Distribution System Reconfiguration with Soft Open Point for Power Loss Reduction in Distribution Systems Based on Hybrid Water Cycle Algorithm

Abstract

In this paper, the role of soft open point (SOP) is investigated with and without system re-configuration (SR) in reducing overall system power losses and improving voltage profile, as well as the effect of increasing the number of SOPs connected to distribution systems under different scenarios using a proposed hybrid water cycle algorithm (HWCA). The HWCA is formulated to enhance the water cycle algorithm (WCA) search performance based on the genetic algorithm (GA) for a complex nonlinear problem with discrete and continuous variables represented in this paper by SOP installation and SR. The WCA is one of the most effective optimization algorithms, however, it may have difficulty striking a balance between exploration and exploitation due to the nature of the proposed nonlinear optimization problem, which mostly causes slow convergence and poor robustness. Consequently, the HWCA proposed in this paper is an efficient solution to improve the balance between exploration and exploitation, which in turn leads to improving the WCA’s overall performance without the possibility of getting trapped in local minima. Several cases are studied and conducted on an IEEE 33-node and the IEEE 69-node to investigate the real benefit gained from using SOPs alone or simultaneously with the SR. Based on the obtained results, the proposed HWCA succeeds in enhancing the performance of the proposed test systems considerably in terms of loss reduction (e.g., 31.1–63.3% for IEEE 33-node and 55.7–82.1% for IEEE 69-node compared to the base case) and voltage profile when compared to the base case while maintaining acceptable voltage magnitudes in most cases. Furthermore, the superiority of the proposed method based on the HWCA is validated when compared with the GA and WCA separately for both test systems. The obtained results show the outperformance of the proposed HWCA in attaining the best optimal solution with the least number of iterations.

1. Introduction

Due to unanticipated load changes and poor configuration of distribution systems (DSs), the likelihood of voltage violations and increased system losses in traditional DSs has become a major challenge for system operators [1]. Because traditional voltage regulators often give a limited reaction and discrete voltage regulation, it is difficult to satisfy the demand for desired voltage regulation and loss reduction utilizing just traditional VAR devices such as on-load tap changers and switchable capacitor banks [2]. Several strategies have been presented by researchers to address these challenges. System reconfiguration (SR) is considered one of the most popular and cost-effective methods to enhance the performance of DSs, which has been proposed by many researchers over the past few decades. These researchers hypothesized that further techniques, such as capacitor installation and cable sizing upgrades, may exceed the DS utilities’ financial constraints. Moreover, DSs are often equipped with switches that can be used to perform SR with the objective to balance loads and hence improve loss reduction. SR can be efficiently conducted by altering the state of sectionalized (closed) and tie (opened) switches while taking the system’s radiality and operational constraints into account [3,4,5,6,7,8]. Regardless of the advantages of SR in terms of improving voltage profile and reducing system losses, further performance improvement in DSs is still restricted due to radial topology and construction limits.

With the evolution of power electronic technology, power electronic devices are nowadays playing a significant role in changing the traditional distribution network into a technologically sophisticated network. The soft open point (SOP), which is an innovative electronic device that generally uses back-to-back voltage source converters (VSC), has a significant influence on the design and functionality of the DS [9,10]. SOP can efficiently modify different operational set points by managing the active and reactive power flowing across nearby branches or feeders in real-time. Additionally, SOP has the ability to inject or absorb reactive power at their terminals to improve system loss reduction and the voltage profile of the network [10]. SOP is also distinguished by its capacity to isolate any voltage and current disturbances and regulate peak currents brought on by faults. As a result, the adoption of SOP will considerably improve the network’s operating conditions, allowing for more operational flexibility and lower operating costs [11].

In this regard, several researchers have investigated the integration of SOP with DSs intending to minimize system losses, improve voltage profile, and increase affordability without affecting DS radiality [11,12,13,14,15]. For example, the taxi-cab and multi-objective particle swarm optimization methods were utilized to discover the optimal set-points of the SOP with the goal of minimizing power loss, improving feeder load balancing, and optimizing voltage profile [12]. The genetic algorithm (GA) was used to determine optimal SOP location and active/reactive power setting points in unbalanced distribution networks, considering the active power losses and voltage unbalance index as objective functions with correlated uncertain distributed generators [13]. In [14], the proposed nonlinear optimization problem was addressed using the mixed-integer second-order cone programming approach in order to save total operating and SOP expenses. Furthermore, the optimal SOP placement was selected utilizing the power flow betweenness index and voltage violation risk index. In [15], the SOP planning problem was handled based on a bi-level strategy to optimize DS resilience. The upper-level problem was solved using genetic algorithms to address the SOP location and capacity. The particle swarm optimization technique was also used to optimize SOP functions at the lower level.

To gain further improvement in the performance of the DS, numerous researchers have adopted simultaneous optimization of SR and SOP size and position to provide considerable advantages to DSs with a variety of objectives. For example, to minimize network power losses, a novel methodology of SR was provided based on the AC-SOP and the DC-SOP [16]. In [17], a novel optimization method known as the Archimedes optimization algorithm was applied to the multi-objective optimization problem in order to maximize DG penetration and decrease system losses through consecutive SR and SOP deployments. In [18], to discover the optimal radial topologies and minimize power loss in DSs, the discrete-continuous hyper-spherical search method was suggested to handle the SR approach while taking into account various distributed generations, SOPs, and SR strategies. In [19], a bi-level multi-objective optimization method was suggested to maximize hosting capacity and reduce overall active losses of DSs with simultaneous SR and SOP allocation, while ensuring operational constraint limitations. In [20], A modified version of the particle swarm optimization was created to solve SR and SOP integration problems in active DSs with the goal to lessen system power losses, improve system efficiency in steady-state operation, and enhance voltage profiles. Furthermore, in [20], choosing the optimal number, placement, and size of DGs was also considered with SR and SOP integration problems in DSs. In [21], with the goals of loss reduction, load balancing, and increasing DG penetration level, the SR and SOP operating problems were presented by a multi-objective framework utilizing the Pareto optimality.

