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Simultaneous siting and sizing of Soft Open Points and the allocation of tie switches in active distribution network considering network reconfiguration

Simultaneous siting and sizing of Soft Open Points and the allocation of tie switches in active... INTRODUCTIONThe increasing integration of renewable energy sources (RES), like wind turbines and photovoltaic panels in today's power grids, is inevitable. These resources can contribute to meeting an increasing need for electricity, enhancing the security of the energy supply, and reducing the dependency on fossil fuels, power losses, and greenhouse gas [1]. Of course, due to the higher investment cost and a longer period of return on capital, distributed generations, especially renewables, are less attractive among investors. Nevertheless, the sustainable expansion of RES to provide the possibility of electric energy storage leads to economic promotion [2]. These resources with high volatility and intermittent nature pose significant challenges, especially in distribution network planning and operation. The main reason is that traditional distribution networks are not designed to accommodate generation sources [3]. Bidirectional power flow caused by high penetration of renewable generation makes operation, control, safety, and flexibility relatively complex and challenging [4]. Equipment overloads and power quality issues comprising voltage and frequency oscillations and harmonic distortion aggregation are other problems when integrating variable energy resources (VERs) [5, 6]. Curtailing the power production of generators is an approach to control the power output of generators [4]. However, curtailment actions are implemented in an ex‐post mode when the system has experienced certain security violations [7]. The active distribution network (ADN) could be equipped with different flexibility options and regulation means to overcome the challenges caused by the uncertainty of increasing penetration of VERs and various demand‐side resources. This proceeding can provide more power adjustment ability by dynamically changing the operational set point to improve the network performance.Literature surveySoft Open Points (SOPs) are novel multi‐functional power electronic switches typically used as soft meshing devices [8]. The SOP can be flexibly connected between feeders to replace traditional circuit‐breaker‐based interconnection switches known as normally open points (NOPs) [9]. Without hard switching operation, SOPs can control the active and reactive power flow, among the feeders, in real‐time and optimize the network voltage profile [10]. Meanwhile, SOP is can isolate any voltage and current disturbances or abnormal conditions and control peak current due to fault [11]. Therefore, the application of SOP will significantly promote the operating conditions of the network with better operational flexibility and a more cost‐effective manner.SOPs consist of voltage source converters (VSCs). These devices have different rating capacities and varying quantities of modules. The major topologies include [12–14]:Back‐to‐back VSC (B2B VSC)Multi‐terminal VSCUnified power flow controller (UPFC)Static series synchronous compensator (SSSC).Some publications have assessed the benefits of using SOPs to improve the performance of the distribution network under steady‐state operating conditions [15–25]. By incorporating SOPs in distribution network architecture, various schemes of optimal network operation can be achieved. Ref. [16] examined the network performance using SOP at several levels of renewable penetration by considering three different optimization problems, each with a separate objective function. Results showed that SOPs significantly improved voltage profile, balanced line utilization of the network, reduced network losses, and increased distributed generation penetration. A similar study to [16] has been conducted in reference [17] in which the voltage profile improvement, feeder load balancing, and power loss reduction have been seen as a multi‐objective function in the optimization problem. Ref. [18] proposed a model for developing a strengthened SOCP‐based approach to maximize distributed generation hosting capacity under the optimal operation of SOPs. An optimal planning model of the presence of distributed energy storage systemsin ADNs is provided in [19] while incorporating the operation of SOPs, reactive power support by DGs, and hourly network reconfiguration. Ref. [23] dealt with the change of network topology in a loop or mesh through SOPs which results in a reduction of network losses. Reference [24] proposed a robust optimization model to obtain operational strategies for SOPs while minimizing voltage violations and power losses. Ref. [25] studied a reconfiguration strategy to determine the optimal number, location, and size of DGs in ADNs while regulating the optimal set points for the operation of SOPs.Considering the flexible power flow control ability and dynamic operation of SOPs, several studies investigated the different control strategies and performance of SOPs in normal and abnormal operating conditions [8, 10, 26–30].Throughout the literature, it can be seen that most of the studies related to the evaluation of optimal network operation with the usage of SOPs consider a fixed number and predetermined installation sites for them. In this case, the optimal network operation may not be achieved. According to [31], the optimal installation location of SOPs has been determined to minimize network power losses and improve feeder load balancing. Here, the limited number of tie switches places is considered for the candidate locations for installing this equipment, and the capacity of each SOP is predetermined. The Optimal installation site and size of SOP were determined considering several operation scenarios of renewable output generated through the Wasserstein distance‐based method in [32].An optimization problem in the form of a bi‐level model was solved in [33] to co‐ordinately allocate DGs, capacitor banks, and SOPs considering the multistate model of DGs and loads. In [6] the simultaneous allocation of soft open points and distributed generations were modelled to obtain the best operation strategy for the distribution network with minimum losses. This study has analyzed several scenarios based on network reconfiguration, different numbers of installed SOPs, and the contribution of SOP's internal losses. The researchers of [34] proposed a hybridization framework for improving the performance of active distribution networks involving hybrid energy resources and SOPs. The best hybrid configuration of the system was obtained while minimizing the net present cost (NPC) and cost of energy (COE) considered the main objectives.It should be noted that there are various methods and techniques to solve the optimization problems of siting and sizing of different equipment, which are mainly categorized as follows [35]:Analytical techniquesClassical optimization techniquesArtificial intelligent (meta‐heuristic) techniquesMiscellaneous techniquesOther techniques for future useThe time complexity of solving optimization processes for siting and sizing problems, especially for large‐scale networks should be considered. Hence, integrating meta‐heuristic approaches with other solution approaches like classical optimization techniques can accelerate the process of problem‐solving and convergence [35]. In this regard, paper [36] has applied a beneficial meta‐heuristic approach called Multi‐Verse Optimizer (MVO) for optimal sizing of a hybrid system planning. Also, the authors of [36] have compared the MVO algorithm with some advanced meta‐heuristic approaches, like grasshopper optimization algorithm (GOA), grey wolf optimization (GWO), dragonfly algorithm (DA), and salp swarm algorithm (SSA). The efficiency of MVO has been promising in terms of convergence, calculation time, cost, and the obtained solutions compared to other approaches. Reference [37] has developed a mixed integer linear programming (MILP) model for integrated energy systems to coordinate different emergency proceedings between power distribution systems and natural gas networks. In this study, the integration of the graph‐based approach with the MILP model has been deployed to alleviate the complexity of solving the optimization problem.We can infer from the above discussion the future work of this paper could be to couple one of the efficient optimization approaches with the optimization method used in the current article. The aim is to reduce the problem's execution time and achieve the best convergence characteristics.Research gapsSome of the specifications of the existing publications have been classified and compared in Table 1. The main goals of using SOPs in ADNs are increasing the operational flexibility of the network with available configuration and load balancing within the connected network feeders. The ability of SOP to control the active and reactive power flow, fast response, and frequent actions can be considered a suitable switching device in the network topology change. Therefore, the coordination of SOPs with other switching devices like tie switches can enhance the adaptability to various operating conditions created due to uncertain renewable generations and other emergency conditions. This is an issue that has been addressed in [31] and [32], and both SOPs and tie switches have been used to perform switching in network reconfiguration. It is necessary to mention that in the reviewed studies, the installation location of the tie switches is predetermined, and only their switching status is evaluated simultaneously with SOPs. However, determining the optimal and simultaneous presence of different equipment to achieve optimal network operation is valuable during network expansion planning studies. This issue has not been investigated in [31] and [32] (the closest available studies to the current work) and other similar studies. Therefore, based on mentioned above and the literature review, the lack of coordinated planning in deciding to install SOPs and tie switches simultaneously to perform flexible manoeuvre operations and change the network configuration is the gap in previous studies.1TABLEThe relevant features of the reviewed studies and the proposed model in the present workConsidered itemsStudy areaLoad and generation modelReferencePlanningOperationModel typeSolving methodDetermination of site and size of SOPsCoordinated planning of the simultaneous presence of SOP with other equipment (other equipment)GenerationLoad[16]–√DeterministicNon‐linear optimization––a weekend day and a weekdaya weekend day and a weekday[17]–√DeterministicMOPSO + Taxi‐cab––––[18]√√DeterministicSOCP––Daily dataDaily data[19]√√DeterministicMISOCP––Yearly dataYearly data[23]–√DeterministicSOP/loop selection algorithm––24 hours24 hours[24]–√RobustSOCP––Daily operation with uncertainty–[25]√√DeterministicMPSO––––[31]√√DeterministicPowell's direct set method√(only site)–––[32]√√StochasticMISOCP√–Stochastic (scenario‐based)–[33]√√DeterministicBi‐level programming√√(DGs and capacitor banks)24 hours24 hours[6]√√DeterministicDiscrete‐continuous HSS√√(DGs)–Different loading levels[34]√√DeterministicHOMER PRO and NEPALN packages√√(PV, WT, Diesel generator)––Present work√√StochasticMISOCP√√(Tie switch)Stochastic(scenario‐based)Different annual loading levelsContributionsThis paper proposes a stochastic scenario‐based model to evaluate the cost‐effectiveness of the simultaneous presence of SOPs and tie switches considering network reconfiguration. The high investment cost of SOPs compared to the tie switches is a determinant factor in distribution network expansion or reinforcement planning. However, considering the cost of network energy losses in a distribution system equipped simultaneously with SOPs and tie switches, reduces the total network expenses to an acceptable level effectively despite the high investment cost of SOPs. This study has been carried out annually, considering the annual three‐level loading profile and different wind power generation scenarios. The problem has been solved for the IEEE 33‐bus and 69‐bus systems to evaluate the efficiency of the proposed model. The major contributions of this paper are summarized as follows:Integrating the optimal allocation problem of new tie switches with the planning problem of siting and sizing of SOPs is the main novel aspect of this paper. Therefore, this paper presents a simultaneous planning model of the presence of SOPs and tie switches during decision‐making for the coordinated investment of this equipment to perform network reconfiguration.The proposed optimization problem becomes more complex in terms of model size and computational time due to the addition of modelling the allocation of new tie switches. So, by using the SOCP approach, the mixed integer non‐linear programming (MINLP) model is converted into mixed‐integer second‐order cone programming (MISOCP) model to moderate the computational time and realize the optimal solution. The conversion procedure is implemented through linearization and conic relaxation to realize convex relaxation.Organization of the paperThe organization of the rest of this paper is as follows: In Section 2, a method of scenario generation is presented by discretizing the renewable power distribution function. Section 3 aims to model the optimal and simultaneous presence of SOPs and tie switches, and then the SOCP approach is applied to the model. In Section 4 the IEEE 33‐bus and 69‐bus systems are adopted, and the results of solving the optimization model on the studied case are described. The conclusions of this study are shown in Section 5.OPERATION SCENARIO GENERATIONDistribution networks will face great uncertainties with the increasing penetration level of renewable energy resources. In order to better achieve the optimal and simultaneous sites and sizes of SOPs and the allocation of tie switches, it is necessary to model the uncertain impact of renewable generation especially for long term planning of electrical distribution networks.One of the approaches to manage the uncertainty of renewable power is to generate several scenarios representing the renewable output power with its probability of occurrence. Using “Point Estimation” method, the continuous probability distribution of renewable power is replaced by a discrete distribution [38, 39].Probabilistic description for wind generationOne general model for probabilistic wind speed description is the Weibull distribution. The Weibull probability density function (PDF) is given by [38]:1f(v|λ,k)=kλvλk−1e−vλk$$\begin{equation}f\;(v|\lambda ,\;k) = \left( {\frac{{\;k}}{\lambda }} \right)\;{\left( {\frac{v}{\lambda }} \right)^{k - 1}}{e^{ - {{\left( {\frac{v}{\lambda }} \right)}^k}}}\end{equation}$$where v is wind speed, k and λ are shape and scale parameters, respectively.The decision variables of the power flow problem are affected by different power injection values of DGs. The active power output of wind turbines depends on the wind speed, which is formulated in the form of linear approximation as shown in (2) [40]:2P=0ifv≤vwc,inorv>vwc,outPwrvwr−vwc,inv−Pwr.vwc,invwr−vwc,inifvwc,in≤v≤vwrPwrifvwr≤v≤vwc,out$$\begin{equation}P{\rm{\;}} = {\rm{\;}}\left\{ \def\eqcellsep{&}\begin{array}{ll} 0 &if\;v \le v_w^{c,{\rm{\;}}in}\;or\;v > v_w^{c,{\rm{\;}}out}\\[8pt] \frac{{P_w^r}}{{v_w^r - v_w^{c,{\rm{\;}}in}{\rm{\;}}}}v - \;\frac{{P_w^r{\rm{\;}}.{\rm{\;}}v_w^{c,{\rm{\;}}in}{\rm{\;}}}}{{v_w^r - v_w^{c,{\rm{\;}}in}{\rm{\;}}}} &if\;v_w^{c,{\rm{\;}}in} \le v \le v_w^r\\[12pt] P_w^r &if\;v_w^r \le v \le v_w^{c,{\rm{\;}}out} \end{array} \right.\end{equation}$$where P is the injected power. Pwr$P_w^r\;$is the rated power of a wind turbine.vwc,in$\;v_w^{c,\;in}$, vwc,out$v_w^{c,\;out}$and vwr$v_w^r\;$denote the cut‐in, cut‐out, and rated wind speed, respectively.Discretization scheme of power distribution functionThe point estimation method is a technique to approximate the output variables of the problem affected by the uncertainty through computing the moments of a random variable. Assuming that the continuous distribution of the input variables is available and using the point estimation method, the discrete distribution can be substituted. The renewable generation scenarios can be built by performing the discretization scheme of the wind power distribution function and obtaining the discretized points [38, 39, 41].Obviously, wind turbine may generate the values of zero power or rated power according to (2) that the associated probabilities of them are calculated as follows [42]:3P1=ProbP=0=Probv≤vwc,in+Probv>vwc,our$$\begin{equation} {{P}_{1}}=Prob\left\{ P=0 \right\}=Prob\left( v\le v_{w}^{c,in} \right)+Prob\left( v>v_{w}^{c,our} \right)\end{equation}$$4=1−e−vwc,inλk+e−vwc,outλkP5=ProbP=Pwr=e−vwrλk−e−vwc,outλk$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} &=& 1 - {e^{\left( { - {{\left( {\frac{{v_w^{c,\;in}}}{\lambda }} \right)}^k}} \right)}} + \;{e^{\left( { - {{\left( {\frac{{v_w^{c,\;out}}}{\lambda }} \right)}^k}} \right)}}\\ {P_5} &=& Prob\left\{ {P = P_w^r} \right\} = {e^{\left( { - {{\left( {\frac{{v_w^r}}{\lambda }} \right)}^k}} \right)}} - \;{e^{\left( { - {{\left( {\frac{{v_w^{c,\;out}}}{\lambda }} \right)}^k}} \right)}} \end{array} \end{equation}$$Then, to determine the remaining three points of the wind power approximation, we must calculate the second, third and fourth central moments of the power distribution as follows [38, 39, 41]:For vwc,in≤v≤vwr$v_w^{c,\;in} \le v \le v_w^r$, redefine PDF of P:5f∼PP|λ,k=1βf(P−αβ|λ,k)1−P1−P5→∫0Pwrf∼PP|λ,kdP=1α=−Pwr.vwc,invwr−vwc,inβ=Pwrvwr−vwc,in$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {{\tilde f}_P}\;\left( {P{\rm{|}}\lambda ,k} \right) &=& \frac{{\frac{1}{\beta }\;f(\frac{{P - \;\alpha }}{\beta }|\lambda ,\;k)}}{{1 - \;{P_1} - \;{P_5}}}\; \to \;\displaystyle\mathop \int \limits_0^{P_w^r} {{\tilde f}_P}\left( {P{\rm{|}}\lambda ,k} \right)dP\; = \;1\;\\ \alpha &=& - \frac{{P_w^r\;.