Chernobyl Disaster Optimizer-Based Optimal Integration of Hybrid Photovoltaic Systems and Network Reconfiguration for Reliable and Quality Power Supply to Nuclear Research Reactors

Abstract

In view of the complexity and importance of nuclear research reactor (NRR) installations, it is imperative to uphold high standards of reliability and quality in the electricity being supplied to them. In this paper, the performance of low-voltage (LV) distribution feeders integrated with NRRs is improved in terms of reduced distribution loss, improved voltage profile, and reduced greenhouse gas (GHG) emissions by determining the optimal location and size of photovoltaic (PV) systems. In the second stage, the power quality of the feeder is optimized by reducing the total harmonic distortion (THD) by optimally allocating D-STATCOM units. In the third and fourth stages, the reliability and resilience aspects of the feeder are optimized using optimal network reconfiguration (ONR) and by integrating an energy storage system (ESS). To solve the non-linear complex optimization problems at all these stages, an efficient meta-heuristic Chernobyl disaster optimizer (CDO) is proposed. Simulations are performed on a modified IEEE 33-bus feeder considering the non-linear characteristics of NRRs, variability of the feeder loading profile, and PV variability. The study reveals that the proposed methodology can significantly improve the service requirements of NRRs for attaining sustainable research activities.

1. Introduction

Nuclear power plants (NPPs) and hydropower plants are major energy sources, producing around 75 percent of low-carbon electricity worldwide. Over the past 50 years, NPPs have cut around 60 gigatons of CO₂, which is equivalent to roughly two years’ worth of global energy-related emissions. However, nuclear power has started to lose popularity in developing nations, with reactors shutting and little new investment undertaken at a time when the world needs more low-carbon electricity [1]. Supporting new nuclear constructions and promoting the development of new nuclear technologies are crucial due to their role in energy security, base-load reliability, and climate change mitigation. In this regard, many of the world’s nuclear research reactors (NRRs) are used for research, material testing, or the production of radioisotopes that are used in both industry and medicine, which may need a reliable and quality power supply. At present, nearly 220 NRRs are operating across 53 countries [2]. In essence, they behave as neutron manufacturers. They are considerably smaller than power reactors or ship propulsion systems and are installed on college campuses. In India, such premier NRR facilities include the Bhabha Atomic Research Center (BARC), Trombay [3], and the Indira Gandhi Center for Atomic Research (IGCAR), Kalpakkam [4].

To ensure effective support for ongoing research, development, training, and teaching activities in nuclear science and engineering, it is necessary to maintain high standards of reliability and quality in the electricity being supplied, given the complexity and importance of NRR installations. The primary connection between utilities and different kinds of consumers including NRR substations is low-voltage (LV) distribution feeders. Most LV feeders have high R/X ratio distribution lines and radial design, which results in high distribution losses, low voltage profiles, low reliability, and low voltage stability margins [5]. To improve the overall performance of these LV feeders and to address a variety of power quality, reliability, and resilience issues, integration of RE-based distribution generation (DG) is necessary [6]. At present, numerous research and development (R&D) facilities, academic institutions, office buildings, and even residential homes, etc., have shade-free rooftop areas that become ideal and potential locations to develop resilient and ecological energy sources by installing rooftop photovoltaic (PV)-type DGs [7]. Accordingly, BARC has recently requested a grid-connected 140 kW rooftop PV system for the hospital sites [8]. Similar to this, IGCAR is requesting to install a 2.1 MWp rooftop PV connected to the grid for its premises [9]. In addition to RE-based DGs, custom power devices (CPDs) [10], distribution-flexible AC transmission system (D-FACTS) devices [11], energy storage systems (ESSs) [12], and network reconfiguration (NR) [13], etc., have been well-accepted prominent solutions. Many researchers have concentrated on these methodologies, either alone or in combination. These techniques may not only support efficient and quality power supply, but may also be appropriate for reliable and resilient supply for managing uncertainty [14].

