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Probabilistic multi‐objective optimization method for interline power flow controller (IPFC) allocation in power systems

Probabilistic multi‐objective optimization method for interline power flow controller (IPFC)... INTRODUCTIONUsually, the maximum capacity of existing transmission lines is used due to the high cost of constructing new ones. Also, reliability, security, and efficiency in the operation and development of power systems are more critical than ever before. The use of flexible AC transmission system (FACTS) devices is a prominent example of trying to improve transmission networks' performance. These devices can control the various power system parameters such as; the lines’ impedance, the active and reactive power flow, and the buses’ voltage magnitude. Optimal management of FACTS devices is essential to reach the proper efficiency of these controllers [1–5]. This managing is implemented by controlling their operating limits to distinguish the practicable capabilities of FACTS devices. Refs. [6–9] have investigated several FACTS devices and some approaches for controlling their constraints.The interline power flow controller (IPFC) device is recognized as one of the most powerful and flexible of the FACTS devices family. Generally, IPFCs are employed to control the power flow of the lines in transmission networks. The fundamental challenge is determining an appropriate location and settings for fixing the IPFC in the power system [10].Improvement of voltage stability and power profile has been studied by the utilization of the unified power flow controller (UPFC), static synchronous compensator (STATCOM), static synchronous series compensator (SSSC), and IPFC in [11]. In [12–15], different models of IPFC are proposed for damping oscillations in power systems. In [16], a new and straightforward approach has been presented for IPFC modelling to control all operating limits by applying the Newton–Raphson method.In [17], it is shown that the placement of the IPFC improves the voltage stability of the power system and the line loading capability. In [18], the optimal allocation of the IPFC and the solar power unit using the line severity index has been investigated to avoid contingencies. The allocation cost of the IPFC has been ignored in both studies, and only operational aspects are considered.In [19], the optimal setting and placement of the IPFC are investigated based on the disparity utilization factor to reduce power congestion. Also, in [20], the optimal location of the IPFC has been determined for improving and controlling the power system parameters. Transmission losses and generation fuel cost have been considered as objective functions in this study. The single objective function with weight coefficients has been used in both studies.Renewable energy sources are preferred due to environmental concerns because they are environmentally friendly. However, the output of these sources is variable. This issue, along with system configuration variation and load fluctuation make challenges in planning and operating of power systems. Therefore, under these conditions, uncertainties handling in studies is very necessary. None of the mentioned studies have included this challenge in their studies related to IPFC allocation.Considering uncertainty in the optimal allocation of other types of FACTS devices has been investigated in numerous studies. Ref. [21] considers the system predictability to optimal allocation of the UPFC. A solving method for the comprehensive problem, including unit commitment and UPFC allocation considering the uncertainty of wind power generation (WPG), is proposed in [22]. Robust control methods are exploited to design the UPFC controllers under parametric uncertainties of the system in [23]. Ref. [24] exploits a probabilistic‐based method to specify the best possible size of multiple FACTS devices, including thyristor‐controlled series capacitor (TSCS), STATCOM, and UPFC, for enhancing the steady‐state voltage profile. However, a comprehensive probabilistic framework has not been presented to study the IPFC impact on improving transmission networks' performance so far. This paper covers this gap.The Monte Carlo simulation (MCS) method is a simple and accurate method to handle uncertainties [25]. This method requires a very large calculation burden. Approximate methods such as data clustering method (DCM) have been introduced to overcome these challenges. The DCM has reasonable speed and presents acceptable results compared to the MCS method based on obtained results in [25] and [26]. In this study k‐means‐based DCM is used for the probabilistic assessment of the mentioned problem for the first time.The main contributions of this paper are as follows:The optimal location and parameters setting (reference set points) of an IPFC are determined with the objectives of decreasing active power losses, improving power flow index (PFI) of the lines, and considering IPFC allocation cost in a probabilistic multi‐objective optimization framework.Uncertain input parameters such as loads and wind speed of wind turbines (WTs) are considered by proper probability density functions (PDFs) to extract robust solutions for the problem.k‐mean‐based DCM is used for probabilistic assessment of optimal IPFC allocation problem in transmission networks.Pareto‐based optimal robust solutions set are extracted by multi‐objective particle swarm optimization (MOPSO) algorithm in a probabilistic framework, and the technique for order of preference by similarity to ideal solution (TOPSIS) is employed to make decisions and select the final solution based on the importance of each objective function for the operators.MATHEMATICAL MODEL OF IPFC IN STEADY‐STATE STUDIESAn IPFC can enhance the power capacity and power transfer capability of the power system. The functional convertibility enables the IPFC to adapt to changing system operating requirements and changing power flow patterns. The expandability of the IPFC is that a number of voltage source converters coupled with a common DC bus can be operated.The IPFC is formed by integrating two or several series‐connected converters operating with each other to enhance the power flow control capability over what is accessible with the usual SSSC [27]. Figure 1 illustrates the simplest IPFC, with FACTS buses i,j,$i,\ j,$ and k which is utilized to show the operation principle [28–30].1FIGUREThe simplest IPFC with two convertersIn the IPFC structure, two transformers are used to series‐connected the two converters with the transmission lines. Furthermore, the sending‐ends of the two transmission lines are connected in series with the FACTS buses j and k, respectively. The mentioned IPFC can independently handle three power flows of the two transmission lines. Figure 2 illustrates the equivalent circuit of the IPFC with two controllable series injected voltage sources.2FIGUREThe equivalent circuit of the simplest IPFC with two convertersThe active power can be transferred between the series converters via the common DC link, while the sum of the transferred active power should be zero. In Figure 2 the controllable injected voltage source is Vsein=Vsein∠θsein(n=j,k)$\bm{Vs}{\bm{e}}_{\bm{in}}=\textit{Vs}{e}_{\textit{in}}\angle \theta s{e}_{\textit{in}}\ (n=j,k)$, and the series transformer impedance is Zsein. Active and reactive power flows of the branches leaving buses i, j, k are obtained by:1Pin=Vi2gin−ViVngincosθin+binsinθin−ViVseingincosθi−θsein+binsinθi−θsein)$$\begin{eqnarray} \ {P}_{in} &=& V_i^2\ {g}_{in} - {V}_i{V}_n\left( {{g}_{in}cos{\theta }_{in} + {b}_{in}sin{\theta }_{in}} \right)\nonumber\\ && - {V}_iVs{e}_{in}\left( {{g}_{in}\cos \left( {{\theta }_i - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{sin}}\left( {{\theta }_i - \theta s{e}_{in}} \right))\end{eqnarray}$$2Qin=−Vi2bin−ViVnginsinθin+bincosθin−ViVseinginsinθi−θsein+bincosθi−θsein)$$\begin{eqnarray} \ {Q}_{in} &=& \ - V_i^2{b}_{in} - {V}_i{V}_n\left( {{g}_{in}sin{\theta }_{in} + {b}_{in}cos{\theta }_{in}} \right)\nonumber\\ && - {V}_iVs{e}_{in}\left( {{g}_{in}\sin \left( {{\theta }_i - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{cos}}\left( {{\theta }_i - \theta s{e}_{in}} \right))\end{eqnarray}$$3Pni=Vn2gin−ViVngincosθn−θi+binsinθn−θi+VnVseingincosθn−θsein+binsinθn−θsein)$$\begin{eqnarray} \ {P}_{ni} &=& V_n^2\ {g}_{in} - {V}_i{V}_n\left( {{g}_{in}\cos \left( {{\theta }_n - {\theta }_i} \right) + {b}_{in}\sin \left( {{\theta }_n - {\theta }_i} \right)} \right)\nonumber\\ && + {V}_nVs{e}_{in}\left( {{g}_{in}\cos \left( {{\theta }_n - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{sin}}\left( {{\theta }_n - \theta s{e}_{in}} \right))\end{eqnarray}$$4Qni=−Vn2bnn−ViVnginsinθn−θi−bincosθn−θi+VnVsein(ginsinθn−θsein−binsinθn−θsein)$$\begin{eqnarray} \ {Q}_{ni} &=& \ - V_n^2{b}_{nn} - {V}_i{V}_n\left( {{g}_{in}\sin \left( {{\theta }_n - {\theta }_i} \right) - {b}_{in}\cos \left( {{\theta }_n - {\theta }_i} \right)} \right)\nonumber\\ && + {V}_nVs{e}_{in}({g}_{in}\sin \left( {{\theta }_n - \theta s{e}_{in}} \right) - {b}_{in}{\rm{sin}}\left( {{\theta }_n - \theta s{e}_{in}} \right))\end{eqnarray}$$where gin=Re(1/Zsein)${g}_{\textit{in}}=\textit{Re}(1/\bm{Z}\bm{s}{\bm{e}}_{\bm{i}\bm{n}})$, bin=Im(1/Zsein)${b}_{\textit{in}}=\textit{Im}(1/\bm{Z}\bm{s}{\bm{e}}_{\bm{i}\bm{n}})$. Pin,Qin(n=j,k)${P}_{\textit{in}},\ {Q}_{\textit{in}}\ (n=j,\ k)$ are the active and reactive power flows of two branches leaving bus i. Pni,Qni(n=j,k)${P}_{\textit{ni}},\ {Q}_{\textit{ni}}\ (n=j,\ k)$ are the active and reactive power flows at the sending‐ends of the two transmission lines. The power mismatches of the IPFC, at buses i,j,k$i,\ j,\ k$ should be:4aΔPm=Pgm−Pdm−Pm=0$$\begin{equation}\Delta {P}_m = P{g}_m\ - P{d}_m - \ {P}_m = \ 0\end{equation}$$5ΔQm=Qgm−Qdm−Qm=0$$\begin{equation}\Delta \ {Q}_m = Q{g}_m\ - Q{d}_m - \ {Q}_m = \ 0\end{equation}$$where, Pgm,Qgm(m=i,j,k)$P{g}_{m},\ Q{g}_{m}\ (m=i,\ j,\ k)$ are the active and reactive power generation entering the bus m, Pdm,Qdm(m=i,j,k)$P{d}_{m},\ Q{d}_{m}\ (m=i,\ j,\ k)$ are the active and reactive power load leaving the bus m. Pm,Qm(m=i,j,k)${P}_{m},\ {Q}_{m}\ (m=i,\ j,\ k)$ are the sum of active and reactive power flows of the circuits connected to bus m.According to the operational principle of the IPFC, the operating limitation indicating the active power transfer between the series converters via the common DC link is as follows:6PEx=−∑PEsein−Pdc=0$$\begin{equation}PEx\ = \ - \sum PEs{e}_{in} - \ {P}_{dc} = \ 0\end{equation}$$where PEsein=Re(VseinIni*)(n=j,k)$\textit{PEs}{e}_{\textit{in}}=\textit{Re}(\bm{Vs}{\bm{e}}_{\bm{in}}{\bm{I}}_{\bm{ni}}^{\ast})\ (n=j,\ k)$. Ini is the series converter current.The IPFC can control the active and reactive power flows of primary line 1 but only the active power of secondary line 2. Corresponding constraints of active and reactive power flow in the IPFC are:7ΔPni=Pni−PniSpec=0$$\begin{equation}\Delta \ {P}_{ni} = {P}_{ni}\ - \ P_{ni}^{Spec} = \ 0\end{equation}$$8ΔQni=Qni−QniSpec=0$$\begin{equation}\Delta \ {Q}_{ni} = {Q}_{ni}\ - \ Q_{ni}^{Spec} = \ 0\end{equation}$$where Pni=Re(VnIni∗)${P}_{ni} = \ Re( {{{\bm{V}}}_{\bm{n}}{\bm{I}}_{{\bm{ni}}}^*} )$, Qni=Im(VnIni∗)${Q}_{ni} = \ Im( {{{\bm{V}}}_{\bm{n}}{\bm{I}}_{{\bm{ni}}}^*} )$ and PniSpec$P_{ni}^{Spec}$, QniSpec$Q_{ni}^{Spec}\ $are control references of active and reactive power flow.Combining Equations (4a), (5), and Equations (6)–(8), the Newton power flow solution is given by:9JΔX=−ΔR$$\begin{equation}J\Delta X = - \Delta R\end{equation}$$where ΔX=[ΔX1,ΔX2]T$\Delta X = {[ {\Delta {X}_1,\Delta {X}_2} ]}^T $, is the incremental vector of state variables;ΔX1=[Δθi,ΔVi,Δθj,ΔVj,Δθk,ΔVk]T$\ \Delta {X}_1 = {[ {\Delta {\theta }_i,\Delta {V}_i,\Delta {\theta }_j,\Delta {V}_j,\ \Delta {\theta }_k,\Delta {V}_k} ]}^T $, is the incremental vector of bus voltage magnitudes and angles, and ΔX2=[Δθseij,ΔVseij,Δθseik,ΔVseik]T$\Delta {X}_2 = {[ {\Delta \theta s{e}_{ij},\Delta Vs{e}_{ij},\Delta \theta s{e}_{ik},\Delta Vs{e}_{ik}} ]}^T\ $, is the incremental vector of state variables of the IPFC.