\;v_w^{c,\;in}\;}}{{v_w^r - v_w^{c,\;in}}}\;\beta \; = \;\frac{{P_w^r}}{{v_w^r - v_w^{c,\;in}\;}} \end{array} \end{equation}$$Define:6μ∼P=∫0PwrPf∼P(P|λ,k)dP$$\begin{equation}{\tilde \mu _P} = \;\mathop \int \limits_0^{P_w^r} P{\tilde f_P}(P|\lambda ,k)dP\end{equation}$$7σ∼P2=∫0PwrP−μ∼P2f∼P(P|λ,k)dP$$\begin{equation}{\rm{\;}}\tilde \sigma _P^2 = \mathop \int \limits_0^{P_w^r} {\left( {P - {{\tilde \mu }_P}} \right)^2}\;{\tilde f_P}(P|\lambda ,k)dP\end{equation}$$8λj=∫0PwrP−μ∼Pσ∼Pjf∼P(P|λ,k)dP$$\begin{equation}{\rm{\;}}{\lambda _j} = \;\mathop \int \limits_0^{P_w^r} {\left( {\frac{{P - {{\tilde \mu }_P}}}{{{{\tilde \sigma }_P}}}} \right)^j}{\tilde f_P}(P|\lambda ,k)dP\;\end{equation}$$μ∼P,σ¯P${\tilde \mu _P},\;{\bar \sigma _P}$ and λj${\lambda _j}$ are the values of mean, standard deviation, and jth central moment of P respectively.The moment equations are given by:9∑i=24pizij=λjforj=1,2,3,4$$\begin{equation}\mathop \sum \limits_{i\; = \;2}^4 {p_i}\;\;z_i^j = \;{\lambda _j}\;for\;j\; = \;1,\;2,\;3,\;4\;\end{equation}$$where pi${p_{i\;}}$is the probability weight corresponding to location zi${z_i}$.Solving for Equation (9), we can obtain:10z2=λ32−λ4−3λ324z3=0z4=λ32+λ4−3λ324$$\begin{equation}\left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\;{z_2} = \frac{{{\lambda _3}}}{2} - \sqrt {{\lambda _4} - \frac{{3\lambda _3^2}}{4}} \;}\\[6pt] {\;{z_3} = 0\;}\\[6pt] {\;{z_4} = \frac{{{\lambda _3}}}{2} + \sqrt {{\lambda _4} - \frac{{3\lambda _3^2}}{4}\;} } \end{array} \right.\end{equation}$$11p2=−1z2z4−z2p3=1−p2−p4p4=1z4z4−z2$$\begin{equation}\left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\;{p_2} = \frac{{ - 1}}{{{z_2}\;\left( {{z_4} - {z_2}} \right)}}\;}\\[6pt] {\;{p_3} = 1 - {p_2} - {p_4}\;}\\[6pt] {\;{p_4} = \frac{1}{{{z_4}\;\left( {{z_4} - {z_2}} \right)}}\;} \end{array} \right.\end{equation}$$It should be noted that one of the three standard values zi${z_i}\;$is set to zero ( z3=0${z_3} = \;0$).Each of the remaining three points of discrete distribution is approximated as a pair composed of a location zi${z_i}\;$and a probability weight pi,${p_{i\;}},\;$as shown in Equations (10) and (11). Then the estimation of points of power output Pi${P_i}\;$and associated probabilities Probi$Pro{b_i}\;$can be obtained as:12zi=Pi−μ∼Pσ∼PP2=μ∼P+σ∼Pz2andProb2=p21−P1−P5P3=μ∼PandProb3=p31−P1−P5P4=μ∼P+σ∼Pz4andProb4=p41−P1−P5$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {z_i} &=& \;\frac{{{P_i} - {{\tilde \mu }_P}\;}}{{{{\tilde \sigma }_P}}}\\[4pt] {\rm{\;}}{P_2} &=& {{\tilde \mu }_P} + {{\tilde \sigma }_P}\;{z_{2\;}}\;and\;\;Prob{\;_2} = {p_2}\;\left( {1 - {P_1} - {P_5}} \right)\\[4pt] {P_3} &=& {{\tilde \mu }_P}\;and\;\;Prob{\;_3} = {p_3}\;\left( {1 - {P_1} - {P_5}} \right)\\[4pt] {\rm{\;}}{P_4} &=& {{\tilde \mu }_P} + {{\tilde \sigma }_P}{z_4}\;and\;\;Prob{\;_4} = {p_4}\;\left( {1 - \;{P_1} - {P_5}} \right) \end{array} \end{equation}$$The parameters of Weibull distribution and wind turbine operating specifications are summarized in Table 2. The injected power distribution calculated through five‐point discrete distribution is given in Table 3 and drawn in Figure 1.2TABLEParameters of Weibull distribution and specifications of wind turbinekλvwc,in(m/s)$v_w^{c,\;in}( {{{{\rm{m}}} / {{\rm{s}}}}} )$vwc,out(m/s)$v_w^{c,\;out}( {{{{\rm{m}}} / {{\rm{s}}}}} )$vwr(m/s)$v_w^r( {{{{\rm{m}}} / {{\rm{s}}}}} )$210325153TABLEApproximate wind power distributionPoint00.10730.45190.85031Probability0.0880.21700.40380.18770.10351FIGUREApproximate wind power distributionPROBLEM FORMULATIONBy enabling the flexible connection between feeders through the SOPs, these devices can adjust different values of operational set points by controlling the active power flowing among the adjacent branches in real‐time and injecting or absorbing the reactive power at their terminals. This paper presents an optimization model to obtain the best simultaneous expansion plan of SOPs and tie switches, including determining the optimal installation site and size of SOP and the best place for installing the tie switch. The network equips with the B2B VSCs‐based topology. Also, the optimal operational strategies of the active power transmission and reactive power support of SOPs are obtained to realize the optimal regulation of the network. After determining the best place for installing the candidate tie switches, optimal switching is needed to achieve optimal network operation.Optimal model for planning the simultaneous presence of SOPs and tie switchesThe distribution network expansion planning (DNEP) problem is one of the topics of interest for decision‐makers in power distribution utilities. The DNEP is a process in which the aim is to decide on reinforcement of existing system elements or install new ones to reliably meet the loads' growth according to the technical and operational limits. One of the main objectives of the DNEP problem is to minimize the annual expense of the distribution system [32]. The objective function of the optimization problem consists of the investment cost of SOPs, new tie switches, and tie lines creation cost, the operation cost of SOPs and the operation and maintenance cost of tie switches and tie lines, and the cost of energy losses. This model is proposed as Equations (13)–(24).Objective function13MinCost=CostINV+CostOPE+CostLOSS$$\begin{equation}Min{\rm{\;}}Cost{\rm{\;}} = {\rm{\;}}Cos{t^{INV}} + {\rm{\;}}Cos{t^{OPE}} + {\rm{\;}}Cos{t^{LOSS}}\end{equation}$$14CostINV=δSOPCostINV−SOP+δTieSwitchCostINV−TieSwitch+δTieLineCostINV−TieLine$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} \;Cos{t^{INV}} &=& {\delta ^{SOP}}\;Cos{t^{INV - SOP}}\\[4pt] &&+ \;{\delta ^{TieSwitch}}Cos{t^{INV - TieSwitch}}\\[4pt] &&+ \;{\delta ^{TieLine}}Cos{t^{INV - TieLine}} \end{array} \end{equation}$$15CostOPE=CostOPE−SOP+CostOPE−TieSwitch+CostOPE−TieLine$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} Cos{t^{OPE}} &=& Cos{t^{OPE - SOP}} + \;Cos{t^{OPE - TieSwitch}}\\[4pt] &&+ \;Cos{t^{OPE - TieLine}} \end{array} \end{equation}$$16CostINV−SOP=costSOP∑ij∈ΩSOPSijSOP$$\begin{equation}{\rm{\;}}Cos{t^{INV - SOP}} = {\rm{\;}}cos{t^{SOP}}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_{SOP}}} {\rm{\;}}S_{ij}^{SOP}\end{equation}$$17CostINV−TieSwitch=costTieSwitch∑ij∈Ωnc×(αij,1)∨(αij,2)∨…∨(αij,NL)$$\begin{align} Cos{{t}^{INV-TieSwitch}} &= cos{{t}^{TieSwitch}}\underset{\left( ij \right)\in ~{{\text{ }\!\!\Omega\!\!\text{ }}_{nc}}}{\mathop \sum }\nonumber\\ &\times\,\begin{matrix} \left( ({{\alpha }_{ij,~1~}})\vee ({{\alpha }_{ij,~2~}})\vee \ldots \vee ({{\alpha }_{ij,~{{N}_{L}}~}}) \right) \end{matrix} \end{align}$$18CostINV−TieLine=costTieLine∑ij∈Ωnc×lengthij(αij,1)∨(αij,2)∨…∨αij,NL$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} Cos{t^{INV - TieLine}} = cos{t^{TieLine}}\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_{nc}}}\\[8pt] \times lengt{h_{ij\;}} \left( ({\alpha _{ij,\;1\;}}) \vee ({\alpha _{ij,\;2\;}}) \vee \ldots \;\left( \vee \left({\alpha _{ij,\;{N_L}\;}} \right) \right)\right. \end{array} \end{equation}$$19δ…=d1+dn…(1+dn…)−1$$\begin{equation} {\delta ^ {\ldots} } = {\rm{\;}}\frac{{d{\rm{\;}}{{\left( {1 + d} \right)}^{{n_ {\ldots} }}}}}{{({{\left( {1 + d} \right)}^{{n_ {\ldots} }}}) - 1}}\end{equation}$$20CostOPE−SOP=ηSOP∑ij∈ΩSOPSijSOP$$\begin{equation}{\rm{\;}}Cos{t^{OPE - SOP}} = {\rm{\;}}{\eta ^{SOP}}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_{SOP}}} {\rm{\;}}S_{ij}^{SOP}\end{equation}$$21CostOPE−TieSwitch=ηTieSwitchCostINV−TieSwitch$$\begin{equation}{\rm{\;}}Cos{t^{OPE - TieSwitch}} = {\rm{\;}}{\eta ^{TieSwitch}}{\rm{\;}}Cos{t^{INV - TieSwitch}}\end{equation}$$22CostOPE−TieLine=ηTieLineCostINV−TieLine$$\begin{equation}{\rm{\;}}Cos{t^{OPE - TieLine}} = {\rm{\;}}{\eta ^{TieLine}}{\rm{\;}}Cos{t^{INV - TieLine}}\end{equation}$$23CostLOSS=costElectricityELOSS$$\begin{equation}{\rm{\;}}Cos{t^{LOSS}} = {\rm{\;}}cos{t^{Electricity}}{\rm{\;}}{E^{LOSS}}\end{equation}$$24ELOSS=∑S=1NS∑LL=1NLDRLL∑ij∈ΩbrijIs,ij,LL2ρ(s)$$\begin{equation}{\rm{\;}}{E^{LOSS}} = \mathop \sum \limits_{S =1}^{{N_S}} \left(\mathop \sum \limits_{LL =1}^{{N_L}} D{R_{LL}}\left( \mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {r_{ij}}{\rm{\;}}I_{s,{\rm{\;}}ij,{\rm{\;}}LL}^2\right) \right)\rho (s)\end{equation}$$Equations (14), (16)–(18) represent the capital expenditures for equipment, including SOPs, tie switches, and tie lines. Equations (15), (20)–(22) give the operation costs of mentioned equipment. Equations (17) and (18) represent a logical summation. In this way, if at least in one of the loading levels, αij,LL=1${\alpha _{ij,\;LL\;}} = \;1$, it means that a new tie switch must be installed in that location. If αij,LL=0${\alpha _{ij,\;LL\;}} = \;0$ at all loading levels, then no new tie switches will be installed in location ij$ij$.Equation (19) is the factor to convert the total investment costs of SOPs and other equipment annually, and n is the economical service life of the component. The annual operation cost of SOP is given in (20), defined as a factor of the SOP's installation capacity. Equation (23) expresses the annual energy loss cost of the distribution system. Equation (24) relates to the expectation of energy losses (ELOSS)$({E^{LOSS}})$ in the studied network.System power flow equationsIn this study, the power flow of distribution networks is modelled using the branch flow deals with currents and powers flowing between connected feeders [43]. The phase angles of voltage and currents must be eliminated to obtain an angle‐relaxed formulation of the power flow problem, as follows [43, 44]:25Ps,i,LL=∑ij∈ΩbPs,ij,LLLine−∑ki∈ΩbPs,ki,LLLine−rki|Is,ki,LL|2+gi|Vi|2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {P_{s,i,LL}} &=& \displaystyle\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_b}} P_{s,\;ij,\;LL}^{Line}\\[16pt] && -\, \displaystyle\sum {_{\left( {ki} \right) \in \;{{{\Omega}}_b}}} \left(P_{s,ki,LL}^{Line} - {r_{ki}}|{I_{s,ki,LL}}{|^2}\right) + {g_i}|{V_i}{|^2} \end{array} \end{equation}$$26Qs,i,LL=∑ij∈ΩbQs,ij,LLLine−∑ki∈ΩbQs,ki,LLLine−xkiIs,ki,LL2+biVi2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {Q_{s,\;i,LL}} &=& \displaystyle\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_b}} Q_{s,\;ij,\;LL}^{Line}\\[16pt] && -\, \displaystyle\;\mathop \sum \limits_{\left( {ki} \right) \in \;{{{\Omega}}_b}} \left(Q_{s,ki,LL}^{Line} - {x_{ki}}{\left| {{I_{s,ki,LL}}} \right|^2}\right) + {b_i}{\left| {{V_i}} \right|^2} \end{array} \end{equation}$$27Vs,j,LL2=Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,\;j,\;LL}^2 &=& V_{s,\;i,\;LL}^2\; - 2\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right)\\[10pt] && + \;\left( {{r_{ij}}^2 + {x_{ij}}^2} \right)\;I_{s,\;ij,\;LL\;}^2\quad \quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$28Is,ij,LL2Vs,i,LL2=Ps,ij,LLLine2+Qs,ij,LLLine2∀ij∈Ωb$$\begin{equation}I_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^2{\rm{\;\;}}V_{s,{\rm{\;}}i,{\rm{\;}}LL}^2 = {\rm{\;}}{\left( {P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}} \right)^2} + {\rm{\;}}{\left( {Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}} \right)^2}{\rm{\;}}\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}{\rm{\;}}\end{equation}$$29Ps,i,LL=Ps,iDG+Ps,i,LLSOP−Pi,LLLOAD$$\begin{equation}{\rm{\;}}{P_{s,{\rm{\;}}i,LL}} = {\rm{\;}}P_{s,{\rm{\;}}i}^{DG} + {\rm{\;}}P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP} - {\rm{\;}}P_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}\end{equation}$$30Qs,i,LL=Qs,iDG+Qs,i,LLSOP−Qi,LLLOAD$$\begin{equation}{\rm{\;}}{Q_{s,{\rm{\;}}i,LL}} = Q_{s,{\rm{\;}}i}^{DG} + {\rm{\;}}Q_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP}{\rm{\;}} - {\rm{\;}}Q_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}\end{equation}$$Reconfiguration formulationThis paper presents the reconfiguration problem based on optimal power flow [44]. Because the purpose is to determine the optimal siting and sizing of SOPs and the allocation of tie switches, the mathematical model is implemented for each renewable generation scenario and loading level. Then the optimal planned capacity of SOP and its installation site are obtained along with the best place for installing the candidate tie switches. 31Vs,j,LL2≤M1−αij,LL+Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,j,LL}^2 &\le& M\left( {1 - {\alpha _{ij,LL}}} \right) + V_{s,i,LL}^2\\[6pt] && - 2\left({r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}\right) + \left({r_{ij}}^2 + {x_{ij}}^2\right)I_{s,ij,LL}^2\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$32Vs,j,LL2≥−M1−αij,LL+Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,j,LL}^2 &\ge& - M\left( {1 - {\alpha _{ij,LL}}} \right) + V_{s,i,LL}^2\\[6pt] && -\, 2\left({r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}\right) + \left({r_{ij}}^2 + {x_{ij}}^2\right)I_{s,ij,LL}^2\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$33∑ij∈Ωbαij,LL=NN−1$$\begin{equation}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {\alpha _{ij,{\rm{\;}}LL}} = {N_N} - 1\end{equation}$$34βij,LL+βji,LL=αij,LL∀ij∈Ωb$$\begin{equation}{\beta _{ij,{\rm{\;}}LL}} + {\beta _{ji,{\rm{\;}}LL}} = {\alpha _{ij,{\rm{\;}}LL{\rm{\;}}}}\quad\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}\end{equation}$$35∑j∈Ωbusiβij,LL=1i∈Ωbus,i≠sb$$\begin{equation}\mathop \sum \limits_{j{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}}\left( i \right)} {\beta _{ij,{\rm{\;}}LL}} = 1\quad i{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}},{\rm{\;}}i{\rm{\;}} \ne sb\end{equation}$$36βsbj,LL=0j∈Ωbussb$$\begin{equation}{\beta _{sbj,{\rm{\;}}LL}} = 0{\rm{\;}}j{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}}\left( {sb} \right)\end{equation}$$37Vsb2=1$$\begin{equation}V_{sb}^2 = {\rm{\;}}1\end{equation}$$Inequality constraints (31) and (32) satisfy Ohm's law over the line between bus i$i\;$and j represented in Equation (27), considering the possibility change the network topology.The binary variable αij,LL${\alpha _{ij,\;LL\;}}$represents the switch status of the line between buses i and j at loading level LL.$LL.$ When the line is closed (αij,LL=1${\alpha _{ij,\;LL}} = 1$), the upper and lower boundaries for Vs,j,LL2$V_{s,\;j,\;LL}^2\;$will be the same values, and Equation (27) will be realized. When the line is open (αij,LL=0${\alpha _{ij,\;LL}} = 0$), the upper and lower boundaries on Vs,j,LL2$V_{s,\;j,\;LL}^2\;$become the large and small values, respectively. So, constraints (31) and (32) are relaxed and always valid.The reconfiguration problem has to be implemented in a way that the radiality and connectivity constraints of the distribution network are satisfied in the new configuration:Radiality constraint: means that all buses except the substation bus are supplied from one direction only, and so no loop is created in the new configuration.Connectivity constraint: Means that the network includes all buses, and so no bus is isolated.Equations (33)–(36) illustrate the above points. Equation (33) expresses that the number of closed switches is exactly one less than the total number of network nodes. According to Equation (34), each switch status of the line is determined by the sum of the two line directional variables βij,LL${\beta _{ij,\;LL}}\;$and βji,LL${\beta _{ji,\;LL}}$. If the power flow direction is from bus j to bus i (βij,LL=1${\beta _{ij,\;LL}} = 1$) or vice versa (βji,LL=1${\beta _{ji,\;LL}} = 1$), the switch of the line between bus i and j is connected (αij,LL=1${\alpha _{ij,\;LL}} = 1$). Obviously, if the directional variables βij,LL${\beta _{ij,\;LL}}\;$and βji,LL${\beta _{ji,\;LL}}\;$have zero values, the line between two buses is open. Equation (35) means that all buses (children buses) except substation bus are supplied only through one bus called the parent bus. Equation (36) means that the substation bus is not the parent bus at all. According to Equation (37), the voltage of the main substation bus equals 1 per unit.System operational constraints38V−2≤Vs,i,LL2≤V¯2$$\begin{equation}{{\underset{\scriptscriptstyle-}{V}}^{2}}\le V_{s,~i,~LL}^{2}\le {{\bar{V}}^{2}}\end{equation}$$39Is,ij,LL2≤I¯2∀ij∈Ωb$$\begin{equation}\;I_{s,\;ij,\;LL\;}^2 \le {\bar I^2}\;\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$Modelling of soft open pointsActive power transmission constraint of SOPsActive and reactive power transmitted through the SOP with its converters is the decision variables in the model. The sum of the active powers provided by the converters and their internal power losses must equal zero for each of the renewable generation scenarios and different loading conditions [24]:40Ps,i,LLSOP+Ps,j,LLSOP+Ps,i,LLSOP,LOSS+Ps,j,LLSOP,LOSS=0$$\begin{equation}P_{s,\;i,\;LL}^{SOP} + \;P_{s,\;j,\;LL}^{SOP} + \;P_{s,\;i,\;LL}^{SOP,\;LOSS} + \;\;P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;0\end{equation}$$41Ps,i,LLSOP,LOSS=AiSOP(Ps,i,LLSOP)2+(Qs,i,LLSOP)2$$\begin{equation}P_{s,\;i,\;LL}^{SOP,\;LOSS} = \;A_i^{SOP}\sqrt {{{(P_{s,\;i,\;LL}^{SOP})}^2} + {{(Q_{s,\;i,\;LL}^{SOP})}^2}\;} \end{equation}$$42Ps,j,LLSOP,LOSS=AjSOP(Ps,j,LLSOP)2+(Qs,j,LLSOP)2$$\begin{equation}P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;A_j^{SOP}\sqrt {{{(P_{s,\;j,\;LL}^{SOP})}^2} + {{(Q_{s,\;j,\;LL}^{SOP})}^2}\;} \end{equation}$$The SOPs have a high operation efficiency, but a