In [15], student psychology-based optimization (SPBO) is introduced for optimizing the weighted-sum multi-objective function with loss, voltage deviation, voltage stability index (VDI), and operating cost while solving the PV-DG allocation problem, which considers different types of load models (i.e., constant power, residential, industrial, commercial, and composite). In [16], multi-objective particle-swarm optimization (MOPSO) is utilized for different kinds of DGs to reduce active and reactive power losses and voltage deviation in EDNs. In [17], single-tuned passive power filters (PPFs) are optimally sized using the water-cycle algorithm (WCA) for an efficient and low-harmonic power supply considering non-linear RE-DGs in the network. The multi-objective function aims to reduce total harmonic distortion (THD), improve voltage profiles, and minimize distribution loss and filter investment costs. In [18], the grey wolf optimizer (GWO) is proposed for optimally allocating the real power filters (APFs) for mitigating the harmonics due to PV-DGs and non-linear loads. In [19], artificial electric field algorithm-pattern search (AEFAPS) is used for optimal network reconfiguration for optimizing distribution losses, voltage unbalance, voltage sag, and the reliability index of energy not supplied (ENS). In [20], a multi-objective covariance matrix adaptation–evolution strategy (MOCMA-ES) is developed for reducing voltage unbalance, greenhouse gas (GHG) emissions, THD, and cost of operation by optimally integrating PV and fuel cell (FC)-based DGs and a network reconfiguration approach. In [21], a techno-economic analysis of photovoltaic (PV) and wind turbine (WT) systems with different types of ESSs is presented. The multi-objective optimization problem is solved using the generalized reduced gradient (GRG) algorithm. In [22], a mixed-integer linear programming (MILP) algorithm-based isolated RE-integrated microgrid (MG) planning study is presented, considering demand response and ESS cost. The goal of [23] is to minimize total cost while solving ESS for grid-connected and islanded MG operations using linear and non-linear programming and GAMS software. In [24], the economic analysis of battery, pumped hydro, and their combination with PV and WT systems are examined for stand-alone applications, and the developed multi-objective function is solved using PSO. In [25], a MILP approach for optimizing the ESS is proposed, considering total operating cost reduction. In the first stage, ESS is formulated as virtual storage for DR, and in the second stage, the actual size of ESS is determined for managing PV penetrations in MG. In [26], a comparative study of generalized reduced gradient (GRG) and HOMER software results is presented on a stand-alone hybrid MG design (with solar, wind, and ESS) for remote areas and grid applications, considering social and environmental benefits. In [27], the coyote optimization algorithm (COA) is employed for designing the flexible photovoltaic system with ESS for stand-alone operation, considering variable EV load penetration. On the other hand, the integration of capacitor banks (CBs) in EDN can improve the voltage profile, resulting in reduced distribution losses and improved voltage stability. There are many researchers focused on various meta-heuristics for solving the optimal allocation of CBs in EDN [28]. In [29], the mayfly algorithm (MA) is proposed for solving ESS with a distribution-static synchronous compensator (D-STATCOM) considering real power-loss reduction in a photovoltaic distribution network. In [30], hunter–prey optimization (HPO) is introduced for solving multiple PV systems in EDNs considering multi-objective functions using loss, voltage profile, maximization of PV penetration, and reduction of greenhouse gas (GHG) emissions.

From these works, optimal integration of RE-based DGs can increase the efficiency of LV-EDNs in terms of loss reduction, voltage profile improvement, VSI enhancement, reduction of GHG emission, and total operating cost. However, RE-based DGs are intermittent in nature and can result in various power quality issues along with non-linear loads. In this connection, the integration of different types of PQ devices such as CBs, passive and real power filters (PPFs and APFs), D-STATCOM, and unified power quality conditioner (UPQC) can be useful for reducing the harmonic effects. On the other side, ONR can suit efficiency increment, solving PQ issues (i.e., voltage unbalance and voltage sag) and various reliability indices improvement under faulty conditions. However, integration of ESS can solve the energy imbalance conditions due to intermittent RE-based DGs, and variable loading conditions, and is suitable for various auxiliary services such as frequency and voltage regulation. In addition, ESS can improve the resilience of EDNs by providing uninterrupted and immediate power supply under faulty and islanding conditions.

Notably, all these works are considered constant power (CP) load modeling, residential (RL), industrial (IL), commercial (CL), and even their composition (Comp). However, most of these works have not considered NRRs and their uninterrupted and quality power supply requirements. A few researchers have focused on this issue recently. The genetic algorithm (GA) is used to integrate several FACTS device types in the best possible way, ensuring the safety and dependability for the feeder power supply of NRRs in the face of various electrical disruptions [10]. On the IEEE 30-bus system, simulations are run for loss reduction, voltage profile improvement, and harmonics reduction. In [31], particle swarm optimization (PSO)-based optimal location for unified power flow controllers (UPFCs) is utilized for ensuring reliable and secure power supply to the NRR centers via reducing distribution losses and improving voltage profile. Simulations are performed on the IEEE 33-bus radial feeder.