ΔR=[ΔR1,ΔR2]T$\Delta R \! =\! {[ {\Delta {R}_1,\Delta {R}_2} ]}^T$; ΔR1=[ΔPi,ΔQi,ΔPj,ΔQj,ΔPk,ΔQk]T$\Delta {R}_1\! =\! {[ {\Delta {P}_i,\Delta {Q}_i,\Delta {P}_j,\Delta {Q}_j,\Delta {P}_k, \Delta {Q}_k} ]}^T$, is the bus power mismatch vector, ΔR2=[Pji−PjiSpec,Qji−QjiSpec,Pki−PkiSpec,PEx]T$\Delta {R}_2 = {[ {{P}_{ji} - P_{ji}^{Spec},{Q}_{ji} - Q_{ji}^{Spec},{P}_{ki} - P_{ki}^{Spec},PEx} ]}^T$, is the operating control mismatch vector of the IPFC. In addition, J=∂ΔR∂X$\ J\ = \frac{{\partial \Delta R}}{{\partial X}}$, is the Jacobian matrix of the system.There is one related active power flow control equation for the secondary series converter in (9). The Jacobian matrix in (9) includes four blocks. The bottom diagonal block has a related formation of traditional power flow, and other blocks are FACTS associated. This matrix can be solved by first eliminating Δθse,ΔVse$\Delta \theta se,\ \Delta Vse$ of the IPFC. Then the resulting reduced bottom diagonal block Newton equation can be solved by block sparse matrix techniques.It should be mentioned that the multi‐control modes of UPFC can be utilized to IPFC. Besides, the methods that are used to handle the violated operative inequalities of SSSC and STATCOM can be utilized to IPFC [31].Vi,Vj,Vk,θi,θj,θk${V}_{i},\ {V}_{j},\ {V}_{k},\ {\theta}_{i},\ {\theta}_{j},\ {\theta}_{k}$ are set to the start values, where Vi=Vj=Vk=1${V}_i = {V}_{j} = {V}_{k} = 1$ and θi=θj=θk=0$\theta_{i} = {\theta}_{j} = {\theta }_{k} = 0$ if buses i,j,k$i, j, k$ are not voltage controlled buses. With solving two simultaneous Equations (7) and (8), the Vseij,θseij$Vs{e}_{ij},\theta s{e}_{ij}$ values for the primary series converter can be determined as:10Vseij=A/gij2+bij2/Vj$$\begin{equation}\ Vs{e}_{ij} = \sqrt {A/\left( {g_{ij}^2 + b_{ij}^2} \right)} \ /{V}_j\end{equation}$$11θseij=tan−1PjiSpec−Vj2gjj+ViVjgijQjiSpec+Vj2bjj−ViVjbij−tan−1−gij/bij$$\begin{equation} \theta s{e}_{\textit{ij}}=\textit{ta}{n}^{-1}\left[\frac{{P}_{\textit{ji}}^{\textit{Spec}}-{V}_{j}^{2}{g}_{\textit{jj}}+{V}_{i}{V}_{j}{g}_{\textit{ij}}}{{Q}_{\textit{ji}}^{\textit{Spec}}+{V}_{j}^{2}{b}_{\textit{jj}}-{V}_{i}{V}_{j}{b}_{\textit{ij}}}\right]-\textit{ta}{n}^{-1}\left(-{g}_{\textit{ij}}/{b}_{\textit{ij}}\right) \end{equation}$$12A=PjiSpec−Vj2gjj+ViVjgij2+(QjiSpec+Vj2bjj−ViVjbij)2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl}{A}& =& {{\left({P}_{\textit{ji}}^{\textit{Spec}}-{V}_{j}^{2}{g}_{\textit{jj}}+{V}_{i}{V}_{j}{g}_{\textit{ij}}\right)}^{2}}\\[6pt] & & {+\,({Q}_{\textit{ji}}^{\textit{Spec}}+{V}_{j}^{2}{b}_{\textit{jj}}-{V}_{i}{V}_{j}{b}_{\textit{ij}})^{2}}\end{array} \end{equation}$$Suppose that Vseik$Vs{e}_{ik}$ is selected to a value between Vseikmin$Vse_{ik}^{min}$, and Vseikmax$Vse_{ik}^{max}\ $. Moreover, θseik$\theta s{e}_{ik}$ can be obtained by using the following equation:13θseik=sin−1PkiSpec−Vki2gik+ViVkgik/VkVsekigik2+bik2−tan−1−gik/bik$$\begin{eqnarray} \ \theta s{e}_{ik} &=& si{n}^{ - 1}\ \left[ {\left( {P_{ki}^{Spec} - V_{ki}^2{g}_{ik} + {V}_i{V}_k{g}_{ik}} \right)/\left( {{V}_kVs{e}_{ki}\sqrt {g_{ik}^2 + b_{ik}^2} } \right)} \right]\nonumber\\ && - \ ta{n}^{ - 1}\left( { - {g}_{ik}/{b}_{ik}} \right)\end{eqnarray}$$PROBABILISTIC LOAD FLOW (PLF)Deterministic load flow (DLF) analyses the power system under a pre‐determined and certain operating condition. By ever‐increasing uncertain resources in power systems, PLF must be used to express the complete range of probable values of the system's output variables. In PLF, statistical information of uncertain output variables is obtained by statistical information of input uncertain variables [32].Generally, there are three main classifications for probabilistic assessment, including MCS, approximate, and analytical methods. The MCS method is precise and flexible and can be employed for many problems. However, this method needs an extra calculation burden. Other approaches are less accurate because of the simplifications in their modelling, and this makes them faster. In this regard, a compromise should be made between reasonable accuracy and speed of the computations. The DCM, with almost precise evaluation ability, is a sampling approach to diminish the number of MCS samples and has been proposed by Michael McKay [33]. The explanation and formulation of the DCM approach are discussed in Section 3.2.Uncertain input variablesIn PLF, the input variable's information is described by PDF. So, it is crucial to consider the precise analytical model to estimate uncertain input variables. PDFs of wind speed and power demands are modelled as follows:Naturally, wind turbine generation relies on wind speed. Various distribution models, namely Gamma, Rayleigh, Lognormal, Gumbel, Generalized Extreme Value, Nakagami, Inverse Gaussian, Burr, and Weibull, have been employed to show wind speed behaviour. Commonly, the Weibull distribution is employed to modell the wind speed. The formulation of the Weibull distribution is represented in Equation (14) [34].14fv=ABvBA−1e−vBAv≥00v<0$$\begin{equation} f\left(v\right)=\left\{ \def\eqcellsep{&}\begin{array}{ll}\frac{A}{B}{\left(\frac{v}{B}\right)}^{A-1}{e}^{-{\left(\frac{v}{B}\right)}^{A}}&v\ge 0\\[10pt] 0&v&lt;0\end{array} \right. \end{equation}$$The WT power generation depends on wind speed that can be determined by Equation (15).15PWTv=0v≤vincorv≥voutcv−vincvrated−vincPrWTvinc<v<vratedPrWTvrated≤v<voutc$$\begin{equation} {P}^{\textit{WT}}\left(v\right)=\left\{ \def\eqcellsep{&}\begin{array}{cc}0& v\le {v}_{\textit{in}}^{c}\ \textit{or}\ v\ge {v}_{\textit{out}}^{c}\\ \frac{v-{v}_{\textit{in}}^{c}}{{v}_{\textit{rated}}-{v}_{\textit{in}}^{c}}{P}_{r}^{\textit{WT}}& {v}_{\textit{in}}^{c}&lt;v&lt;{v}_{\textit{rated}}\\[10pt] {P}_{r}^{\textit{WT}}& {v}_{\textit{rated}}\le v&lt;{v}_{\textit{out}}^{c}\end{array} \right. \end{equation}$$where PrWT${P}_{r}^{\textit{WT}}$ is the rated output power of the WT; v is the wind speed, vinc$v_{in}^c$, voutc$v_{out}^c$, and vrated${v}_{rated}$ are cut in, cut out, and the rated speeds of the WT, respectively.Various distribution models, namely Rayleigh, Weibull, and normal have been employed to model the demand load [35]. In this paper, the normal distribution is used. The normal distribution formulation is represented in Equation (16) [36].16fx=1σx2π×e−x−Ex22σx2$$\begin{equation}f\ \left( x \right) = \frac{1}{{\left( {{{\sigma}}\left[ x \right]} \right)\sqrt {2\pi } }}\ \times {e}^{ - \frac{{{{\left( {x - {\rm{E}}\left[ x \right]} \right)}}^2}}{{2{{\sigma}}{{\left[ x \right]}}^2}}}\end{equation}$$where σ[] and E[] are the standard deviation and expected value operators, respectively.k‐means‐based DCMThe k‐means‐based DCM was presented by Mac Queen in 1967 [37]. This approach has a simple process of classifying the data set into a determined quantity of clusters. In this regard, the distance of each cluster point is reduced to its center [38]. Concisely, the k‐means method steps are according to below [39]:The number of clusters (K) is designated.K agents or centers are initialized randomly (ak,k=1,2,…,K${a}_{k},\ k=1,2,\text{\ensuremath{\ldots}},K$).Data is assigned to the clusters based on the minimum distance with K centers (17):17ifdn−ak<dn−al⇒dn∈Gk$$\begin{equation} \textit{if}\ \left|{d}_{n}-{a}_{k}\right|&lt;\left|{d}_{n}-{a}_{l}\right|\Rightarrow {d}_{n}\in {G}_{k} \end{equation}$$n=1,2,…,N$n=1,\ 2,\ \text{\ensuremath{\ldots}},N$,k=1,2,…,K$k=1,\ 2,\ \text{\ensuremath{\ldots}},\ K$,l=1,2,…,K$l=1,\ 2,\ \text{\ensuremath{\ldots}},\ K$,(l≠k)$( {l \ne k} )$N and dn${d}_n$ are the number of data and the nth data, respectively. ak${a}_k$ and al${a}_l$are the kth and lth agents, respectively. Also, Gk${G}_k$ is the kth cluster.d.The centers of clusters are updated using the average of all data points of each cluster as (18):18ak=∑n∈GkdnNGk,k=1,2,…,K$$\begin{equation}\ {a}_k = \frac{{\mathop \sum \nolimits_{n \in {G}_k} {d}_n}}{{{N}_{{G}_k}}}\ ,\ k\ = \ 1,2, \ldots ,\ K\end{equation}$$NGk${N}_{{G}_k}$ is the number of members in the kth cluster.e.The steps c and d are repeated until the centers' value no longer changes.f.A probability is determined for each center as (19):19Pak=NGkN$$\begin{equation} \mathrm{P}\left({a}_{k}\right)=\frac{{N}_{{G}_{k}}}{N} \end{equation}$$Moreover, the ith statistical moment of F is calculated by (20).20EFi=∑k=1KPak.Faki$$\begin{equation} \mathrm{E}\left[{F}^{i}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).F{\left({a}_{k}\right)}^{i} \end{equation}$$So, it can be said that each cluster center (ak${a}_k$) has a probability of P(ak)$\mathrm{P}({a}_{k})$. The centers ak(k=1,2,…,K)${a}_{k}\ (k=1,\ 2,\text{\ensuremath{\ldots}},\ K)$ are obtained from all data points, and after calculating the F(ak)$F( {{a}_k} )$, the ith statistical moment of F is determined using Equation (20).PROBLEM FORMULATION AND SOLVING TOOLS IN THE PROPOSED STUDY METHODIn this study, various objective functions are considered, which are presented in the following subsection. Also, the power system and IPFC constraints are discussed. In multi‐objective optimization problems, a proper optimization algorithm must be employed. In this regard, the MOPSO algorithm is employed, and the TOPSIS is used to trade‐off between the objective functions and select the final solution.Objective functionsObjective functions are one of the main components that must be correctly defined for the proposed study method. In this paper, three objective functions are considered for the optimal allocation of IPFC. The first objective is active power losses, which is one of the essential elements in the optimization of power systems' operating. The second objective is related to the nature of IPFC, which is intended to control the power flow of the lines. This objective is a security‐based index. Finally, the economic aspect of IPFC allocation is considered as the third objective function.The expected value of active power lossesThis objective function can be calculated as:EPLosses=∑k=1KPak.PLossesk$$\begin{equation*} \mathrm{E}\left[{P}_{\textit{Loss}\textit{es}}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).