the most crucial point in RP was determining the proper location of the Var equipment in order to reduce calculation time and ensure voltage stability. In this paper, interactive fuzzy classification is proposed to obtain FACTS location devices. Fuzzy classification combines standard fuzzy classification and regression [49]. In the proposed fuzzy method, the distance (d_ij) between each data (x_i) and the center of the class (v) is expressed based on the standard fuzzy distance and regression:

$$d^2((x_j,y_j),(v_i,{\beta }_i))=\underset{fuzzy \mathrm{distance}}{\underbrace{\left|\right|x_i-v_i\left|\right|}}+\underset{fuzzy-regression \mathrm{distance}}{\underbrace{{(y_j-{\beta }_i^T{x}_j)}^2}}$$ (10)

where $x_j\in X$ and $y_j\in Y$ are input and output data, respectively. nd in the index j = 1, …, nd is the data number in a vector for training.

c in the index i = 1, …, c is the number of centers of the classes. ${\widehat{y}}_j={\beta }_i^{T}x$ is the polynomial estimation, where β is a polynomial coefficient. Based on the above definitions, the membership function u_j(x_i) can be expressed as follows:

$$u_{ik}={({\displaystyle \sum _{j=1}^c\frac{d_{ik}^2-(\min d_{ik}^2-\eta )}{d_{jk}^2-(\min d_{jk}^2-\eta )}})}^{-1},\eta >0$$ (11)

where $\eta >0$ is considered as a constant value. In order to optimize fuzzy members, the error of the regression is considered in the following objective function:

$$J_m^{M\mathrm{FC}}=\underset{FCM}{\underbrace{{\displaystyle \sum _{i=1}^c{\displaystyle \sum _{k=1}^{nd}{\mu }_{ik}^m{d}_{ik}^2}}}}+{\displaystyle \sum _{i=1}^c{\displaystyle \sum _{j=1}^{nd}{\mu }_{ik}^m\underset{error \mathrm{of} \mathrm{fuzzy} \mathrm{regression}}{\underbrace{{(y_k-f_i({\tau }_{ik}))}^2}}}}$$ (12)

where ${\mu }_{ik}$ is the undefined function from the kth input vector in the ith fuzzy category. m is the degree of fuzzification that expresses the amount of overlap between the categories. As the abovementioned relationship shows, $J_m^{M\mathrm{FC}}$ consider two parts. The first part, which is similar to the standard fuzzy classification mechanism (FCM), controls the categorization accuracy. The second part is the sum of the error squares ( $f({\tau }_i)$ ) derived from the ith category. This section checks the error rate due to the fuzzy classification based on the fuzzy function, the matrix ( $\tau$ ), and the “or” and “and” constraints. Based on the new fuzzy distance defined in the above relation, we can introduce the new fuzzy membership function as follows:

$$\underset{\begin{array}{l}1<i<c\\ 1<k<nd\end{array}}{{\mu }_{ik}}={({\displaystyle \sum _{j=1}^c[\frac{d_{ik}^2+{(y_k-f_i({\tau }_{ik}))}^2}{d_{ik}^2+{(y_k-f_i({\tau }_{jk}))}^2}}]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}{)}^{-1}$$ (13)

In the new membership function, the centrality of each category can be expressed as follows:

$$\underset{1<i<c}{\forall }{v}_i=\frac{{\displaystyle {\sum }_{k=1}^{nd}{({\mu }_{ik})}^m(x_k{y}_k)}}{{\displaystyle {\sum }_{k=1}^n{({\mu }_{ik})}^m}}$$ (14)

The above formula states that the membership function for x_i depends on the distance between that variable and all v_i categories. The categories created in this paper can be defined based on the loss factor, voltage stability, and the cost of installing FACTS equipment.

2.2. RP with FACTS and Wind Unit

Rapid progress in wind power generation has led to a significant increase in the installation of wind turbines in the world. Research centers worldwide are extensively studying and researching various aspects of the optimal utilization of this energy resource. Simulation of wind turbines by software is a research tool for investigating the behavior of wind turbines. In addition, increasing the simulation parameters and considering the system details increase the accuracy of the simulations and the simulation time. Therefore, various dynamic models of generators for power generation from wind have been studied, one of which is electricity generation from the double-fed induction generator (DFIG) [6]. DFIG can be a good option for systems with variable speed (approximately 30% synchronous speed). Figure 1 shows the schema of a DFIG [91].

Figure 1.

Variable speed wind turbine with a DFIG.

In a DFIG, the injection of RP for the network depends on the control system and the size of the converter. Figure 2 shows the P-Q characteristic for wind turbines (WTs) used in this paper. Specifications of this turbine (Gamesa WT G80-2.0MW) can be obtained from the reference [92]. This turbine provides a 0.98 power factor in the pre-phase mode and 0.96 in the post-phase mode. Therefore, its reactivity depends on the generation of active power, as shown in Figure 3.

Figure 2.

Q of WT in G80 (-2.0 MW).

Figure 3.

Q of WF in twelve G80 (-2.0MWWTs).

The green line in Figure 3 shows the P-Q of the wind turbine (WT) at the power factor 1. For power less than 10 megawatts, the cables have a more significant effect than the transformer, while the transformer is more influential in the high-power WT. In addition, if the wind farm (WF) receives the RP of a capacitor lower than the range of the active power generation, then WTs are set to the selfie mode. Based on the above explanations, since WFs can inject or receive RP, we need to adjust them optimally. In order to optimize RP, STATCOM or capacitor is used. The ultimate target function, by taking into account the loss function, is as follows:

$${\mathrm{Minimize} \mathrm{J}(\mathrm{Var}}_{\mathrm{x}}{,\mathrm{Var}}_{\mathrm{y}}{)=\mathrm{Min} \mathrm{P}}_{\mathrm{losses}}$$ (15)

where Var_y shows the location of the tapped transformer. The optimal configuration of STATCOM and capacitor is performed independently, which includes the following constraints:

$$Q_{W{T}_i}^{\min }\le Q_{W{T}_i}\le Q_{W{T}_i}^{\max },i=1,2,\dots ,N_G$$ (16)

$$T_i^{\min }\le T_i\le T_i^{\max }$$ (17)

$$Q_{Statcom}^{\min }\le Q_{Statcom}\le Q_{Statcom}^{\max }$$ (18)