In previously mentioned studies, various approaches were utilized to discover the appropriate allocation and size of SOP with SR by limiting SOPs’ location to only tie switches. However, these may not achieve the optimal solution, particularly in large DSs. Unlike the other mentioned studies, this paper proposes that all sectionalized and tie switches could be efficient candidate locations for SOP installations. Additionally, for better improvement in DS performance, this paper proposes concurrent integration of SR and SOP by improving the search performance of the original water cycle algorithm (WCA) based on GA to handle the discrete–continuous search space. The WCA has been effectively employed in a variety of fields to solve constrained as well as unconstrained complex linear and nonlinear optimization problems with high accuracy and rapid convergence speed. The WCA is primarily designed to solve optimization problems with a continuous search space [22]. However, this conventional WCA often struggles to maintain an efficient balance between exploitation and exploration tendencies when it comes to specific optimization problems, such as a mixed-variables nonlinear optimization problem, which mostly leads to premature convergence. As a result, several researchers enhanced the conventional WCA to address the discrete and continuous search space in complex combinatorial optimization problems [23,24,25,26,27].

The main contribution of this paper is to propose a hybrid water cycle algorithm (HWCA) for concurrently solving SR and SOP installation problems to provide the best operation of DSs in terms of system loss and voltage profile. This hybridization eliminates the disadvantages of the original WCA, especially for the mixed-variables nonlinear optimization problem (i.e., limited exploration features due to the discrete–continuous nature of the proposed problem). It also emphasizes the benefits of combining the GA with WCA to deal with both discrete and continuous search spaces and improve the balance between exploration and exploitation phases. As a result, this efficient approach improves the search performance of the original WCA, allowing it to find the optimum solution without being trapped in local minima. The HWCA mainly aims to optimize the SR and SOP integration simultaneously for minimizing system power losses and improving the voltage profile in the DS while considering all operational constraints. Furthermore, unlike previous research, this paper provides a comprehensive evaluation of potential SOP locations that include not just tie switches, but also sectionalizing switches. Additionally, in this paper, the impact of increasing the number of SOPs coupled to distribution networks is evaluated with and without SR in lowering total active losses and improving voltage profiles under various scenarios. Finally, to validate the proposed method, the HWCA is compared to the WCA and the GA. In the comparison with the WCA, a sigmoid function is used to control the type of search space and its boundaries in order to avoid the continuous random movement of stream-to-river and stream-to-sea flows [28]. The main advantage of the sigmoid function is that it changes the values of the random movement of control variables selected by the original WCA to fall within the boundaries of the selective search space due to the discrete nature of SR (i.e., certain switches that are already present in the DS).

The remainder of this paper is organized as follows: Section 2 presents the formulation of the proposed problem. Section 3 presents a full description of the proposed HWCA. The numerical results and discussions are provided in Section 4. The conclusion is then provided in Section 5.

2. Problem Formulation

In this paper, a complex optimization problem with discrete and continuous variables represented by the SOP location and size along with SR is solved using the proposed HWCA to minimize system losses and improve the system voltage profile in radial DSs. The primary objective function taken into consideration in this paper is total power system losses, considering the operational and SOP constraints. To demonstrate the viability of the suggested method for reducing system losses as much as possible, eight cases, including the base case, will be used as examples. A description of the SOP modeling, objective function, and operational constraints is provided below:

2.1. SOP Modeling

SOP is a promising power electronic device recently used in radial distribution networks to replace sectionalizing switches as shown in Figure 1, or tie switches as shown in Figure 2. This cutting-edge power electronic device can effectively provide superior system performance in terms of reducing system losses, improving the system voltage profile, and balancing load flow [9]. This may be accomplished in an effective manner by providing quick, dynamic, and continuous active and reactive power flow control among nodes or feeders during normal network operating conditions, as well as fast fault isolation and supply restoration during abnormal events [10]. Figure 1 and Figure 2 depict the simple structure of two terminals’ VSCs placed in sectionalizing and tie switches as considered in this paper. The output reactive power of the two converters is independent since there is a DC bus connecting them [11]. The following equations can be used to simulate the proposed SOP topology:

$$P_n^{SOP}+P_m^{SOP}+P_n^{SO{P}_{loss}}+P_m^{SO{P}_{loss}}=0,$$

(1)

where $P_n^{SOP},P_m^{SOP}$ are the injected active powers of SOP at the $n^{th}$ and $m^{th}$ nodes, respectively, and $P_n^{SO{P}_{loss}}$ and $P_m^{SO{P}_{loss}}$ denote internal power losses of SOP converters at the $n^{th}$ and $m^{th}$ nodes, respectively, which can be calculated by multiplying the loss coefficient and the injected apparent power of SOP at the $n^{th}$ and $m^{th}$ nodes, respectively. Given that SOP has sufficient operational efficiency, created internal power losses from a small-scale power transfer can be ignored [18]. Lossless SOP is therefore taken into account in this paper. In the case of lossless SOP installation, the summation of the injected SOP powers to the $m^{th}$ and $n^{th}$ nodes is equal to zero [12]. The active power boundaries for SOP are described in Equation (2):

$$P_n^{SOP}+P_m^{SOP}=0,$$

(2)

The total reactive powers that SOP injects into the DS should not be more than the total reactive powers of the system loads, which may be described in Equation (3):

$${\sum }_{k=1}^{N_{SOP}}(Q_n^{SOP}\left(k\right)+Q_m^{SOP}\left(k\right)\le {\sum }_{u=1}^{N_{load}}{Q}_u^l, \forall u\in N_{load}, \forall k\in N_{SOP},$$