{{P}_{\textit{Loss}\textit{es}}}_{k}\ \end{equation*}$$21PLosses=∑i=1NLinesRiIi2$$\begin{equation} {P}_{\textit{Loss}\textit{es}}=\sum _{i=1}^{{N}_{\textit{Lines}}}{R}_{i}{\left|{I}_{i}\right|}^{2} \end{equation}$$where K is the number of the DCM clusters. NLines${N}_{\textit{Lines}}$ is the number of transmission lines. Ij${I}_j$ is the current of line j, and Rj${R}_j$ is the resistance of the jth line.The expected value of PFI of the linesThis security‐based objective function is defined to minimize the congestion of the lines. For this purpose, the following equation is represented:EPFI=∑k=1KPak.PFIk$$\begin{equation*} \mathrm{E}\left[\textit{PFI}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).\textit{PF}{I}_{k} \end{equation*}$$22PFI=∑i=1NLinesSi|Simax|2$$\begin{equation} \textit{PFI}=\sum _{i=1}^{{N}_{\textit{Lines}}}{\left(\frac{\left|{S}_{i}\right|}{|{S}_{i}^{\max}|}\right)}^{2} \end{equation}$$where Sj${S}_j$ and Sjmax$S_j^{max}$ are the complex power and the complex power rate of line j. It should be mentioned that power flow can be raised in some lines due to the operation of the IPFC.The expected value of the IPFC costAccording to the Section 2, the IPFC has two converters that are connected in series with the transmission lines. Thus, the cost functions are defined for both converters as follows:EIPFCCost=∑k=1KPak.IPFCCostk$$\begin{equation*} \mathrm{E}\left[\textit{IPF}{C}_{\textit{Cost}}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).\textit{IPF}{{C}_{\textit{Cost}}}_{k} \end{equation*}$$IPFCCost=CIPFC−A+CIPFC−B$$\begin{equation*} \textit{IPF}{C}_{\textit{Cost}}={C}_{\textit{IPFC}-A}+{C}_{\textit{IPFC}-B} \end{equation*}$$CIPFC−A=0.00015Sj2−0.1345Sj+94.11$$\begin{equation*} {C}_{IPFC - A} = 0.00015S_j^2 - 0.1345{{\rm{S}}}_j + 94.11\end{equation*}$$23CIPFC−B=0.00015Sk2−0.1345Sk+94.11$$\begin{equation} {C}_{IPFC - B} = 0.00015S_k^2 - 0.1345{{\rm{S}}}_k + 94.11\end{equation}$$where CIPFC−AandCIPFC−B${C}_{\textit{IPFC}-A}\ \mathrm{and}\ {C}_{\textit{IPFC}-B}$ are the cost functions for converters connected to lines j$j\ $ and k, respectively. Sj and Sk are the operating range of converters connected to lines j and k, respectively[40].ConstraintsIn this paper, main purpose is the optimal allocation of the IPFC. The constraints of buses' voltage, lines' apparent power flow, and active power generation are as follows:24Vimin≤E[Vi]≤Vimax$$\begin{equation} {V}_{i}^{\min}\le \mathrm{E}[{V}_{i}]\le {V}_{i}^{\max} \end{equation}$$25E[Si]≤|Simax|$$\begin{equation} \mathrm{E}\big[\left|{S}_{i}\right|\big]\le |{S}_{i}^{\max}| \end{equation}$$26PGimin≤E[PGi]≤PGimax$$\begin{equation} {P}_{{G}_{i}}^{\min}\le \mathrm{E}[{P}_{{G}_{i}}]\le {P}_{{G}_{i}}^{\max} \end{equation}$$The active power exchange between or among the converters via the common DC link, voltage, and phase angle constraints of the series converter are as follows:27−PEsemax≤E[PEse]≤PEsemax$$\begin{equation} -P{E}_{\textit{se}}^{\max}\le \mathrm{E}[P{E}_{\textit{se}}]\le P{E}_{\textit{se}}^{\max} \end{equation}$$28Vsemin≤EVse≤Vsemax$$\begin{equation} {V}_{\textit{se}}^{\min}\le \mathrm{E}\left[{V}_{\textit{se}}\right]\le {V}_{\textit{se}}^{\max} \end{equation}$$29θsemin≤Eθse≤θsemax$$\begin{equation} {\theta}_{\textit{se}}^{\mathrm{\min}}\le \mathrm{E}\left[{\theta}_{\textit{se}}\right]\le {\theta}_{\textit{se}}^{\max} \end{equation}$$The optimal location and settings of IPFC are determined using the MOPSO algorithm while all constraints are satisfied.Multi‐objective optimizationA multi‐objective optimization problem has two or more objectives with equality and inequality restraints that can be defined as: [41] 30OF=Fix;i=1,2,…,Nof$$\begin{equation} \textit{OF}={F}_{i}\left(x\right);\ \ i=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{of}} \end{equation}$$31Hjx=0;j=1,2,…,Neq$$\begin{equation} {H}_{j}\left(x\right)=0;\ \ j=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{eq}} \end{equation}$$32Mkx≤0;k=1,2,…,Nineq$$\begin{equation} {M}_{k}\left(x\right)\le 0;\ \ k=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{ineq}} \end{equation}$$In (30‐32), OF$OF$ denotes the multi‐objective function where i is the number of objectives. Also, x is the control variables vector, Hj${H}_{j}$, and Mk${M}_{k}$ are equality and inequality constraints.The MOPSO algorithm has been used for solving various problems, including the operating of power systems [42, 43]. A set of feasible solutions is obtained in the multi‐objective optimization framework. Then, the TOPSIS can be used to sort the solutions based on some criteria.MOPSO algorithmIn this algorithm, a search space is considered with d dimension and n particles, where ith particle is in position Xi(xi1,xi2,…,xid)${X}_{i}\ ({x}_{i1},\ {x}_{i2},\text{\ensuremath{\ldots}},\ {x}_{\textit{id}})$, and moves with velocity Vi(vi1,vi2,…,vid)${V}_{i}\ ({v}_{i1},\ {v}_{i2},\text{\ensuremath{\ldots}},\ {v}_{\textit{id}})$. There is personal best position pbesti$\textit{pbes}{t}_{i}$ for each particle, which is specified by its own best performance in the population. Also, global best gbest$\textit{gbest}$ is defined according to the general best performance of all particles. Each particle modifies its velocity and location as (33) and (34), respectively.33Vit+1=wVit+c1r1xpbest−Xit+wVit+c2r2xgbest−Xit$$\begin{equation}\ V_i^{t + 1} = \ wV_i^t + {c}_1{r}_1\left( {{x}_{pbest} - X_i^t} \right) + wV_i^t + {c}_2{r}_2\left( {{x}_{gbest} - X_i^t} \right)\end{equation}$$34Xit+1=Xit+Vit+1$$\begin{equation}\ X_i^{t + 1} = X_i^t\ + V_i^{t + 1}\end{equation}$$where c1 and c2${c}_2\ $are cognitive acceleration and social acceleration coefficients, w is inertia weight, xpbest${x}_{pbest}$ and xgbest${x}_{gbest}$ are the personal and global best of the particle, r1 and r2 are the random variables. Also, Xit,$\ X_i^t,\ $and Vit$V_i^t$are current position and velocity of ith particle in iteration t.The particles' velocity and position are renewed similarly at the PSO and MOPSO. The PSO and MOPSO algorithms are different in some aspects: The selection and update procedures of the global and individual leaders are different. Besides, the Pareto optimal solution set is employed to obtain the solution in the MOPSO algorithm. Beyond, the MOPSO algorithm needs a repository for saving the non‐dominated solutions obtained by the input variables to be established. Several researchers have studied the selection and updating of global and individual leaders, the parameter settings, and the establishment of a repository and have achieved some results [44, 45]. In addition, this algorithm is still encountered challenges such as the ineffectiveness of exploring the gbest$gbest$ solution and prone to premature convergence. The flowchart of the MOPSO algorithm according to dominance criteria is illustrated in Figure 3.3FIGUREFlowchart of the MOPSO algorithmTOPSISThe number of results has to be compared and deliberated by some criteria in multi‐objective optimization problems. The purpose of the TOPSIS is to assist decision‐makers in the trade‐off between objective functions. So, possible results are usually specified by different criteria, and maybe none of the results satisfies all the criteria. Hence, the result is a relative result based on the decision‐makers' preference. TOPSIS was proposed by Chen in 1992 [46]. The basic concept is that the determined result must have the maximum interval from the negative‐ideal result and the minimum interval from the ideal result. The following steps are the process of the TOPSIS:Compute the rij${r}_{ij}$ as a normalized decision matrix by Equation (35).35rij=fij/∑j=1Jfij2j=1,…,J;i=1,…,n.$$\begin{equation} {r}_{\textit{ij}}={f}_{\textit{ij}}/\sqrt{\sum _{j=1}^{J}{f}_{\textit{ij}}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J;\ \ i=1,\text{\ensuremath{\ldots}},n. \end{equation}$$Compute the vij$v_{ij}$ as weighted normalized decision matrix by Equation (36).36vij=wirijj=1,…,J;i=1,…,n.$$\begin{equation} {v}_{\textit{ij}}={w}_{i}{r}_{\textit{ij}}\ \ \ j=1,\text{\ensuremath{\ldots}},J;\ \ i=1,\text{\ensuremath{\ldots}},n. \end{equation}$$Specify the negative‐ideal and ideal result.37A−=v1−,…,vn−=minviji∈I′,maxviji∈I′′$$\begin{equation}\ {A}^ - = \left\{ {v_1^ - , \ldots ,v_n^ - } \right\}\ = \left\{ {\left( {\min {v}_{ij}\left| {i \in I^{\prime}} \right.} \right),\left( {\max \,\ {v}_{ij}\left| {i \in I^{\prime\prime}} \right.} \right)} \right\}\ \end{equation}$$38A∗=v1∗,…,vn∗=maxviji∈I′,minviji∈I′′$$\begin{equation}\ {A}^* = \left\{ {v_1^*, \ldots ,v_n^*} \right\}\ = \left\{ {\left( {\max {v}_{ij}\left| {i \in I^{\prime}} \right.} \right),\left( {\min \ {v}_{ij}\left| {i \in I^{\prime\prime}} \right.} \right)} \right\}\ \end{equation}$$where I′$I^{\prime}$ and I′′$I^{\prime\prime}$ are associated with benefit criteria, and cost criteria, respectively.4.Compute the distance measures, by the n‐dimensional Euclidean distance. The distance of every result from the ideal result is calculated as:39Dj*=∑i=1nvij−vi*2j=1,…,J.$$\begin{equation} {D}_{j}^{\ast}=\sqrt{\sum _{i=1}^{n}{\left({v}_{\textit{ij}}-{v}_{i}^{\ast}\right)}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$Uniformly, the distance from the negative‐ideal result is calculated as:40Dj−=∑i=1nvij−vi−2j=1,…,J.$$\begin{equation} {D}_{j}^{-}=\sqrt{\sum _{i=1}^{n}{\left({v}_{\textit{ij}}-{v}_{i}^{-}\right)}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$5.The relative closeness of the related resultis determined by:41Cj*=Dj−/Dj*+Dj−j=1,…,J.