There is also a non-linear constraint. RP demand for PCC to adjust voltage is modeled as follows:

$$Q_{PCC}^{\ast }=Q_{PCC}^{meas}$$ (19)

In this paper, the condition of obtaining the possible response is given in Equation (20) to guarantee that the set of solutions covers the limits imposed on the system:

$$S_i^{k+1}=\{\begin{array}{l}{S}_i^k+v_i^{k+1},S_i^{\min }\le S_i^k+v_i^{k+1}\le S_i^{\max }\\ S_i^{\min } ,S_i^{\min }>S_i^k+v_i^{k+1}\\ S_i^{\max } ,S_i^k+v_i^{k+1}>S_i^{\max }\end{array}$$ (20)

Since Constraint (19) is an equality constraint, to apply it non-linearly, we can use the following equation based on error (ε):

$$Q_{PCC}^{*}-Q_{PCC}^{meas}|<\epsilon$$ (21)

Based on the explanations mentioned earlier, Figure 4 shows the strategy used in this paper.

Figure 4.

Solution search process.

3. The Hybrid Algorithm of Virus Colony Search and Modified Particle Swarm Optimization

3.1. Virus Colony Search Algorithm

• Definitions and Basic Concepts

VCS algorithm form on viruses population, and it was founded for two virus behaviors. Viruses require host cells to ensure their survival, production, and plurality. In what follows, the strategy of viruses has been described, including host cell contamination [22].

The function of the immune system is also considered as the opposite relationship between these two stages. These stages are explained in more detail as follows:

• (1)Virus replication: Viruses randomly scan the cells to find the raw materials needed for survival. In this process, a random walk method can be the best mathematical model for expressing the function of viruses to detect host cells.
• (2)The infection or influence of host cells: When a virus selects a cell, it tries to replicate itself in the best manner. In other words, the virus replicates itself based on the essential materials in the host cell until the host cell dies and acts like a virus. To model this behavior, the CMA-ES mathematical model is a covariance matrix-based method.
• (3)Function of the immune system: The immune system has the essential task of protecting the cell against the replication and spread of the virus. In addition, more powerful viruses save themselves from reproducing in the proper position.

• Matching mathematical models

Virus algorithm has three models: (1) Gaussian model to virus propagation that utilized for recognition improving; (2) CMA-ES model to cell damage modeling that utilized for social behavior improving; and (3) evolution strategy to resist a virus that utilized for improving two other strategies [93]. Therefore, the virus colony algorithms have five steps:

1. Vpop virus and Hpop cells were utilized in the VCS model.

2. Virus for transferring has unique behavior.

3. Viruses can desire to infect the cell.

4. Virus reproduction was formed on cell destruction to find survival.

5. Best viruses can remain for replication.

• Virus transmission

Gaussian model is formulated via this equation:

$$V_{po{p}_i^{’}}=Gaussian\left(G_{best}^g,\tau \right)+(r_1.G_{best}^g-r_2.V_{po{p}_i})$$ (22)

The ɩ standard deviation was calculated via $\log \left(g\right)/g.(V_{po{p}_i}-G_{best}^g)$ _the $\log \left(g\right)/g$ is used to increase virus search. The $\left(G_{best}^g,\tau \right)$ is produced the best answer for virus searching via vector $(r_1.G_{best}^g-r_2.V_{po{p}_i})$ .

• The cell influence

The cell was infected, and it can be modeled via CMA-ES in these steps:

Step 1: Update the Hpop via:

$$H_{po{p}_i^g}=X_{mean}^g+{\sigma }_i^g\times N_i(0,{\mathit{C}}^g)$$ (23)

where σ^g > 0. $X_{mean}^g$ was obtained via this equation:

$$X_{mean}^0=\frac{1}{N}{\sum }_{i=1}^N{V}_{po{p}_i}$$ (24)

Step 2: Select vector γ and obtain parents vector via this equation:

$$X_{mean}^{g+1}=\frac{1}{\gamma }\sum _{i=1}^{\gamma }{\omega }_i.V_{po{p}_i^{\gamma best}}|{\omega }_i=\ln \left(\gamma +1\right)/(\sum _{j=1}^{\gamma }(\ln \left(\gamma +1\right)-\ln (j))$$ (25)

In the above relation, $\gamma =[N/2]$ , w_i is the coefficient of the combination, and two mentioned paths are obtained via:

$$P_{\sigma }^{g+1}=\left(1-C_{\sigma }\right)P_{\sigma }^g+\sqrt{C_a\left(2-C_{\sigma }\right){\gamma }_{\omega }}\frac{1}{{\sigma }^g}{\left({\mathit{C}}^g\right)}^{-\frac{1}{2}}(X_{mean}^{g+1}-X_{mean}^g)$$ (26)

$$P_C^{g+1}=\left(1-C_c\right)P_C^g+h_{\sigma }\sqrt{C_C\left(2-C_C\right){\gamma }_{\omega }}\frac{1}{{\sigma }^g}(X_{mean}^{g+1}-X_{mean}^g)$$ (27)

The parameters were set via $C_{\sigma }=({\gamma }_{\omega }+2)/(N+{\gamma }_{\omega }+3)$ , s.