(3)

where $Q_u^l$ represents the load reactive power at the $u^{th}$ node; $N_{load}$ and $N_{SOP}$ represent the total number of loads and SOPs, respectively; and $Q_n^{SOP}$ and $Q_m^{SOP}$ are the injected reactive powers of SOP at the $n^{th}$ and $m^{th}$ nodes, respectively. The SOP capacity limits can be expressed in Equations (4) and (5):

$$\sqrt{{\left(P_n^{SOP}\right)}^2+{\left(Q_n^{SOP}\right)}^2}\le S_{rated}^{SOP},$$

(4)

$$\sqrt{{\left(P_m^{SOP}\right)}^2+{\left(Q_m^{SOP}\right)}^2}\le S_{rated}^{SOP},$$

(5)

where $S_{rated}^{SOP}$ is the rated size of SOP.

2.2. Objective Function (OF)

The proposed OF is formulated in this paper based on the active power losses in a DS while maintaining all operational and SOP constraints within acceptable limits [12]. The OF is expressed in Equation (6):

$$\underset{X}{\min }{P}_{loss}=\sum _{d=1}^{N_{br}}{\alpha }_d\left(\frac{P_d^2+Q_d^2}{{\left|V_d\right|}^2}.r_d\right),$$

(6)

where $X$ indicates the decision vector made up of the sectionalizing and tie switches status and SOP sizing and sitting; $N_{br}$ represents the number of total branches; the value of ${\alpha }_d$ denotes the connection (${\alpha }_d=1$) or disconnection $({\alpha }_d=0)$ of the $d^{th}$ branch; active power, reactive power, and the voltage at sending of the $d^{th}$ branch are represented by $P_d$, $Q_d$, $V_d$, respectively; and $r_d$ is the resistance of the $d^{th}$ branch. To solve Equation (6), equality and inequality operational and SOP constraints of DSs are considered [18,19]. The problem constraints are given as follows:

• The voltage magnitude of each node in the DS must be maintained within the permitted boundaries for the inequality constraints specified in Equation (7):

  $$V_{lower}\le \left|{\tilde{V}}_n\right|\le V_{upper},\forall n\in N_{node},$$

  (7)

  where ${\tilde{V}}_n$ is the voltage at the node $n$ and $V_{lower}$ and $V_{upper}$ are lower and upper acceptable voltage limits (i.e., $V_{lower}=0.95 p.u$ and $V_{upper}=1.05 p.u$), respectively, and $N_{node}$ represents the total number of system nodes.
• Another inequality constraint that is taken into account is the branch’s current limitation, which is expressed in Equation (8):

  $$\left|I_d\right|\le I_d^{max}, \forall d\in N_{br},$$

  (8)

  where $I_d$ and $I_d^{max}$ represent the current and the maximum current permitted for the $d^{th}$ branch.
• To ensure that all loads are kept connected and powered by the main substation during network reconfiguration, a radiality of the DS must be maintained. As a result, the radiality constraint is added as an equality constraint to the proposed objective function and is represented in Equation (9):

  $$N_{br}=N_{node}-1$$

  (9)
• SOP constraints can be expressed in Equations (2)–(5), as well as in Equations (10) and (11):

  $$Q_{min}^{SOP-n}\le Q_n^{SOP}\le Q_{max}^{SOP-n},$$

  (10)

  $$Q_{min}^{SOP-m}\le Q_m^{SOP}\le Q_{max}^{SOP-m},$$

  (11)

  where $Q_{min}^{SOP-n}$ and $Q_{max}^{SOP-n}$ represent the minimum and maximum reactive power constraints of SOP injected to the $n^{th}$ node, respectively; $Q_{min}^{SOP-m}$ and $Q_{max}^{SOP-m}$ indicate the minimum and maximum reactive power constraints of SOP injected to the $m^{th}$ node, respectively; and $Q_n^{SOP}$ and $Q_m^{SOP}$ are the injected reactive powers of SOP at the $n^{th}$ and $m^{th}$ nodes, respectively.

3. Description of Proposed HWCA

Many concepts of metaheuristic algorithms are inspired by nature. As a result, many researchers have drawn inspiration for their approaches from observations of these fascinating natural events, which are being controlled by a well-organized system. One of these natural systems that can be used to solve a wide range of difficult optimization problems is the water cycle process. The WCA was first developed by Eskandar et al. in 2012 [22] to solve constrained nonlinear optimization problems. Then, it was further improved by taking into account mixed-integer nonlinear optimization problems [29,30]. The WCA was created by researching how water circulates naturally, starting with raindrops produced by the evaporation process. Then raindrops gradually gathered to form streams, several of which were thereafter either moved toward rivers or sea. The following are the steps of the HWCA [22]:

3.1. Initial Population

An initial population matrix M of size $N_p\times N_v$ is represented in Equation (12). Each row in the matrix M consists of raindrops, taking into account the upper and lower constraints of the control variables (i.e., UC and LC), and can be considered an initial candidate solution for the proposed optimization problem [22].

$$M=\left[\begin{array}{ccc}{R}_{11}& \cdots & R_{1,Nv}\\ ⋮& \ddots & ⋮\\ R_{Np,1}& \cdots & R_{Np,Nv}\end{array}\right]=\left[\begin{array}{c}{R}_{D1}\\ ⋮\\ R_{D,Np}\end{array}\right],$$

(12)

where $N_p$ and $N_v$ indicate the population size and the total number of control variables, respectively, and $R_{D1}$ represents a raindrop that contains a vector of each candidate solution.