$$\begin{equation} {C}_{j}^{\ast}={D}_{j}^{-}/\left({D}_{j}^{\ast}+{D}_{j}^{-}\right)\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$6.Rate the preference result.Proposed solution methodAs shown in Figure 4, first of all, control variables values are generated, and the optimization process begins. The uncertain data set is generated, and the desired data are determined using the k‐means‐based DCM. The generated values for control variables are applied to the input data of the power system, and the wanted information is determined by Newton–Raphson load‐flow [47]. The values of objective functions are calculated, and the end requirements of the optimization process are investigated. At the end of the process and after extracting the Pareto‐based solutions set, the ranking of solutions will be performed based on the weight coefficient of each objective function by TOPSIS.4FIGUREThe proposed solution method flowchartSIMULATION RESULTSThe proposed solution method is examined on the IEEE 30‐bus test power system. For this purpose, firstly, the expected value of the objective functions is calculated in the base condition of the test system without any IPFC. Then, the IPFC is added to the system, and optimization is performed using the MOPSO algorithm. In continue, the obtained optimal results are compared with the base condition.AssumptionsThe IEEE 30‐bus test system, including 30 buses and 41 transmission lines, is illustrated in Figure 5. This system represents a Midwest region of the American electrical power system, and its information is collected from [48].5FIGUREThe one‐line diagram of the IEEE 30‐bus test systemA WT unit with a rated output equal to 20 MW has been located at bus No. 5. Table 1 presents additional information about this unit.1TABLEThe information of WTParameterValueBus No.5PWTr(MW)${\bm{P}}_{\bm{W}\bm{T}}^{\bm{r}}\ \mathbf{(}\mathbf{M}\mathbf{W}\mathbf{)}$20vrated(m/s)${\bm{v}}_{\bm{r}\bm{a}\bm{t}\bm{e}\bm{d}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$13vcout(m/s)${\bm{v}}_{\bm{c}\ }^{\bm{o}\bm{u}\bm{t}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$25vcin(m/s)${\bm{v}}_{\bm{c}}^{\bm{i}\bm{n}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$3A Weibull distribution with A and B parameters equal to 3 and 8, respectively, is considered for the statistical behavior of wind speed. Similarly, the power demands have an uncertain nature that are modeled by a normal distribution. The mean value of active power demands equals their nominal value of deterministic data of the test system. The standard deviation is considered to be 10% of the mean value. The k‐means‐based DCM is used for probabilistic assessment of the problem. The uncertain variables’ samplingsize is equal to 1000, which are grouped into 10 clusters, and the centers of each cluster are selected asrepresentatives of that cluster. The scatter plot of active power demand samples in buses No. 8 and 29 with their histograms is shown in Figure 6. In addition, Figure 7 shows the scatter plot of active power demand samples and wind speed of WT samples in bus No. 5. It should be mentioned that the high limit of apparent power flow of lines is assumed to be 200 MVA. The MOPSO's population size and its repository size are equal to 100 and 25, respectively. Also, the maximum iteration number of this algorithm is considered to be 100. In order to decision making and choose the final solution, the weight coefficients of TOPSIS for each objectivefunction are considered to be the same.6FIGUREThe scatter plot of the load samples in buses No. 8 and 297FIGUREThe scatter plot of the load samples and wind speed of WT samples in bus No. 5It is assumed the IPFC's series converters can be placed in every line except for the lines that are connected to the P‐V buses on both sides. Also, it should be mentioned that both of the series converters cannot be connected to the same Line. The constraints of the IPFC, including reference setting of active and reactive power flow ofrelated lines as a ratio of the value in the base condition, RatioP, and RatioQ, as well as voltage magnitude and voltage phasor of series converters, are represented in Table 2. It should be noted that the second converter has no control over the reactive power of the line.2TABLEThe constraints of the IPFC's convertersParameterMinMaxRatioPLine1,RatioPLine2(p.u.)$\bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{P}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{1}}}\ \mathbf{,}\ \bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{P}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{2}}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$12RatioQLine1(p.u.)$\bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{Q}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{1}}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$0.012Vse1,Vse2(p.u.)${\bm{V}}_{\bm{s}\bm{e}\mathbf{1}}\ \mathbf{,}\ {\bm{V}}_{\bm{s}\bm{e}\mathbf{2}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$−0.50.5θse1,θse2(Degree)${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}\ \mathbf{,}\ {\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}\ (\mathbf{D}\mathbf{e}\mathbf{g}\mathbf{r}\mathbf{e}\mathbf{e})$−180180Obtained resultsIn order to inform the base condition of the WT‐included test system without any IPFC, Table 3 presents the expected value of objective functions by DCM and MCS. The results given in this table are used to reveal the efficiency of the proposed solution method.3TABLEThe expected value of objective functions in base condition without any IPFCObjective functionsMCSDCME [PLosses] (MW)15.5615.52E [PFI] (p.u.)1.60261.5987E [IPFCCost] ($/kVAr)188.22188.22According to Table 3, it is apparent that the outcomes of these methods are significantly similar. For example, the error of the expected value of active power losses and PFI in the DCM compared to the MCS are 0.2571% and 0.2434%, respectively. It should be noted that the cost of IPFC remains constant because it has not been assigned to the power system.In the following, the proposed solution method for optimal allocation of IPFC considering uncertainties and in a multi‐objective optimization framework is performed, and the optimal solution is obtained for the case study system. According to the obtained solution, the IPFC’s converters are connected to lines 3‐4 and 2‐4. Table 4 presents the optimal settings of each converter. In order to know the output parameters of the IPFC for each cluster, by applying the results provided in Table 4, Table 5 is presented.4TABLEThe optimal obtained location and settings of IPFC's convertersConverter No.LocationRatio of P LinesRatio of Q Lines1Line 3‐41.4281.1672Line 2‐41.305–5TABLEThe IPFC's parameters for each clusterCluster No.IPFC's parametersUnit12345678910Sse1MVA0119.26124.76124.88133.20129.57119.73122.54115.63124.06Ss2MVA065.0465.8366.1668.3267.5264.8166.0564.4766.96Vse1p.u.00.47950.47950.48000.47900.47950.48050.48000.48050.4800Vse2p.u.00.49670.49710.49710.49710.49670.49670.49670.49670.4967θse1${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}$Degree0114.09113.08112.63113.41112.07112.18112.07112.18111.84θse2${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}$Degree0180180176.60180175.50166.01164.70156.60162.90According to Table 5, the IPFC has to be turned off for data of cluster No. 1. Based on the results obtained from this table, the expected value of IPFC’ parameters can be calculated. These values are given in Table 6. It should be mentioned that under this condition and considering the obtained optimal values for control variables of the problem, the DCM’s performance has been compared with the MCS method.6TABLEThe expected value of the IPFC's parametersIPFC's parametersMCSDCME [Sse1] (MVA)111.85111.37E [Sse2] (MVA)59.5959.52E [Vse1] (p.u.)0.43670.4319E [Vse2] (p.u.)0.44760.4472E [θse1${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}$] (Degree)100.41101.35E [θse2${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}$] (Degree)158.42154.23Table 7 presents the expected value and standard deviation of the objective functions by DCM and MCS after the optimal allocation of the IPFC in the test system.7TABLEStatistical information of the objective functions with optimal allocation of IPFCObjective functionsMCSDCME [PLosses] (MW)14.6714.62σ[PLosses](MW)$\bm{\sigma}{\bm [}{\bm{P}}_{\bm{L}\bm{o}\bm{s}\bm{s}\bm{e}\bm{s}}{\bm ]}\ \mathbf{(}\mathbf{M}\mathbf{W}\mathbf{)}$1.65091.5048E [PFI] (p.u.)1.42591.4606σ [PFI](p.u.)0.17400.1496E [IPFCCost] ($/kVAr)212.57210.06σ [IPFCCost] ($/kVAr)19.2718.01The expected value of the active power losses is decreased from 15.52 MW to 14.62 MW after the optimal allocation of the IPFC in the test system. Also, under this condition, the expected value of the PFI is reached from 1.5987 p.u. to 1.4606 p.u. However, the expected value of IPFC's cost is equal to 210.06 $/kVAr.The DCM’s performance for probabilistic assessment of the problem is also acceptable considering the IPFC in the test system. The statistical information error of the objective functions by DCM is low compared with MCS. This issue proves the validity of the DCM in the problem. For instance, the error of the expected value of active power losses and PFI in the DCM compared to the MCS are 0.3408% and 2.43%, respectively.By utilizing the cumulative distribution function (CDF) of output variables, the results would be more understandable. Figures8 to 10 illustrate the CDF of the active power losses, PFI, and IPFC cost, respectively. Notice that the blue line is obtained by DCM. Also, the red line denotes CDF by the MCS method. The results show that the performance of the two probabilistic assessment methods is very close to each other.8FIGUREThe CDF of the active power losses by DCM and MCS9FIGUREThe CDF of the PFI by DCM and MCS10FIGUREThe CDF of the IPFC cost by DCM and MCSCONCLUSIONThis paper proposed a probabilistic multi‐objective optimization method for optimal allocation of IPFC to reduce the active power losses, PFI of the lines, and IPFC cost considering uncertainties of the loads and wind speed of WTs. The k‐means‐based DCM was used as the probabilistic assessment, and its performance was compared with the MCS method. Furthermore, the simulation of the optimization process was implemented on the IEEE 30‐bus test system by employing the MOPSO algorithm. Results represented that by optimal allocation of IPFC, the expected value of active power losses and the expected value of PFI were decreased by 5.80% and 8.64%, respectively. However, the expected value of IPFC cost was calculated at 210.06 $/kVAr. Finally, based on the results obtained can be concluded that the performance of DCM for probabilistic assessment of the IPFC‐included transmission network is very close to the MCS method.AUTHOR CONTRIBUTIONSAll of the authors have contributed to preparing this paper. The main role of the authors can be summarized as follows: S.R.‐M.: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing—original draft, Writing—review and editing. H.E.: Data curation, Investigation, Resources, Writing—original draft. B.T.: Project administration, Supervision, Writing—review and editing.CONFLICT OF INTERESTThe authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent‐licensing arrangements), or non‐financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.FUNDING INFORMATIONThe authors received no specific funding for this work.DATA AVAILABILITY STATEMENTResearch data are not shared.REFERENCESGitizadeh, M., Kalantar, M.: A novel approach for optimum allocation of FACTS devices using multi objective function. 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Probabilistic multi‐objective optimization method for interline power flow controller (IPFC) allocation in power systems

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© 2022 The Institution of Engineering and Technology.