Step 3: Update ${\sigma }^{g+1}$ and $C^{g+1}$ via:

$${\sigma }^{g+1}={\sigma }^g\times \exp (\frac{C_{\sigma }}{d_{\sigma }}(\frac{\left|\right|P_{\sigma }^{g+1}\left|\right|}{E\left|\left|N\left(0,1\right)\right|\right|}-1))$$ (28)

$${\mathit{C}}^{g+1}=\left(1-C_1-C_{\gamma }\right){\mathit{C}}^g+C_1{P}_C^{g+1}{(P_C^{g+1})}^T+C_{\gamma }\sum _{i=1}^{\gamma }{w}_i\frac{(V_{po{p}_i^{\gamma best}}-X_{mean}^g)}{{\sigma }^g}.\frac{{(V_{po{p}_i^{\gamma best}}-X_{mean}^g)}^T}{{\sigma }^g}$$ (29)

where $C_{\gamma }$ is acted via this equation:

$$C_1=\frac{1}{{\gamma }_w}((1-\frac{1}{{\gamma }_w})\min \{1,\frac{2{\gamma }_w-1}{{\left(N+2\right)}^2+{\gamma }_w}\}+\frac{1}{{\gamma }_w}\frac{2}{{\left(N+\sqrt{2}\right)}^2}),C_{\gamma }=({\gamma }_w-1)C_1$$ (30)

• Immune system

The evolutionary movement is obtained via these steps:

Step 1: P_r to V_pop population:

$${\mathrm{Pr}}_{rank(i)}=\frac{N-i+1}{N}$$ (31)

Step 2: Each individual grows individually to V_pop via this equation:

$$\{\begin{array}{l}{V}_{po{p}_{i,j}^{\prime }}=V_{po{p}_{k,j}}-rand.(V_{po{p}_{h,j}}-V_{po{p}_{i,j}})\\ {\mathrm{if} \mathrm{r}>\mathrm{Pr}}_{rank(i)}\\ V_{po{p}_{i,j}^{\prime }}=V_{po{p}_{i,j}}\\ \mathrm{otherwise}\end{array}$$ (32)

The response is tried to save the best stage value, and other unusual response can be obtained via this equation:

$$x_{ij}=lo{w}_{ij}+rand\times (u{p}_{ij}-lo{w}_{ij})$$ (33)

3.2. Modified Particle Swarm Optimization

Eberhart has introduced a swarm optimization model by the group search for food by birds or fish [94]. In the past decade, a variety of topologies has been proposed for the PSO algorithm. Topologies used to exchange information between particles in the PSO algorithm include star, ring, and square topologies [95]. Figure 5 shows the relationship between particles in different topologies.

Figure 5.

Topologies used in the PSO algorithm.

In the star topology for D-dimensional space, the best personal status for particle i is indicated by ${\overrightarrow{p}}_i=(p_{i1},p_{i2},\dots ,p_{iD})$ and the group’s best position $\overrightarrow{g}=(g_1,g_2,\dots ,g_D)$ . The relations between the velocity and movement of particle i in the next moment or repetition are obtained by the following relations [29].

$$v_{id}(t+1)=\omega v_{id}(t)+c_{1}ran{d}_1(p_{id}(t)-x_{id}(t))+c_{2}ran{d}_2(g_d(t)-x_{id}(t))$$ (34)

$$\overrightarrow{x}(t+1)=\overrightarrow{x}(t)+\overrightarrow{v}(t+1)$$ (35)

c₁ and c₂ are called the cognitive and the social parameters, respectively, which are usually set to 2. To randomize the nature of velocity, the coefficients c₁ and c₂ are multiplied by the random numbers rand₁ and rand₂. Usually, in the implementation of the PSO, the value of ω decreases linearly from one to zero. Generally, the inertia coefficient ω is governed by the following equation.

$$\omega ={\omega }_{\max }-\frac{{\omega }_{\max }-{\omega }_{\min }}{ite{r}_{\max }}.iter$$ (36)

The value of v_max is usually chosen in such a manner that v_max = kx_max and 0.1 < k < 1. x_max specifies the length of the search. As is clear from Formula (34), the coefficients c₁ and c₂ are usually considered constant, which is the dark point in the local and final search for the particle swarm model. These coefficients can be suggested to improve the efficiency of virus search:

$$v_{id}(t+1)=\omega v_{id}(t)+c_{1i}ran{d}_1(p_{id}(t)-x_{id}(t))+c_{2i}ran{d}_2(g_d(t)-x_{id}(t))$$ (37)

The correlation coefficient $c_{1i}$ is updated in each replication. If $c_{1i}$ has a small value, then $c_{1i}{r}_1$ is small and local search is strengthened. Conversely, it has an immense value if it has a significant value, thus improving the overall search. To select the best value for $c_{1i}$ , two thresholds of T₁ < 0 and T₂ > 0 and two variables of R₁ in the range (0, T₁) and R₂ in the range (0, T₂) are used and defined as $c_{1i}=2^{R_1}$ and $c_{2i}=2^{R_2}$ . As a result, the two vector populations with coefficients of $c_{1i}$ and $c_{2i}$ are generated. When T₁ < 0, then R₁ is negative, and $c_{1i}$ has a small value, thus improving local search.

3.3. The Proposed Hybrid Algorithm

This section introduces the structure of the hybrid modified particle swarm optimization and HMPSO-VCS model. Figure 6 shows a schema of the HMPSO-VCS. It also shows that the hybrid VCS and particle swarm optimization algorithm is started for an initial population. When N-dimensional problem is obtained, then the hybrid algorithm has 4N members generated entirely randomly. Next, 4N members are sorted according to their eligibility, and the upper 2N members enter the VCS as a virus. VCS formulations create the new 2N member population. The mechanism of optimizing the particle swarm is applied to the 2N lower members as particles. In applying the mechanism of particle swarm optimization, the population is obtained via VCS. The P_gbest is the corresponding member is used in Equation (37). The population generated by applying the mechanism of the particle swarm optimization to the population obtained via virus colony search algorithm was merged, and the 4N new member population is sorted according to the eligibility, and the previous process is repeated until convergence is achieved.

Figure 6.

Combining the VCS (⟶) and swarm optimization (-→).

The steps for the HMPSO-VCS algorithm are as follows:

• (1)Random generation of the initial population with 4N members as initial responses
• (2)Evaluating and sorting the population according to its eligibility
• (3)
  Applying the VCS algorithm to the 2N upper members of the population based on the mutation and crossover of the generations

  • Selection: For the target population, the best 2N members are selected based on their eligibility.
  • Crossover: For a well-chosen population, we use the crossover of two parents to produce a new generation.
  • Mutations: 20% of the new population is mutated.
• (4)The particle swarm optimization algorithm is applied to another 2N population based on the production relations of the new population, and the new population is produced. Next, 2N population is combined with the 2N population generated by the virus colony search algorithm.
• (5)Repeat the previous steps from Step (2) until the convergence or termination requirements are met.