3.2. Evaluation of Each Candidate’s Solution

The cost value of each candidate solution represented by $R_{Di}$ in the matrix $M$ is evaluated using a fitness function $F_i$ considering problem constraints, which can be expressed in Equation (13):

$$F_i=f\left(R_{Di}\right), i\in N_p$$

(13)

The initial population and evaluation of each candidate’s solution are represented in block 1 of the main flowchart shown in Figure 3.

3.3. Raindrops Classification

In this step, after the cost value of each candidate solution has been evaluated using Equation (13), evaluated raindrops $R_{Ds}$ are categorized in ascending order. The best $R_{Di}$ with the lowest cost value is picked to be the sea and the best $R_{Ds}$ number is chosen to be the rivers, $N_r$. Finally, the remaining $R_{Ds}$ are assumed to be streams $N_{sr}$ that may flow into the sea or the rivers [22]. The classification matrix $R_{Ds}$ can be expressed in Equation (14):

$$R_{Ds}=\left[\begin{array}{c}{R}_{\mathrm{D},sea}\\ R_{\mathrm{D},{\mathrm{river}}_1}\\ \begin{array}{c}⋮\\ R_{\mathrm{D},{\mathrm{river}}_{N_r}}\\ \begin{array}{c}{R}_{\mathrm{D},{\mathrm{stream}}_1}\\ ⋮\\ R_{\mathrm{D},{\mathrm{stream}}_{N_{sr}}}\end{array}\end{array}\end{array}\right]$$

(14)

The total of rivers combined with a single sea can be expressed in Equation (15):

$$N_{sn}=Sea+N_{r }$$

(15)

The streams can also be calculated using Equation (16)

$$N_{sr}=N_p-N_{sn}$$

(16)

3.4. Determining the Stream’s Maximum Flow Rate

Equation (17) is used to calculate the flow rate of streams that flow into rivers or directly into the sea [22]:

$$N_{sk}=round\left\{\left|\frac{F_k}{{\sum }_{i=1}^{N_{sn}}{F}_i}\right|\times N_{sr}\right\}, k\in N_{sn}$$

(17)

The raindrops classification and calculation of the flow intensity of streams are demonstrated in Block 2 of the main flowchart shown in Figure 3.

3.5. Update the Location of Streams and Rivers

In the original version of the WCA [22], the stream-to-river flow can be determined using a random distance given in Equation (18). Furthermore, the rivers can also flow to the sea with the same concept in Equation (18), so the new position for the streams and rivers can be expressed using Equations (19) and (20):

$$R\in \left(0,H\times d\right), H>1$$

(18)

$$R_{\mathrm{D},{\mathrm{stream}}_{i+1}}=R_{\mathrm{D},{\mathrm{stream}}_i}+rand\times H\times \left(R_{\mathrm{D},{\mathrm{river}}_i}-R_{\mathrm{D},{\mathrm{stream}}_i}\right)$$

(19)

$$R_{\mathrm{D},{\mathrm{river}}_{i+1}}=R_{\mathrm{D},{\mathrm{river}}_i}+rand\times H\times \left(R_{\mathrm{D},{\mathrm{sea}}_i}-R_{\mathrm{D},{\mathrm{river}}_i}\right),$$

(20)

where $H$ is a constant and can be chosen as 2, $d$ represents the distance between stream and river, and $rand$ is a random number between 0 and 1.

What can be deduced from Equations (19) and (20) is that the positions of streams and rivers are constantly adjusted in order to diversify the population and prevent becoming locked in a local optimum. These adjustments are based on a continuous search space. Therefore, a nonlinear optimization problem with either continuous or discrete control variables has been addressed by several modified WCAs [31,32,33,34] with different objectives. In this paper, the proposed problem aimed to be solved is also a nonlinear optimization problem with continuous and discrete control variables. These variables are categorized as discrete line numbers for the SR problem, discrete node numbers for SOP locations, and continuous SOP size limits. Therefore, to avoid becoming locked in a local optimum and handle continuous-discrete decision variables, the GA-based WCA is utilized to update the placement of intended streams and rivers.

As shown in Figure 3, the optimal new locations of assigned streams toward the sea and rivers as well as rivers toward the sea are illustrated in Blocks 3, 4, and 5. These optimal new locations can be efficiently discovered using GA, represented in Block 7.

Running WCA or GA optimizers, especially in big systems, may result in suboptimal results due to the nonlinear nature of the proposed problem. The proposed hybridization technique enhances the WCA generating and searching process, prevents becoming stuck at a local optimum point, and eventually solves this problem and yields the optimal solution. The GA is suggested in this paper to update the position of assigned streams and rivers because it is an evolutionary algorithm derived from the natural evolution and selection of human genetics and it can efficiently work in any search space [35]. Moreover, the GA increases the possibility of locating the best movement of streams and rivers because of its random nature. The GA procedure used in this paper can be summarized in the steps below [36]:

3.5.1. Initialization

In the general concept of the GA, the candidate solutions are randomly generated from an initial population. However, in this paper, candidate solutions are represented by designated streams and rivers, including the sea, selected by HWCA. The structure indicated in Figure 4 is categorized into two parts. The first part is integer control variables, which represent the discrete line numbers for the SR problem and SOP locations, and the second part is real control variables, which represent continuous SOP size limits. In this paper, when the GA is used to update the designated streams and river locations, we deal with each part of this initial structure separately in order to avoid selecting non-discrete numbers that could be located outside the range of a test system node, especially for SR and SOP location problems. As a result, each part of this structure (i.e., SOP location, SOP active power injection, SOP reactive power injection, and reconfiguration) has its own crossover and mutation operators used to combine candidate solutions to create new solutions.