large‐scale transmission of active power through VSCs in the SOP unavoidably produces losses. The Equations (41)–(42) are replaced by the following equations to reduce the computational burden of the optimization problem [32]:43Ps,i,LLSOP,LOSS=AiSOPPs,i,LLSOP$$\begin{equation}P_{s,\;i,\;LL}^{SOP,\;LOSS} = \;A_i^{SOP}\;\left| {P_{s,\;i,\;LL}^{SOP}} \right|\end{equation}$$44Ps,j,LLSOP,LOSS=AjSOPPs,j,LLSOP$$\begin{equation}P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;A_j^{SOP}\;\left| {P_{s,\;j,\;LL}^{SOP}} \right|\;\end{equation}$$SOP capacity constraints45(Ps,i,LLSOP)2+(Qs,i,LLSOP)2≤SijSOP∀ij∈Ωb$$\begin{equation}\sqrt {{{(P_{s,\;i,\;LL}^{SOP})}^2} + {{(Q_{s,\;i,\;LL}^{SOP})}^2}\;} \le \;S_{ij}^{SOP}{\rm{\;}}\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$46(Ps,j,LLSOP)2+(Qs,j,LLSOP)2≤SijSOP∀ij∈Ωb$$\begin{equation}\sqrt {{{(P_{s,\;j,\;LL}^{SOP})}^2} + {{(Q_{s,\;j,\;LL}^{SOP})}^2}\;} \le \;S_{ij}^{SOP}\;\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$The reactive power injected or absorbed by the converter terminals is independent of each other because of the DC bus and only meets the converter's capacity constraint as described using Equations (45) and (46) [24]. The location and capacity of an SOP are formulated as follows [32]:47SijSOP=mijSmodule(1−αij,LL)∀ij∈ΩSOP$$\begin{equation}S_{ij}^{SOP} = \;{m_{ij}}\;{S^{module}}\;(1 - {\alpha _{ij,\;LL}}){\rm{\;}}\forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}}\end{equation}$$Conversion to an MISOCP modelThe optimal sites and sizes of SOPs with the allocation of tie switches is a non‐convex non‐linear programming (NLP) model.Considering the stochastic operation scenarios of renewable generation and different loading conditions, the optimization model becomes complicated, and so it is difficult to find the optimal global solution. As a consequence, the original non‐convex NLP model is reformulated as a second‐order cone programming (SOCP) model that tractably can be solved. The SOCP is a mathematically convex optimization approach that can solve minimum linear objective functions on a feasible region composed of linear equality constraints and convex cone constraints [32]. Therefore, the mentioned optimization model with non‐linear function has to be changed in linear form and then the second‐order cone programming approach must be applied. The conversion procedure consists of the following steps:Substitute new variables vs,i,LL${v_{s,\;i,\;LL}}\;$and ls,ij,LL${l_{s,\;ij,\;LL}}\;$for the square of voltage and current, that is, Vs,i,LL2$V_{s,\;i,\;LL}^2\;$andIs,ij,LL2$\;I_{s,\;ij,\;LL\;}^2\;$in constraints (24)–(26), (28), (31), (32), (38) and (39) to linearize them [24].48ELOSS=∑S=1NS(∑LL=1NLDRLL∑ij∈Ωbrijls,ij,LL)ρs$$\begin{equation}{E^{LOSS}} = {\rm{\;}}\mathop \sum \limits_{S{\rm{\;}} = {\rm{\;}}1}^{{N_S}} (\mathop \sum \limits_{LL{\rm{\;}} = {\rm{\;}}1}^{{N_L}} D{R_{LL}}\left( {\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {r_{ij}}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}})} \right)\rho \left( s \right)\end{equation}$$49Ps,i,LL=∑ij∈ΩbPs,ij,LLLine−∑ki∈Ωb(Ps,ki,LLLine−rkils,ki,LL)+givs,i,LL$$\begin{eqnarray}{P_{s,{\rm{\;}}i,LL}} &=& \mathop \sum \limits_{\left( {ij} \right) \in {{{\Omega}}_b}} P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line} - \mathop \sum \limits_{\left( {ki} \right) \in {\rm{\;}}{{{\Omega}}_b}} (P_{s,{\rm{\;}}ki,{\rm{\;}}LL}^{Line} - {r_{ki}}{l_{s,{\rm{\;}}ki,{\rm{\;}}LL}})\nonumber\\ && +\, {g_i}{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$50Qs,i,LL=∑ij∈ΩbQs,ij,LLLine−∑ki∈Ωb(Qs,ki,LLLine−xkils,ki,LL)+bivs,i,LL$$\begin{eqnarray}{Q_{s,{\rm{\;}}i,LL}} &=& \mathop \sum \limits_{\left( {ij} \right) \in {{{\Omega}}_b}} Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line} - \mathop \sum \limits_{\left( {ki} \right) \in {{{\Omega}}_b}} (Q_{s,{\rm{\;}}ki,{\rm{\;}}LL}^{Line} - {x_{ki}}{l_{s,{\rm{\;}}ki,{\rm{\;}}LL}})\nonumber\\ && +\, {b_i}{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$51vs,j,LL≤M1−αij,LL+vs,i,LL−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2ls,ij,LL∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {v_{s,j,LL}} &\le& M\left( {1 - \;{\alpha _{ij,LL\;}}} \right) + {v_{s,i,LL}}\\[6pt] &&- 2\;\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right) + \left( {{r_{ij}}^2 + {x_{ij}}^2} \right){l_{s,ij,LL}}\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$52vs,j,LL≥−M1−αij,LL+vs,i,LL−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2ls,ij,LL∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {v_{s,j,LL}} &\ge& - M\left( {1 - \;{\alpha _{ij,LL\;}}} \right) + {v_{s,i,LL}}\\[6pt] && - 2\;\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right) + \left( {{r_{ij}}^2 + {x_{ij}}^2} \right){l_{s,ij,LL}}\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$53V−2≤vs,i,LL≤V¯2$$\begin{equation}{{\underset{\scriptscriptstyle-}{V}}^{2}}\le {{v}_{s,\text{ }\!\!~\!\!\text{ }i,\text{ }\!\!~\!\!\text{ }LL}}\le {{\bar{V}}^{2}}\end{equation}$$54ls,ij,LL≤I¯2∀ij∈Ωb$$\begin{equation}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} \le {\bar I^2}{\rm{\;}}\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}\end{equation}$$For the absolute terms |Ps,i,LLSOP|$| {P_{s,\;i,\;LL}^{SOP}} |$ and |Ps,j,LLSOP|$| {P_{s,\;j,\;LL}^{SOP}} |$ in Equations (43) and (44), auxiliary variables Ms,i,LLSOP$M_{s,\;i,\;LL}^{SOP}\;$and Ms,j,LLSOP$M_{s,\;j,\;LL}^{SOP}\;$are introduced to represent and linearize them as follows [32]:55Ms,i,LLSOP≥0,Ms,j,LLSOP≥0Ms,i,LLSOP≥Ps,i,LLSOP,Ms,i,LLSOP≥−Ps,i,LLSOPMs,j,LLSOP≥PS,j,LLSOP,Ms,j,LLSOP≥−Ps,j,LLSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} M_{s,i,LL}^{SOP} \ge 0,\quad M_{s,j,LL}^{SOP} \ge 0\\[6pt] M_{s,i,LL}^{SOP} \ge \;P_{s,i,LL}^{SOP},\;\quad M_{s,i,LL}^{SOP} \ge \; - P_{s,i,LL}^{SOP}\\[6pt] \! M_{s,j,LL}^{SOP} \ge \;P_{S,j,LL}^{SOP},\;\quad M_{s,j,LL}^{SOP} \ge \; - P_{s,j,LL}^{SOP} \end{array} \end{equation}$$In Equations (43) and (44), the terms |Ps,i,LLSOP|$| {P_{s,\;i,\;LL}^{SOP}} |$ and |Ps,j,LLSOP|$| {P_{s,\;j,\;LL}^{SOP}} |\;$are replaced by Ms,i,LLSOP$M_{s,\;i,\;LL}^{SOP}{\rm{\;}}$and Ms,j,LLSOP,$M_{s,\;j,\;LL}^{SOP},$ respectively, and then (55) is added to the set of constraints.Relax the non‐linear equality constraint (28) into the quadratic cone form (56) after linearizing [43]:562Ps,ij,LLLine2Qs,ij,LLLinels,ij,LL−vs,i,LLT2≤ls,ij,LL+vs,i,LL$$\begin{eqnarray}&&{\left\| {{{\left[ {2P_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^{Line}{\rm{\;}}2Q_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^{Line}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} - {\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}} \right]}^T}} \right\|_2}\nonumber\\ &&\quad \le {\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} + {\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$Transform the capacity constraints of an SOP in (45) and (46) into the following rotated quadratic cone constraints [24]:57Ps,i,LLSOP2+Qs,i,LLSOP2≤2SijSOP2SijSOP2∀ij∈ΩSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\left( {P_{s,\;i,\;LL}^{SOP}} \right)^2} + {\left( {Q_{s,\;i,\;LL}^{SOP}} \right)^2}\; \le 2\;\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\\[12pt] \forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}} \end{array} \end{equation}$$58Ps,j,LLSOP2+Qs,j,LLSOP2≤2SijSOP2SijSOP2∀ij∈ΩSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\left( {P_{s,\;j,\;LL}^{SOP}} \right)^2} + {\left( {Q_{s,\;j,\;LL}^{SOP}} \right)^2}\; \le 2\;\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\\[12pt] \forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}} \end{array} \end{equation}$$By relaxing the equality constraint (28) to the inequality constraint (56), the relaxation deviation (59) is formulated in the form of the infinite norm to evaluate the accuracy of the convex relaxation. If the gap value is small enough, the convex cone relaxation can be regarded as accurate for model conversion [28].59gap=ls,ij,LL−Ps,ij,LLLine2+Qs,ij,LLLine2Vs,i,LL∞$$\begin{equation}gap\; = \;{\left\| {{l_{s,\;ij,\;LL}} - \;\frac{{{{\left( {P_{s,\;ij,\;LL\;}^{Line}} \right)}^2} + \;{{\left( {Q_{s,\;ij,\;LL\;}^{Line}} \right)}^2}}}{{{V_{s,\;i,\;LL}}}}} \right\|_\infty }\end{equation}$$CASE STUDIES AND ANALYSISIn this section, the MISOCP model for simultaneous sites and sizes of SOPs and the allocation of tie switches is applied to a distribution network. The IEEE 33‐bus test system is adopted to investigate the effectiveness of the proposed model. The obtained optimization model is implemented in the YALMIP optimization toolbox (version 20210331) using MATLAB R2018a and solved by GUROBI (version 9.5.0) optimizer interface. Also, CPLEX or MOSEK software packages can be used to solve the convex model efficiently.Test networkIn this study, the results are obtained for the IEEE 33‐bus and 69‐bus systems under different loading conditions. The modified IEEE 33‐bus and 69‐bus system are presented in Figures A1 and A2. The rated voltage level is 12.66 kV. The IEEE 33‐bus system consists of 32 sectionalized lines and 5 tie‐lines. The IEEE 69‐bus system has 68 sectionalized lines and 5 tie‐lines. In this networks, 6 candidate locations for installing the new tie switches are suggested and shown in Table B1. The resistance and reactance of all the candidate tie lines for two networks are the same and equal 0.5 ohms. The location and capacity of wind generation are given in Table B2. All the DGs have a unity power factor. In this study, loads are classified as residential, commercial, and industrial customers, which are labelled in Figures A1 and A2. Different loading conditions are considered according to Table B3. The detailed parameters of the test system are shown in [45, 46]. The load data given in [45, 46] is considered as medium loading level and the other two loading levels are constructed according to Table B3. The other values of test distribution network parameters are shown in Table B4.Results and analysisIn this section, the optimal presence planning of SOPs and tie switches are determined on the standard IEEE 33‐node and 69‐node systems. The following schemes are evaluated in this study, and network reconfiguration, renewable generation scenarios given in Table 3, and different loading levels are considered in all schemes:-Scheme I: It is the result of network study without installing SOPs and tie switches.-Scheme II: It is the result of network study considering the optimal allocation of tie switches.-Scheme III: It is the result of network study considering the optimal siting and sizing of SOPs.-Scheme IV: The optimal sites and sizes of SOPs and the optimal placement of tie switches are determined simultaneously.In schemes III and IV, all sectionalized and tie switches are considered the candidate locations in the 33‐bus (43 candidates) and 69‐bus (79 candidates) systems to install the SOPs. It is assumed the direction of power injection of SOP into the node is the positive direction.The optimization results of scheme II indicate that the candidate locations of T1 (4–25), T2 (11–18), and T6 (5–10) are selected to invest and install the switches in the 33‐bus test system. For the 69‐bus system, the candidate locations of T5 (36–50) and T6 (20–67) are selected to invest and install the switches during scheme II. The best installation plan for the SOPs and the best topology obtained for the network after reconfiguration in scheme III are reported in Tables 4 and 5 for IEEE 33‐bus and 69‐bus systems, respectively. In scheme III, two SOPs are installed with the capacities of 300 and 100 kVA between nodes (31–32) and (25–29), respectively, for the IEEE 33‐bus system. For the 69‐bus network, the result of scheme III is to obtain two SOPs with a capacity of 400 kVA in locations (56–57) and (63–64). Also, the results of scheme IV are presented in Tables 6 and 7. In scheme IV, two candidate locations of T1 (4–25) and T5 (20–14) for installing new tie switches are selected in the 33‐bus system, and only one SOP device with a capacity of 200 kVA is decided to install between nodes (31–32).4TABLESite and size of SOPs and configuration of the 33‐bus network (scheme III)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(7–8) (8–9)(7–8) (11–12)(7–8) (10–11)(14–15) (31–32)(14–15) (31–32)(14–15) (31–32)(25–29)(25–29)(25–29)The site and size of SOPSite(31‐32)(25‐29)Size (KVA)3001005TABLESite and size of SOPs and configuration of the 69‐bus network (scheme III)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(12–13) (13–14)(56–57) (63–64)(11–43)(12–13) (56–57)(63–64) (11–43)(13–21)(10–11) (13–14)(20–21) (56–57)(63–64)The site and size of SOPSite(56–57)(63–64)Size (KVA)4004006TABLESite and size of SOP, the location for installing the tie switches and configuration of the 33‐bus network (scheme IV)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(10–11) (12–13) (24–25) (28–29) (31–32) (21–8) (9–15) (11–18) (11–15) (30–16) (5–10)(7‐8) (8‐9) (12‐13) (24–25) (28–29) (31–32) (9–15) (11–18) (11–15) (30–16) (5–10)(10–11) (12–13) (24–25) (28–29) (31–32) (21–8) (9–15) (11–18) (11–15) (30–16) (5–10)The location for installing new tie switches(4–25)The site and size of SOPSite(31–32)(20–14)Size (KVA)2007TABLESite and size of SOP, the location for installing the tie switches and configuration of the 69‐bus network (scheme IV)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(15–16) (41–42) (45–46) (49–50) (56–57) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)(10–11) (14–15) (17‐18) (49‐50(49–50) (55–56) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)(10–11) (17–18) (44–45) (49–50) (55–56) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)The location for installing new tie switches(23–41)The site and size of SOPSite(63–64)(36–50)Size (KVA)300The objective function terms are shown in Tables 8 and 9 for all schemes.8TABLEAnnual costs of Schemes I‐IV and total energy losses of the 33‐bus networkScheme I(Without SOP and tie switches)Scheme IIScheme IIIScheme IVCostINV−SOP($)$Cos{t^{INV - SOP}}\;( \$ )$––12,5816290CostINV−TieSwitch+CostINV−TieLine($)$Cos{t^{INV - TieSwitch}} + \;Cos{t^{INV - TieLine}}\;( \$ )$–14,775–9850CostOPE−SOP($)$Cos{t^{OPE - SOP}}\;( \$ )$––24701235CostOPE−TieSwitch+CostOPE−TieLine($)$Cos{t^{OPE - TieSwitch}} + \;Cos{t^{OPE - TieLine}}( \$ )$–1812–1208CostLOSS($)$Cos{t^{LOSS}}\;( \$ )$8493559,79761,92151,099ELOSS(kWh)${E^{LOSS}}\;( {{\rm{kWh}}} )$106169074,746677,401963,8730Cost($)$Cost\;( \$ )$8493576,38476,97369,6829TABLEAnnual costs of Schemes I–IV and total energy losses of the 69‐bus networkScheme I(Without SOP and tie switches)Scheme IIScheme IIIScheme IVCostINV−SOP($)$Cos{t^{INV - SOP}}\;( \$ )$––25,1629436CostINV−TieSwitch+CostINV−TieLine($)$Cos{t^{INV - TieSwitch}} + \;Cos{t^{INV - TieLine}}\;( \$ )$–9850–9850CostOPE−SOP($)$Cos{t^{OPE - SOP}}\;( \$ )$––49411853CostOPE−TieSwitch+CostOPE−TieLine($)$Cos{t^{OPE - TieSwitch}} + \;Cos{t^{OPE - TieLine}}( \$ )$–1208–1208CostLOSS($)$Cos{t^{LOSS}}\;( \$ )$78,02664,24441,08645,992ELOSS(kWh)${E^{LOSS}}\;( {{\rm{kWh}}} )$975,330803,050513,578574,900Cost($)$Cost\;( \$ )$78,02675,30271,18968,339The results of Table 8 show that scheme IV is more economical than the other schemes. The annual cost of network expansion in scheme IV is 15253 $ (18%) less than that in scheme I. The effect of installing new tie switches in scheme IV compared to scheme III is to reduce the different objective function values as well as total energy losses of the network. With the simultaneous and optimal presence of SOPs and tie switches, the annual cost of energy losses is decreased by 33,836 $ (a reduction of 40%). Also, the amount of the total network energy losses reduction is 42,2960 KWH and 40%. All these comparisons have been made between scheme IV and the case without SOP and tie switches. In this way, it is obvious the optimal and simultaneous use of SOPs and tie switches will improve the operation of the network economically. Also, the final network configuration for the medium loading level in schemes III and IV is illustrated in Figures 2 and 3, respectively. Notably, the final connections of the test networks after solving the problem are presented in the relevant tables, and Figures 2 and 3 are drawn for the 33‐bus system, only to show an example of the allocation of SOPs in schemes III and IV.2FIGURELocation of SOPs in scheme III (medium loading level)3FIGURELocation of SOPs in scheme IV (medium loading level)Figures 4 and 5 show the expected values for the voltage and current of network buses, respectively in schemes II, III, and IV in the 33‐bus network. As can be seen, the simultaneous presence of SOPs and tie switches generally improves the average network voltage, and by reducing the average current level of the network, it further reduces energy losses. According to the results of Table 8, the optimal installation of tie switches and changing the network topology only through them (scheme II) reduce the energy losses of the network more than the optimal presence of SOPs in the network (scheme III).4FIGUREExpected values of 33‐bus network voltage in schemes II–IV5FIGUREExpected values of 33‐bus network current in schemes II–IV. (a) Active power transmission of SOP installed between nodes (31)–(32). (b) Reactive power compensation of SOP installed between nodes (31)–(32). (c) Active power transmission of SOP installed between nodes (25)–(29). (d) Reactive power compensation of SOP installed between nodes (25)–(29)Figure 6a,c illustrates that the operation strategy of SOPs installed in scheme III is adjusted according to the operating conditions of the network, including the supply and demand level of the distribution system. The diagrams related to the operation