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10.1049/gtd2.12645
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Abstract

INTRODUCTIONUsually, the maximum capacity of existing transmission lines is used due to the high cost of constructing new ones. Also, reliability, security, and efficiency in the operation and development of power systems are more critical than ever before. The use of flexible AC transmission system (FACTS) devices is a prominent example of trying to improve transmission networks' performance. These devices can control the various power system parameters such as; the lines’ impedance, the active and reactive power flow, and the buses’ voltage magnitude. Optimal management of FACTS devices is essential to reach the proper efficiency of these controllers [1–5]. This managing is implemented by controlling their operating limits to distinguish the practicable capabilities of FACTS devices. Refs. [6–9] have investigated several FACTS devices and some approaches for controlling their constraints.The interline power flow controller (IPFC) device is recognized as one of the most powerful and flexible of the FACTS devices family. Generally, IPFCs are employed to control the power flow of the lines in transmission networks. The fundamental challenge is determining an appropriate location and settings for fixing the IPFC in the power system [10].Improvement of voltage stability and power profile has been studied by the utilization of the unified power flow controller (UPFC), static synchronous compensator (STATCOM), static synchronous series compensator (SSSC), and IPFC in [11]. In [12–15], different models of IPFC are proposed for damping oscillations in power systems. In [16], a new and straightforward approach has been presented for IPFC modelling to control all operating limits by applying the Newton–Raphson method.In [17], it is shown that the placement of the IPFC improves the voltage stability of the power system and the line loading capability. In [18], the optimal allocation of the IPFC and the solar power unit using the line severity index has been investigated to avoid contingencies. The allocation cost of the IPFC has been ignored in both studies, and only operational aspects are considered.In [19], the optimal setting and placement of the IPFC are investigated based on the disparity utilization factor to reduce power congestion. Also, in [20], the optimal location of the IPFC has been determined for improving and controlling the power system parameters. Transmission losses and generation fuel cost have been considered as objective functions in this study. The single objective function with weight coefficients has been used in both studies.Renewable energy sources are preferred due to environmental concerns because they are environmentally friendly. However, the output of these sources is variable. This issue, along with system configuration variation and load fluctuation make challenges in planning and operating of power systems. Therefore, under these conditions, uncertainties handling in studies is very necessary. None of the mentioned studies have included this challenge in their studies related to IPFC allocation.Considering uncertainty in the optimal allocation of other types of FACTS devices has been investigated in numerous studies. Ref. [21] considers the system predictability to optimal allocation of the UPFC. A solving method for the comprehensive problem, including unit commitment and UPFC allocation considering the uncertainty of wind power generation (WPG), is proposed in [22]. Robust control methods are exploited to design the UPFC controllers under parametric uncertainties of the system in [23]. Ref. [24] exploits a probabilistic‐based method to specify the best possible size of multiple FACTS devices, including thyristor‐controlled series capacitor (TSCS), STATCOM, and UPFC, for enhancing the steady‐state voltage profile. However, a comprehensive probabilistic framework has not been presented to study the IPFC impact on improving transmission networks' performance so far. This paper covers this gap.The Monte Carlo simulation (MCS) method is a simple and accurate method to handle uncertainties [25]. This method requires a very large calculation burden. Approximate methods such as data clustering method (DCM) have been introduced to overcome these challenges. The DCM has reasonable speed and presents acceptable results compared to the MCS method based on obtained results in [25] and [26]. In this study k‐means‐based DCM is used for the probabilistic assessment of the mentioned problem for the first time.The main contributions of this paper are as follows:The optimal location and parameters setting (reference set points) of an IPFC are determined with the objectives of decreasing active power losses, improving power flow index (PFI) of the lines, and considering IPFC allocation cost in a probabilistic multi‐objective optimization framework.Uncertain input parameters such as loads and wind speed of wind turbines (WTs) are considered by proper probability density functions (PDFs) to extract robust solutions for the problem.k‐mean‐based DCM is used for probabilistic assessment of optimal IPFC allocation problem in transmission networks.Pareto‐based optimal robust solutions set are extracted by multi‐objective particle swarm optimization (MOPSO) algorithm in a probabilistic framework, and the technique for order of preference by similarity to ideal solution (TOPSIS) is employed to make decisions and select the final solution based on the importance of each objective function for the operators.MATHEMATICAL MODEL OF IPFC IN STEADY‐STATE STUDIESAn IPFC can enhance the power capacity and power transfer capability of the power system. The functional convertibility enables the IPFC to adapt to changing system operating requirements and changing power flow patterns. The expandability of the IPFC is that a number of voltage source converters coupled with a common DC bus can be operated.The IPFC is formed by integrating two or several series‐connected converters operating with each other to enhance the power flow control capability over what is accessible with the usual SSSC [27]. Figure 1 illustrates the simplest IPFC, with FACTS buses i,j,$i,\ j,$ and k which is utilized to show the operation principle [28–30].1FIGUREThe simplest IPFC with two convertersIn the IPFC structure, two transformers are used to series‐connected the two converters with the transmission lines. Furthermore, the sending‐ends of the two transmission lines are connected in series with the FACTS buses j and k, respectively. The mentioned IPFC can independently handle three power flows of the two transmission lines. Figure 2 illustrates the equivalent circuit of the IPFC with two controllable series injected voltage sources.2FIGUREThe equivalent circuit of the simplest IPFC with two convertersThe active power can be transferred between the series converters via the common DC link, while the sum of the transferred active power should be zero. In Figure 2 the controllable injected voltage source is Vsein=Vsein∠θsein(n=j,k)$\bm{Vs}{\bm{e}}_{\bm{in}}=\textit{Vs}{e}_{\textit{in}}\angle \theta s{e}_{\textit{in}}\ (n=j,k)$, and the series transformer impedance is Zsein. Active and reactive power flows of the branches leaving buses i, j, k are obtained by:1Pin=Vi2gin−ViVngincosθin+binsinθin−ViVseingincosθi−θsein+binsinθi−θsein)$$\begin{eqnarray} \ {P}_{in} &=& V_i^2\ {g}_{in} - {V}_i{V}_n\left( {{g}_{in}cos{\theta }_{in} + {b}_{in}sin{\theta }_{in}} \right)\nonumber\\ && - {V}_iVs{e}_{in}\left( {{g}_{in}\cos \left( {{\theta }_i - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{sin}}\left( {{\theta }_i - \theta s{e}_{in}} \right))\end{eqnarray}$$2Qin=−Vi2bin−ViVnginsinθin+bincosθin−ViVseinginsinθi−θsein+bincosθi−θsein)$$\begin{eqnarray} \ {Q}_{in} &=& \ - V_i^2{b}_{in} - {V}_i{V}_n\left( {{g}_{in}sin{\theta }_{in} + {b}_{in}cos{\theta }_{in}} \right)\nonumber\\ && - {V}_iVs{e}_{in}\left( {{g}_{in}\sin \left( {{\theta }_i - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{cos}}\left( {{\theta }_i - \theta s{e}_{in}} \right))\end{eqnarray}$$3Pni=Vn2gin−ViVngincosθn−θi+binsinθn−θi+VnVseingincosθn−θsein+binsinθn−θsein)$$\begin{eqnarray} \ {P}_{ni} &=& V_n^2\ {g}_{in} - {V}_i{V}_n\left( {{g}_{in}\cos \left( {{\theta }_n - {\theta }_i} \right) + {b}_{in}\sin \left( {{\theta }_n - {\theta }_i} \right)} \right)\nonumber\\ && + {V}_nVs{e}_{in}\left( {{g}_{in}\cos \left( {{\theta }_n - \theta s{e}_{in}} \right)} \right) + {b}_{in}{\rm{sin}}\left( {{\theta }_n - \theta s{e}_{in}} \right))\end{eqnarray}$$4Qni=−Vn2bnn−ViVnginsinθn−θi−bincosθn−θi+VnVsein(ginsinθn−θsein−binsinθn−θsein)$$\begin{eqnarray} \ {Q}_{ni} &=& \ - V_n^2{b}_{nn} - {V}_i{V}_n\left( {{g}_{in}\sin \left( {{\theta }_n - {\theta }_i} \right) - {b}_{in}\cos \left( {{\theta }_n - {\theta }_i} \right)} \right)\nonumber\\ && + {V}_nVs{e}_{in}({g}_{in}\sin \left( {{\theta }_n - \theta s{e}_{in}} \right) - {b}_{in}{\rm{sin}}\left( {{\theta }_n - \theta s{e}_{in}} \right))\end{eqnarray}$$where gin=Re(1/Zsein)${g}_{\textit{in}}=\textit{Re}(1/\bm{Z}\bm{s}{\bm{e}}_{\bm{i}\bm{n}})$, bin=Im(1/Zsein)${b}_{\textit{in}}=\textit{Im}(1/\bm{Z}\bm{s}{\bm{e}}_{\bm{i}\bm{n}})$. Pin,Qin(n=j,k)${P}_{\textit{in}},\ {Q}_{\textit{in}}\ (n=j,\ k)$ are the active and reactive power flows of two branches leaving bus i. Pni,Qni(n=j,k)${P}_{\textit{ni}},\ {Q}_{\textit{ni}}\ (n=j,\ k)$ are the active and reactive power flows at the sending‐ends of the two transmission lines. The power mismatches of the IPFC, at buses i,j,k$i,\ j,\ k$ should be:4aΔPm=Pgm−Pdm−Pm=0$$\begin{equation}\Delta {P}_m = P{g}_m\ - P{d}_m - \ {P}_m = \ 0\end{equation}$$5ΔQm=Qgm−Qdm−Qm=0$$\begin{equation}\Delta \ {Q}_m = Q{g}_m\ - Q{d}_m - \ {Q}_m = \ 0\end{equation}$$where, Pgm,Qgm(m=i,j,k)$P{g}_{m},\ Q{g}_{m}\ (m=i,\ j,\ k)$ are the active and reactive power generation entering the bus m, Pdm,Qdm(m=i,j,k)$P{d}_{m},\ Q{d}_{m}\ (m=i,\ j,\ k)$ are the active and reactive power load leaving the bus m. Pm,Qm(m=i,j,k)${P}_{m},\ {Q}_{m}\ (m=i,\ j,\ k)$ are the sum of active and reactive power flows of the circuits connected to bus m.According to the operational principle of the IPFC, the operating limitation indicating the active power transfer between the series converters via the common DC link is as follows:6PEx=−∑PEsein−Pdc=0$$\begin{equation}PEx\ = \ - \sum PEs{e}_{in} - \ {P}_{dc} = \ 0\end{equation}$$where PEsein=Re(VseinIni*)(n=j,k)$\textit{PEs}{e}_{\textit{in}}=\textit{Re}(\bm{Vs}{\bm{e}}_{\bm{in}}{\bm{I}}_{\bm{ni}}^{\ast})\ (n=j,\ k)$. Ini is the series converter current.The IPFC can control the active and reactive power flows of primary line 1 but only the active power of secondary line 2. Corresponding constraints of active and reactive power flow in the IPFC are:7ΔPni=Pni−PniSpec=0$$\begin{equation}\Delta \ {P}_{ni} = {P}_{ni}\ - \ P_{ni}^{Spec} = \ 0\end{equation}$$8ΔQni=Qni−QniSpec=0$$\begin{equation}\Delta \ {Q}_{ni} = {Q}_{ni}\ - \ Q_{ni}^{Spec} = \ 0\end{equation}$$where Pni=Re(VnIni∗)${P}_{ni} = \ Re( {{{\bm{V}}}_{\bm{n}}{\bm{I}}_{{\bm{ni}}}^*} )$, Qni=Im(VnIni∗)${Q}_{ni} = \ Im( {{{\bm{V}}}_{\bm{n}}{\bm{I}}_{{\bm{ni}}}^*} )$ and PniSpec$P_{ni}^{Spec}$, QniSpec$Q_{ni}^{Spec}\ $are control references of active and reactive power flow.Combining Equations (4a), (5), and Equations (6)–(8), the Newton power flow solution is given by:9JΔX=−ΔR$$\begin{equation}J\Delta X = - \Delta R\end{equation}$$where ΔX=[ΔX1,ΔX2]T$\Delta X = {[ {\Delta {X}_1,\Delta {X}_2} ]}^T $, is the incremental vector of state variables;ΔX1=[Δθi,ΔVi,Δθj,ΔVj,Δθk,ΔVk]T$\ \Delta {X}_1 = {[ {\Delta {\theta }_i,\Delta {V}_i,\Delta {\theta }_j,\Delta {V}_j,\ \Delta {\theta }_k,\Delta {V}_k} ]}^T $, is the incremental vector of bus voltage magnitudes and angles, and ΔX2=[Δθseij,ΔVseij,Δθseik,ΔVseik]T$\Delta {X}_2 = {[ {\Delta \theta s{e}_{ij},\Delta Vs{e}_{ij},\Delta \theta s{e}_{ik},\Delta Vs{e}_{ik}} ]}^T\ $, is the incremental vector of state variables of the IPFC.