3.4. The Proposed Multi-Objective Algorithm

The proposed algorithm has been used based on the entropy accumulation to label each objective function. Entropy optimization is based on the Pareto criterion. In this method, for optimization in each step, a set is labeled as a better response and enters the next step. The criterion for the multi-objective problem can be obtained via:

The X₁ was no worse than X_2, and X₁ were better than X_2, and mathematical terms are stated via:

$$\begin{array}{l}{X}_1\prec X_2\iff ({\forall }_i\in \{1,2,\dots ,n\}:f_i(X_1)\le \\ f_i(X_2))^({\exists }_i\in \{1,2,\dots ,n\}:f_i(X_1)\le f_i(X_2))\end{array}$$ (38)

In addition, the $X\in X_f$ is stated non-dominated of relation to $A\subseteq X_f$ if

$${\exists }_a\in A:X\prec a$$ (39)

Therefore, we can assume that the vector X is optimal [96]. The non-dominated decision vectors sets is obtained via:

$$P(A)=\{a\in A|a\mathrm{is}\mathrm{Non}-\mathrm{dominated}A\}$$ (40)

Set A was equal to X_f, then P(A) is produced Pareto optimal front. An ideal point and anti-ideal point, objective functions are minimal and maximal, respectively.

3.5. Combining Fuzzy Logic with the Proposed Algorithm

Figure 7 indicates membership μ_c [97] for fuzzy that states cost concept.

Figure 7.

The membership function for the cost and loss function and the voltage synchronization.

The f_i(P_gi) is stated via μ_i(P_gi) in this equation:

$${\mu }_i=\frac{f_i^{\max }-f_i}{f_i^{\max }-f_i^{\min }}$$ (41)

The lower limits of ith objective function:

$$FD{M}_i=\{\begin{array}{c}0\\ {\mu }_i\\ 1\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}{\mu }_i\le 0\\ 0<{\mu }_i<1\\ {\mu }_i\ge 1\end{array}$$ (42)

The FDM^k function is obtained via:

$$FD{M}^k=\frac{{\displaystyle \sum _{i=1}^{2}FD{M}_i^k}}{{\displaystyle \sum _{j=1}^M{\displaystyle \sum _{i=1}^{2}FD{M}_i^j}}}$$ (43)

The decision maker is selected the best solution by considering the function in actual conditions.

4. Implementation of Problem of RP Planning

To solve RP with FACTS presence, the following steps are performed:

Step 1: The set initial parameters of the suggested algorithm are studied. The initial population with N units is produced as follows:

$$P=[Va{r}_1,Va{r}_2,\dots ,Va{r}_n{ \mathrm{Q}}_1{,\mathrm{Q}}_2{,\mathrm{Q}}_3,\dots {,\mathrm{Q}}_m{,\mathrm{T}}_1,{\mathrm{T}}_2,\dots ,{\mathrm{T}}_o]$$ (44)

In this regard, the number of FACTS devices, tap transformers, and shunt capacitors is encoded as mentioned above. Subnets n, m, and o represent the number of variables used. Variables are also generated based on the upper and lower bounds:

$$\begin{array}{l}{T}_i=T_{i,\min }+rand\times (T_{i,\max }-T_{i,\min }),i=1,2,3,\dots ,o\\ Q_i=Q_{i,\min }+rand\times (Q_{i,\max }-Q_{i,\min }),i=1,2,3,\dots ,m\\ Va{r}_i=Va{r}_{i,\min }+rand\times (Va{r}_{i,\max }-Va{r}_{i,\min }),i=1,2,3,\dots ,n\end{array}$$ (45)

The initial responses should be placed within the constraints to create a better search for the algorithm at this stage.

Step 2: Apply the limitations on the problem, arrange them on the cost function.

Step 3: Use the proposed fuzzy cluster algorithm to select the best load bus bars for installing the FACTS devices.

Step 4: Choose the best response, and these sets of responses are replaced in initial responses that are considered null.

Step 5: Use the target function introduced for the loss function, voltage, and cost of FACTs installation, calculate the fitness of responses and select the best response.

Step 6: Compare good response and stored response.

Step 7: Sort data based on the Pareto criterion and choose the best response sets as the best current response.

Step 8: Replace the worst responses by random data and use a modified particle swarm to find the best response.

Step 9: Is termination completed? In this case, leave the program; if not, go to Step 3.

Figure 8 shows the flowchart of the proposed algorithm.

Figure 8.

Applying the proposed algorithm and FACTS presence.

5. Results and Simulation Analysis

5.1. Determination of Optimal Location by the Proposed Fuzzy Method

First, we propose a fuzzy method for obtaining FACTS installation. The purpose of this action is to determine the most talented buses for installing these resources. To determine these locations, criteria such as L-index, U/U₀, and voltage stability have been used.

• (A).U/U₀ criterion: This criterion is considered based on the voltage range for all buses as follows:

$$\frac{U}{U_0}=[\frac{U_1}{U_{01}},\dots ,\frac{U_k}{U_{0k}}]$$ (46)

The U and U₀ are the voltage range vector of the normal state for all buses and the voltage range vector of all bass when all loads are zero.

• (B).L-Index criterion: Generally, increasing load and the optimal performance of the power system are considered more than before. A voltage failure can be created suddenly in the system, resulting from weaker buses. For the jth bus, the rate of voltage drop or failure is expressed as follows:

$$\{\begin{array}{l}{L}_j=|1-{\displaystyle {\sum }_{i=1}^{N_{PV}}{F}_{ji}{V}_i{V}_j^{-1}},j=1,2,\dots ,N_{PQ}\\ F_{ji}=-{[Y_1]}^{-1}[Y_2]\end{array}$$ (47)

Y₁ and Y₂ matrices were below the invertible submatrices for Y_BUS matrix, which are obtained by sorting them according to the following matrix:

$$\left[\begin{array}{l}{I}_{PQ}\\ I_{PV}\end{array}\right]=\left[\begin{array}{l}\begin{array}{cc}{Y}_1& Y_2\end{array}\\ \begin{array}{cc}{Y}_3& Y_4\end{array}\end{array}\right]\left[\begin{array}{l}{V}_{PQ}\\ V_{PV}\end{array}\right]$$ (48)

• (C).Voltage stability criterion is expressed in Formula (2).