On the other hand, most of these planning and operational studies are formulated as multi-objective, non-linear, non-convex, multi-variable optimization problems with various equal and unequal constraints. From the literature, it can also be seen explicitly that the usage of heuristic algorithms predominate in all approaches. Numerous heuristic techniques are presented in the literature, but the no-free-lunch (NFL) theorem [32] states that not all algorithms are suitable for all types of optimization problems because of either their local or premature convergence. In this regard, researchers continue to be motivated to create new, simple, and efficient algorithms or to modify/hybridize current ones to enhance their convergence properties [33].

Considering these works, a combined optimization approach using RE-based DGs, D-FACTS, ESSs, and ONR can further improve the performance, power quality, reliability, and resilience matrices for NRRs, and for developing sustainable LV-EDNs. From this motivation, the following are the major contributions of this paper.

• A novel and recent meta-heuristic Chernobyl disaster optimizer (CDO) [34] is proposed for uninterrupted quality power supply to the NRRs even under faulty conditions.
• At the first stage, the feeder performance is improved in terms of reduced distribution loss, improved voltage profile, and reduced GHG emission via determining the optimal location and sizes of PV systems.
• In the second stage, the power quality of the feeder is optimized by reducing the total harmonic distortion (THD) and individual harmonic distortion (IHD) via optimally allocating D-STATCOM units.
• In the third stage, the reliability indices, namely, system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), and customer average interruption duration index (CAIDI), are optimized by optimal network reconfiguration (ONR).
• In the fourth stage, the resilience of the feeder is optimized in terms of average service unavailability index (ASUI), expected energy not served (ENS), and average energy not supplied (AENS) via optimally sizing the energy storage systems.
• Simulations are performed on a modified IEEE 33-bus feeder considering non-linear characteristics of NRRs, and the variability of feeder loading profile and PV variability.

The rest of the paper is organized as follows: In Section 2, the modeling of load with multiple types of consumers, PV system, D-STATCOM, and NRRs is explained. Section 3 describes the multi-objective optimization problem for different stages. In Section 4, the modeling of the proposed KOA and its application procedure is explained. In Section 5, simulation results of KOA for different stages are explicated and compared with benchmark literature works. Finally, the research findings and major contributions of this paper are comprehensively discussed in Section 6.

2. Modeling of Hybrid Photovoltaic System Components

A photovoltaic (PV) system is an intermittent energy source that operates exclusively during daylight hours. As a result, these sources cannot be classified as dispatchable DG units. An ESS must be integrated for them to be converted into dispatchable DG units. On the other hand, both PV and ESS are primarily intended for actual power supply, whereas LV-EDNs require reactive power assistance. To develop the final proposed hybrid PV (HPV) system, a voltage source inverter (VSI)-based D-STATCOM arrangement is incorporated. Figure 1 shows a schematic diagram of this system. In this HPV system, a buck-boost DC-DC converter is employed to stabilize the DC bus voltage regardless of PV power and voltage changes, allowing the ESS to maintain a constant DC voltage. D-STATCOM may import and export real and/or reactive power from and to the LV-EDN based on operational and control needs. This section describes the mathematical modeling of each component in the HPV system.

2.1. Photovoltaic System

Electric power generation by a photovoltaic system is dependent on mainly solar radiation and cell temperature. These two parameters are variable depending on the geographical location of the PV system and the day of the time. The following mathematical model is used for determining the hourly generation profile of the PV system [35]:

$$P_{pv\left(h\right)}={PV}_{cap}\times {\displaystyle \frac{G_{\left(h\right)}}{G_{\left(r\right)}}}\times \left[1+{\tau }_{pv}\left(T_{c\left(h\right)}-T_{c\left(h\right)}\right)\right]\times {\eta }_{in}\times {\kappa }_{loss}$$

(1)

$$T_{c\left(h\right)}=T_{a\left(h\right)}+{\displaystyle \frac{P_{d\left(h\right)}}{800}}\times \left(T_{noct}-20\right)$$

(2)

where $P_{pv\left(h\right)}$ is the power generation by PV system at time-h; $P{V}_{cap}$ is the PV plant installed capacity; $T_{c\left(h\right)}$, $T_{a\left(h\right)}$, and $T_{noct}$ are the cell temperature and ambient temperatures at time-h and nominal operating cell temperature, respectively; $P_{d\left(h\right)}$ is the power density at time-h, $G_{\left(h\right)}$, $G_{\left(r\right)}$, and $T_{\left(r\right)}$ are the incident radiation at time-h, reference radiation, and temperature, respectively; ${\tau }_{pv}$ is the temperature loss coefficient; and ${\eta }_{in}$ and ${\kappa }_{loss}$ are the inverter efficiency and efficiency factor to account for cable resistance, dust, and other losses, respectively.