ΔR=[ΔR1,ΔR2]T$\Delta R \! =\! {[ {\Delta {R}_1,\Delta {R}_2} ]}^T$; ΔR1=[ΔPi,ΔQi,ΔPj,ΔQj,ΔPk,ΔQk]T$\Delta {R}_1\! =\! {[ {\Delta {P}_i,\Delta {Q}_i,\Delta {P}_j,\Delta {Q}_j,\Delta {P}_k, \Delta {Q}_k} ]}^T$, is the bus power mismatch vector, ΔR2=[Pji−PjiSpec,Qji−QjiSpec,Pki−PkiSpec,PEx]T$\Delta {R}_2 = {[ {{P}_{ji} - P_{ji}^{Spec},{Q}_{ji} - Q_{ji}^{Spec},{P}_{ki} - P_{ki}^{Spec},PEx} ]}^T$, is the operating control mismatch vector of the IPFC. In addition, J=∂ΔR∂X$\ J\ = \frac{{\partial \Delta R}}{{\partial X}}$, is the Jacobian matrix of the system.There is one related active power flow control equation for the secondary series converter in (9). The Jacobian matrix in (9) includes four blocks. The bottom diagonal block has a related formation of traditional power flow, and other blocks are FACTS associated. This matrix can be solved by first eliminating Δθse,ΔVse$\Delta \theta se,\ \Delta Vse$ of the IPFC. Then the resulting reduced bottom diagonal block Newton equation can be solved by block sparse matrix techniques.It should be mentioned that the multi‐control modes of UPFC can be utilized to IPFC. Besides, the methods that are used to handle the violated operative inequalities of SSSC and STATCOM can be utilized to IPFC [31].Vi,Vj,Vk,θi,θj,θk${V}_{i},\ {V}_{j},\ {V}_{k},\ {\theta}_{i},\ {\theta}_{j},\ {\theta}_{k}$ are set to the start values, where Vi=Vj=Vk=1${V}_i = {V}_{j} = {V}_{k} = 1$ and θi=θj=θk=0$\theta_{i} = {\theta}_{j} = {\theta }_{k} = 0$ if buses i,j,k$i, j, k$ are not voltage controlled buses. With solving two simultaneous Equations (7) and (8), the Vseij,θseij$Vs{e}_{ij},\theta s{e}_{ij}$ values for the primary series converter can be determined as:10Vseij=A/gij2+bij2/Vj$$\begin{equation}\ Vs{e}_{ij} = \sqrt {A/\left( {g_{ij}^2 + b_{ij}^2} \right)} \ /{V}_j\end{equation}$$11θseij=tan−1PjiSpec−Vj2gjj+ViVjgijQjiSpec+Vj2bjj−ViVjbij−tan−1−gij/bij$$\begin{equation} \theta s{e}_{\textit{ij}}=\textit{ta}{n}^{-1}\left[\frac{{P}_{\textit{ji}}^{\textit{Spec}}-{V}_{j}^{2}{g}_{\textit{jj}}+{V}_{i}{V}_{j}{g}_{\textit{ij}}}{{Q}_{\textit{ji}}^{\textit{Spec}}+{V}_{j}^{2}{b}_{\textit{jj}}-{V}_{i}{V}_{j}{b}_{\textit{ij}}}\right]-\textit{ta}{n}^{-1}\left(-{g}_{\textit{ij}}/{b}_{\textit{ij}}\right) \end{equation}$$12A=PjiSpec−Vj2gjj+ViVjgij2+(QjiSpec+Vj2bjj−ViVjbij)2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl}{A}& =& {{\left({P}_{\textit{ji}}^{\textit{Spec}}-{V}_{j}^{2}{g}_{\textit{jj}}+{V}_{i}{V}_{j}{g}_{\textit{ij}}\right)}^{2}}\\[6pt] & & {+\,({Q}_{\textit{ji}}^{\textit{Spec}}+{V}_{j}^{2}{b}_{\textit{jj}}-{V}_{i}{V}_{j}{b}_{\textit{ij}})^{2}}\end{array} \end{equation}$$Suppose that Vseik$Vs{e}_{ik}$ is selected to a value between Vseikmin$Vse_{ik}^{min}$, and Vseikmax$Vse_{ik}^{max}\ $. Moreover, θseik$\theta s{e}_{ik}$ can be obtained by using the following equation:13θseik=sin−1PkiSpec−Vki2gik+ViVkgik/VkVsekigik2+bik2−tan−1−gik/bik$$\begin{eqnarray} \ \theta s{e}_{ik} &=& si{n}^{ - 1}\ \left[ {\left( {P_{ki}^{Spec} - V_{ki}^2{g}_{ik} + {V}_i{V}_k{g}_{ik}} \right)/\left( {{V}_kVs{e}_{ki}\sqrt {g_{ik}^2 + b_{ik}^2} } \right)} \right]\nonumber\\ && - \ ta{n}^{ - 1}\left( { - {g}_{ik}/{b}_{ik}} \right)\end{eqnarray}$$PROBABILISTIC LOAD FLOW (PLF)Deterministic load flow (DLF) analyses the power system under a pre‐determined and certain operating condition. By ever‐increasing uncertain resources in power systems, PLF must be used to express the complete range of probable values of the system's output variables. In PLF, statistical information of uncertain output variables is obtained by statistical information of input uncertain variables [32].Generally, there are three main classifications for probabilistic assessment, including MCS, approximate, and analytical methods. The MCS method is precise and flexible and can be employed for many problems. However, this method needs an extra calculation burden. Other approaches are less accurate because of the simplifications in their modelling, and this makes them faster. In this regard, a compromise should be made between reasonable accuracy and speed of the computations. The DCM, with almost precise evaluation ability, is a sampling approach to diminish the number of MCS samples and has been proposed by Michael McKay [33]. The explanation and formulation of the DCM approach are discussed in Section 3.2.Uncertain input variablesIn PLF, the input variable's information is described by PDF. So, it is crucial to consider the precise analytical model to estimate uncertain input variables. PDFs of wind speed and power demands are modelled as follows:Naturally, wind turbine generation relies on wind speed. Various distribution models, namely Gamma, Rayleigh, Lognormal, Gumbel, Generalized Extreme Value, Nakagami, Inverse Gaussian, Burr, and Weibull, have been employed to show wind speed behaviour. Commonly, the Weibull distribution is employed to modell the wind speed. The formulation of the Weibull distribution is represented in Equation (14) [34].14fv=ABvBA−1e−vBAv≥00v<0$$\begin{equation} f\left(v\right)=\left\{ \def\eqcellsep{&}\begin{array}{ll}\frac{A}{B}{\left(\frac{v}{B}\right)}^{A-1}{e}^{-{\left(\frac{v}{B}\right)}^{A}}&v\ge 0\\[10pt] 0&v&lt;0\end{array} \right. \end{equation}$$The WT power generation depends on wind speed that can be determined by Equation (15).15PWTv=0v≤vincorv≥voutcv−vincvrated−vincPrWTvinc<v<vratedPrWTvrated≤v<voutc$$\begin{equation} {P}^{\textit{WT}}\left(v\right)=\left\{ \def\eqcellsep{&}\begin{array}{cc}0& v\le {v}_{\textit{in}}^{c}\ \textit{or}\ v\ge {v}_{\textit{out}}^{c}\\ \frac{v-{v}_{\textit{in}}^{c}}{{v}_{\textit{rated}}-{v}_{\textit{in}}^{c}}{P}_{r}^{\textit{WT}}& {v}_{\textit{in}}^{c}&lt;v&lt;{v}_{\textit{rated}}\\[10pt] {P}_{r}^{\textit{WT}}& {v}_{\textit{rated}}\le v&lt;{v}_{\textit{out}}^{c}\end{array} \right. \end{equation}$$where PrWT${P}_{r}^{\textit{WT}}$ is the rated output power of the WT; v is the wind speed, vinc$v_{in}^c$, voutc$v_{out}^c$, and vrated${v}_{rated}$ are cut in, cut out, and the rated speeds of the WT, respectively.Various distribution models, namely Rayleigh, Weibull, and normal have been employed to model the demand load [35]. In this paper, the normal distribution is used. The normal distribution formulation is represented in Equation (16) [36].16fx=1σx2π×e−x−Ex22σx2$$\begin{equation}f\ \left( x \right) = \frac{1}{{\left( {{{\sigma}}\left[ x \right]} \right)\sqrt {2\pi } }}\ \times {e}^{ - \frac{{{{\left( {x - {\rm{E}}\left[ x \right]} \right)}}^2}}{{2{{\sigma}}{{\left[ x \right]}}^2}}}\end{equation}$$where σ[] and E[] are the standard deviation and expected value operators, respectively.k‐means‐based DCMThe k‐means‐based DCM was presented by Mac Queen in 1967 [37]. This approach has a simple process of classifying the data set into a determined quantity of clusters. In this regard, the distance of each cluster point is reduced to its center [38]. Concisely, the k‐means method steps are according to below [39]:The number of clusters (K) is designated.K agents or centers are initialized randomly (ak,k=1,2,…,K${a}_{k},\ k=1,2,\text{\ensuremath{\ldots}},K$).Data is assigned to the clusters based on the minimum distance with K centers (17):17ifdn−ak<dn−al⇒dn∈Gk$$\begin{equation} \textit{if}\ \left|{d}_{n}-{a}_{k}\right|&lt;\left|{d}_{n}-{a}_{l}\right|\Rightarrow {d}_{n}\in {G}_{k} \end{equation}$$n=1,2,…,N$n=1,\ 2,\ \text{\ensuremath{\ldots}},N$,k=1,2,…,K$k=1,\ 2,\ \text{\ensuremath{\ldots}},\ K$,l=1,2,…,K$l=1,\ 2,\ \text{\ensuremath{\ldots}},\ K$,(l≠k)$( {l \ne k} )$N and dn${d}_n$ are the number of data and the nth data, respectively. ak${a}_k$ and al${a}_l$are the kth and lth agents, respectively. Also, Gk${G}_k$ is the kth cluster.d.The centers of clusters are updated using the average of all data points of each cluster as (18):18ak=∑n∈GkdnNGk,k=1,2,…,K$$\begin{equation}\ {a}_k = \frac{{\mathop \sum \nolimits_{n \in {G}_k} {d}_n}}{{{N}_{{G}_k}}}\ ,\ k\ = \ 1,2, \ldots ,\ K\end{equation}$$NGk${N}_{{G}_k}$ is the number of members in the kth cluster.e.The steps c and d are repeated until the centers' value no longer changes.f.A probability is determined for each center as (19):19Pak=NGkN$$\begin{equation} \mathrm{P}\left({a}_{k}\right)=\frac{{N}_{{G}_{k}}}{N} \end{equation}$$Moreover, the ith statistical moment of F is calculated by (20).20EFi=∑k=1KPak.Faki$$\begin{equation} \mathrm{E}\left[{F}^{i}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).F{\left({a}_{k}\right)}^{i} \end{equation}$$So, it can be said that each cluster center (ak${a}_k$) has a probability of P(ak)$\mathrm{P}({a}_{k})$. The centers ak(k=1,2,…,K)${a}_{k}\ (k=1,\ 2,\text{\ensuremath{\ldots}},\ K)$ are obtained from all data points, and after calculating the F(ak)$F( {{a}_k} )$, the ith statistical moment of F is determined using Equation (20).PROBLEM FORMULATION AND SOLVING TOOLS IN THE PROPOSED STUDY METHODIn this study, various objective functions are considered, which are presented in the following subsection. Also, the power system and IPFC constraints are discussed. In multi‐objective optimization problems, a proper optimization algorithm must be employed. In this regard, the MOPSO algorithm is employed, and the TOPSIS is used to trade‐off between the objective functions and select the final solution.Objective functionsObjective functions are one of the main components that must be correctly defined for the proposed study method. In this paper, three objective functions are considered for the optimal allocation of IPFC. The first objective is active power losses, which is one of the essential elements in the optimization of power systems' operating. The second objective is related to the nature of IPFC, which is intended to control the power flow of the lines. This objective is a security‐based index. Finally, the economic aspect of IPFC allocation is considered as the third objective function.The expected value of active power lossesThis objective function can be calculated as:EPLosses=∑k=1KPak.PLossesk$$\begin{equation*} \mathrm{E}\left[{P}_{\textit{Loss}\textit{es}}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).