5.2. IEEE 30-Bus Standard

The 30-bus system is the most popular in power engineering. This system consists of 6 power plants and 46 transmission lines. Table 1 presents the results of applying the fuzzy method and determines the most talented bus for SVC. As the numerical results show, according to the L-Index, the most talented buses are {3, 7, 9, 14, 15, 17, 23, and 24}. According to the U/U₀ criterion, these talented buses are {4, 7, 15, 17, 20, 21, and 25}. Finally, based on the voltage criterion, the best buses for installation are {4, 7, 15, 16, 17, 21, and 24}. Based on the criteria described, the candidate buses are not the same in all three modes. By fuzzy classification, the best buses are classified as follows: {7, 15, 17, 21}, {4, 24}, and {9, 16}. Similarly, fuzzy classification is performed for TCSC, and the best candidates are lines 5, 25, 28, and 41. Now, we should consider the RP planning with the presence FACTS devices in the system under study. Table 2, Table 3 and Table 4 shows these results with the presence of FACTS devices. As mentioned previously, two types of SVC equipment have been used as shunt and TCSC series in line with the present study. Table 3 indicates a comparison of the costs and losses of the 30-bus standard system in RP planning. According to this table, it is clear that when FACTS equipment operates in sync with RP devices such as tap transformers and RP generators, the system operation costs and losses considerably improve compared with when FACTS devices are not present. In addition, the results reveal that the mentioned model has better performance than another models.

Table 1.

The results of determining the most talented bus in SVC installation.

PQ Bus No.	L-Index	U/U0	Voltage Fluctuation (%)
3	1.2222	1.227	1.1108
4	1.1307	1.189	1.0667
6	0.9732	0.9599	1.1627
7	0.996	1.1638	1.1512
9	1.2546	1.1074	1.0168
10	0.9115	1.2892	1.0727
12	1.096	1.1596	0.9062
14	0.9672	1.2201	1.2936
15	1.2915	1.0815	0.9669
16	1.1851	1.073	0.9425
17	1.1002	1.2301	1.049
18	1.0884	0.9334	0.9792
20	0.9238	0.9533	1.0959
21	1.1728	0.9694	1.0358
23	0.917	1.0564	1.2807
24	0.9286	1.2326	1.2681
25	1.1087	1.2213	0.9211
28	0.9387	0.9242	1.1951
30	1.2273	1.0597	1.0076

Table 2.

The optimum range for RP resources for the 30-bus standard system without the presence of FACTS devices.

Method	Generation of RP with Generators (Perunit)	Adjustment of Tap Transformer (Perunit)	Shunt Capacitor (Perunit)
The proposed method	0.1601 (2) 0.2611 (5) 0.3034 (8) 0.0105 (11) 0.2243 (13)	0.9046 (11) 0.9014 (12) 0.9032 (15) 0.8732 (36)	0.0243 (7) 0.0167 (15) 0.0065 (17) 0.0134 (21)
SPSO [98]	0.1705 (2) 0.2655 (5) 0.3028 (8) −0.0121 (11) 0.2467 (13)	0.9 (11) 0.9 (12) 0.9019 (15) 0.9 (36)	0.0330 (7) 0.0527 (15) 0.0 (17) 0.0 (21)
APSO [98]	0.1629 (2) 0.2671 (5) 0.2964 (8) 0.0563 (11) 0.1797 (13)	0.9 (11) 0.9133 (12) 0.9 (15) 0.9010 (36)	0.0174 (7) 0.0468 (15) 0.0025 (17) 0.0209 (21)
EPSO [98]	0.1462 (2) 0.2642 (5) 0.2977 (8) 0.1061 (11) 0.2213 (13)	0.9026 (11) 0.9 (12) 0.9 (15) 0.9 (36)	0.0 (7) 0.0 (15) 0.0295 (17) 0.0 (21)

Table 3.

The optimal range for RP sources for the 30-bus system with the presence of FACTS devices.

Method	Generation of RP with Generators (Perunit)	Adjustment of Tap Transformer (Perunit)	SVC (Perunit)	TCSC
The proposed method	0.532 (2) 0.065 (5) 0.032 (8) 0.344 (11) 0.002 (13)	0.843 (11) 0.703 (12) 0.984 (15) 0.898 (36)	0.0 (7) 0.0 (15) 0.0 (17) 0.04 (21)	0.1298 (25) 0.0837 (41) 0.2312 (28) 0.2109 (5)
SPSO [98]	0.6 (2) 0 (5) 0 (8) 0.4 (11) 0 (13)	0.9 (11) 0.9 (12) 0.9 (15) 0.9223 (36)	0.0 (7) 0.0 (15) 0.0 (17) 0.0840 (21)	0.1463 (25) 0.0419 (41) 0.1049 (28) 0.1388 (5)
APSO [98]	0 (2) 0 (5) 0 (8) 0.4 (11) 0 (13)	0.9 (11) 0.9501 (12) 0.9180 (15) 0.9330 (36)	0.0 (7) 0.0 (15) 0.0 (17) 0.0768 (21)	0.1463 (25) 0.0419 (41) 0.1049 (28) 0.1388 (5)
EPSO [98]	0.6 (2) 0 (5) 0 (8) 0.4 (11) 0 (13)	0.9439 (11) 0.9 (12) 0.9 (15) 0.9326 (36)	0.0 (7) 0.0 (15) 0.0 (17) 0.0 (21)	0.1463 (25) 0.0419 (41) 0.1049 (28) 0.1368 (5)

Table 4.

Active power losses and operation costs for the 30-bus system with different optimization methods.

FACTS	Method	Active Power Losses	Operation Costs after RP Planning (×106)	Reduction in Active Power Losses	Reduction in Operation Costs (×106)
without the presence of FACTS devices	The proposed method	0.0756	3.403	0.0024	1.5103
	SPSO [98]	0.0684	3.5951	0.0027	1.41916
	APSO [98]	0.0684	3.5966	0.0027	1.40416
	EPSO [98]	0.0685	3.6000	0.0026	1.37016
with the presence of FACTS devices	The proposed method	0.0431	2.3543	0.0279	1.38283
	SPSO [98]	0.0435	2.3622	0.0276	1.37481
	APSO [98]	0.0434	2.3558	0.0277	1.38121
	EPSO [98]	0.0438	2.3671	0.0273	1.36991

As can be seen from the results of Table 2, Table 3 and Table 4, the proposed method has been able to obtain far fewer answers compared to other methods in terms of cost criteria, loss rate, and operating cost. On average, it can be seen that the proposed method worked about 5% better.