2.2. Energy Storage System

The purpose of the ESS in this work is to store total PV generation in the daytime at NRRs and to discharge during either islanding conditions or peak loading conditions on the main grid. Considering autonomy, the following model is used for the battery ESS (BESS) [36]:

$$BE{S}_{cap}={\displaystyle \frac{n_{d,a}\times \left\{\left(\frac{1}{24}\times L{F}_d\times \overline{P_{d(\max )}}\right)-{\displaystyle \sum _{h=1}^{24}{P}_{pv\left(h\right)}}\right\}}{\left({\varsigma }_{tc}\times {\sigma }_{af}\times {\rho }_{dod}\times V_{bn}\times {\eta }_{bc}\right)}}$$

(3)

where $BE{S}_{cap}$ is the BESS capacity in Ah; $n_{d,a}$ is the number of days autonomy required; $L{F}_d$ is the daily load factor of the network; $\overline{P_{d(\max )}}$ is the total peak demand of the network; $P_{pv\left(h\right)}$ is the hourly power generation by the PV system; ${\varsigma }_{tc}$ and ${\sigma }_{af}$ are the temperature correction and aging factors, respectively; ${\rho }_{dod}$ and $V_{bn}$ is the allowable depth of discharge (DoD) and battery nominal voltage, respectively; and ${\eta }_{bc}$ is the battery efficiency.

2.3. D-STATCOM

In the proposed HPV system, the role of D-STATCOM is to exchange power flow between the LV-EDN, PV, and ESS by maintaining AC bus voltage at the desired constant value. By regulating the voltage magnitude and phase angle of VSI with respect to the AC bus, the real and reactive power flows can be controlled effectively by relating the following mathematical modeling [37].

The complex power flow that can be controlled between D-STATCOM buses and AC buses is given by

$$S_{ds}=V_{ds}{I}_{ds}^{\ast }=V_{ds}\times {\left[Y_{sc}\left(V_{ds}-V_{\left(k\right)}\right)\right]}^{*}$$

(4)

Here, the AC bus voltage and D-STATCOM bus voltages are considered as $V_{\left(k\right)}=\left|V_{\left(k\right)}\right|\angle {\delta }_{\left(k\right)}$ and $V_{ds}=\left|V_{ds}\right|\angle {\delta }_{ds}$, respectively. The transfer admittance between the D-STATCOM and AC bus is taken as $Y_{ds}=Z_{ds}^{-1}=G_{ds}+j{B}_{ds}$. By expanding and separating the real and imaginary parts on both sides, the real and reactive power flows are given by

$$P_{HPV}={\left|V_{ds}\right|}^2{G}_{ds}-\left|V_{ds}\right|\left|V_{\left(k\right)}\right|\left\{G_{ds}\cos \left({\delta }_{ds}-{\delta }_{\left(k\right)}\right)+B_{ds}\sin \left({\delta }_{ds}-{\delta }_{\left(k\right)}\right)\right\}$$

(5)

$$Q_{HPV}=-{\left|V_{ds}\right|}^2{B}_{ds}-\left|V_{ds}\right|\left|V_{\left(k\right)}\right|\left\{G_{ds}\sin \left({\delta }_{ds}-{\delta }_{\left(k\right)}\right)-B_{ds}\sin \left({\delta }_{ds}-{\delta }_{\left(k\right)}\right)\right\}$$

(6)

For ${\delta }_{ds}>{\delta }_{\left(k\right)}$, $P_{HPV}$ becomes negative and thus HPV can be able to supply real power (i.e., source mode). On the other hand, for ${\delta }_{ds}<{\delta }_{\left(k\right)}$, $P_{HPV}$ becomes positive and thus HPV can be able to consume reactive power (i.e., inductive mode of D-STATCOM).