{{P}_{\textit{Loss}\textit{es}}}_{k}\ \end{equation*}$$21PLosses=∑i=1NLinesRiIi2$$\begin{equation} {P}_{\textit{Loss}\textit{es}}=\sum _{i=1}^{{N}_{\textit{Lines}}}{R}_{i}{\left|{I}_{i}\right|}^{2} \end{equation}$$where K is the number of the DCM clusters. NLines${N}_{\textit{Lines}}$ is the number of transmission lines. Ij${I}_j$ is the current of line j, and Rj${R}_j$ is the resistance of the jth line.The expected value of PFI of the linesThis security‐based objective function is defined to minimize the congestion of the lines. For this purpose, the following equation is represented:EPFI=∑k=1KPak.PFIk$$\begin{equation*} \mathrm{E}\left[\textit{PFI}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).\textit{PF}{I}_{k} \end{equation*}$$22PFI=∑i=1NLinesSi|Simax|2$$\begin{equation} \textit{PFI}=\sum _{i=1}^{{N}_{\textit{Lines}}}{\left(\frac{\left|{S}_{i}\right|}{|{S}_{i}^{\max}|}\right)}^{2} \end{equation}$$where Sj${S}_j$ and Sjmax$S_j^{max}$ are the complex power and the complex power rate of line j. It should be mentioned that power flow can be raised in some lines due to the operation of the IPFC.The expected value of the IPFC costAccording to the Section 2, the IPFC has two converters that are connected in series with the transmission lines. Thus, the cost functions are defined for both converters as follows:EIPFCCost=∑k=1KPak.IPFCCostk$$\begin{equation*} \mathrm{E}\left[\textit{IPF}{C}_{\textit{Cost}}\right]=\sum _{k=1}^{K}\mathrm{P}\left({a}_{k}\right).\textit{IPF}{{C}_{\textit{Cost}}}_{k} \end{equation*}$$IPFCCost=CIPFC−A+CIPFC−B$$\begin{equation*} \textit{IPF}{C}_{\textit{Cost}}={C}_{\textit{IPFC}-A}+{C}_{\textit{IPFC}-B} \end{equation*}$$CIPFC−A=0.00015Sj2−0.1345Sj+94.11$$\begin{equation*} {C}_{IPFC - A} = 0.00015S_j^2 - 0.1345{{\rm{S}}}_j + 94.11\end{equation*}$$23CIPFC−B=0.00015Sk2−0.1345Sk+94.11$$\begin{equation} {C}_{IPFC - B} = 0.00015S_k^2 - 0.1345{{\rm{S}}}_k + 94.11\end{equation}$$where CIPFC−AandCIPFC−B${C}_{\textit{IPFC}-A}\ \mathrm{and}\ {C}_{\textit{IPFC}-B}$ are the cost functions for converters connected to lines j$j\ $ and k, respectively. Sj and Sk are the operating range of converters connected to lines j and k, respectively[40].ConstraintsIn this paper, main purpose is the optimal allocation of the IPFC. The constraints of buses' voltage, lines' apparent power flow, and active power generation are as follows:24Vimin≤E[Vi]≤Vimax$$\begin{equation} {V}_{i}^{\min}\le \mathrm{E}[{V}_{i}]\le {V}_{i}^{\max} \end{equation}$$25E[Si]≤|Simax|$$\begin{equation} \mathrm{E}\big[\left|{S}_{i}\right|\big]\le |{S}_{i}^{\max}| \end{equation}$$26PGimin≤E[PGi]≤PGimax$$\begin{equation} {P}_{{G}_{i}}^{\min}\le \mathrm{E}[{P}_{{G}_{i}}]\le {P}_{{G}_{i}}^{\max} \end{equation}$$The active power exchange between or among the converters via the common DC link, voltage, and phase angle constraints of the series converter are as follows:27−PEsemax≤E[PEse]≤PEsemax$$\begin{equation} -P{E}_{\textit{se}}^{\max}\le \mathrm{E}[P{E}_{\textit{se}}]\le P{E}_{\textit{se}}^{\max} \end{equation}$$28Vsemin≤EVse≤Vsemax$$\begin{equation} {V}_{\textit{se}}^{\min}\le \mathrm{E}\left[{V}_{\textit{se}}\right]\le {V}_{\textit{se}}^{\max} \end{equation}$$29θsemin≤Eθse≤θsemax$$\begin{equation} {\theta}_{\textit{se}}^{\mathrm{\min}}\le \mathrm{E}\left[{\theta}_{\textit{se}}\right]\le {\theta}_{\textit{se}}^{\max} \end{equation}$$The optimal location and settings of IPFC are determined using the MOPSO algorithm while all constraints are satisfied.Multi‐objective optimizationA multi‐objective optimization problem has two or more objectives with equality and inequality restraints that can be defined as: [41] 30OF=Fix;i=1,2,…,Nof$$\begin{equation} \textit{OF}={F}_{i}\left(x\right);\ \ i=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{of}} \end{equation}$$31Hjx=0;j=1,2,…,Neq$$\begin{equation} {H}_{j}\left(x\right)=0;\ \ j=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{eq}} \end{equation}$$32Mkx≤0;k=1,2,…,Nineq$$\begin{equation} {M}_{k}\left(x\right)\le 0;\ \ k=1,\ 2,\text{\ensuremath{\ldots}},\ {N}_{\textit{ineq}} \end{equation}$$In (30‐32), OF$OF$ denotes the multi‐objective function where i is the number of objectives. Also, x is the control variables vector, Hj${H}_{j}$, and Mk${M}_{k}$ are equality and inequality constraints.The MOPSO algorithm has been used for solving various problems, including the operating of power systems [42, 43]. A set of feasible solutions is obtained in the multi‐objective optimization framework. Then, the TOPSIS can be used to sort the solutions based on some criteria.MOPSO algorithmIn this algorithm, a search space is considered with d dimension and n particles, where ith particle is in position Xi(xi1,xi2,…,xid)${X}_{i}\ ({x}_{i1},\ {x}_{i2},\text{\ensuremath{\ldots}},\ {x}_{\textit{id}})$, and moves with velocity Vi(vi1,vi2,…,vid)${V}_{i}\ ({v}_{i1},\ {v}_{i2},\text{\ensuremath{\ldots}},\ {v}_{\textit{id}})$. There is personal best position pbesti$\textit{pbes}{t}_{i}$ for each particle, which is specified by its own best performance in the population. Also, global best gbest$\textit{gbest}$ is defined according to the general best performance of all particles. Each particle modifies its velocity and location as (33) and (34), respectively.33Vit+1=wVit+c1r1xpbest−Xit+wVit+c2r2xgbest−Xit$$\begin{equation}\ V_i^{t + 1} = \ wV_i^t + {c}_1{r}_1\left( {{x}_{pbest} - X_i^t} \right) + wV_i^t + {c}_2{r}_2\left( {{x}_{gbest} - X_i^t} \right)\end{equation}$$34Xit+1=Xit+Vit+1$$\begin{equation}\ X_i^{t + 1} = X_i^t\ + V_i^{t + 1}\end{equation}$$where c1 and c2${c}_2\ $are cognitive acceleration and social acceleration coefficients, w is inertia weight, xpbest${x}_{pbest}$ and xgbest${x}_{gbest}$ are the personal and global best of the particle, r1 and r2 are the random variables. Also, Xit,$\ X_i^t,\ $and Vit$V_i^t$are current position and velocity of ith particle in iteration t.The particles' velocity and position are renewed similarly at the PSO and MOPSO. The PSO and MOPSO algorithms are different in some aspects: The selection and update procedures of the global and individual leaders are different. Besides, the Pareto optimal solution set is employed to obtain the solution in the MOPSO algorithm. Beyond, the MOPSO algorithm needs a repository for saving the non‐dominated solutions obtained by the input variables to be established. Several researchers have studied the selection and updating of global and individual leaders, the parameter settings, and the establishment of a repository and have achieved some results [44, 45]. In addition, this algorithm is still encountered challenges such as the ineffectiveness of exploring the gbest$gbest$ solution and prone to premature convergence. The flowchart of the MOPSO algorithm according to dominance criteria is illustrated in Figure 3.3FIGUREFlowchart of the MOPSO algorithmTOPSISThe number of results has to be compared and deliberated by some criteria in multi‐objective optimization problems. The purpose of the TOPSIS is to assist decision‐makers in the trade‐off between objective functions. So, possible results are usually specified by different criteria, and maybe none of the results satisfies all the criteria. Hence, the result is a relative result based on the decision‐makers' preference. TOPSIS was proposed by Chen in 1992 [46]. The basic concept is that the determined result must have the maximum interval from the negative‐ideal result and the minimum interval from the ideal result. The following steps are the process of the TOPSIS:Compute the rij${r}_{ij}$ as a normalized decision matrix by Equation (35).35rij=fij/∑j=1Jfij2j=1,…,J;i=1,…,n.$$\begin{equation} {r}_{\textit{ij}}={f}_{\textit{ij}}/\sqrt{\sum _{j=1}^{J}{f}_{\textit{ij}}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J;\ \ i=1,\text{\ensuremath{\ldots}},n. \end{equation}$$Compute the vij$v_{ij}$ as weighted normalized decision matrix by Equation (36).36vij=wirijj=1,…,J;i=1,…,n.$$\begin{equation} {v}_{\textit{ij}}={w}_{i}{r}_{\textit{ij}}\ \ \ j=1,\text{\ensuremath{\ldots}},J;\ \ i=1,\text{\ensuremath{\ldots}},n. \end{equation}$$Specify the negative‐ideal and ideal result.37A−=v1−,…,vn−=minviji∈I′,maxviji∈I′′$$\begin{equation}\ {A}^ - = \left\{ {v_1^ - , \ldots ,v_n^ - } \right\}\ = \left\{ {\left( {\min {v}_{ij}\left| {i \in I^{\prime}} \right.} \right),\left( {\max \,\ {v}_{ij}\left| {i \in I^{\prime\prime}} \right.} \right)} \right\}\ \end{equation}$$38A∗=v1∗,…,vn∗=maxviji∈I′,minviji∈I′′$$\begin{equation}\ {A}^* = \left\{ {v_1^*, \ldots ,v_n^*} \right\}\ = \left\{ {\left( {\max {v}_{ij}\left| {i \in I^{\prime}} \right.} \right),\left( {\min \ {v}_{ij}\left| {i \in I^{\prime\prime}} \right.} \right)} \right\}\ \end{equation}$$where I′$I^{\prime}$ and I′′$I^{\prime\prime}$ are associated with benefit criteria, and cost criteria, respectively.4.Compute the distance measures, by the n‐dimensional Euclidean distance. The distance of every result from the ideal result is calculated as:39Dj*=∑i=1nvij−vi*2j=1,…,J.$$\begin{equation} {D}_{j}^{\ast}=\sqrt{\sum _{i=1}^{n}{\left({v}_{\textit{ij}}-{v}_{i}^{\ast}\right)}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$Uniformly, the distance from the negative‐ideal result is calculated as:40Dj−=∑i=1nvij−vi−2j=1,…,J.$$\begin{equation} {D}_{j}^{-}=\sqrt{\sum _{i=1}^{n}{\left({v}_{\textit{ij}}-{v}_{i}^{-}\right)}^{2}}\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$5.The relative closeness of the related resultis determined by:41Cj*=Dj−/Dj*+Dj−j=1,…,J.