5.3. IEEE 57-Bus Standard System

Here, to illustrate the proposed model, a more extensive 57-bus system is selected with the information mentioned in other papers. Similar to the previous section, in this section, the fuzzy classification method is used first. Based on this method, buses 25, 38, and 49 are the best option for SVC installation, and lines 13, 37, 57, and 61 are the best candidates for TCSC installation. Table 5 shows the comparison between system costs and losses of the 57-bus standard system in RP planning. The results indicate mentioned model is more successful than other models. This represents the better local and overall search of this algorithm to find the final result. This algorithm has shown better performance despite the increasing number of input variables.

Table 5.

Active power losses and operation costs for the 57-bus system with various optimization methods.

FACTS	Method	Active Power Losses	Operation Costs after RP Planning (×107)	Reduction in Active Power Losses	Reduction in Operation Costs (×106)
without the presence of FACTS devices	The proposed method	0.222	1.413	0.0821	1. 43
	SPSO [98]	0.2522	1.325	0.0277	1.46
	APSO [98]	0.2495	1.311	0.0304	1.60
	EPSO [98]	0.2526	1.327	0.0273	1.44
with the presence of FACTS devices	The proposed method	0.2205	1.163	0.0798	3.05
	SPSO [98]	0.2210	1.168	0.0589	3.03
	APSO [98]	0.2231	1.179	0.0568	2.92
	EPSO [98]	0.2275	1.203	0.0524	2.68

5.4. RP Planning for Wind Turbines (WTs)

RP planning performance in the Mongolia system is studied. There are approximately 200 wind turbines installed in this system. Capacity is over 300 megawatts. With the help of T₂ transmissions, the output of these turbines is increased to 35 kV and is transmitted through the transmission lines to the distribution site. This wind farm contains 403 nodes, and more information is available in reference [99]. Figure 9 shows a section of this network. To compensate for RP, the SVC compensator is installed in the nodes. The reference power in this system is 100 megawatts, and the most significant revenue generated from RP is approximately $500 million. Table 6 provides the comparison between the performance of the mentioned model and other models for RP planning.

Figure 9.

The structure of the system under study with the presence of wind turbines.

Table 6.

Numerical comparison of the losses of the system under study with the presence of wind resources.

Project	Investment of RP Compensation (Million Yuan)	The System Loss (kW)
		V = 4 m/s	V = 8 m/s	V = 12 m/s
TGA [100]	338	1872	2480	3129
IGA [100]	336	1731	2292	2892
PSO	332	1729	2290	2687
VCS	327	1723	2288	2676
The proposed method	311	1698	2261	2623

As the results presented in the following tables demonstrate, the proposed method has produced significant improvements in responses to the problem. It is also clear from Table 6 that when the wind speed changes from 4 to 12 m/s, the loss rate increases significantly, but compared to other optimization methods, the proposed method has fewer losses.

5.5. RP Planning Based on the Possibility of Controlling Wind Turbines (WTs)

As the last analysis of the system under study, this section investigates RP planning by fully covering Section 2.2. RP planning is applied to the wind farm (WF) for this model, according to Figure 10. The short circuit current for the external network is 400 megawatts. In the wind farm, 12 connected wind turbines are considered.

Figure 10.

Structure of the tested wind farm.

In addition, in this system, the STATCOM compensator is used at PCC. The objective in the scenario is to reduce the losses and keep the system error in the safe range based on setting the PCC setting point. Usually, such a function is considered for the demand for a large amount of RP. In addition, the tapped transformer and RP compensator are used as other control items. In order to control RP, six different control models are applied to control RP in the wind farm. Figure 11 shows the structure of these control systems. The maximum RP error based on the PCC adjustment system is approximately 5%. The strategies adopted are as follows:

Figure 11.

RP strategies, the proposed HMPSO-VCS, and basic VCS.

Strategy 1: The variables are RP for wind turbine Q_WTi, location of the tapped transformer for PCC, and capacitor banks. The capacitor capacity was 0.96 MVAr. Figure 11a shows the structure of this strategy.

Strategy 2: RP of wind turbine Q_WTi is the only control parameter. Power production maximum, amount of RP was 4 MVAr. Figure 11b shows the structure of this strategy.

Strategy 3: In this case, RP planning was formed on the RP amount of wind turbine Q_WTi injected into the capacitor system and bank. Power production maximum, amount of RP was 4 MVAr and Q^*_PCC = 4 MVAr. Figure 11c shows the structure of this strategy.

Strategy 4: In this case, RP planning is based on adjusting the tapped transformer and the generation of RP from the wind turbine. Therefore, two control parameters in this system are Q_WTi and tap transformer. RP value is equal to Q^*_PCC = 4 MVAr. Figure 11d shows the structure of this strategy.

Strategy 5: STATCOM compensator was installed for PCC. Therefore, in addition to the two previous variables, the STATCOM compensator is considered a new control parameter. Figure 11e shows the structure of this strategy. The STATCOM system with 30% of the total power and wind turbine capacity is selected. Therefore, the capacity of RP with the STATCOM compensator is approximately 1.22 MVAr.

Strategy 6 is similar to Strategy 5, but the difference is that the constraint Q^*_PCC is not considered. Virus search is employed, and Figure 11 shows the structure of this strategy.

Table 7 and Table 8 show RP data by the proposed algorithm based on the possible response under different working conditions and the corresponding compensation. It is found that different individual adjustment points are considered for WT because the distance and feature of PCC vary from one wind turbine to another.

Table 7.

Data of option I.