strategy of SOPs are drawn only for scheme III as an instance.6FIGUREOperation strategy of SOPs in scheme III. (a) Maximum gap values in each renewable generation scenario of scheme III (33‐bus test system). (b) Maximum gap values in each renewable generation scenario of scheme IV (33‐bus test system)When the renewable generation is low, such as in scenarios 1 and 2, SOP transmits the active power from node 31 into node 32 and from node 25 into node 29. In another hand, as the output of the wind turbine increasessuch as in scenarios 4 and 5, by adjusting the transmission of the active power into node 31 and node 25 through the SOPs installed between nodes (31–32) and (25–29), the fluctuations of wind turbine power output are mitigated. Also, the higher the loading level of the network, the lower the absolute amount of active power transmission of SOPs to help reduce power losses of the network.Figure 6b,d shows the reactive power support of SOPs installed between nodes (31–32) and (25–29) within their own capacity constraints. The higher the loading level of network, the higher the amount of reactive power compensation of SOPs to help meet the required reactive power of loads.The gap values in each renewable generation scenario of scheme III and IV are shown in Figure 7a,b. The maximum gap values in scheme III and IV are some with a 10−6 level and another some with a 10−5 level which are small enough to be considered as accurate7FIGUREMaximum gap values in each renewable generation scenario of scheme III and IVCONCLUSIONThis paper presented an optimal model for planning the simultaneous presence of SOPs and tie switches in active distribution networks. The optimal site and capacity of the SOPs and installation sites of the tie switches were determined simultaneously with consideration of the renewable generation characteristics, different loading conditions, and network topology changes.After linearization and convexification, the original optimization problem is solved in the form of the MISOCP model to moderate the execution time of the program. Results showed the application of SOPs could improve the operational economy, reduce the annual energy losses of distribution networks, and improve the voltage profile. Another suitable extension way for enabling network reconfiguration is to coordinate SOPs with conventional switching devices such as tie switches, considering the relatively high investment cost of power electronics. Optimal installation of tie switches simultaneously with the optimal presence of SOPs led to an additional reduction in the total network expansion costs compared to the case in which only SOPs are used. Also, the results show that planning the presence of SOPs in the network alone will reduce energy losses by 27% during the year, while the optimal presence of SOPs with the tie switches reduces energy losses in the network by up to 40%. These percentages are reported for the 33‐bus network.Thus, although the simultaneous investment of SOPs and tie switches at first glance has increased the network costs, regarding reducing the cost of energy losses, the total network expansion costs are decreased.NOMENCLATURESetsΩb${\Omega _b}$Set of all branchesΩnc${\Omega _{nc}}$Set of all new candidate tie linesΩSOP${\Omega _{SOP}}$Set of all candidate locations of SOPsΩbus${\Omega _{bus}}$Set of busesΩbus(i)${\Omega _{bus}}( i )$Set of all adjacent buses of bus iIndicesi,j$i,{\rm{\;}}j$Indices of buses, from 1 to NN${N_N}$LL$LL$Indices of loading level, from 1 to NL${N_L}$sIndices of the operation scenarios of renewable generation from 1 to Ns${N_s}$ij$ij$Indices of branchesVariablesPs,ij,LLLine,Qs,ij,LLLine$P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line},{\rm{\;}}Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}$Active and reactive power flow of branch ij$ij$ in scenario s and loading level LL$LL$Is,ij,LL,ls,ij,LL${I_{s,{\rm{\;}}ij,{\rm{\;}}LL}},{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}}{\rm{\;}}$Current magnitude and its square of branch ij$ij$ in scenario s and loading level LL$LL$Vs,i,LL,vs,i,LL${V_{s,{\rm{\;}}i,{\rm{\;}}LL}},{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}$Voltage magnitude and its square at busi${\rm{\;}}i$ in scenario s and loading level LL$LL$Ps,i,LL,Qs,i,LL${P_{s,{\rm{\;}}i,LL}},{\rm{\;}}{Q_{s,{\rm{\;}}i,LL}}$Total active and reactive power injection at node i in scenario s and loading level LL$LL$Ps,iDG,Qs,iDG$P_{s,{\rm{\;}}i}^{DG},{\rm{\;}}Q_{s,{\rm{\;}}i}^{DG}$Active and reactive power generation of DG at node i in scenario sPs,i,LLSOP,Qs,i,LLSOP$P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP},{\rm{\;}}Q_{s,{\rm{\;}}i,{\rm{\;}}LL}^{SOP}$Active and reactive power injection by SOP at node i in scenario s and loading level LL$LL$Ps,i,LLSOP,LOSS$P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{SOP,{\rm{\;}}LOSS}$Active power losses of SOP at nodei${\rm{\;}}i$ in scenario s and loading level LL$LL$SijSOP$S_{ij}^{SOP}$Maximum planned capacity of the SOP in ijth$i{j^{th{\rm{\;}}}}$branchαij,LL${\alpha _{ij,{\rm{\;}}LL}}$Binary variable expressing the branch status takes value of 1 if the branch ij$ij$ is closed and 0 otherwiseβij,LL${\beta _{ij,{\rm{\;}}LL}}$Binary variable which is equal to 1 if bus j is the parent of bus i and to 0 otherwisemij${m_{ij}}$The number of modules that SOP consists of themParametersNN${N_N}$Total number of the busesNL${N_L}$Total number of the loading levelsNs${N_s}$Total number of the operation scenariosPi,LLLOAD,Qi,LLLOAD$P_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD},\;Q_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}$Active and reactive power consumption in loading levels LLat node irij,xij${r_{ij}},{\rm{\;}}{x_{ij}}$Resistance and reactance of branch ij$ij$V¯,V$\bar V,V$Lower and upper boundaries of system voltageI¯$\bar I$Maximum current magnitude of branch ij$ij$dDiscount ratenSOP${n_{SOP}}$SOP economical service lifenTieSwitch${n_{TieSwitch}}$Tie switch economical service lifenTieLine${n_{TieLine}}$Tie line economical service lifecostSOP$cos{t^{SOP}}$SOP unit capital costcostTieSwitch$cos{t^{TieSwitch}}$Investment cost of tie switchescostTieLine$cos{t^{TieLine}}$Investment cost of tie linesηSOP${\eta ^{SOP}}$Coefficient of the annual operational costs of SOPηTieSwitch,ηTieLine${\eta ^{TieSwitch}},{\rm{\;}}{\eta ^{TieLine}}$Coefficient of operation and maintenance cost of tie switches and tie linesδ…${\delta ^ {\ldots} }$Annuity factorcostElectricity$cos{t^{Electricity}}$Electricity pricelengthij$lengt{h_{ij{\rm{\;}}}}$Length of the tie line ij$ij$DRLL$D{R_{LL}}$Duration of each loading levelSmodule${S^{module}}$Capacity of each module used in SOPAiSOP$A_i^{SOP}$Loss coefficient of SOP at node iρ(s)$\rho ( s )$The probability corresponding to the sth scenariosb$sb$Substation busbarAUTHOR CONTRIBUTIONSM.E.: Conceptualization, Investigation, Methodology, Software, Writing—original draft, Writing—review and editing. M.S.S.: Conceptualization, Investigation, Methodology, Supervision, Writing—review and editing.FUNDING INFORMATIONThere is no funding to report for this submission.CONFLICT OF INTERESTThe authors declare that there are no conflicts of interest regarding the publication of this paper.DATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESLopes, J.A.P., Hatziargyriou, N., Mutale, J., Djapic, P., Jenkins, N.: Integrating distributed generation into electric power systems: A review of drivers, challenges and opportunities. Electr. Power Syst. Res. 77(9), 1189–1203 (2007)Allan, G., Eromenko, I., Gilmartin, M., Kockar, I., McGregor, P.: The economics of distributed energy generation: A literature review. Renewable Sustainable Energy Rev. 42, 543–556 (2015)Trebolle, D., Gómez, T., Cossent, R., Frías, P.: Distribution planning with reliability options for distributed generation. Electr. Power Syst. Res. 80(2), 222–229 (2010)Cruz, M.R.M.: Stochastic management framework of distribution network systems featuring large‐scale variable renewable energy sources and flexibility options. Ph.D. dissertation, University of Beira Interior, Covilhã, Portugal (2019)Liang, X.: Emerging power quality challenges due to integration of renewable energy sources. IEEE Trans. Ind. Appl. 53(2), 855–866 (2017)Diaaeldin, I., Aleem, S.A., El‐Rafei, A., Abdelaziz, A., Zobaa, A.F.: Optimal network reconfiguration in active distribution networks with soft open points and distributed generation. Energies 12(21), 4172 (2019)Zhao, J., Zheng, T., Member, S., Litvinov, E.: Variable resource dispatch through do‐not‐exceed limit. IEEE Trans. Power Syst. 30(2), 820–828 (2015)Aithal, A.: Operation of soft open point in a distribution network under faulted network conditions. Ph.D. dissertation, Cardiff University, Wales, (2018)Sun, F., Ma, J., Yu, M., Wei, W.: Optimized two‐time scale robust dispatching method for the multi‐terminal soft open point in unbalanced active distribution networks. IEEE Trans. Sustainable Energy 12(1), 587–598 (2021)Cao, W., Wu, J., Jenkins, N., Wang, C., Green, T.: Operating principle of soft open points for electrical distribution network operation. Appl. Energy 164, 245–257 (2016)Hafezi, H., Laaksonen, H.: Autonomous soft open point control for active distribution network voltage level management. In: 2019 IEEE Milan PowerTech, Milan (2019)Bloemink, J.M., Green, T.C.: Benefits of distribution‐level power electronics for supporting distributed generation growth. IEEE Trans. Power Delivery 28(2), 911–919 (2013)Cao, W., Wu, J., Jenkins, N.: Feeder load balancing in MV distribution networks using soft normally‐open points. In: IEEE PES Innovative Smart Grid Technologies, Europe, Istanbul, Turkey (2015)Cao, W.: Soft open points for the operation of medium voltage distribution networks. Ph.D dissertation, Cardiff University, Wales (2015)Qi, Q., Wu, J., Zhang, L., Cheng, M.: Multi‐objective optimization of electrical distribution network operation considering reconfiguration and soft open points. Energy Procedia 103(April), 141–146 (2016)Long, C., Wu, J., Thomas, L., Jenkins, N.: Optimal operation of soft open points in medium voltage electrical distribution networks with distributed generation. Appl. Energy 184, 427–437 (2016)Qi, Q., Wu, J., Long, C.: Multi‐objective operation optimization of an electrical distribution network with soft open point. Appl. Energy 208(May), 734–744 (2017)Ji, H., Li, P., Wang, C., et al.: A strengthened SOCP‐based approach for evaluating the distributed generation hosting capacity with soft open points. Energy Procedia 142, 1947–1952 (2017)Bai, L., Jiang, T., Li, F., Chen, H., Li, X.: Distributed energy storage planning in soft open point based active distribution networks incorporating network reconfiguration and DG reactive power capability. Appl. Energy 210, 1082–1091 (2018)Qi, Q., Wu, J.: Increasing distributed generation penetration using network reconfiguration and soft open points. Energy Procedia 105, 2169–2174 (2017)Yao, C., Zhou, C., Yu, J., Xu, K., Li, P., Song, G.: A sequential optimization method for soft open point integrated with energy storage in active distribution networks. Energy Procedia 145, 528–533 (2018)Ji, H., Wang, C., Li, P., et al.: An enhanced SOCP‐based method for feeder load balancing using the multi‐terminal soft open point in active distribution networks. Appl. Energy 208(August), 986–995 (2017)Point, S.O.: Minimization of network power losses in the AC‐DC hybrid distribution network through network reconfiguration using soft open point. Electronics 10(3), 326 (2021)Ji, H., Wang, C., Li, P., Ding, F., Wu, J.: Robust operation of soft open points in active distribution networks with high penetration of photovoltaic integration. IEEE Trans. Sustainable Energy 10(1), 280–289 (2019)Shafik, M.B., Rashed, G.I., Chen, H., Elkadeem, M.R., Wang, S.: Reconfiguration strategy for active distribution networks with soft open points. In: Proceedings of the 14th IEEE Conference on Industrial Electronics and Applications (ICIEA), Xi'an, China, pp. 330—334 (2019)Li, P., Song, G., Ji, H., Zhao, J., Wang, C., Wu, J.: A supply restoration method of distribution system based on Soft Open Point. In: IEEE PES Innovative Smart Grid Technologies ‐ Asia (ISGT‐Asia), Melbourne, Australia, pp. 535–539 (2016)Li, P., Ji, H., Wang, C., et al.: Optimal operation of soft open points in active distribution networks under three‐phase unbalanced conditions. IEEE Trans. Smart Grid 10(1), 380–391 (2019)Li, P., Ji, H., Wang, C., et al.: A coordinated control method of voltage and reactive power for active distribution networks based on soft open point. IEEE Trans. Sustainable Energy 8(4), 1430–1442 (2017)Li, P., Ji, H., Yu, H., et al.: Combined decentralized and local voltage control strategy of soft open points in active distribution networks. Appl. Energy 241(March), 613–624 (2019)Zhao, J., Yao, M., Yu, H., Song, G., Ji, H., Li, P.: Decentralized voltage control strategy of soft open points in active distribution networks based on sensitivity analysis. Electron 9(2), 295 (2020)Cao, W., Wu, J., Jenkins, N., Wang, C., Green, T.: Benefits analysis of Soft Open Points for electrical distribution network operation. Appl. Energy 165(March), 36–47 (2016)Wang, C., Song, G., Li, P., Ji, H., Zhao, J., Wu, J.: Optimal siting and sizing of soft open points in active electrical distribution networks. Appl. Energy 189, 301–309 (2017)Zhang, L., Shen, C., Chen, Y., Huang, S., Tang, W.: Coordinated optimal allocation of DGs, capacitor banks and SOPs in active distribution network considering dispatching results through bi‐level programming. Energy Procedia 142, 2065–2071 (2017)Shafik, M.B., Rashed, G.I., Chen, H.: Optimizing energy savings and operation of active distribution networks utilizing hybrid energy resources and soft open points: Case study in Sohag, Egypt. IEEE Access 8, 28704–28717 (2020)Prakash, P., Khatod, D.K.: Optimal sizing and siting techniques for distributed generation in distribution systems: A review. Renewable Sustainable Energy Rev. 57, 111–130 (2016)Abazari, A., Soleymani, M.M., Kamwa, I., et al.: A reliable and cost‐effective planning framework of rural area hybrid system considering intelligent weather forecasting. Energy Rep. 7, 5647–5666 (2021)Zadsar, M., et al.: Central situational awareness system for resiliency enhancement of integrated energy systems. In: 2021 IEEE 4th International Conference on Computing, Power and Communication Technologies (GUCON), Kuala Lumpur, Malaysia, pp. 1–6 (2021)Fu, Q., Yu, D., Ghorai, J.: Probabilistic load flow analysis for power systems with multi‐correlated wind sources. In: IEEE Power and Energy Society General Meeting, Detroit, MI, USA, pp. 1–6 (2011)Wen, S., Lan, H., Fu, Q., Yu, D.C., Zhang, L.: Economic allocation for energy storage system considering wind power distribution. IEEE Trans. Power Syst. 30(2), 644–652 (2015)Atwa, Y.M., El‐Saadany, E.F.: Probabilistic approach for optimal allocation of wind‐based distributed generation in distribution systems. IET Renewable Power Gener. 5(1), 79–88 (2011)Morales, J.M., Pérez‐Ruiz, J.: Point estimate schemes to solve the probabilistic power flow. IEEE Trans. Power Syst. 22(4), 1594–1601 (2007)Wu, L., Jiang, L., Hao, X.: Optimal scenario generation algorithm for multi‐objective optimization operation of active distribution network. In: Chinese Control Conference CCC, Dalian, China, pp. 2680–2685 (2017)Farivar, M., Low, S.H.: Branch flow model: Relaxations and convexification‐part i. IEEE Trans. Power Syst. 28(3), 2554–2564 (2013)Dorostkar‐Ghamsari, M.R., Fotuhi‐Firuzabad, M., Lehtonen, M., Safdarian, A.: Value of distribution network reconfiguration in presence of renewable energy resources. IEEE Trans. Power Syst. 31(3), 1879–1888 (2016)Baran, M.E., Wu, F.F.: Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans. Power Delivery 4(2), 1401–1407 (1989)Savier, J. S., Das, D.: Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans. Power Delivery 22(4), 2473–2480 (2007)AAPPENDIXTEST DISTRIBUTION NETWORKSSee Figure A1 and Figure A2.A1FIGURE33‐bus test systemA2FIGURE69‐bus test systemBAPPENDIXCHARACTERISTICS OF TEST DISTRIBUTION NETWORKSSee Table B1, Table B2, Table B3, and Table B4.B1TABLECandidate locations of tie switches (TS)33‐bus69‐busTS candidateFrom busTo busFrom busTo busT1(4)(25)(23)(41)T2(11)(18)(18)(44)T3(11)(15)(10)(48)T4(30)(16)(24)(57)T5(20)(14)(36)(50)T6(5)(10)(20)(67)B2TABLEParameters of wind generatorsLocation101617303233‐busCapacity (kVA)500200150200300Location2133466269‐busCapacity (kVA)300100200400B3TABLEMultilevel modeling of load in optimization problemLoading level(1) High(2) Medium(3) LowTypePercentage of peak loadDuration (h)Percentage of peak loadDuration (h)Percentage of peak loadDuration (h)Residential100219057.53650302920Commercial100219087.53650302920Industrial100219079.03650722920B4TABLEValues of the studied network parametersParametersValueDiscount rate0.08SOP economical service life, year20Tie switch economical service life, year15Tie line economical service life, year35SOP unit capital cost, $/kVA308.8Coefficient of the annual operational costs of SOP, $/KVA6.17Investment cost of tie switches, K$4.7Coefficient of operation and maintenance cost of tie switches0.02The investment cost of tie lines, K$/km60Coefficient of operation and maintenance cost of tie lines0.01Length of the tie line (km)0.85Electricity price, $/kWh0.08Minimum optimum capacity of SOP, kVA100Loss coefficient of SOP0.02 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "IET Generation, Transmission & Distribution" Wiley

Simultaneous siting and sizing of Soft Open Points and the allocation of tie switches in active distribution network considering network reconfiguration