The structure of each stream and river, including the sea, is shown in Figure 4:

Where $L$ represents SOP location; $L\in \left(N_s, N_{tie}\right)$, $N_s$ and $N_{tie}$ are the total number of sectionalizing and tie switches, respectively; $P$ and $Q$ are SOP active and reactive power injections, respectively; $S_w$ represents sectionalizing (open) and tie (closed) switches; and $S_w\in \left(N_s, N_{tie}\right)$.

3.5.2. Selection

In this stage, candidate individuals are chosen based on their cost functions, and individuals with better solutions have a higher chance of selection. In this paper, the best-evaluated individuals are already arranged by the HWCA in ascending order (i.e., minimum value) and categorized as sea, rivers, and streams.

3.5.3. Crossover

This operator combines two candidate individuals (parents) to create new individuals (offspring). If the new individual inherits the best features from both parents, it may outperform both. The simplest method used in this paper to perform crossover is to select a few crossover places based on a probability of crossover, which is selected in this paper as $p_c=0.9$.

3.5.4. Mutation

After a crossover operator is performed, mutation takes place in the offspring generated from the crossover operation, with the probability of mutation selected in this paper to be $p_m=0.01$. The use of mutation in the GA prevents falling into a local optimum and maintains the desired level of diversity in the population.

3.5.5. Evaluate the Candidate’s Solutions

The offspring produced by crossover and mutation operators are evaluated using the proposed cost-function while taking operational restrictions into account in this step of the GA process. The position of streams and rivers that flow towards the sea will be changed to the one with the best cost-evaluation.

3.6. Check the Evaporation and Raining Process

The major purpose of the evaporation condition in Equations (21) and (22) is to lessen the chances of the algorithm getting stuck in local solutions and to identify whether or not streams or rivers have arrived, or are sufficiently close to the sea. If the evaporation requirement is met, it signifies that the distance between streams and rivers, or rivers and the sea [22], is less than $d_{max}$.

$$\Vert R_{\mathrm{D},{\mathrm{sea}}_i}-R_{\mathrm{D},{\mathrm{river}}_i}\Vert <d_{max}$$

(21)

$$\Vert R_{\mathrm{D},{\mathrm{river}}_i}-R_{\mathrm{D},{\mathrm{stream}}_i}\Vert <d_{max},$$

(22)

where $d_{max}$ is a preset number that is close to zero, $i\in N_{iter}$, and $N_{iter}$ is the total number of iterations. In order to find the best solution, the value of $d_{max}$ is used to adjust the search intensity close to the sea. When iterations occur, this predefined value is incrementally updated and is defined using Equation (23):

$$d_{ma{x}_{i+1}}=d_{ma{x}_i}-\frac{d_{ma{x}_i}}{N_{iter}}$$

(23)

The raining process begins immediately to generate new raindrops whenever either one, or both, of the evaporation requirements in Equations (21) and (22) are satisfied [22]. Once it rains, new streams are created in various places. The raining process in this paper is carried out using a mutation operator and is based on the following random probability defined in Equation (24):

$$R_{\mathrm{D}}^{new}=LC+\mathrm{rand}\times \left(UC-LC\right)$$

(24)

The evaporation condition and raining process are demonstrated in Block 6 of the main flowchart shown in Figure 3.

The main structure of the proposed HWCA used in this paper is demonstrated using the flow chart shown in Figure 3.

4. Numerical Results and Discussions

In this paper, eight cases are considered and applied to two modified versions of the IEEE 33-Node and IEEE 69-node DSs utilizing the proposed HWCA to show and assess the effectiveness of the HWCA in solving SR and installation of SOP unit problems [37,38]. All tie and sectionalizing switches are examined as candidate switches for solving the SR and SOP allocation problem in both test systems. The maximum number of SOPs that can be placed on the provided test systems is two. However, the proposed approach may be used for any number of SOPs. Furthermore, in order to demonstrate that the HWCA is superior to other approaches in addressing the proposed problem with the goal of enhancing the voltage profile and reducing system loss, simulation results of the HWCA for case 7 are compared to those obtained by the WCA and GA, while the HWCA parameters that are initialized (e.g., population size of 80 and maximum iteration of 300) are shared with all cases and other compared optimization algorithms for both test systems. Seven different cases, including the base case, are simulated and categorized as follows:

• Base case: A power-flow solution without considering SOP installation and SR;
• Case 1: Only optimal SR without SOP installation;
• Case 2: One optimal SOP installation without SR;
• Case 3: One optimal SOP installation only at Tie switches without SR;
• Case 4: Two optimal SOP installations without SR.
• Case 5: Two optimal SOP installations only at Tie switches without SR.
• Case 6: Simultaneous optimal SR and one SOP installation (proposed method).
• Case 7: Simultaneous optimal SR and two SOP installations (proposed method).

In this paper, the proposed cost function indicated in Equation (6), operational limitations, and features of the system after SOP power injection and SR are evaluated using a modified version of the ladder-iterative power-flow technique [39]. Furthermore, evaluation of the proposed eight cases is performed using MATLAB and run on a personal computer with an Intel Core i7-3770 processor running at 3.40 GHz with 16 GB of RAM. The descriptions and optimal simulation results for IEEE 33-Node and 69-Node DSs are presented as follows:

4.1. Numerical Results of IEEE 33-Node Test System

The initial configuration system of the IEEE 33-bus test system includes 33 nodes, 37 branches, 32 normally-close sectionalized switches, and five normally-closed tie switches (e.g., $T_{33}$ to $T_{37}$), with a 12.6 kV (or 1 p.u) base system voltage as shown in Figure 5. The total real and reactive power loads are 3.72 MW and 2.3 MVAR, respectively, and the base active power loss is 202.67 kW. The limits of real and reactive power injected by SOPs are 0 to 2.5 MW and 0 to 2.5 MVAR, respectively, and the voltage magnitude constraints of all buses are set at 0.95 and 1.05 p.u.