Q*PCC	PWF 100%		PWF 80%	PWF 50%		PWF 20%	PWF 10%
	4	3.5	3.5	2	3	1	0.5
QWT1	0.0316	0.0203	0.0232	0.0077	0.0469	0.0089	0.0802
QWT2	0.0845	0.0154	0.0368	0.0236	0.0208	0.0236	0.0301
QWT3	0.0788	0.0205	0.0535	0.0721	0.044	0.0302	0.0629
QWT4	0.0495	0.0392	0.0236	0.0026	0.0562	0.0612	0.0178
QWT5	0.056	0.028	0.0543	0.0836	0.0611	0.0123	0.0027
QWT6	0.0528	0.0831	0.064	0.0657	0.0356	0.0649	0.067
QWT7	0.0187	0.0387	0.022	0.044	0.0331	0.0096	0.045
QWT8	0.0271	0.0166	0.0106	0.0521	0.0889	0.0588	0.0432
QWT9	0.0424	0.0814	0.0267	0.0214	0.0034	0.0445	0.0814
QWT10	0.0207	0.0882	0.0287	0.0413	0.0797	0.0701	0.0549
QWT11	0.076	0.0395	0.0382	0.0867	0.0822	0.0644	0.0556
QWT12	0.0175	0.01	0.0457	0.0492	0.0717	0.0813	0.0773
Tab	−2	−2	−2	−2	−2	−2	−2
Comp	ON	ON	ON	ON	ON	OFF	OFF

Table 8.

Data of strategy I.

PWF	Proportional Distribution (PD)
	Q*PCC	Plosses (MVAr)	${\mathit{Q}}_{\mathit{P}\mathit{C}\mathit{C}}^{*}-{\mathit{Q}}_{\mathit{P}\mathit{C}\mathit{C}}^{\mathit{m}\mathit{e}\mathit{a}\mathit{s}}$ (%)
100%	4	0.0222	14.287
100%	3.5	0.1743	10.725
80%	3.5	0.1239	8.109
50%	2	0.1478	2.312
50%	3	0.1623	33.029
20%	1	0.0820	11.023
10%	0.5	0.0046	28.635
PWF	HMPSO-VCS
	Plosses (MVAr)	$Q_{PCC}^{*}-Q_{PCC}^{meas}$ (%)	Reduction Plosses %
100%	0.1102	4.2122	0.0723
100%	0.1101	4.2355	0.7019
80%	0.0712	4.2197	1.7081
50%	0.0266	4.2283	2.0564
50%	0.0285	4.1091	6.1498
20%	0.0023	4.2401	10.216
10%	0.0010	4.2222	6.522

Table 9 represents the simulation results for strategies 2 to 6 (Figure 11). It is clear that lower power is obtained via STATCOM, and error is lower with Strategy 5 using the STATCOM and the proposed HMPSO-VCS. In Strategies 5 and 6, the results depict that RP error is more significant than PSO than the proposed HMPSO-VCS method.

Table 9.

Data for an option I.

WT Units	Strategy
	2	3	4	5	6
QWT1	0.3756	0.4071	0.4586	0.2844	0.3446
QWT2	0.1275	0.1218	0.1429	0.2347	0.3741
QWT3	0.253	0.4646	0.3786	0.006	0.2253
QWT4	0.3495	0.175	0.3769	0.1686	0.0419
QWT5	0.4455	0.0983	0.1902	0.0811	0.1145
QWT6	0.4796	0.1255	0.2839	0.3971	0.4567
QWT7	0.2736	0.308	0.0379	0.1556	0.0762
QWT8	0.0693	0.2366	0.027	0.2643	0.4129
QWT9	0.0746	0.1758	0.2654	0.0828	0.2692
QWT10	0.1288	0.4154	0.3896	0.301	0.4981
QWT11	0.4204	0.2926	0.467	0.1315	0.0391
QWT12	0.1271	0.2749	0.065	0.327	0.2213
Comp	–	1	–	–	–
Tab	–	–	−2	−2	−2
QST	–	–	–	1.218	1.219
Plosses	0.1209	0.1212	0.1021	0.1118	0.1119
$Q_{PCC}^{*}-Q_{PCC}^{meas}$ (%)	4.9602	4.9401	4.9316	4.0345	40.721

In the following, the proposed algorithms with other optimization methods are compared. For this purpose, Table 10 reports the list of different test functions which used in this simulation and Table 11 shows the results of the simulation. The proposed method is compared with many other methods and results shown that the proposed algorithm has worked much better than other methods [101,102,103,104,105,106,107].

Table 10.

The mathematical detailed of employed benchmark functions, D: dimension, [L,U]: lower and upper bands, Fun: function name, No: number, Min: minimum value.

Many Local Minima Group
No	Fun	[L,U]	D	Formulation	Min
1	Ackley	[−32, 32]	30	$f_1(x)=-20\exp \left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D{x}_i^2}}\right)-\exp \left(\frac{1}{D}{\displaystyle \sum _{i=1}^D\cos (2\pi x_i)}\right)+20+\exp (1)$	0
2	Bukin N. 6	$\begin{array}{l}{x}_1\in \left[-15,-5\right]\\ x_2\in \left[-3,3\right]\end{array}$	2	$f_2(x)=100\sqrt{\left|x_2-0.01x_1^2\right|}+0.01\left|x_1+10\right|$	0
3	Cross-in-tray function	[−10, 10]	2	$f_3(x)=-0.0001{\left(\left|\sin (x_1)\sin (x_2)\exp \left(\left|100-\frac{\sqrt{x_1^2+x_2^2}}{\pi }\right|\right)\right|+1\right)}^{0.1}$	−2.06261
4	Drop-wave	[−5.12, 5.12]	2	$f_4(x)=\frac{1+\cos \left(12\sqrt{x_1^2+x_2^2}\right)}{0.5\left(x_1^2+x_2^2\right)+2}$	−1
5	Eggholder	[−512, 512]	2	$f_5(x)=-(x_2+47)\sin \left(\sqrt{\left|x_2+\frac{x_1}{2}+47\right|}\right)-x_1\sin \left(\sqrt{\left|x_1-x_2-47\right|}\right)$	−959.6407
6	Gramacy and Lee (2012)	[0.5, 2.5]	1	$f_6(x)=\frac{\sin (10\pi x)}{2x}+{(x-1)}^4$	−0.8690
7	Griewank	[−600, 600]	30	$f_7(x)={\displaystyle \sum _{i=1}^D\frac{x_i^2}{4000}}-{\displaystyle \prod _{i=1}^D\cos \left(\frac{x_i}{\sqrt{i}}\right)}+1$	0

Table 11.