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Abstract

INTRODUCTIONThe increasing integration of renewable energy sources (RES), like wind turbines and photovoltaic panels in today's power grids, is inevitable. These resources can contribute to meeting an increasing need for electricity, enhancing the security of the energy supply, and reducing the dependency on fossil fuels, power losses, and greenhouse gas [1]. Of course, due to the higher investment cost and a longer period of return on capital, distributed generations, especially renewables, are less attractive among investors. Nevertheless, the sustainable expansion of RES to provide the possibility of electric energy storage leads to economic promotion [2]. These resources with high volatility and intermittent nature pose significant challenges, especially in distribution network planning and operation. The main reason is that traditional distribution networks are not designed to accommodate generation sources [3]. Bidirectional power flow caused by high penetration of renewable generation makes operation, control, safety, and flexibility relatively complex and challenging [4]. Equipment overloads and power quality issues comprising voltage and frequency oscillations and harmonic distortion aggregation are other problems when integrating variable energy resources (VERs) [5, 6]. Curtailing the power production of generators is an approach to control the power output of generators [4]. However, curtailment actions are implemented in an ex‐post mode when the system has experienced certain security violations [7]. The active distribution network (ADN) could be equipped with different flexibility options and regulation means to overcome the challenges caused by the uncertainty of increasing penetration of VERs and various demand‐side resources. This proceeding can provide more power adjustment ability by dynamically changing the operational set point to improve the network performance.Literature surveySoft Open Points (SOPs) are novel multi‐functional power electronic switches typically used as soft meshing devices [8]. The SOP can be flexibly connected between feeders to replace traditional circuit‐breaker‐based interconnection switches known as normally open points (NOPs) [9]. Without hard switching operation, SOPs can control the active and reactive power flow, among the feeders, in real‐time and optimize the network voltage profile [10]. Meanwhile, SOP is can isolate any voltage and current disturbances or abnormal conditions and control peak current due to fault [11]. Therefore, the application of SOP will significantly promote the operating conditions of the network with better operational flexibility and a more cost‐effective manner.SOPs consist of voltage source converters (VSCs). These devices have different rating capacities and varying quantities of modules. The major topologies include [12–14]:Back‐to‐back VSC (B2B VSC)Multi‐terminal VSCUnified power flow controller (UPFC)Static series synchronous compensator (SSSC).Some publications have assessed the benefits of using SOPs to improve the performance of the distribution network under steady‐state operating conditions [15–25]. By incorporating SOPs in distribution network architecture, various schemes of optimal network operation can be achieved. Ref. [16] examined the network performance using SOP at several levels of renewable penetration by considering three different optimization problems, each with a separate objective function. Results showed that SOPs significantly improved voltage profile, balanced line utilization of the network, reduced network losses, and increased distributed generation penetration. A similar study to [16] has been conducted in reference [17] in which the voltage profile improvement, feeder load balancing, and power loss reduction have been seen as a multi‐objective function in the optimization problem. Ref. [18] proposed a model for developing a strengthened SOCP‐based approach to maximize distributed generation hosting capacity under the optimal operation of SOPs. An optimal planning model of the presence of distributed energy storage systemsin ADNs is provided in [19] while incorporating the operation of SOPs, reactive power support by DGs, and hourly network reconfiguration. Ref. [23] dealt with the change of network topology in a loop or mesh through SOPs which results in a reduction of network losses. Reference [24] proposed a robust optimization model to obtain operational strategies for SOPs while minimizing voltage violations and power losses. Ref. [25] studied a reconfiguration strategy to determine the optimal number, location, and size of DGs in ADNs while regulating the optimal set points for the operation of SOPs.Considering the flexible power flow control ability and dynamic operation of SOPs, several studies investigated the different control strategies and performance of SOPs in normal and abnormal operating conditions [8, 10, 26–30].Throughout the literature, it can be seen that most of the studies related to the evaluation of optimal network operation with the usage of SOPs consider a fixed number and predetermined installation sites for them. In this case, the optimal network operation may not be achieved. According to [31], the optimal installation location of SOPs has been determined to minimize network power losses and improve feeder load balancing. Here, the limited number of tie switches places is considered for the candidate locations for installing this equipment, and the capacity of each SOP is predetermined. The Optimal installation site and size of SOP were determined considering several operation scenarios of renewable output generated through the Wasserstein distance‐based method in [32].An optimization problem in the form of a bi‐level model was solved in [33] to co‐ordinately allocate DGs, capacitor banks, and SOPs considering the multistate model of DGs and loads. In [6] the simultaneous allocation of soft open points and distributed generations were modelled to obtain the best operation strategy for the distribution network with minimum losses. This study has analyzed several scenarios based on network reconfiguration, different numbers of installed SOPs, and the contribution of SOP's internal losses. The researchers of [34] proposed a hybridization framework for improving the performance of active distribution networks involving hybrid energy resources and SOPs. The best hybrid configuration of the system was obtained while minimizing the net present cost (NPC) and cost of energy (COE) considered the main objectives.It should be noted that there are various methods and techniques to solve the optimization problems of siting and sizing of different equipment, which are mainly categorized as follows [35]:Analytical techniquesClassical optimization techniquesArtificial intelligent (meta‐heuristic) techniquesMiscellaneous techniquesOther techniques for future useThe time complexity of solving optimization processes for siting and sizing problems, especially for large‐scale networks should be considered. Hence, integrating meta‐heuristic approaches with other solution approaches like classical optimization techniques can accelerate the process of problem‐solving and convergence [35]. In this regard, paper [36] has applied a beneficial meta‐heuristic approach called Multi‐Verse Optimizer (MVO) for optimal sizing of a hybrid system planning. Also, the authors of [36] have compared the MVO algorithm with some advanced meta‐heuristic approaches, like grasshopper optimization algorithm (GOA), grey wolf optimization (GWO), dragonfly algorithm (DA), and salp swarm algorithm (SSA). The efficiency of MVO has been promising in terms of convergence, calculation time, cost, and the obtained solutions compared to other approaches. Reference [37] has developed a mixed integer linear programming (MILP) model for integrated energy systems to coordinate different emergency proceedings between power distribution systems and natural gas networks. In this study, the integration of the graph‐based approach with the MILP model has been deployed to alleviate the complexity of solving the optimization problem.We can infer from the above discussion the future work of this paper could be to couple one of the efficient optimization approaches with the optimization method used in the current article. The aim is to reduce the problem's execution time and achieve the best convergence characteristics.Research gapsSome of the specifications of the existing publications have been classified and compared in Table 1. The main goals of using SOPs in ADNs are increasing the operational flexibility of the network with available configuration and load balancing within the connected network feeders. The ability of SOP to control the active and reactive power flow, fast response, and frequent actions can be considered a suitable switching device in the network topology change. Therefore, the coordination of SOPs with other switching devices like tie switches can enhance the adaptability to various operating conditions created due to uncertain renewable generations and other emergency conditions. This is an issue that has been addressed in [31] and [32], and both SOPs and tie switches have been used to perform switching in network reconfiguration. It is necessary to mention that in the reviewed studies, the installation location of the tie switches is predetermined, and only their switching status is evaluated simultaneously with SOPs. However, determining the optimal and simultaneous presence of different equipment to achieve optimal network operation is valuable during network expansion planning studies. This issue has not been investigated in [31] and [32] (the closest available studies to the current work) and other similar studies. Therefore, based on mentioned above and the literature review, the lack of coordinated planning in deciding to install SOPs and tie switches simultaneously to perform flexible manoeuvre operations and change the network configuration is the gap in previous studies.1TABLEThe relevant features of the reviewed studies and the proposed model in the present workConsidered itemsStudy areaLoad and generation modelReferencePlanningOperationModel typeSolving methodDetermination of site and size of SOPsCoordinated planning of the simultaneous presence of SOP with other equipment (other equipment)GenerationLoad[16]–√DeterministicNon‐linear optimization––a weekend day and a weekdaya weekend day and a weekday[17]–√DeterministicMOPSO + Taxi‐cab––––[18]√√DeterministicSOCP––Daily dataDaily data[19]√√DeterministicMISOCP––Yearly dataYearly data[23]–√DeterministicSOP/loop selection algorithm––24 hours24 hours[24]–√RobustSOCP––Daily operation with uncertainty–[25]√√DeterministicMPSO––––[31]√√DeterministicPowell's direct set method√(only site)–––[32]√√StochasticMISOCP√–Stochastic (scenario‐based)–[33]√√DeterministicBi‐level programming√√(DGs and capacitor banks)24 hours24 hours[6]√√DeterministicDiscrete‐continuous HSS√√(DGs)–Different loading levels[34]√√DeterministicHOMER PRO and NEPALN packages√√(PV, WT, Diesel generator)––Present work√√StochasticMISOCP√√(Tie switch)Stochastic(scenario‐based)Different annual loading levelsContributionsThis paper proposes a stochastic scenario‐based model to evaluate the cost‐effectiveness of the simultaneous presence of SOPs and tie switches considering network reconfiguration. The high investment cost of SOPs compared to the tie switches is a determinant factor in distribution network expansion or reinforcement planning. However, considering the cost of network energy losses in a distribution system equipped simultaneously with SOPs and tie switches, reduces the total network expenses to an acceptable level effectively despite the high investment cost of SOPs. This study has been carried out annually, considering the annual three‐level loading profile and different wind power generation scenarios. The problem has been solved for the IEEE 33‐bus and 69‐bus systems to evaluate the efficiency of the proposed model. The major contributions of this paper are summarized as follows:Integrating the optimal allocation problem of new tie switches with the planning problem of siting and sizing of SOPs is the main novel aspect of this paper. Therefore, this paper presents a simultaneous planning model of the presence of SOPs and tie switches during decision‐making for the coordinated investment of this equipment to perform network reconfiguration.The proposed optimization problem becomes more complex in terms of model size and computational time due to the addition of modelling the allocation of new tie switches. So, by using the SOCP approach, the mixed integer non‐linear programming (MINLP) model is converted into mixed‐integer second‐order cone programming (MISOCP) model to moderate the computational time and realize the optimal solution. The conversion procedure is implemented through linearization and conic relaxation to realize convex relaxation.Organization of the paperThe organization of the rest of this paper is as follows: In Section 2, a method of scenario generation is presented by discretizing the renewable power distribution function. Section 3 aims to model the optimal and simultaneous presence of SOPs and tie switches, and then the SOCP approach is applied to the model. In Section 4 the IEEE 33‐bus and 69‐bus systems are adopted, and the results of solving the optimization model on the studied case are described. The conclusions of this study are shown in Section 5.OPERATION SCENARIO GENERATIONDistribution networks will face great uncertainties with the increasing penetration level of renewable energy resources. In order to better achieve the optimal and simultaneous sites and sizes of SOPs and the allocation of tie switches, it is necessary to model the uncertain impact of renewable generation especially for long term planning of electrical distribution networks.One of the approaches to manage the uncertainty of renewable power is to generate several scenarios representing the renewable output power with its probability of occurrence. Using “Point Estimation” method, the continuous probability distribution of renewable power is replaced by a discrete distribution [38, 39].Probabilistic description for wind generationOne general model for probabilistic wind speed description is the Weibull distribution. The Weibull probability density function (PDF) is given by [38]:1f(v|λ,k)=kλvλk−1e−vλk$$\begin{equation}f\;(v|\lambda ,\;k) = \left( {\frac{{\;k}}{\lambda }} \right)\;{\left( {\frac{v}{\lambda }} \right)^{k - 1}}{e^{ - {{\left( {\frac{v}{\lambda }} \right)}^k}}}\end{equation}$$where v is wind speed, k and λ are shape and scale parameters, respectively.The decision variables of the power flow problem are affected by different power injection values of DGs. The active power output of wind turbines depends on the wind speed, which is formulated in the form of linear approximation as shown in (2) [40]:2P=0ifv≤vwc,inorv>vwc,outPwrvwr−vwc,inv−Pwr.vwc,invwr−vwc,inifvwc,in≤v≤vwrPwrifvwr≤v≤vwc,out$$\begin{equation}P{\rm{\;}} = {\rm{\;}}\left\{ \def\eqcellsep{&}\begin{array}{ll} 0 &if\;v \le v_w^{c,{\rm{\;}}in}\;or\;v > v_w^{c,{\rm{\;}}out}\\[8pt] \frac{{P_w^r}}{{v_w^r - v_w^{c,{\rm{\;}}in}{\rm{\;}}}}v - \;\frac{{P_w^r{\rm{\;}}.{\rm{\;}}v_w^{c,{\rm{\;}}in}{\rm{\;}}}}{{v_w^r - v_w^{c,{\rm{\;}}in}{\rm{\;}}}} &if\;v_w^{c,{\rm{\;}}in} \le v \le v_w^r\\[12pt] P_w^r &if\;v_w^r \le v \le v_w^{c,{\rm{\;}}out} \end{array} \right.\end{equation}$$where P is the injected power. Pwr$P_w^r\;$is the rated power of a wind turbine.vwc,in$\;v_w^{c,\;in}$, vwc,out$v_w^{c,\;out}$and vwr$v_w^r\;$denote the cut‐in, cut‐out, and rated wind speed, respectively.Discretization scheme of power distribution functionThe point estimation method is a technique to approximate the output variables of the problem affected by the uncertainty through computing the moments of a random variable. Assuming that the continuous distribution of the input variables is available and using the point estimation method, the discrete distribution can be substituted. The renewable generation scenarios can be built by performing the discretization scheme of the wind power distribution function and obtaining the discretized points [38, 39, 41].Obviously, wind turbine may generate the values of zero power or rated power according to (2) that the associated probabilities of them are calculated as follows [42]:3P1=ProbP=0=Probv≤vwc,in+Probv>vwc,our$$\begin{equation} {{P}_{1}}=Prob\left\{ P=0 \right\}=Prob\left( v\le v_{w}^{c,in} \right)+Prob\left( v>v_{w}^{c,our} \right)\end{equation}$$4=1−e−vwc,inλk+e−vwc,outλkP5=ProbP=Pwr=e−vwrλk−e−vwc,outλk$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} &=& 1 - {e^{\left( { - {{\left( {\frac{{v_w^{c,\;in}}}{\lambda }} \right)}^k}} \right)}} + \;{e^{\left( { - {{\left( {\frac{{v_w^{c,\;out}}}{\lambda }} \right)}^k}} \right)}}\\ {P_5} &=& Prob\left\{ {P = P_w^r} \right\} = {e^{\left( { - {{\left( {\frac{{v_w^r}}{\lambda }} \right)}^k}} \right)}} - \;{e^{\left( { - {{\left( {\frac{{v_w^{c,\;out}}}{\lambda }} \right)}^k}} \right)}} \end{array} \end{equation}$$Then, to determine the remaining three points of the wind power approximation, we must calculate the second, third and fourth central moments of the power distribution as follows [38, 39, 41]:For vwc,in≤v≤vwr$v_w^{c,\;in} \le v \le v_w^r$, redefine PDF of P:5f∼PP|λ,k=1βf(P−αβ|λ,k)1−P1−P5→∫0Pwrf∼PP|λ,kdP=1α=−Pwr.vwc,invwr−vwc,inβ=Pwrvwr−vwc,in$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {{\tilde f}_P}\;\left( {P{\rm{|}}\lambda ,k} \right) &=& \frac{{\frac{1}{\beta }\;f(\frac{{P - \;\alpha }}{\beta }|\lambda ,\;k)}}{{1 - \;{P_1} - \;{P_5}}}\; \to \;\displaystyle\mathop \int \limits_0^{P_w^r} {{\tilde f}_P}\left( {P{\rm{|}}\lambda ,k} \right)dP\; = \;1\;\\ \alpha &=& - \frac{{P_w^r\;.