For $\left|V_{ds}\right|>\left|V_{\left(k\right)}\right|$, $Q_{HPV}$ becomes negative and thus HPV can be able to supply reactive power (i.e., capacitive mode of D-STATCOM). On the other hand, for $\left|V_{ds}\right|<\left|V_{\left(k\right)}\right|$, $Q_{HPV}$ becomes positive and thus HPV can be able to consume reactive power (i.e., inductive mode of D-STATCOM).

In addition, for ${\delta }_{ds}={\delta }_{\left(k\right)}$, $P_{HPV}$ becomes zero, and thus the HPV system can be able to support only reactive power control as per the working principle of ideal D-STATCOM. In this case, by maintaining $\left|V_{ds}\right|=\left|V_{\left(k\right)}\right|$, $Q_{HPV}$ becomes zero, and thus the HPV system can be able to support only real power control as per the working principle of an ideal PV or ESS. In this way, the proposed HPV system can be treated as dispatchable DG in load flow studies.

2.4. Load Modeling

Most of the loads in LV-EDNs are voltage-sensitive. Thus, a voltage-dependent composite load modeling is proposed in this network considering residential (RL), industrial (IL), and commercial loads (CL) [15].

$$\overline{P_{d\left(k\right)}}=P_{d\left(k\right)}\left\{{\kappa }_R{\left|V_{\left(k\right)}\right|}^{{\alpha }_R}+{\kappa }_C{\left|V_{\left(k\right)}\right|}^{{\alpha }_C}+{\kappa }_I{\left|V_{\left(k\right)}\right|}^{{\alpha }_I}\right\}, \forall k=1:nbus$$

(7)

$$\overline{Q_{d\left(k\right)}}=Q_{d\left(k\right)}\left\{{\kappa }_R{\left|V_{\left(k\right)}\right|}^{{\beta }_R}+{\kappa }_C{\left|V_{\left(k\right)}\right|}^{{\beta }_C}+{\kappa }_I{\left|V_{\left(k\right)}\right|}^{{\beta }_I}\right\}, \forall k=1:nbus$$

(8)

where ${\kappa }_R$, ${\kappa }_C$, and ${\kappa }_I$ are the proportional factors for representing the mix of RL, CL, and ILs at a bus-k, respectively; (${\alpha }_R$, ${\beta }_R$), (${\alpha }_C$, ${\beta }_C$), and (${\alpha }_I$, ${\beta }_I$) are the exponents of real and reactive power loads as per voltage-dependent load modeling, respectively; $V_{\left(k\right)}$ is the voltage magnitude of bus-k, $P_{d\left(k\right)}$ and $Q_{d\left(k\right)}$ are the real and reactive power loads of bus-k at nominal voltage magnitudes, respectively; and $\overline{P_{d\left(k\right)}}$ and $\overline{Q_{d\left(k\right)}}$ are the composite real and reactive power loads at bus-k, respectively.

In addition, some locations in the network are considered NRRs. As per [31], three different types of loads can be found in NRR centers: important and interruptible, reconnectable after interruptions, and admits interruptions. Thus, the net-effective loading of NRRs is modeled as a combination of constant power (CP), constant current (CI), and constant impedance (CZ). Mathematically,

$$\overline{P_{d\left(k\right)}}=\overline{P_{d\left(k\right)}}+P_{nrr,d\left(k\right)}\left\{{\kappa }_p{\left|V_{\left(k\right)}\right|}^{{\alpha }_p}+{\kappa }_i{\left|V_{\left(k\right)}\right|}^{{\alpha }_i}+{\kappa }_z{\left|V_{\left(k\right)}\right|}^{{\alpha }_z}\right\}, \forall k=1:nnrr$$

(9)

$$\overline{Q_{d\left(k\right)}}=\overline{Q_{d\left(k\right)}}+P_{nrr,d\left(k\right)}\times \tan \left({\varphi }_{nrr}\right)\times \left\{{\kappa }_p{\left|V_{\left(k\right)}\right|}^{{\beta }_p}+{\kappa }_i{\left|V_{\left(k\right)}\right|}^{{\beta }_i}+{\kappa }_z{\left|V_{\left(k\right)}\right|}^{{\beta }_z}\right\}, \forall k=1:nnrr$$

(10)

where $P_{nrr,d\left(k\right)}$ and $\cos \left({\varphi }_{nrr}\right)$ are the real power demand of NRR at bus-k and its operating power factor, respectively; where ${\kappa }_p$, ${\kappa }_i$, and ${\kappa }_z$ are the proportional factors for representing the mix of CP, CI, and CZ type of loads at a bus-k, respectively; (${\alpha }_p$, ${\beta }_p$), (${\alpha }_i$, ${\beta }_i$), and (${\alpha }_z$, ${\beta }_z$) are the exponents of real and reactive power loads as per voltage-dependent load modeling, respectively; and $nbus$ and $nnrr$ are the number of buses, and NRR locations, respectively.