Statistical results obtained by GA, PSO, GSA, ABC, HBMO, GWO, COA, LCA, SSA, and VCS through 30 independent runs on mentioned benchmark functions in Table 1, Best, Worst, Mean, and STD denotes the best solution, the worst solution, the mean solution, and the standard deviation, respectively.

Fun. No		Algorithms	GA	PSO	GSA	ABC	COA	Proposed
	Indices
f1	Best		2.142	7.053 × 10−3	6.983 × 10−6	5.982 × 10−10	7.873 × 10−6	2.948 × 10−12
	Worst		3.525	1.279 × 10−2	3.982 × 10−2	3.043 × 10−6	5.563 × 10−2	3.128 × 10−5
	Mean		3.240	9.858 × 10−1	0.195 × 10−1	0.989 × 10−9	2.098 × 10−4	1.029 × 10−9
	STD		3.058 × 10−1	1.586 × 10−2	1.329 × 10−1	1.093 × 10−2	9.340 × 10−2	1.928 × 10−2
f2	Best		7.837 × 10−1	5.38	2.098	4.652	5.127 × 10−2	3.822 × 10−3
	Worst		9.651 × 10+2	9.043 × 10−1	8.675	8.341	3.428	5.938 × 10−1
	Mean		8.564 × 10−1	7.542 × 10−1	3.342	6.783	1.090 × 10−1	1.625 × 10−2
	STD		1.243 × 10−1	1.054 × 10+1	1.764 × 10−1	3.762	7.893	2.736 × 10−2
f3	Best		−0.323	−0.645	−1.972	−1.234	−1.867	−1.948
	Worst		1.248	1.121	−1.146	0.653	−1.023	−1.226
	Mean		1.032	0.342 × 10−1	−1.345	−8.837 × 10−1	−1.648	−1.873
	STD		8.637 × 10−1	9.748	1.097 × 10−1	2.019 × 10−1	3.265 × 10−2	2.039 × 10−2
f4	Best		−1.425 × 10−1	−4.837 × 10−1	−1.000	−8.948 × 10−1	−9.763 × 10−1	−1.000
	Worst		3.243	−1.093 × 10−1	−9.849 × 10−1	−2.192 × 10−1	−1.039 × 10−1	−0.827
	Mean		1.907 × 10−1	−1.623 × 10−1	−9.992 × 10−1	−7.765 × 10−1	−6.543 × 10−1	−0.928
	STD		1.783 × 10−1	5.425 × 10−1	−6.938 × 10−5	1.492 × 10-2	2.327 × 10−1	1.413 × 10−4
f5	Best		−7.653 × 10−2	−9.321 × 10+2	−9.585 × 10+2	−9.596 × 10-2	−9.564 × 10+2	−9.596 × 10+2
	Worst		−6.536 × 10−2	−9.025 × 10+2	−9.413 × 10+2	−9.513 × 10-2	−9.323 × 10+2	−9.151 × 10+2
	Mean		−7.192 × 10+1	−9.294 × 10+2	−9.526 × 10+2	−9.576 × 10-2	−9.488 × 10+2	−9.029 × 10+2
	STD		3.201	3.052 × 10−1	1.0451 × 10-2	3.656 × 10-2	2.114 × 10−4	1.983 × 10−4
f6	Best		−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1
	Worst		−8.689 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1
	Mean		−8.690 × 10−1	−8.6900 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1	−8.690 × 10−1
	STD		0	0	0	0	0	0
f7	Best		1.208 × 10−1	0	0	4.837 × 10−8	2.897 × 10−8	0
	Worst		2.685 × 10−1	1.029	2.546 × 10−2	2.039 × 10−1	8.675 × 10−1	0
	Mean		1.952 × 10−1	1.938 × 10−2	1.645 × 10−5	5.038 × 10−7	2.653 × 10−4	0
	STD		3.915 × 10−2	1.523 × 10−3	5.320 × 10−4	9.456 × 10−14	3.332 × 10−5	0

6. Conclusions

RP distribution in power systems is optimized by adjusting the voltage of generators, tap transformers with the ability to change under load, and the size of parallel capacitors. With the expansion of power networks and the development of renewable resources, considerable attention is devoted to RP resources in the network. With this end in view, objective functions based on losses, voltage, and cost of RP sources were applied in this paper. This paper presents a new model for optimal reactive power distribution. To bring the issue to a more realistic level, renewable resources and facsimiles have also been considered. On the other hand, by increasing the numbers of these types of equipment, if it is not possible to make an optimal and appropriate choice from them, the system’s stability is not adequately established, but it also leads to more errors. For this reason, solving this problem has become an optimization, and a new hybrid algorithm based on virus search and particle clustering has been used to solve it. Using this hybrid model has been able to improve local and global search dramatically. The fuzzy theory was employed to install RP resources of shunt and the SVC and TCSC series. In this model, various criteria were used to determine the best position. In the method used in this scheme, the discrete variables are considered continuous and treated like any other continuous variables in the algorithm’s space. At each stage, after reaching the new position, the discrete variables that have become continuous are now rounded off to the appropriate number before applying to the load program. For example, tapping tap changers is a discrete quantity. In the process of running the algorithm, when a value between the two tap changers is obtained for this quantity, this value is rounded off to the nearest step and then applied to the load program. Finally, the problem of RP planning was solved using a novel algorithm from on virus search. Appropriate search, choosing the best response at each stage, improving the set of responses, and the crossover of generations to give the best responses are among prominent features for the proposed method. The results obtained in different sections show that the proposed method has an average performance of 5% better.

Author Contributions

Conceptualization, R.S. and R.H.; methodology, P.K.M.; software, M.E.; validation, N.A. and D.R.; formal analysis, N.A.; investigation, A.D.; resources, R.H.; data curation, P.K.M.; writing—original draft preparation, D.R.; writing—review and editing, A.D.; visualization, R.H.; supervision, A.D.; project administration, R.S.; funding acquisition, M.E. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.