\;v_w^{c,\;in}\;}}{{v_w^r - v_w^{c,\;in}}}\;\beta \; = \;\frac{{P_w^r}}{{v_w^r - v_w^{c,\;in}\;}} \end{array} \end{equation}$$Define:6μ∼P=∫0PwrPf∼P(P|λ,k)dP$$\begin{equation}{\tilde \mu _P} = \;\mathop \int \limits_0^{P_w^r} P{\tilde f_P}(P|\lambda ,k)dP\end{equation}$$7σ∼P2=∫0PwrP−μ∼P2f∼P(P|λ,k)dP$$\begin{equation}{\rm{\;}}\tilde \sigma _P^2 = \mathop \int \limits_0^{P_w^r} {\left( {P - {{\tilde \mu }_P}} \right)^2}\;{\tilde f_P}(P|\lambda ,k)dP\end{equation}$$8λj=∫0PwrP−μ∼Pσ∼Pjf∼P(P|λ,k)dP$$\begin{equation}{\rm{\;}}{\lambda _j} = \;\mathop \int \limits_0^{P_w^r} {\left( {\frac{{P - {{\tilde \mu }_P}}}{{{{\tilde \sigma }_P}}}} \right)^j}{\tilde f_P}(P|\lambda ,k)dP\;\end{equation}$$μ∼P,σ¯P${\tilde \mu _P},\;{\bar \sigma _P}$ and λj${\lambda _j}$ are the values of mean, standard deviation, and jth central moment of P respectively.The moment equations are given by:9∑i=24pizij=λjforj=1,2,3,4$$\begin{equation}\mathop \sum \limits_{i\; = \;2}^4 {p_i}\;\;z_i^j = \;{\lambda _j}\;for\;j\; = \;1,\;2,\;3,\;4\;\end{equation}$$where pi${p_{i\;}}$is the probability weight corresponding to location zi${z_i}$.Solving for Equation (9), we can obtain:10z2=λ32−λ4−3λ324z3=0z4=λ32+λ4−3λ324$$\begin{equation}\left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\;{z_2} = \frac{{{\lambda _3}}}{2} - \sqrt {{\lambda _4} - \frac{{3\lambda _3^2}}{4}} \;}\\[6pt] {\;{z_3} = 0\;}\\[6pt] {\;{z_4} = \frac{{{\lambda _3}}}{2} + \sqrt {{\lambda _4} - \frac{{3\lambda _3^2}}{4}\;} } \end{array} \right.\end{equation}$$11p2=−1z2z4−z2p3=1−p2−p4p4=1z4z4−z2$$\begin{equation}\left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\;{p_2} = \frac{{ - 1}}{{{z_2}\;\left( {{z_4} - {z_2}} \right)}}\;}\\[6pt] {\;{p_3} = 1 - {p_2} - {p_4}\;}\\[6pt] {\;{p_4} = \frac{1}{{{z_4}\;\left( {{z_4} - {z_2}} \right)}}\;} \end{array} \right.\end{equation}$$It should be noted that one of the three standard values zi${z_i}\;$is set to zero ( z3=0${z_3} = \;0$).Each of the remaining three points of discrete distribution is approximated as a pair composed of a location zi${z_i}\;$and a probability weight pi,${p_{i\;}},\;$as shown in Equations (10) and (11). Then the estimation of points of power output Pi${P_i}\;$and associated probabilities Probi$Pro{b_i}\;$can be obtained as:12zi=Pi−μ∼Pσ∼PP2=μ∼P+σ∼Pz2andProb2=p21−P1−P5P3=μ∼PandProb3=p31−P1−P5P4=μ∼P+σ∼Pz4andProb4=p41−P1−P5$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {z_i} &=& \;\frac{{{P_i} - {{\tilde \mu }_P}\;}}{{{{\tilde \sigma }_P}}}\\[4pt] {\rm{\;}}{P_2} &=& {{\tilde \mu }_P} + {{\tilde \sigma }_P}\;{z_{2\;}}\;and\;\;Prob{\;_2} = {p_2}\;\left( {1 - {P_1} - {P_5}} \right)\\[4pt] {P_3} &=& {{\tilde \mu }_P}\;and\;\;Prob{\;_3} = {p_3}\;\left( {1 - {P_1} - {P_5}} \right)\\[4pt] {\rm{\;}}{P_4} &=& {{\tilde \mu }_P} + {{\tilde \sigma }_P}{z_4}\;and\;\;Prob{\;_4} = {p_4}\;\left( {1 - \;{P_1} - {P_5}} \right) \end{array} \end{equation}$$The parameters of Weibull distribution and wind turbine operating specifications are summarized in Table 2. The injected power distribution calculated through five‐point discrete distribution is given in Table 3 and drawn in Figure 1.2TABLEParameters of Weibull distribution and specifications of wind turbinekλvwc,in(m/s)$v_w^{c,\;in}( {{{{\rm{m}}} / {{\rm{s}}}}} )$vwc,out(m/s)$v_w^{c,\;out}( {{{{\rm{m}}} / {{\rm{s}}}}} )$vwr(m/s)$v_w^r( {{{{\rm{m}}} / {{\rm{s}}}}} )$210325153TABLEApproximate wind power distributionPoint00.10730.45190.85031Probability0.0880.21700.40380.18770.10351FIGUREApproximate wind power distributionPROBLEM FORMULATIONBy enabling the flexible connection between feeders through the SOPs, these devices can adjust different values of operational set points by controlling the active power flowing among the adjacent branches in real‐time and injecting or absorbing the reactive power at their terminals. This paper presents an optimization model to obtain the best simultaneous expansion plan of SOPs and tie switches, including determining the optimal installation site and size of SOP and the best place for installing the tie switch. The network equips with the B2B VSCs‐based topology. Also, the optimal operational strategies of the active power transmission and reactive power support of SOPs are obtained to realize the optimal regulation of the network. After determining the best place for installing the candidate tie switches, optimal switching is needed to achieve optimal network operation.Optimal model for planning the simultaneous presence of SOPs and tie switchesThe distribution network expansion planning (DNEP) problem is one of the topics of interest for decision‐makers in power distribution utilities. The DNEP is a process in which the aim is to decide on reinforcement of existing system elements or install new ones to reliably meet the loads' growth according to the technical and operational limits. One of the main objectives of the DNEP problem is to minimize the annual expense of the distribution system [32]. The objective function of the optimization problem consists of the investment cost of SOPs, new tie switches, and tie lines creation cost, the operation cost of SOPs and the operation and maintenance cost of tie switches and tie lines, and the cost of energy losses. This model is proposed as Equations (13)–(24).Objective function13MinCost=CostINV+CostOPE+CostLOSS$$\begin{equation}Min{\rm{\;}}Cost{\rm{\;}} = {\rm{\;}}Cos{t^{INV}} + {\rm{\;}}Cos{t^{OPE}} + {\rm{\;}}Cos{t^{LOSS}}\end{equation}$$14CostINV=δSOPCostINV−SOP+δTieSwitchCostINV−TieSwitch+δTieLineCostINV−TieLine$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} \;Cos{t^{INV}} &=& {\delta ^{SOP}}\;Cos{t^{INV - SOP}}\\[4pt] &&+ \;{\delta ^{TieSwitch}}Cos{t^{INV - TieSwitch}}\\[4pt] &&+ \;{\delta ^{TieLine}}Cos{t^{INV - TieLine}} \end{array} \end{equation}$$15CostOPE=CostOPE−SOP+CostOPE−TieSwitch+CostOPE−TieLine$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} Cos{t^{OPE}} &=& Cos{t^{OPE - SOP}} + \;Cos{t^{OPE - TieSwitch}}\\[4pt] &&+ \;Cos{t^{OPE - TieLine}} \end{array} \end{equation}$$16CostINV−SOP=costSOP∑ij∈ΩSOPSijSOP$$\begin{equation}{\rm{\;}}Cos{t^{INV - SOP}} = {\rm{\;}}cos{t^{SOP}}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_{SOP}}} {\rm{\;}}S_{ij}^{SOP}\end{equation}$$17CostINV−TieSwitch=costTieSwitch∑ij∈Ωnc×(αij,1)∨(αij,2)∨…∨(αij,NL)$$\begin{align} Cos{{t}^{INV-TieSwitch}} &= cos{{t}^{TieSwitch}}\underset{\left( ij \right)\in ~{{\text{ }\!\!\Omega\!\!\text{ }}_{nc}}}{\mathop \sum }\nonumber\\ &\times\,\begin{matrix} \left( ({{\alpha }_{ij,~1~}})\vee ({{\alpha }_{ij,~2~}})\vee \ldots \vee ({{\alpha }_{ij,~{{N}_{L}}~}}) \right) \end{matrix} \end{align}$$18CostINV−TieLine=costTieLine∑ij∈Ωnc×lengthij(αij,1)∨(αij,2)∨…∨αij,NL$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} Cos{t^{INV - TieLine}} = cos{t^{TieLine}}\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_{nc}}}\\[8pt] \times lengt{h_{ij\;}} \left( ({\alpha _{ij,\;1\;}}) \vee ({\alpha _{ij,\;2\;}}) \vee \ldots \;\left( \vee \left({\alpha _{ij,\;{N_L}\;}} \right) \right)\right. \end{array} \end{equation}$$19δ…=d1+dn…(1+dn…)−1$$\begin{equation} {\delta ^ {\ldots} } = {\rm{\;}}\frac{{d{\rm{\;}}{{\left( {1 + d} \right)}^{{n_ {\ldots} }}}}}{{({{\left( {1 + d} \right)}^{{n_ {\ldots} }}}) - 1}}\end{equation}$$20CostOPE−SOP=ηSOP∑ij∈ΩSOPSijSOP$$\begin{equation}{\rm{\;}}Cos{t^{OPE - SOP}} = {\rm{\;}}{\eta ^{SOP}}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_{SOP}}} {\rm{\;}}S_{ij}^{SOP}\end{equation}$$21CostOPE−TieSwitch=ηTieSwitchCostINV−TieSwitch$$\begin{equation}{\rm{\;}}Cos{t^{OPE - TieSwitch}} = {\rm{\;}}{\eta ^{TieSwitch}}{\rm{\;}}Cos{t^{INV - TieSwitch}}\end{equation}$$22CostOPE−TieLine=ηTieLineCostINV−TieLine$$\begin{equation}{\rm{\;}}Cos{t^{OPE - TieLine}} = {\rm{\;}}{\eta ^{TieLine}}{\rm{\;}}Cos{t^{INV - TieLine}}\end{equation}$$23CostLOSS=costElectricityELOSS$$\begin{equation}{\rm{\;}}Cos{t^{LOSS}} = {\rm{\;}}cos{t^{Electricity}}{\rm{\;}}{E^{LOSS}}\end{equation}$$24ELOSS=∑S=1NS∑LL=1NLDRLL∑ij∈ΩbrijIs,ij,LL2ρ(s)$$\begin{equation}{\rm{\;}}{E^{LOSS}} = \mathop \sum \limits_{S =1}^{{N_S}} \left(\mathop \sum \limits_{LL =1}^{{N_L}} D{R_{LL}}\left( \mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {r_{ij}}{\rm{\;}}I_{s,{\rm{\;}}ij,{\rm{\;}}LL}^2\right) \right)\rho (s)\end{equation}$$Equations (14), (16)–(18) represent the capital expenditures for equipment, including SOPs, tie switches, and tie lines. Equations (15), (20)–(22) give the operation costs of mentioned equipment. Equations (17) and (18) represent a logical summation. In this way, if at least in one of the loading levels, αij,LL=1${\alpha _{ij,\;LL\;}} = \;1$, it means that a new tie switch must be installed in that location. If αij,LL=0${\alpha _{ij,\;LL\;}} = \;0$ at all loading levels, then no new tie switches will be installed in location ij$ij$.Equation (19) is the factor to convert the total investment costs of SOPs and other equipment annually, and n is the economical service life of the component. The annual operation cost of SOP is given in (20), defined as a factor of the SOP's installation capacity. Equation (23) expresses the annual energy loss cost of the distribution system. Equation (24) relates to the expectation of energy losses (ELOSS)$({E^{LOSS}})$ in the studied network.System power flow equationsIn this study, the power flow of distribution networks is modelled using the branch flow deals with currents and powers flowing between connected feeders [43]. The phase angles of voltage and currents must be eliminated to obtain an angle‐relaxed formulation of the power flow problem, as follows [43, 44]:25Ps,i,LL=∑ij∈ΩbPs,ij,LLLine−∑ki∈ΩbPs,ki,LLLine−rki|Is,ki,LL|2+gi|Vi|2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {P_{s,i,LL}} &=& \displaystyle\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_b}} P_{s,\;ij,\;LL}^{Line}\\[16pt] && -\, \displaystyle\sum {_{\left( {ki} \right) \in \;{{{\Omega}}_b}}} \left(P_{s,ki,LL}^{Line} - {r_{ki}}|{I_{s,ki,LL}}{|^2}\right) + {g_i}|{V_i}{|^2} \end{array} \end{equation}$$26Qs,i,LL=∑ij∈ΩbQs,ij,LLLine−∑ki∈ΩbQs,ki,LLLine−xkiIs,ki,LL2+biVi2$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {Q_{s,\;i,LL}} &=& \displaystyle\mathop \sum \limits_{\left( {ij} \right) \in \;{{{\Omega}}_b}} Q_{s,\;ij,\;LL}^{Line}\\[16pt] && -\, \displaystyle\;\mathop \sum \limits_{\left( {ki} \right) \in \;{{{\Omega}}_b}} \left(Q_{s,ki,LL}^{Line} - {x_{ki}}{\left| {{I_{s,ki,LL}}} \right|^2}\right) + {b_i}{\left| {{V_i}} \right|^2} \end{array} \end{equation}$$27Vs,j,LL2=Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,\;j,\;LL}^2 &=& V_{s,\;i,\;LL}^2\; - 2\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right)\\[10pt] && + \;\left( {{r_{ij}}^2 + {x_{ij}}^2} \right)\;I_{s,\;ij,\;LL\;}^2\quad \quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$28Is,ij,LL2Vs,i,LL2=Ps,ij,LLLine2+Qs,ij,LLLine2∀ij∈Ωb$$\begin{equation}I_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^2{\rm{\;\;}}V_{s,{\rm{\;}}i,{\rm{\;}}LL}^2 = {\rm{\;}}{\left( {P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}} \right)^2} + {\rm{\;}}{\left( {Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}} \right)^2}{\rm{\;}}\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}{\rm{\;}}\end{equation}$$29Ps,i,LL=Ps,iDG+Ps,i,LLSOP−Pi,LLLOAD$$\begin{equation}{\rm{\;}}{P_{s,{\rm{\;}}i,LL}} = {\rm{\;}}P_{s,{\rm{\;}}i}^{DG} + {\rm{\;}}P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP} - {\rm{\;}}P_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}\end{equation}$$30Qs,i,LL=Qs,iDG+Qs,i,LLSOP−Qi,LLLOAD$$\begin{equation}{\rm{\;}}{Q_{s,{\rm{\;}}i,LL}} = Q_{s,{\rm{\;}}i}^{DG} + {\rm{\;}}Q_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP}{\rm{\;}} - {\rm{\;}}Q_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}\end{equation}$$Reconfiguration formulationThis paper presents the reconfiguration problem based on optimal power flow [44]. Because the purpose is to determine the optimal siting and sizing of SOPs and the allocation of tie switches, the mathematical model is implemented for each renewable generation scenario and loading level. Then the optimal planned capacity of SOP and its installation site are obtained along with the best place for installing the candidate tie switches. 31Vs,j,LL2≤M1−αij,LL+Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,j,LL}^2 &\le& M\left( {1 - {\alpha _{ij,LL}}} \right) + V_{s,i,LL}^2\\[6pt] && - 2\left({r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}\right) + \left({r_{ij}}^2 + {x_{ij}}^2\right)I_{s,ij,LL}^2\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$32Vs,j,LL2≥−M1−αij,LL+Vs,i,LL2−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2Is,ij,LL2∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} V_{s,j,LL}^2 &\ge& - M\left( {1 - {\alpha _{ij,LL}}} \right) + V_{s,i,LL}^2\\[6pt] && -\, 2\left({r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}\right) + \left({r_{ij}}^2 + {x_{ij}}^2\right)I_{s,ij,LL}^2\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$33∑ij∈Ωbαij,LL=NN−1$$\begin{equation}\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {\alpha _{ij,{\rm{\;}}LL}} = {N_N} - 1\end{equation}$$34βij,LL+βji,LL=αij,LL∀ij∈Ωb$$\begin{equation}{\beta _{ij,{\rm{\;}}LL}} + {\beta _{ji,{\rm{\;}}LL}} = {\alpha _{ij,{\rm{\;}}LL{\rm{\;}}}}\quad\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}\end{equation}$$35∑j∈Ωbusiβij,LL=1i∈Ωbus,i≠sb$$\begin{equation}\mathop \sum \limits_{j{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}}\left( i \right)} {\beta _{ij,{\rm{\;}}LL}} = 1\quad i{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}},{\rm{\;}}i{\rm{\;}} \ne sb\end{equation}$$36βsbj,LL=0j∈Ωbussb$$\begin{equation}{\beta _{sbj,{\rm{\;}}LL}} = 0{\rm{\;}}j{\rm{\;}} \in {\rm{\;}}{{{\Omega}}_{bus}}\left( {sb} \right)\end{equation}$$37Vsb2=1$$\begin{equation}V_{sb}^2 = {\rm{\;}}1\end{equation}$$Inequality constraints (31) and (32) satisfy Ohm's law over the line between bus i$i\;$and j represented in Equation (27), considering the possibility change the network topology.The binary variable αij,LL${\alpha _{ij,\;LL\;}}$represents the switch status of the line between buses i and j at loading level LL.$LL.$ When the line is closed (αij,LL=1${\alpha _{ij,\;LL}} = 1$), the upper and lower boundaries for Vs,j,LL2$V_{s,\;j,\;LL}^2\;$will be the same values, and Equation (27) will be realized. When the line is open (αij,LL=0${\alpha _{ij,\;LL}} = 0$), the upper and lower boundaries on Vs,j,LL2$V_{s,\;j,\;LL}^2\;$become the large and small values, respectively. So, constraints (31) and (32) are relaxed and always valid.The reconfiguration problem has to be implemented in a way that the radiality and connectivity constraints of the distribution network are satisfied in the new configuration:Radiality constraint: means that all buses except the substation bus are supplied from one direction only, and so no loop is created in the new configuration.Connectivity constraint: Means that the network includes all buses, and so no bus is isolated.Equations (33)–(36) illustrate the above points. Equation (33) expresses that the number of closed switches is exactly one less than the total number of network nodes. According to Equation (34), each switch status of the line is determined by the sum of the two line directional variables βij,LL${\beta _{ij,\;LL}}\;$and βji,LL${\beta _{ji,\;LL}}$. If the power flow direction is from bus j to bus i (βij,LL=1${\beta _{ij,\;LL}} = 1$) or vice versa (βji,LL=1${\beta _{ji,\;LL}} = 1$), the switch of the line between bus i and j is connected (αij,LL=1${\alpha _{ij,\;LL}} = 1$). Obviously, if the directional variables βij,LL${\beta _{ij,\;LL}}\;$and βji,LL${\beta _{ji,\;LL}}\;$have zero values, the line between two buses is open. Equation (35) means that all buses (children buses) except substation bus are supplied only through one bus called the parent bus. Equation (36) means that the substation bus is not the parent bus at all. According to Equation (37), the voltage of the main substation bus equals 1 per unit.System operational constraints38V−2≤Vs,i,LL2≤V¯2$$\begin{equation}{{\underset{\scriptscriptstyle-}{V}}^{2}}\le V_{s,~i,~LL}^{2}\le {{\bar{V}}^{2}}\end{equation}$$39Is,ij,LL2≤I¯2∀ij∈Ωb$$\begin{equation}\;I_{s,\;ij,\;LL\;}^2 \le {\bar I^2}\;\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$Modelling of soft open pointsActive power transmission constraint of SOPsActive and reactive power transmitted through the SOP with its converters is the decision variables in the model. The sum of the active powers provided by the converters and their internal power losses must equal zero for each of the renewable generation scenarios and different loading conditions [24]:40Ps,i,LLSOP+Ps,j,LLSOP+Ps,i,LLSOP,LOSS+Ps,j,LLSOP,LOSS=0$$\begin{equation}P_{s,\;i,\;LL}^{SOP} + \;P_{s,\;j,\;LL}^{SOP} + \;P_{s,\;i,\;LL}^{SOP,\;LOSS} + \;\;P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;0\end{equation}$$41Ps,i,LLSOP,LOSS=AiSOP(Ps,i,LLSOP)2+(Qs,i,LLSOP)2$$\begin{equation}P_{s,\;i,\;LL}^{SOP,\;LOSS} = \;A_i^{SOP}\sqrt {{{(P_{s,\;i,\;LL}^{SOP})}^2} + {{(Q_{s,\;i,\;LL}^{SOP})}^2}\;} \end{equation}$$42Ps,j,LLSOP,LOSS=AjSOP(Ps,j,LLSOP)2+(Qs,j,LLSOP)2$$\begin{equation}P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;A_j^{SOP}\sqrt {{{(P_{s,\;j,\;LL}^{SOP})}^2} + {{(Q_{s,\;j,\;LL}^{SOP})}^2}\;} \end{equation}$$The SOPs have a high operation efficiency, but a large‐scale transmission of active power through VSCs in the SOP unavoidably produces losses. The Equations (41)–(42) are replaced by the following equations