3. Problem Formulation

This section presents the different objective functions formulated for developing sustainable and uninterrupted quality power supply to the NRRs using PV, D-STATCOM, and ESSs.

3.1. Objective Functions

Objective Function 1 ($f_1$): The main objective function while allocating the PV system is to minimize the total GHG emission from conventional power plants in the main grid. This can be achieved by maximizing the PV penetration in LV-EDNs, reducing distribution losses, and net-effective loading on the main grid.

$$f_1=\min \left(GH{G}_{em}\right)=\min \left[P_{d\left(grid\right)}\times \left(C{O}_2+S{O}_2+N{O}_x\right)\right]$$

(11)

$$P_{d\left(grid\right)}=\left(∆P_{d\left(grid\right)}+P_{loss}\right)=\left\{\left({\displaystyle \sum _{k=1}^{nbus}\overline{P_{d\left(k\right)}}}-{\displaystyle \sum _{j=1}^{npv}{P}_{pv\left(j\right)}}\right)+{\displaystyle \sum _{k=1}^{nbr}{I}_{ij\left(k\right)}^2{r}_{ij\left(k\right)}}\right\}$$

(12)

where $GH{G}_{em}$ is the total GHG emission from the main grid; $S{O}_2$ and $N{O}_x$ are the major pollutants from the conventional power plants; $∆P_{d\left(grid\right)}$ is the net-effective loading after PV installation; $P_{d\left(grid\right)}$ is the total real power demand on the main grid; $P_{loss}$ is the distribution losses; $P_{pv\left(j\right)}$ and $\overline{P_{d\left(k\right)}}$ are the real power support by PV and net-effective reactive loading by composite load model and NRRs, respectively; $I_{ij\left(k\right)}$ and $r_{ij\left(k\right)}$ are the current flow and resistance of branch-k, connected between buses i and j, respectively; and $npv$ and $nbr$ are the number of PV locations and number of branches (including sectionalizers and tie-lines), respectively.

Objective Function 2 ($f_2$): The voltage profile improvement and minimization of THD are treated as major objectives while designing the D-STATCOM by extracting optimal VAr support. This also can reduce the overall VAr burden on the main grid.

$$\begin{gathered} f_2=\min \left(Q_{d\left(grid\right)}+TVD+TH{D}_v\right) \\ =\min \left\{\left({\displaystyle \sum _{k=1}^{nbus}\overline{Q_{d\left(k\right)}}}-{\displaystyle \sum _{j=1}^{nds}{Q}_{ds\left(j\right)}}\right)+{\displaystyle \sum _{k=1}^{nbus}{\left(\left|V_{\left(r\right)}\right|-\left|V_{\left(k\right)}\right|\right)}^2}+\left({\displaystyle \frac{1}{\left|V_{\left(k\right)}^1\right|}}\sqrt{{\displaystyle \sum _{h=2}^H{\left|V_{\left(k\right)}^h\right|}^2}}\right)\times 100\right\} \end{gathered}$$

(13)

where $Q_{d\left(grid\right)}$ is the net-effective reactive loading on the main grid after D-STATCOM installation; $TVD$ and $TH{D}_v$ are the total voltage deviation and total harmonic voltage distortion level, respectively; $Q_{ds\left(j\right)}$ and $\overline{Q_{d\left(k\right)}}$ are the reactive power support by D-STATCOM and net-effective reactive loading by composite load model and NRRs, respectively; $\left|V_{\left(r\right)}\right|$ is the reference/sub-station voltage magnitude; $\left|V_{\left(k\right)}^1\right|$ and $\left|V_{\left(k\right)}^h\right|$ are the voltage magnitudes of bus-k for fundamental and other harmonic frequency levels, respectively; and h and H are the specific and maximum harmonic frequency levels under consideration, respectively.