The validity of the proposed HWCA in addressing SR and SOP installation problems is demonstrated in

Table 1. Numerical results for IEEE 33-Node with and without SOP installation.

Case	No. of SOP	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)
Base Case	---	---	---	---	---	202.67	0.9131/0.9970
SR (Case 1)	---	---	OS: 7, 9, 14, 32 CS: 33, 34, 35, 36	---	---	139.55	0.9378/0.9971
SOP without SR (Case 2)	1	5–6	OS: 5 CS: 33	−1.3758/1.3758	0.3324/1.6926	114.59	0.9512/0.9977
SOP without SR (Tie-switches) (Case 3)	1	8–21	---	1.0813/−1.0813	1.3660/0.3268	119.39	0.9454/0.9976
SOP without SR (Case 4)	2	5–6 30–31	OS: 5, 30 CS: 33, 36	−1.5554/1.5554, −0.5889/0.5889	0.3401/0.4996, 0.5000/0.4323	93.67	0.9599/0.9976
SOP without SR (Tie-switches) (Case 5)	2	25–29 12–22	---	−0.4190/0.4190 0.7439/−0.7439	0.4983/0.4956 0.4991/0.1397	94.71	0.9561/0.9976
SOP with SR (Case 6)	1	24–25	OS: 7, 9, 14, 17, 24 CS: 33, 34, 35, 36, 37	−0.9708/0.9708	0.3767/1.1359	92.22	0.9631/0.9976
SOP with SR (Case 7)	2	24–25 19–20	OS: 9, 14, 19, 24, 32 CS: 33, 34, 35, 36, 37	−0.8205/0.8205 −1.3458/1.3458	0.3424/1.1164 0.1742/0.5273	74.32	0.9670/0.9983

with eight cases, including the base case. In this table, the base case illustrates the worst-case scenario, in which there is no performance improvement in the test system in terms of loss reduction and voltage limitations. Case 1 is also presented in Table 1. In this case, the optimal SR obtained by HWCA is to open and close some sectionalized and tie switches as shown in Table 1. This indicated that the system loss is improved by 31.144% compared with the base case, while the minimum voltage magnitude limit is still violated. This is expected because of the limitations of search space and radial topology. In case 2 as shown in Table 1, one SOP installation is optimally integrated to improve the system’s performance without considering SR. In this case, it is observed that the HWCA made an optimal decision to select the best active and reactive setting and location for SOP as shown in Table 1. This decision resulted in a significant system loss reduction of nearly 43.45% compared with the base case with no voltage magnitude violation. Additionally, compared to scenario 1, the possibility of improving the system performance is increased by injecting reactive power along with active power into the test system. Case 3 is also included in Table 1. This case is similar to case 2, with the exception that the tie switch locations are the only viable options for the HWCA to place the SOP. In this case, it is clear that system performance is improved in terms of loss reduction, being roughly 41.1% compared to the base case, however, voltage magnitude limitation is violated. This happens because the search space of HWCA is restricted by only tie switch locations. Case 4 is also shown in Table 1. This case is similar to case 2, which suggests installing two SOPs rather than one. In this case, according to Table 1, the HWCA is successful in selecting the optimal SOP settings and locations, which considerably improves the system reduction by roughly 53.78% compared to the base case, with no voltage magnitude violations. In this case, the increase in loss reduction is expected compared to mentioned cases because the rising number of SOPs installed into the test system can bring further improvement to DSs in terms of loss reduction and voltage profile improvement. Table 1 also shows case 5. The only difference between this case and case 4 is that the tie switch locations are the only viable sites to install the SOPs. Consequently, as seen from Table 1, there is less improvement in loss reduction of about 53.27% compared to case 4, with no violation in voltage magnitude of all system nodes. Table 1 also includes case 6. Unlike cases 1 and 2, where the SOP and SR are integrated independently, in this case, the HWCA optimizes the SR and one SOP installation simultaneously. From Table 1, the HWCA is effective in determining the optimal SOP setting, position, as well as SR, resulting in a considerable system loss reduction of about 54.50% compared to the base case with no voltage magnitude violations. Finally, case 7 is also depicted in Table 1. This case is similar to case 6 except that it suggests installing two SOPs rather than one. As indicated in Table 1, the HWCA is capable of choosing the best settings for the two SOPs and the SR in this case, as shown in Figure 5, reducing system loss by around 63.33% compared to the base case with no voltage magnitude violations. Additionally, Figure 6 demonstrates the stages of loss reduction improvement from the base case to case 7. From this figure, it is obvious that case 7 is the best in system loss reduction.

Figure 7 compares and presents the voltage profile curves for all cases, including the base case. This figure demonstrates that the voltage magnitudes at each test system node are within permissible bounds for all cases except the base case, case 1, and case 3. These cases are unable to find the best solution while keeping all system nodes within allowed limits because the search space is severely constrained by the test system’s radial topology and tie switch positions.

4.2. Numerical Results of the IEEE 69-Node Test System

This test system consists of 69 nodes, 73 branches, 68 normally-close sectionalized switches, and five normally-closed tie switches (e.g., $T_{69}$ to $T_{73}$), with a 12.6 kV (or 1 p.u) base system voltage, as shown in Figure 8. The total real and reactive power loads are 3.80 MW and 2.70 MVAR, respectively, and the base active power loss is 224.69 kW. The limits of real and reactive power injected by SOPs are 0 to 2.5 MW and 0 to 2.5 MVAR, respectively, and the voltage magnitude limits of all buses are 0.95 to 1.05 p.u.

Table 2. Numerical results for IEEE 69-Node with and without SOP installation.