to reduce the computational burden of the optimization problem [32]:43Ps,i,LLSOP,LOSS=AiSOPPs,i,LLSOP$$\begin{equation}P_{s,\;i,\;LL}^{SOP,\;LOSS} = \;A_i^{SOP}\;\left| {P_{s,\;i,\;LL}^{SOP}} \right|\end{equation}$$44Ps,j,LLSOP,LOSS=AjSOPPs,j,LLSOP$$\begin{equation}P_{s,\;j,\;LL}^{SOP,\;LOSS} = \;A_j^{SOP}\;\left| {P_{s,\;j,\;LL}^{SOP}} \right|\;\end{equation}$$SOP capacity constraints45(Ps,i,LLSOP)2+(Qs,i,LLSOP)2≤SijSOP∀ij∈Ωb$$\begin{equation}\sqrt {{{(P_{s,\;i,\;LL}^{SOP})}^2} + {{(Q_{s,\;i,\;LL}^{SOP})}^2}\;} \le \;S_{ij}^{SOP}{\rm{\;}}\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$46(Ps,j,LLSOP)2+(Qs,j,LLSOP)2≤SijSOP∀ij∈Ωb$$\begin{equation}\sqrt {{{(P_{s,\;j,\;LL}^{SOP})}^2} + {{(Q_{s,\;j,\;LL}^{SOP})}^2}\;} \le \;S_{ij}^{SOP}\;\forall \;\left( {ij} \right) \in \;{{{\Omega}}_b}\end{equation}$$The reactive power injected or absorbed by the converter terminals is independent of each other because of the DC bus and only meets the converter's capacity constraint as described using Equations (45) and (46) [24]. The location and capacity of an SOP are formulated as follows [32]:47SijSOP=mijSmodule(1−αij,LL)∀ij∈ΩSOP$$\begin{equation}S_{ij}^{SOP} = \;{m_{ij}}\;{S^{module}}\;(1 - {\alpha _{ij,\;LL}}){\rm{\;}}\forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}}\end{equation}$$Conversion to an MISOCP modelThe optimal sites and sizes of SOPs with the allocation of tie switches is a non‐convex non‐linear programming (NLP) model.Considering the stochastic operation scenarios of renewable generation and different loading conditions, the optimization model becomes complicated, and so it is difficult to find the optimal global solution. As a consequence, the original non‐convex NLP model is reformulated as a second‐order cone programming (SOCP) model that tractably can be solved. The SOCP is a mathematically convex optimization approach that can solve minimum linear objective functions on a feasible region composed of linear equality constraints and convex cone constraints [32]. Therefore, the mentioned optimization model with non‐linear function has to be changed in linear form and then the second‐order cone programming approach must be applied. The conversion procedure consists of the following steps:Substitute new variables vs,i,LL${v_{s,\;i,\;LL}}\;$and ls,ij,LL${l_{s,\;ij,\;LL}}\;$for the square of voltage and current, that is, Vs,i,LL2$V_{s,\;i,\;LL}^2\;$andIs,ij,LL2$\;I_{s,\;ij,\;LL\;}^2\;$in constraints (24)–(26), (28), (31), (32), (38) and (39) to linearize them [24].48ELOSS=∑S=1NS(∑LL=1NLDRLL∑ij∈Ωbrijls,ij,LL)ρs$$\begin{equation}{E^{LOSS}} = {\rm{\;}}\mathop \sum \limits_{S{\rm{\;}} = {\rm{\;}}1}^{{N_S}} (\mathop \sum \limits_{LL{\rm{\;}} = {\rm{\;}}1}^{{N_L}} D{R_{LL}}\left( {\mathop \sum \limits_{\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}} {r_{ij}}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}})} \right)\rho \left( s \right)\end{equation}$$49Ps,i,LL=∑ij∈ΩbPs,ij,LLLine−∑ki∈Ωb(Ps,ki,LLLine−rkils,ki,LL)+givs,i,LL$$\begin{eqnarray}{P_{s,{\rm{\;}}i,LL}} &=& \mathop \sum \limits_{\left( {ij} \right) \in {{{\Omega}}_b}} P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line} - \mathop \sum \limits_{\left( {ki} \right) \in {\rm{\;}}{{{\Omega}}_b}} (P_{s,{\rm{\;}}ki,{\rm{\;}}LL}^{Line} - {r_{ki}}{l_{s,{\rm{\;}}ki,{\rm{\;}}LL}})\nonumber\\ && +\, {g_i}{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$50Qs,i,LL=∑ij∈ΩbQs,ij,LLLine−∑ki∈Ωb(Qs,ki,LLLine−xkils,ki,LL)+bivs,i,LL$$\begin{eqnarray}{Q_{s,{\rm{\;}}i,LL}} &=& \mathop \sum \limits_{\left( {ij} \right) \in {{{\Omega}}_b}} Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line} - \mathop \sum \limits_{\left( {ki} \right) \in {{{\Omega}}_b}} (Q_{s,{\rm{\;}}ki,{\rm{\;}}LL}^{Line} - {x_{ki}}{l_{s,{\rm{\;}}ki,{\rm{\;}}LL}})\nonumber\\ && +\, {b_i}{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$51vs,j,LL≤M1−αij,LL+vs,i,LL−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2ls,ij,LL∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {v_{s,j,LL}} &\le& M\left( {1 - \;{\alpha _{ij,LL\;}}} \right) + {v_{s,i,LL}}\\[6pt] &&- 2\;\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right) + \left( {{r_{ij}}^2 + {x_{ij}}^2} \right){l_{s,ij,LL}}\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$52vs,j,LL≥−M1−αij,LL+vs,i,LL−2rijPs,ij,LLLine+xijQs,ij,LLLine+rij2+xij2ls,ij,LL∀ij∈Ωb$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {v_{s,j,LL}} &\ge& - M\left( {1 - \;{\alpha _{ij,LL\;}}} \right) + {v_{s,i,LL}}\\[6pt] && - 2\;\left( {{r_{ij}}P_{s,ij,LL}^{Line} + {x_{ij}}Q_{s,ij,LL}^{Line}} \right) + \left( {{r_{ij}}^2 + {x_{ij}}^2} \right){l_{s,ij,LL}}\\[6pt] &&\quad \forall \left( {ij} \right) \in {{{\Omega}}_b} \end{array} \end{equation}$$53V−2≤vs,i,LL≤V¯2$$\begin{equation}{{\underset{\scriptscriptstyle-}{V}}^{2}}\le {{v}_{s,\text{ }\!\!~\!\!\text{ }i,\text{ }\!\!~\!\!\text{ }LL}}\le {{\bar{V}}^{2}}\end{equation}$$54ls,ij,LL≤I¯2∀ij∈Ωb$$\begin{equation}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} \le {\bar I^2}{\rm{\;}}\forall {\rm{\;}}\left( {ij} \right) \in {\rm{\;}}{{{\Omega}}_b}\end{equation}$$For the absolute terms |Ps,i,LLSOP|$| {P_{s,\;i,\;LL}^{SOP}} |$ and |Ps,j,LLSOP|$| {P_{s,\;j,\;LL}^{SOP}} |$ in Equations (43) and (44), auxiliary variables Ms,i,LLSOP$M_{s,\;i,\;LL}^{SOP}\;$and Ms,j,LLSOP$M_{s,\;j,\;LL}^{SOP}\;$are introduced to represent and linearize them as follows [32]:55Ms,i,LLSOP≥0,Ms,j,LLSOP≥0Ms,i,LLSOP≥Ps,i,LLSOP,Ms,i,LLSOP≥−Ps,i,LLSOPMs,j,LLSOP≥PS,j,LLSOP,Ms,j,LLSOP≥−Ps,j,LLSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} M_{s,i,LL}^{SOP} \ge 0,\quad M_{s,j,LL}^{SOP} \ge 0\\[6pt] M_{s,i,LL}^{SOP} \ge \;P_{s,i,LL}^{SOP},\;\quad M_{s,i,LL}^{SOP} \ge \; - P_{s,i,LL}^{SOP}\\[6pt] \! M_{s,j,LL}^{SOP} \ge \;P_{S,j,LL}^{SOP},\;\quad M_{s,j,LL}^{SOP} \ge \; - P_{s,j,LL}^{SOP} \end{array} \end{equation}$$In Equations (43) and (44), the terms |Ps,i,LLSOP|$| {P_{s,\;i,\;LL}^{SOP}} |$ and |Ps,j,LLSOP|$| {P_{s,\;j,\;LL}^{SOP}} |\;$are replaced by Ms,i,LLSOP$M_{s,\;i,\;LL}^{SOP}{\rm{\;}}$and Ms,j,LLSOP,$M_{s,\;j,\;LL}^{SOP},$ respectively, and then (55) is added to the set of constraints.Relax the non‐linear equality constraint (28) into the quadratic cone form (56) after linearizing [43]:562Ps,ij,LLLine2Qs,ij,LLLinels,ij,LL−vs,i,LLT2≤ls,ij,LL+vs,i,LL$$\begin{eqnarray}&&{\left\| {{{\left[ {2P_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^{Line}{\rm{\;}}2Q_{s,{\rm{\;}}ij,{\rm{\;}}LL{\rm{\;}}}^{Line}{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} - {\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}} \right]}^T}} \right\|_2}\nonumber\\ &&\quad \le {\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}} + {\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}\end{eqnarray}$$Transform the capacity constraints of an SOP in (45) and (46) into the following rotated quadratic cone constraints [24]:57Ps,i,LLSOP2+Qs,i,LLSOP2≤2SijSOP2SijSOP2∀ij∈ΩSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\left( {P_{s,\;i,\;LL}^{SOP}} \right)^2} + {\left( {Q_{s,\;i,\;LL}^{SOP}} \right)^2}\; \le 2\;\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\\[12pt] \forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}} \end{array} \end{equation}$$58Ps,j,LLSOP2+Qs,j,LLSOP2≤2SijSOP2SijSOP2∀ij∈ΩSOP$$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\left( {P_{s,\;j,\;LL}^{SOP}} \right)^2} + {\left( {Q_{s,\;j,\;LL}^{SOP}} \right)^2}\; \le 2\;\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\left( {\frac{{S_{ij}^{SOP}}}{{\sqrt 2 }}} \right)\\[12pt] \forall \;\left( {ij} \right) \in \;{{{\Omega}}_{SOP}} \end{array} \end{equation}$$By relaxing the equality constraint (28) to the inequality constraint (56), the relaxation deviation (59) is formulated in the form of the infinite norm to evaluate the accuracy of the convex relaxation. If the gap value is small enough, the convex cone relaxation can be regarded as accurate for model conversion [28].59gap=ls,ij,LL−Ps,ij,LLLine2+Qs,ij,LLLine2Vs,i,LL∞$$\begin{equation}gap\; = \;{\left\| {{l_{s,\;ij,\;LL}} - \;\frac{{{{\left( {P_{s,\;ij,\;LL\;}^{Line}} \right)}^2} + \;{{\left( {Q_{s,\;ij,\;LL\;}^{Line}} \right)}^2}}}{{{V_{s,\;i,\;LL}}}}} \right\|_\infty }\end{equation}$$CASE STUDIES AND ANALYSISIn this section, the MISOCP model for simultaneous sites and sizes of SOPs and the allocation of tie switches is applied to a distribution network. The IEEE 33‐bus test system is adopted to investigate the effectiveness of the proposed model. The obtained optimization model is implemented in the YALMIP optimization toolbox (version 20210331) using MATLAB R2018a and solved by GUROBI (version 9.5.0) optimizer interface. Also, CPLEX or MOSEK software packages can be used to solve the convex model efficiently.Test networkIn this study, the results are obtained for the IEEE 33‐bus and 69‐bus systems under different loading conditions. The modified IEEE 33‐bus and 69‐bus system are presented in Figures A1 and A2. The rated voltage level is 12.66 kV. The IEEE 33‐bus system consists of 32 sectionalized lines and 5 tie‐lines. The IEEE 69‐bus system has 68 sectionalized lines and 5 tie‐lines. In this networks, 6 candidate locations for installing the new tie switches are suggested and shown in Table B1. The resistance and reactance of all the candidate tie lines for two networks are the same and equal 0.5 ohms. The location and capacity of wind generation are given in Table B2. All the DGs have a unity power factor. In this study, loads are classified as residential, commercial, and industrial customers, which are labelled in Figures A1 and A2. Different loading conditions are considered according to Table B3. The detailed parameters of the test system are shown in [45, 46]. The load data given in [45, 46] is considered as medium loading level and the other two loading levels are constructed according to Table B3. The other values of test distribution network parameters are shown in Table B4.Results and analysisIn this section, the optimal presence planning of SOPs and tie switches are determined on the standard IEEE 33‐node and 69‐node systems. The following schemes are evaluated in this study, and network reconfiguration, renewable generation scenarios given in Table 3, and different loading levels are considered in all schemes:-Scheme I: It is the result of network study without installing SOPs and tie switches.-Scheme II: It is the result of network study considering the optimal allocation of tie switches.-Scheme III: It is the result of network study considering the optimal siting and sizing of SOPs.-Scheme IV: The optimal sites and sizes of SOPs and the optimal placement of tie switches are determined simultaneously.In schemes III and IV, all sectionalized and tie switches are considered the candidate locations in the 33‐bus (43 candidates) and 69‐bus (79 candidates) systems to install the SOPs. It is assumed the direction of power injection of SOP into the node is the positive direction.The optimization results of scheme II indicate that the candidate locations of T1 (4–25), T2 (11–18), and T6 (5–10) are selected to invest and install the switches in the 33‐bus test system. For the 69‐bus system, the candidate locations of T5 (36–50) and T6 (20–67) are selected to invest and install the switches during scheme II. The best installation plan for the SOPs and the best topology obtained for the network after reconfiguration in scheme III are reported in Tables 4 and 5 for IEEE 33‐bus and 69‐bus systems, respectively. In scheme III, two SOPs are installed with the capacities of 300 and 100 kVA between nodes (31–32) and (25–29), respectively, for the IEEE 33‐bus system. For the 69‐bus network, the result of scheme III is to obtain two SOPs with a capacity of 400 kVA in locations (56–57) and (63–64). Also, the results of scheme IV are presented in Tables 6 and 7. In scheme IV, two candidate locations of T1 (4–25) and T5 (20–14) for installing new tie switches are selected in the 33‐bus system, and only one SOP device with a capacity of 200 kVA is decided to install between nodes (31–32).4TABLESite and size of SOPs and configuration of the 33‐bus network (scheme III)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(7–8) (8–9)(7–8) (11–12)(7–8) (10–11)(14–15) (31–32)(14–15) (31–32)(14–15) (31–32)(25–29)(25–29)(25–29)The site and size of SOPSite(31‐32)(25‐29)Size (KVA)3001005TABLESite and size of SOPs and configuration of the 69‐bus network (scheme III)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(12–13) (13–14)(56–57) (63–64)(11–43)(12–13) (56–57)(63–64) (11–43)(13–21)(10–11) (13–14)(20–21) (56–57)(63–64)The site and size of SOPSite(56–57)(63–64)Size (KVA)4004006TABLESite and size of SOP, the location for installing the tie switches and configuration of the 33‐bus network (scheme IV)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(10–11) (12–13) (24–25) (28–29) (31–32) (21–8) (9–15) (11–18) (11–15) (30–16) (5–10)(7‐8) (8‐9) (12‐13) (24–25) (28–29) (31–32) (9–15) (11–18) (11–15) (30–16) (5–10)(10–11) (12–13) (24–25) (28–29) (31–32) (21–8) (9–15) (11–18) (11–15) (30–16) (5–10)The location for installing new tie switches(4–25)The site and size of SOPSite(31–32)(20–14)Size (KVA)2007TABLESite and size of SOP, the location for installing the tie switches and configuration of the 69‐bus network (scheme IV)Open switches in reconfiguration processLoading level: (1) HighLoading level: (2) MediumLoading level: (3) Low(15–16) (41–42) (45–46) (49–50) (56–57) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)(10–11) (14–15) (17‐18) (49‐50(49–50) (55–56) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)(10–11) (17–18) (44–45) (49–50) (55–56) (63–64) (13–21) (18–44) (10–48) (24–57) (20–67)The location for installing new tie switches(23–41)The site and size of SOPSite(63–64)(36–50)Size (KVA)300The objective function terms are shown in Tables 8 and 9 for all schemes.8TABLEAnnual costs of Schemes I‐IV and total energy losses of the 33‐bus networkScheme I(Without SOP and tie switches)Scheme IIScheme IIIScheme IVCostINV−SOP($)$Cos{t^{INV - SOP}}\;( \$ )$––12,5816290CostINV−TieSwitch+CostINV−TieLine($)$Cos{t^{INV - TieSwitch}} + \;Cos{t^{INV - TieLine}}\;( \$ )$–14,775–9850CostOPE−SOP($)$Cos{t^{OPE - SOP}}\;( \$ )$––24701235CostOPE−TieSwitch+CostOPE−TieLine($)$Cos{t^{OPE - TieSwitch}} + \;Cos{t^{OPE - TieLine}}( \$ )$–1812–1208CostLOSS($)$Cos{t^{LOSS}}\;( \$ )$8493559,79761,92151,099ELOSS(kWh)${E^{LOSS}}\;( {{\rm{kWh}}} )$106169074,746677,401963,8730Cost($)$Cost\;( \$ )$8493576,38476,97369,6829TABLEAnnual costs of Schemes I–IV and total energy losses of the 69‐bus networkScheme I(Without SOP and tie switches)Scheme IIScheme IIIScheme IVCostINV−SOP($)$Cos{t^{INV - SOP}}\;( \$ )$––25,1629436CostINV−TieSwitch+CostINV−TieLine($)$Cos{t^{INV - TieSwitch}} + \;Cos{t^{INV - TieLine}}\;( \$ )$–9850–9850CostOPE−SOP($)$Cos{t^{OPE - SOP}}\;( \$ )$––49411853CostOPE−TieSwitch+CostOPE−TieLine($)$Cos{t^{OPE - TieSwitch}} + \;Cos{t^{OPE - TieLine}}( \$ )$–1208–1208CostLOSS($)$Cos{t^{LOSS}}\;( \$ )$78,02664,24441,08645,992ELOSS(kWh)${E^{LOSS}}\;( {{\rm{kWh}}} )$975,330803,050513,578574,900Cost($)$Cost\;( \$ )$78,02675,30271,18968,339The results of Table 8 show that scheme IV is more economical than the other schemes. The annual cost of network expansion in scheme IV is 15253 $ (18%) less than that in scheme I. The effect of installing new tie switches in scheme IV compared to scheme III is to reduce the different objective function values as well as total energy losses of the network. With the simultaneous and optimal presence of SOPs and tie switches, the annual cost of energy losses is decreased by 33,836 $ (a reduction of 40%). Also, the amount of the total network energy losses reduction is 42,2960 KWH and 40%. All these comparisons have been made between scheme IV and the case without SOP and tie switches. In this way, it is obvious the optimal and simultaneous use of SOPs and tie switches will improve the operation of the network economically. Also, the final network configuration for the medium loading level in schemes III and IV is illustrated in Figures 2 and 3, respectively. Notably, the final connections of the test networks after solving the problem are presented in the relevant tables, and Figures 2 and 3 are drawn for the 33‐bus system, only to show an example of the allocation of SOPs in schemes III and IV.2FIGURELocation of SOPs in scheme III (medium loading level)3FIGURELocation of SOPs in scheme IV (medium loading level)Figures 4 and 5 show the expected values for the voltage and current of network buses, respectively in schemes II, III, and IV in the 33‐bus network. As can be seen, the simultaneous presence of SOPs and tie switches generally improves the average network voltage, and by reducing the average current level of the network, it further reduces energy losses. According to the results of Table 8, the optimal installation of tie switches and changing the network topology only through them (scheme II) reduce the energy losses of the network more than the optimal presence of SOPs in the network (scheme III).4FIGUREExpected values of 33‐bus network voltage in schemes II–IV5FIGUREExpected values of 33‐bus network current in schemes II–IV. (a) Active power transmission of SOP installed between nodes (31)–(32). (b) Reactive power compensation of SOP installed between nodes (31)–(32). (c) Active power transmission of SOP installed between nodes (25)–(29). (d) Reactive power compensation of SOP installed between nodes (25)–(29)Figure 6a,c illustrates that the operation strategy of SOPs installed in scheme III is adjusted according to the operating conditions of the network, including the supply and demand level of the distribution system. The diagrams related to