$$\begin{equation} {C}_{j}^{\ast}={D}_{j}^{-}/\left({D}_{j}^{\ast}+{D}_{j}^{-}\right)\ \ \ j=1,\text{\ensuremath{\ldots}},J. \end{equation}$$6.Rate the preference result.Proposed solution methodAs shown in Figure 4, first of all, control variables values are generated, and the optimization process begins. The uncertain data set is generated, and the desired data are determined using the k‐means‐based DCM. The generated values for control variables are applied to the input data of the power system, and the wanted information is determined by Newton–Raphson load‐flow [47]. The values of objective functions are calculated, and the end requirements of the optimization process are investigated. At the end of the process and after extracting the Pareto‐based solutions set, the ranking of solutions will be performed based on the weight coefficient of each objective function by TOPSIS.4FIGUREThe proposed solution method flowchartSIMULATION RESULTSThe proposed solution method is examined on the IEEE 30‐bus test power system. For this purpose, firstly, the expected value of the objective functions is calculated in the base condition of the test system without any IPFC. Then, the IPFC is added to the system, and optimization is performed using the MOPSO algorithm. In continue, the obtained optimal results are compared with the base condition.AssumptionsThe IEEE 30‐bus test system, including 30 buses and 41 transmission lines, is illustrated in Figure 5. This system represents a Midwest region of the American electrical power system, and its information is collected from [48].5FIGUREThe one‐line diagram of the IEEE 30‐bus test systemA WT unit with a rated output equal to 20 MW has been located at bus No. 5. Table 1 presents additional information about this unit.1TABLEThe information of WTParameterValueBus No.5PWTr(MW)${\bm{P}}_{\bm{W}\bm{T}}^{\bm{r}}\ \mathbf{(}\mathbf{M}\mathbf{W}\mathbf{)}$20vrated(m/s)${\bm{v}}_{\bm{r}\bm{a}\bm{t}\bm{e}\bm{d}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$13vcout(m/s)${\bm{v}}_{\bm{c}\ }^{\bm{o}\bm{u}\bm{t}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$25vcin(m/s)${\bm{v}}_{\bm{c}}^{\bm{i}\bm{n}}\ \mathbf{(}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}$3A Weibull distribution with A and B parameters equal to 3 and 8, respectively, is considered for the statistical behavior of wind speed. Similarly, the power demands have an uncertain nature that are modeled by a normal distribution. The mean value of active power demands equals their nominal value of deterministic data of the test system. The standard deviation is considered to be 10% of the mean value. The k‐means‐based DCM is used for probabilistic assessment of the problem. The uncertain variables’ samplingsize is equal to 1000, which are grouped into 10 clusters, and the centers of each cluster are selected asrepresentatives of that cluster. The scatter plot of active power demand samples in buses No. 8 and 29 with their histograms is shown in Figure 6. In addition, Figure 7 shows the scatter plot of active power demand samples and wind speed of WT samples in bus No. 5. It should be mentioned that the high limit of apparent power flow of lines is assumed to be 200 MVA. The MOPSO's population size and its repository size are equal to 100 and 25, respectively. Also, the maximum iteration number of this algorithm is considered to be 100. In order to decision making and choose the final solution, the weight coefficients of TOPSIS for each objectivefunction are considered to be the same.6FIGUREThe scatter plot of the load samples in buses No. 8 and 297FIGUREThe scatter plot of the load samples and wind speed of WT samples in bus No. 5It is assumed the IPFC's series converters can be placed in every line except for the lines that are connected to the P‐V buses on both sides. Also, it should be mentioned that both of the series converters cannot be connected to the same Line. The constraints of the IPFC, including reference setting of active and reactive power flow ofrelated lines as a ratio of the value in the base condition, RatioP, and RatioQ, as well as voltage magnitude and voltage phasor of series converters, are represented in Table 2. It should be noted that the second converter has no control over the reactive power of the line.2TABLEThe constraints of the IPFC's convertersParameterMinMaxRatioPLine1,RatioPLine2(p.u.)$\bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{P}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{1}}}\ \mathbf{,}\ \bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{P}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{2}}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$12RatioQLine1(p.u.)$\bm{R}\bm{a}\bm{t}\bm{i}{\bm{o}}_{{\bm{Q}}_{\bm{L}\bm{i}\bm{n}\bm{e}\mathbf{1}}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$0.012Vse1,Vse2(p.u.)${\bm{V}}_{\bm{s}\bm{e}\mathbf{1}}\ \mathbf{,}\ {\bm{V}}_{\bm{s}\bm{e}\mathbf{2}}\ (\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.})$−0.50.5θse1,θse2(Degree)${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}\ \mathbf{,}\ {\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}\ (\mathbf{D}\mathbf{e}\mathbf{g}\mathbf{r}\mathbf{e}\mathbf{e})$−180180Obtained resultsIn order to inform the base condition of the WT‐included test system without any IPFC, Table 3 presents the expected value of objective functions by DCM and MCS. The results given in this table are used to reveal the efficiency of the proposed solution method.3TABLEThe expected value of objective functions in base condition without any IPFCObjective functionsMCSDCME [PLosses] (MW)15.5615.52E [PFI] (p.u.)1.60261.5987E [IPFCCost] ($/kVAr)188.22188.22According to Table 3, it is apparent that the outcomes of these methods are significantly similar. For example, the error of the expected value of active power losses and PFI in the DCM compared to the MCS are 0.2571% and 0.2434%, respectively. It should be noted that the cost of IPFC remains constant because it has not been assigned to the power system.In the following, the proposed solution method for optimal allocation of IPFC considering uncertainties and in a multi‐objective optimization framework is performed, and the optimal solution is obtained for the case study system. According to the obtained solution, the IPFC’s converters are connected to lines 3‐4 and 2‐4. Table 4 presents the optimal settings of each converter. In order to know the output parameters of the IPFC for each cluster, by applying the results provided in Table 4, Table 5 is presented.4TABLEThe optimal obtained location and settings of IPFC's convertersConverter No.LocationRatio of P LinesRatio of Q Lines1Line 3‐41.4281.1672Line 2‐41.305–5TABLEThe IPFC's parameters for each clusterCluster No.IPFC's parametersUnit12345678910Sse1MVA0119.26124.76124.88133.20129.57119.73122.54115.63124.06Ss2MVA065.0465.8366.1668.3267.5264.8166.0564.4766.96Vse1p.u.00.47950.47950.48000.47900.47950.48050.48000.48050.4800Vse2p.u.00.49670.49710.49710.49710.49670.49670.49670.49670.4967θse1${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}$Degree0114.09113.08112.63113.41112.07112.18112.07112.18111.84θse2${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}$Degree0180180176.60180175.50166.01164.70156.60162.90According to Table 5, the IPFC has to be turned off for data of cluster No. 1. Based on the results obtained from this table, the expected value of IPFC’ parameters can be calculated. These values are given in Table 6. It should be mentioned that under this condition and considering the obtained optimal values for control variables of the problem, the DCM’s performance has been compared with the MCS method.6TABLEThe expected value of the IPFC's parametersIPFC's parametersMCSDCME [Sse1] (MVA)111.85111.37E [Sse2] (MVA)59.5959.52E [Vse1] (p.u.)0.43670.4319E [Vse2] (p.u.)0.44760.4472E [θse1${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{1}}$] (Degree)100.41101.35E [θse2${\bm{\theta}}_{\bm{s}\bm{e}\mathbf{2}}$] (Degree)158.42154.23Table 7 presents the expected value and standard deviation of the objective functions by DCM and MCS after the optimal allocation of the IPFC in the test system.7TABLEStatistical information of the objective functions with optimal allocation of IPFCObjective functionsMCSDCME [PLosses] (MW)14.6714.62σ[PLosses](MW)$\bm{\sigma}{\bm [}{\bm{P}}_{\bm{L}\bm{o}\bm{s}\bm{s}\bm{e}\bm{s}}{\bm ]}\ \mathbf{(}\mathbf{M}\mathbf{W}\mathbf{)}$1.65091.5048E [PFI] (p.u.)1.42591.4606σ [PFI](p.u.)0.17400.1496E [IPFCCost] ($/kVAr)212.57210.06σ [IPFCCost] ($/kVAr)19.2718.01The expected value of the active power losses is decreased from 15.52 MW to 14.62 MW after the optimal allocation of the IPFC in the test system. Also, under this condition, the expected value of the PFI is reached from 1.5987 p.u. to 1.4606 p.u. However, the expected value of IPFC's cost is equal to 210.06 $/kVAr.The DCM’s performance for probabilistic assessment of the problem is also acceptable considering the IPFC in the test system. The statistical information error of the objective functions by DCM is low compared with MCS. This issue proves the validity of the DCM in the problem. For instance, the error of the expected value of active power losses and PFI in the DCM compared to the MCS are 0.3408% and 2.43%, respectively.By utilizing the cumulative distribution function (CDF) of output variables, the results would be more understandable. Figures8 to 10 illustrate the CDF of the active power losses, PFI, and IPFC cost, respectively. Notice that the blue line is obtained by DCM. Also, the red line denotes CDF by the MCS method. The results show that the performance of the two probabilistic assessment methods is very close to each other.8FIGUREThe CDF of the active power losses by DCM and MCS9FIGUREThe CDF of the PFI by DCM and MCS10FIGUREThe CDF of the IPFC cost by DCM and MCSCONCLUSIONThis paper proposed a probabilistic multi‐objective optimization method for optimal allocation of IPFC to reduce the active power losses, PFI of the lines, and IPFC cost considering uncertainties of the loads and wind speed of WTs. The k‐means‐based DCM was used as the probabilistic assessment, and its performance was compared with the MCS method. Furthermore, the simulation of the optimization process was implemented on the IEEE 30‐bus test system by employing the MOPSO algorithm. Results represented that by optimal allocation of IPFC, the expected value of active power losses and the expected value of PFI were decreased by 5.80% and 8.64%, respectively. However, the expected value of IPFC cost was calculated at 210.06 $/kVAr. Finally, based on the results obtained can be concluded that the performance of DCM for probabilistic assessment of the IPFC‐included transmission network is very close to the MCS method.AUTHOR CONTRIBUTIONSAll of the authors have contributed to preparing this paper. The main role of the authors can be summarized as follows: S.R.‐M.: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing—original draft, Writing—review and editing. H.E.: Data curation, Investigation, Resources, Writing—original draft. B.T.: Project administration, Supervision, Writing—review and editing.CONFLICT OF INTERESTThe authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent‐licensing arrangements), or non‐financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.FUNDING INFORMATIONThe authors received no specific funding for this work.DATA AVAILABILITY STATEMENTResearch data are not shared.REFERENCESGitizadeh, M., Kalantar, M.: A novel approach for optimum allocation of FACTS devices using multi objective function. 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Journal

"IET Generation, Transmission & Distribution" – Wiley

Published: Dec 1, 2022

Keywords: flexible alternating current transmission system (FACTS); particle swarm optimisation; TOPSIS; uncertainty handling

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