Case	No. of SOP	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)
Base Case	---	---	---	---	---	224.96	0.9092/1.000
SR (Case 1)	---	---	OS: 14, 57, 61 CS: 71, 72, 73	−−−	−−−	99.60	0.9428/1.0000
SOP without SR (Case 2)	1	50–59	---	−1.5433/1.5433	0.5455/1.3852	60.43	0.9709/1.0000
SOP without SR (Tie-switches) (Case 3)	1	50–59	---	−1.5433/1.5433	0.5455/1.3852	60.43	0.9709/1.0000
SOP without SR (Case 4)	2	15–46 50–59	---	−1.5926/1.5926 0.4490/−0.4490	0.5503/1.2818 0.3574/0.0952	45.95	0.9794/1.0000
SOP without SR (Tie-switches) (Case 5)	2	15–46 50–59	---	−1.5926/1.5926 0.4490/−0.4490	0.5503/1.2818 0.3574/0.0952	45.95	0.9794/1.0000
SOP with SR (Case 6)	1	50–59	OS: 12, 64 CS: 71, 73	−1.5492/1.5492	0.5478/1.2654	47.89	0.9808/1.0000
SOP with SR (Case 7)	2	61–62 50–59	OS: 12, 61 CS: 71, 73	−1.4630/1.4630 −0.1812/0.1812	0.5549/0.2244 0.8557/0.4057	40.23	0.9852/1.0000

shows the performance of the proposed HWCA in handling SR and SOP installation problems in the 69-node system using eight cases, including the base case. According to Table 2, the worst-case scenario is the base case with no performance improvement in the test system with regards to loss reduction and voltage limit violation. Case 1 is also presented in Table 2. In this case, the HWCA determines an optimal sectionalized and tie switch as illustrated in Table 2. Additionally, the system loss reduction is improved by 55.7% compared to the base case, with a slight violation in the minimum voltage magnitude limit. This is expected because the search space of HWCA is constrained by the radial topology of the test system. Case 2 and case 3 are also shown in Table 2. In these cases, it is observed that the HWCA chooses an optimal SOP location at the same tie switch $T_{72}$ with the best active and reactive settings without considering SR. This decision results in a significant system loss reduction of nearly 73.1% compared with the base case, with no voltage magnitude violation. Table 2 also depicts cases 4 and 5. In these cases, the results obtained by the HWCA exhibit high system performance, which can be explained by a loss reduction improvement of nearly 79.6% compared with the based case and no voltage magnitude violation. Furthermore, the HWCA selects the optimum SOP locations for both cases, which are $T_{72}$ and $T_{71}$ with the best active and reactive settings without taking SR into account, as shown in Table 2. Case 6 is also included in Table 2. In this case, the HWCA concurrently optimizes one SOP installation and SR. As shown in Table 2, the HWCA is effective in determining the optimal SOP setting and position, as well as the SR. This results in a considerable system loss reduction of around 78.7% with no voltage magnitude violations, as opposed to cases 1 and 2 where the SOP and SR are integrated individually. Case 7 is also shown in Table 2 as the last case. This case is comparable to case 6 with the exception that it recommends implementing two SOPs as opposed to one. Figure 8 illustrates how the HWCA is able to select the optimal configurations and placements for the two SOPs and the SR in this case, as given in Table 2. This results in a system loss reduction of around 82.1% compared to the base case, with no voltage magnitude violations. Furthermore, Table 2 shows that case 7 exhibits the greatest power loss reduction and voltage magnitude improvement when compared to the other cases. Finally, Figure 9 is used to visualize the improvement of loss reduction from the base case to case 7. This figure indicates that the HWCA offers the best system loss reduction in case 7.

The voltage profile curve for each case, including the basic case, is compared and shown in Figure 10. Except for the base case and case 1, this figure shows that the voltage magnitudes for all system nodes are all within acceptable boundaries. Notwithstanding, as a result of the search space being heavily confined by the radial topology of the test system in case 1, the HWCA is unable to find the optimal solution while keeping all system nodes within permitted bounds.

4.3. Comparison of the HWCA Efficacy with Other Optimization Algorithms

The efficacy of the proposed method, represented by cases 6 and 7, based on the HWCA reducing system power loss and enhancing the voltage profile is further proven by comparison with two well-known optimization algorithms, GA and WCA, utilizing the same metrics from Table 1 and Table 2 in addition to adding the number of iterations as indicated in

Table 3. Compared numerical results of HWCA for case 6 with WCA and GA for the 33-node system.

Case	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)	No. of iter.
Base Case	---	---	---	---	202.67	0.9131/0.9970	---
GA	24–25	OS: 7, 9, 14, 17, 24 CS: 33, 34, 35, 36, 37	−1.0800/1.088	0.477427/1.1129	93.02	0.9631/0.9976	172
WCA	25–29	OS: 7, 9, 14, 17 CS: 33, 34, 35, 36	−0.4205/0.4205	0.3830/1.1411	93.47	0.9631/0.9976	291
HWCA	24–25	OS: 7, 9, 14, 17, 24 CS: 33, 34, 35, 36, 37	−0.9708/0.9708	0.3767/1.1359	92.24	0.9631/0.9976	122

,

Table 4. Compared numerical results of HWCA for case 6 with WCA and GA for the 69-node system.

Case	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)	No. of Iter.
Base Case	---	---	---	---	202.67	0.9131/0.9970	---
GA	50–59	OS: 12 CS: 71	−1.6418/1.6418	0.3670/1.4570	49.66	0.9820/1.0000	204
WCA	50–59	OS: 12, 64 CS: 71, 73	−1.5501/1.5501	0.5592/1.2864	47.90	0.9811/1.0000	253
HWCA	50–59	OS: 12, 64 CS: 71, 73	−1.5492/1.5492	0.5478/1.2654	47.89	0.9810/1.0000	81

,

Table 5. Compared numerical results of HWCA for case 7 with WCA and GA for the 33-node system.