the operation strategy of SOPs are drawn only for scheme III as an instance.6FIGUREOperation strategy of SOPs in scheme III. (a) Maximum gap values in each renewable generation scenario of scheme III (33‐bus test system). (b) Maximum gap values in each renewable generation scenario of scheme IV (33‐bus test system)When the renewable generation is low, such as in scenarios 1 and 2, SOP transmits the active power from node 31 into node 32 and from node 25 into node 29. In another hand, as the output of the wind turbine increasessuch as in scenarios 4 and 5, by adjusting the transmission of the active power into node 31 and node 25 through the SOPs installed between nodes (31–32) and (25–29), the fluctuations of wind turbine power output are mitigated. Also, the higher the loading level of the network, the lower the absolute amount of active power transmission of SOPs to help reduce power losses of the network.Figure 6b,d shows the reactive power support of SOPs installed between nodes (31–32) and (25–29) within their own capacity constraints. The higher the loading level of network, the higher the amount of reactive power compensation of SOPs to help meet the required reactive power of loads.The gap values in each renewable generation scenario of scheme III and IV are shown in Figure 7a,b. The maximum gap values in scheme III and IV are some with a 10−6 level and another some with a 10−5 level which are small enough to be considered as accurate7FIGUREMaximum gap values in each renewable generation scenario of scheme III and IVCONCLUSIONThis paper presented an optimal model for planning the simultaneous presence of SOPs and tie switches in active distribution networks. The optimal site and capacity of the SOPs and installation sites of the tie switches were determined simultaneously with consideration of the renewable generation characteristics, different loading conditions, and network topology changes.After linearization and convexification, the original optimization problem is solved in the form of the MISOCP model to moderate the execution time of the program. Results showed the application of SOPs could improve the operational economy, reduce the annual energy losses of distribution networks, and improve the voltage profile. Another suitable extension way for enabling network reconfiguration is to coordinate SOPs with conventional switching devices such as tie switches, considering the relatively high investment cost of power electronics. Optimal installation of tie switches simultaneously with the optimal presence of SOPs led to an additional reduction in the total network expansion costs compared to the case in which only SOPs are used. Also, the results show that planning the presence of SOPs in the network alone will reduce energy losses by 27% during the year, while the optimal presence of SOPs with the tie switches reduces energy losses in the network by up to 40%. These percentages are reported for the 33‐bus network.Thus, although the simultaneous investment of SOPs and tie switches at first glance has increased the network costs, regarding reducing the cost of energy losses, the total network expansion costs are decreased.NOMENCLATURESetsΩb${\Omega _b}$Set of all branchesΩnc${\Omega _{nc}}$Set of all new candidate tie linesΩSOP${\Omega _{SOP}}$Set of all candidate locations of SOPsΩbus${\Omega _{bus}}$Set of busesΩbus(i)${\Omega _{bus}}( i )$Set of all adjacent buses of bus iIndicesi,j$i,{\rm{\;}}j$Indices of buses, from 1 to NN${N_N}$LL$LL$Indices of loading level, from 1 to NL${N_L}$sIndices of the operation scenarios of renewable generation from 1 to Ns${N_s}$ij$ij$Indices of branchesVariablesPs,ij,LLLine,Qs,ij,LLLine$P_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line},{\rm{\;}}Q_{s,{\rm{\;}}ij,{\rm{\;}}LL}^{Line}$Active and reactive power flow of branch ij$ij$ in scenario s and loading level LL$LL$Is,ij,LL,ls,ij,LL${I_{s,{\rm{\;}}ij,{\rm{\;}}LL}},{\rm{\;}}{l_{s,{\rm{\;}}ij,{\rm{\;}}LL}}{\rm{\;}}$Current magnitude and its square of branch ij$ij$ in scenario s and loading level LL$LL$Vs,i,LL,vs,i,LL${V_{s,{\rm{\;}}i,{\rm{\;}}LL}},{\rm{\;}}{v_{s,{\rm{\;}}i,{\rm{\;}}LL}}$Voltage magnitude and its square at busi${\rm{\;}}i$ in scenario s and loading level LL$LL$Ps,i,LL,Qs,i,LL${P_{s,{\rm{\;}}i,LL}},{\rm{\;}}{Q_{s,{\rm{\;}}i,LL}}$Total active and reactive power injection at node i in scenario s and loading level LL$LL$Ps,iDG,Qs,iDG$P_{s,{\rm{\;}}i}^{DG},{\rm{\;}}Q_{s,{\rm{\;}}i}^{DG}$Active and reactive power generation of DG at node i in scenario sPs,i,LLSOP,Qs,i,LLSOP$P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}SOP},{\rm{\;}}Q_{s,{\rm{\;}}i,{\rm{\;}}LL}^{SOP}$Active and reactive power injection by SOP at node i in scenario s and loading level LL$LL$Ps,i,LLSOP,LOSS$P_{s,{\rm{\;}}i,{\rm{\;}}LL}^{SOP,{\rm{\;}}LOSS}$Active power losses of SOP at nodei${\rm{\;}}i$ in scenario s and loading level LL$LL$SijSOP$S_{ij}^{SOP}$Maximum planned capacity of the SOP in ijth$i{j^{th{\rm{\;}}}}$branchαij,LL${\alpha _{ij,{\rm{\;}}LL}}$Binary variable expressing the branch status takes value of 1 if the branch ij$ij$ is closed and 0 otherwiseβij,LL${\beta _{ij,{\rm{\;}}LL}}$Binary variable which is equal to 1 if bus j is the parent of bus i and to 0 otherwisemij${m_{ij}}$The number of modules that SOP consists of themParametersNN${N_N}$Total number of the busesNL${N_L}$Total number of the loading levelsNs${N_s}$Total number of the operation scenariosPi,LLLOAD,Qi,LLLOAD$P_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD},\;Q_{{\rm{\;}}i,{\rm{\;}}LL}^{{\rm{\;}}LOAD}$Active and reactive power consumption in loading levels LLat node irij,xij${r_{ij}},{\rm{\;}}{x_{ij}}$Resistance and reactance of branch ij$ij$V¯,V$\bar V,V$Lower and upper boundaries of system voltageI¯$\bar I$Maximum current magnitude of branch ij$ij$dDiscount ratenSOP${n_{SOP}}$SOP economical service lifenTieSwitch${n_{TieSwitch}}$Tie switch economical service lifenTieLine${n_{TieLine}}$Tie line economical service lifecostSOP$cos{t^{SOP}}$SOP unit capital costcostTieSwitch$cos{t^{TieSwitch}}$Investment cost of tie switchescostTieLine$cos{t^{TieLine}}$Investment cost of tie linesηSOP${\eta ^{SOP}}$Coefficient of the annual operational costs of SOPηTieSwitch,ηTieLine${\eta ^{TieSwitch}},{\rm{\;}}{\eta ^{TieLine}}$Coefficient of operation and maintenance cost of tie switches and tie linesδ…${\delta ^ {\ldots} }$Annuity factorcostElectricity$cos{t^{Electricity}}$Electricity pricelengthij$lengt{h_{ij{\rm{\;}}}}$Length of the tie line ij$ij$DRLL$D{R_{LL}}$Duration of each loading levelSmodule${S^{module}}$Capacity of each module used in SOPAiSOP$A_i^{SOP}$Loss coefficient of SOP at node iρ(s)$\rho ( s )$The probability corresponding to the sth scenariosb$sb$Substation busbarAUTHOR CONTRIBUTIONSM.E.: Conceptualization, Investigation, Methodology, Software, Writing—original draft, Writing—review and editing. M.S.S.: Conceptualization, Investigation, Methodology, Supervision, Writing—review and editing.FUNDING INFORMATIONThere is no funding to report for this submission.CONFLICT OF INTERESTThe authors declare that there are no conflicts of interest regarding the publication of this paper.DATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESLopes, J.A.P., Hatziargyriou, N., Mutale, J., Djapic, P., Jenkins, N.: Integrating distributed generation into electric power systems: A review of drivers, challenges and opportunities. Electr. Power Syst. Res. 77(9), 1189–1203 (2007)Allan, G., Eromenko, I., Gilmartin, M., Kockar, I., McGregor, P.: The economics of distributed energy generation: A literature review. Renewable Sustainable Energy Rev. 42, 543–556 (2015)Trebolle, D., Gómez, T., Cossent, R., Frías, P.: Distribution planning with reliability options for distributed generation. Electr. Power Syst. Res. 80(2), 222–229 (2010)Cruz, M.R.M.: Stochastic management framework of distribution network systems featuring large‐scale variable renewable energy sources and flexibility options. Ph.D. dissertation, University of Beira Interior, Covilhã, Portugal (2019)Liang, X.: Emerging power quality challenges due to integration of renewable energy sources. IEEE Trans. Ind. Appl. 53(2), 855–866 (2017)Diaaeldin, I., Aleem, S.A., El‐Rafei, A., Abdelaziz, A., Zobaa, A.F.: Optimal network reconfiguration in active distribution networks with soft open points and distributed generation. Energies 12(21), 4172 (2019)Zhao, J., Zheng, T., Member, S., Litvinov, E.: Variable resource dispatch through do‐not‐exceed limit. IEEE Trans. Power Syst. 30(2), 820–828 (2015)Aithal, A.: Operation of soft open point in a distribution network under faulted network conditions. Ph.D. dissertation, Cardiff University, Wales, (2018)Sun, F., Ma, J., Yu, M., Wei, W.: Optimized two‐time scale robust dispatching method for the multi‐terminal soft open point in unbalanced active distribution networks. IEEE Trans. Sustainable Energy 12(1), 587–598 (2021)Cao, W., Wu, J., Jenkins, N., Wang, C., Green, T.: Operating principle of soft open points for electrical distribution network operation. Appl. Energy 164, 245–257 (2016)Hafezi, H., Laaksonen, H.: Autonomous soft open point control for active distribution network voltage level management. In: 2019 IEEE Milan PowerTech, Milan (2019)Bloemink, J.M., Green, T.C.: Benefits of distribution‐level power electronics for supporting distributed generation growth. IEEE Trans. Power Delivery 28(2), 911–919 (2013)Cao, W., Wu, J., Jenkins, N.: Feeder load balancing in MV distribution networks using soft normally‐open points. In: IEEE PES Innovative Smart Grid Technologies, Europe, Istanbul, Turkey (2015)Cao, W.: Soft open points for the operation of medium voltage distribution networks. Ph.D dissertation, Cardiff University, Wales (2015)Qi, Q., Wu, J., Zhang, L., Cheng, M.: Multi‐objective optimization of electrical distribution network operation considering reconfiguration and soft open points. Energy Procedia 103(April), 141–146 (2016)Long, C., Wu, J., Thomas, L., Jenkins, N.: Optimal operation of soft open points in medium voltage electrical distribution networks with distributed generation. Appl. Energy 184, 427–437 (2016)Qi, Q., Wu, J., Long, C.: Multi‐objective operation optimization of an electrical distribution network with soft open point. Appl. Energy 208(May), 734–744 (2017)Ji, H., Li, P., Wang, C., et al.: A strengthened SOCP‐based approach for evaluating the distributed generation hosting capacity with soft open points. Energy Procedia 142, 1947–1952 (2017)Bai, L., Jiang, T., Li, F., Chen, H., Li, X.: Distributed energy storage planning in soft open point based active distribution networks incorporating network reconfiguration and DG reactive power capability. Appl. Energy 210, 1082–1091 (2018)Qi, Q., Wu, J.: Increasing distributed generation penetration using network reconfiguration and soft open points. Energy Procedia 105, 2169–2174 (2017)Yao, C., Zhou, C., Yu, J., Xu, K., Li, P., Song, G.: A sequential optimization method for soft open point integrated with energy storage in active distribution networks. Energy Procedia 145, 528–533 (2018)Ji, H., Wang, C., Li, P., et al.: An enhanced SOCP‐based method for feeder load balancing using the multi‐terminal soft open point in active distribution networks. Appl. Energy 208(August), 986–995 (2017)Point, S.O.: Minimization of network power losses in the AC‐DC hybrid distribution network through network reconfiguration using soft open point. Electronics 10(3), 326 (2021)Ji, H., Wang, C., Li, P., Ding, F., Wu, J.: Robust operation of soft open points in active distribution networks with high penetration of photovoltaic integration. IEEE Trans. Sustainable Energy 10(1), 280–289 (2019)Shafik, M.B., Rashed, G.I., Chen, H., Elkadeem, M.R., Wang, S.: Reconfiguration strategy for active distribution networks with soft open points. In: Proceedings of the 14th IEEE Conference on Industrial Electronics and Applications (ICIEA), Xi'an, China, pp. 330—334 (2019)Li, P., Song, G., Ji, H., Zhao, J., Wang, C., Wu, J.: A supply restoration method of distribution system based on Soft Open Point. In: IEEE PES Innovative Smart Grid Technologies ‐ Asia (ISGT‐Asia), Melbourne, Australia, pp. 535–539 (2016)Li, P., Ji, H., Wang, C., et al.: Optimal operation of soft open points in active distribution networks under three‐phase unbalanced conditions. IEEE Trans. Smart Grid 10(1), 380–391 (2019)Li, P., Ji, H., Wang, C., et al.: A coordinated control method of voltage and reactive power for active distribution networks based on soft open point. IEEE Trans. Sustainable Energy 8(4), 1430–1442 (2017)Li, P., Ji, H., Yu, H., et al.: Combined decentralized and local voltage control strategy of soft open points in active distribution networks. Appl. Energy 241(March), 613–624 (2019)Zhao, J., Yao, M., Yu, H., Song, G., Ji, H., Li, P.: Decentralized voltage control strategy of soft open points in active distribution networks based on sensitivity analysis. Electron 9(2), 295 (2020)Cao, W., Wu, J., Jenkins, N., Wang, C., Green, T.: Benefits analysis of Soft Open Points for electrical distribution network operation. Appl. Energy 165(March), 36–47 (2016)Wang, C., Song, G., Li, P., Ji, H., Zhao, J., Wu, J.: Optimal siting and sizing of soft open points in active electrical distribution networks. Appl. Energy 189, 301–309 (2017)Zhang, L., Shen, C., Chen, Y., Huang, S., Tang, W.: Coordinated optimal allocation of DGs, capacitor banks and SOPs in active distribution network considering dispatching results through bi‐level programming. Energy Procedia 142, 2065–2071 (2017)Shafik, M.B., Rashed, G.I., Chen, H.: Optimizing energy savings and operation of active distribution networks utilizing hybrid energy resources and soft open points: Case study in Sohag, Egypt. IEEE Access 8, 28704–28717 (2020)Prakash, P., Khatod, D.K.: Optimal sizing and siting techniques for distributed generation in distribution systems: A review. Renewable Sustainable Energy Rev. 57, 111–130 (2016)Abazari, A., Soleymani, M.M., Kamwa, I., et al.: A reliable and cost‐effective planning framework of rural area hybrid system considering intelligent weather forecasting. Energy Rep. 7, 5647–5666 (2021)Zadsar, M., et al.: Central situational awareness system for resiliency enhancement of integrated energy systems. In: 2021 IEEE 4th International Conference on Computing, Power and Communication Technologies (GUCON), Kuala Lumpur, Malaysia, pp. 1–6 (2021)Fu, Q., Yu, D., Ghorai, J.: Probabilistic load flow analysis for power systems with multi‐correlated wind sources. In: IEEE Power and Energy Society General Meeting, Detroit, MI, USA, pp. 1–6 (2011)Wen, S., Lan, H., Fu, Q., Yu, D.C., Zhang, L.: Economic allocation for energy storage system considering wind power distribution. IEEE Trans. Power Syst. 30(2), 644–652 (2015)Atwa, Y.M., El‐Saadany, E.F.: Probabilistic approach for optimal allocation of wind‐based distributed generation in distribution systems. IET Renewable Power Gener. 5(1), 79–88 (2011)Morales, J.M., Pérez‐Ruiz, J.: Point estimate schemes to solve the probabilistic power flow. IEEE Trans. Power Syst. 22(4), 1594–1601 (2007)Wu, L., Jiang, L., Hao, X.: Optimal scenario generation algorithm for multi‐objective optimization operation of active distribution network. In: Chinese Control Conference CCC, Dalian, China, pp. 2680–2685 (2017)Farivar, M., Low, S.H.: Branch flow model: Relaxations and convexification‐part i. IEEE Trans. Power Syst. 28(3), 2554–2564 (2013)Dorostkar‐Ghamsari, M.R., Fotuhi‐Firuzabad, M., Lehtonen, M., Safdarian, A.: Value of distribution network reconfiguration in presence of renewable energy resources. IEEE Trans. Power Syst. 31(3), 1879–1888 (2016)Baran, M.E., Wu, F.F.: Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans. Power Delivery 4(2), 1401–1407 (1989)Savier, J. S., Das, D.: Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans. Power Delivery 22(4), 2473–2480 (2007)AAPPENDIXTEST DISTRIBUTION NETWORKSSee Figure A1 and Figure A2.A1FIGURE33‐bus test systemA2FIGURE69‐bus test systemBAPPENDIXCHARACTERISTICS OF TEST DISTRIBUTION NETWORKSSee Table B1, Table B2, Table B3, and Table B4.B1TABLECandidate locations of tie switches (TS)33‐bus69‐busTS candidateFrom busTo busFrom busTo busT1(4)(25)(23)(41)T2(11)(18)(18)(44)T3(11)(15)(10)(48)T4(30)(16)(24)(57)T5(20)(14)(36)(50)T6(5)(10)(20)(67)B2TABLEParameters of wind generatorsLocation101617303233‐busCapacity (kVA)500200150200300Location2133466269‐busCapacity (kVA)300100200400B3TABLEMultilevel modeling of load in optimization problemLoading level(1) High(2) Medium(3) LowTypePercentage of peak loadDuration (h)Percentage of peak loadDuration (h)Percentage of peak loadDuration (h)Residential100219057.53650302920Commercial100219087.53650302920Industrial100219079.03650722920B4TABLEValues of the studied network parametersParametersValueDiscount rate0.08SOP economical service life, year20Tie switch economical service life, year15Tie line economical service life, year35SOP unit capital cost, $/kVA308.8Coefficient of the annual operational costs of SOP, $/KVA6.17Investment cost of tie switches, K$4.7Coefficient of operation and maintenance cost of tie switches0.02The investment cost of tie lines, K$/km60Coefficient of operation and maintenance cost of tie lines0.01Length of the tie line (km)0.85Electricity price, $/kWh0.08Minimum optimum capacity of SOP, kVA100Loss coefficient of SOP0.02

Journal

"IET Generation, Transmission & Distribution"Wiley

Published: Jan 1, 2023

Keywords: distribution planning and operation; mathematical programming; mathematics computing; power distribution economics; renewable energy sources; resource allocation

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