Case	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)	No. of Iter.
Base Case	---	---	---	---	202.67	0.9131/ 0.9970	---
GA	19–20 25–29	OS: 9, 14, 19, 32 CS: 33, 34, 35, 36	−1.4914/1.4914 −0.2672/0.2672	0.4183/0.6429 0.4660/0.8353	76.37	0.9686/1.0043	202
WCA	19–20 23–24	OS: 9, 14, 19, 23 CS: 33, 34, 35, 37	−1.0528/1.0528 −1.5134/1.5134	1.0059/0.5078 0.5700/1.1248	79.54	0.9675/0.9981	266
HWCA	19–20 24–25	OS: 14, 32, 9, 19, 24 CS: 33, 34, 35, 36, 37	−0.8205/0.8205 −1.3458/1.3458	0.3424/1.1164 0.1742/0.5273	74.33	0.9670/0.9983	194

and

Table 6. Compared numerical results of HWCA for case 7 with WCA and GA for the 69-node system.

Case	SOP Location (Node-Node)	Sec. and Tie Switch Status Opened Switch (OS) Closed Switch (CS)	Optimal SOP Active Power Injection (MW)	Optimal SOP Reactive Power Injection (MVAr)	Ploss (kW)	Vmin/Vmax (p.u)	No. of Iter.
Base Case	---	---	---	---	224.96	0.9092/1.0000	---
GA	9–10 50–59	OS: 9, 13, 20, 63 CS: 69, 70, 71, 73	−1.4084/1.4084 −0.6483/0.6483	0.5526/0.9699 0.0811/0.7696	45.41	0.9755/1.0000	188
WCA	16–17 50–59	OS: 12, 16, 64 CS: 70, 71, 73	−0.2903/0.2903 −1.5520/1.5520	0.2029/0.1863 0.5508/1.1467	46.11	0.9795/1.0000	251
HWCA	61–62 50–59	OS: 12, 61 CS: 71, 73	−1.4630/1.4630 −0.1812/0.1812	0.5549/0.2244 0.8557/0.4057	40.23	0.9852/1.0000	205

. Cases 6 and 7 are applied to these optimization algorithms with the same initial optimization settings to both the IEEE 33-node and 69-node DSs as shown in

Table 7. Optimization parameter settings of GA, WCA, and HWCA for cases 6 and 7.

Algorithm	33-Node and 69-Node Systems
GA	Population of chromosomes = 50, maximum iteration = 300, number of genes (number of variables) = 13, probability of crossover = 0.9, probability of mutation = 0.01.
WCA	Population of raindrops = 50, maximum iteration = 300, raindrops (number of variables) = 13, number of rivers = 6, a constant H = 2, $d_{max}={10}^{-16}.$
HWCA	Population of raindrops = 50, maximum iteration = 300, raindrops (number of variables) = 13, number of rivers = 6, $d_{max}={10}^{-16}.$ probability of crossover = 0.9, probability of mutation = 0.01.

. Table 3, Table 4, Table 5 and Table 6 demonstrate that the HWCA surpassed the other approaches in terms of loss reduction when compared to the base case. Moreover, the proposed HWCA shows fast convergence speed with the least iteration numbers as indicated in Figure 11, Figure 12 and Figure 13 for both cases 6 and 7, except in Figure 14. In this figure, the proposed HWCA takes more iterations than the GA to attain the optimal solution; meanwhile, the GA failed to attain this optimal solution, as shown in Figure 14. The key reason for the HWCA’s superiority is that the HWCA takes a distinctive indirect strategy to explore the most optimal solution based on updating the position of streams and rivers toward the sea by utilizing the GA, which is regarded as the temporary optimum solution. The HWCA can also efficiently prevent getting caught in a locally optimal solution or quick convergence thanks to the usage of the GA in updating the streams and rivers locations.

5. Conclusions

In this paper, the proposed hybrid water cycle algorithm (HWCA) was developed to address a complex nonlinear problem that is exemplified by simultaneously addressing the placement and size of the soft open point (SOP) and system reconfiguration (SR), with the objective to minimize system losses and improve the voltage profile while considering operational constraints in DSs. The water cycle algorithm (WCA) and genetic algorithm (GA) formed the basis of the proposed hybrid optimization algorithm to reduce the drawbacks of each compared method when utilized separately, deal with the discrete and continuous search space, and avoid getting trapped in local minima.

Eight cases, including the base case, were conducted on the IEEE 33-node and IEEE 69-node to evaluate the performance of the HWCA and investigate the real benefit gained from using SOPs alone or simultaneously with the SR, as well as show the effect of increasing the number of SOPs in improving system loss reduction and system voltage profiles in DSs under different scenarios. In each of these cases, the HWCA could efficiently improve the system loss reduction (e.g., 31.1–63.3% for IEEE 33-node and 55.7–82.1% for IEEE 69-node compared to the base case) while maintaining acceptable voltage magnitudes in most cases. Moreover, to demonstrate the efficacy of HWCA, case 7 was selected as a comparison case conducted on the modified IEEE 33-node and IEEE 69-node test systems with two other well-known metaheuristic optimization algorithms, WCA and GA. The simulation results show that the proposed method based on HWCA outperforms other optimization algorithms in terms of system loss reduction.

In future work, we will extend our method to simultaneously integrate the placement and size of the soft open point (SOP) and distributed generators (DGs) along with system reconfiguration (SR) to increase the DG hosting capacity and reduce system loss in DSs, while considering load uncertainty and system operational limits.