Objective Function 3 ($f_3$): The ONR is proposed not only to eliminate faulty locations in the network, but also to ensure improved reliability indices. The major reliability indices are SAIFI, SAIDI, and CAIDI, which are considered while altering the configuration. These indices are mathematically defined by [38]:

$$SAIFI={\displaystyle \sum _{j=1}^{nbr}{\mu }_j{n}_i}/{\displaystyle \sum _{j=1}^{nbr}{n}_i}$$

(14)

$$SAIDI={\displaystyle \sum _{j=1}^{nbr}{\lambda }_j{n}_i}/{\displaystyle \sum _{j=1}^{nbr}{n}_i}$$

(15)

$$CAIDI={\displaystyle \sum _{j=1}^{nbr}{\lambda }_i{n}_i}/{\displaystyle \sum _{j=1}^{nbr}{\mu }_i{n}_i}={\displaystyle \frac{SAIDI}{SAIFI}}$$

(16)

Objective Function 4 ($f_4$): Integration of ESSs in this work is to ensure improved resilience operation even under faulty and islanding conditions. To ensure uninterrupted power supply, particularly to NRRs, the following resilience indices are considered while optimizing the ESS capacity.

$$ASUI=1-\left\{\left[{\displaystyle \sum _{j=1}^{nbr}\left(8760\times n_i\right)-{\displaystyle \sum _{j=1}^{nbr}{\lambda }_i{n}_i}}\right]/{\displaystyle \sum _{j=1}^{nbr}\left(8760\times n_i\right)}\right\}$$

(17)

$$AENS={\displaystyle \frac{1}{nbus-1}}\left({\displaystyle \sum _{j=1}^{nbr}{\lambda }_j{L}_{a,i}}\right)$$

(18)

where ASUI and AENS are defined as the average service unavailability index (ASUI) and average energy expected not served (AENS), respectively; ${\mu }_j$ is the failure rate of branch-j; $n_i$ is the number of consumers associated with bus-i for the failure of branch-j; ${\lambda }_j$ is the yearly unavailability of branch-j; $L_{a,j}$ is the average load associated with bus-i; and $nbr$ is the number of switched-on branches (including sectionalizers and tie-lines) in the network.

3.2. Operational Constraints

The following bus voltage magnitude, branch current, real and reactive power balance, BESS capacity, DG location, radiality, and harmonic level constraints are handled suitably while solving the objective function.

$$\mathrm{Bus} \mathrm{voltage} \mathrm{magnitude} \mathrm{limits}: \left|V_{\min }\right|\le \left|V_{\left(k\right)}\right|\le \left|V_{\max }\right|$$

(19)

$$\mathrm{Branch} \mathrm{current} \mathrm{limits}: I_{ij\left(k\right)}\le I_{ij\left(k\right),\max }$$

(20)

$$\mathrm{Real} \mathrm{power} \mathrm{balance} \mathrm{constraint}: {\displaystyle \sum _{j=1}^{npv}{P}_{pv\left(j\right)}}\le {\displaystyle \sum _{k=1}^{nbus}\overline{P_{d\left(k\right)}}}$$

(21)

$$\mathrm{Reactive} \mathrm{power} \mathrm{balance} \mathrm{constraint}: {\displaystyle \sum _{j=1}^{nds}{Q}_{ds\left(j\right)}}\le {\displaystyle \sum _{k=1}^{nbus}\overline{Q_{d\left(k\right)}}}$$

(22)

$$\mathrm{BESS} \mathrm{Capacity} \mathrm{limit}: BE{S}_{cap}\ge n_{d,a}\times \left({\displaystyle \sum _{k=1}^{nbus}\overline{P_{d\left(k\right)}}}-{\displaystyle \sum _{j=1}^{npv}{P}_{pv\left(j\right)}}\right)$$

(23)

$$\mathrm{DG} \mathrm{location} \mathrm{constraint}: 2\le l_{dg}\le nbus$$

(24)

$$\mathrm{Radiality} \mathrm{constraint}:nbr=nbus-1 \& \left|A\right|\ne 0$$

(25)

$$\mathrm{IEEE} 519 \mathrm{harmonic} \mathit{limits} TH{D}_v<5\% \& IH{D}_v<3\%$$

(26)

where $\left|V_{\min }\right|$ and $\left|V_{\max }\right|$ are the minimum and maximum limits of bus voltage magnitudes, respectively; $I_{ij\left(k\right),\max }$ is the maximum current limit of branch-k, connected between buses i and j, respectively; $l_{dg}$ is the location of DG; $\left|A\right|$ is the determinant of bus-incident matrix; and $IH{D}_v$ is the individual harmonic distortion.

4. Solution Methodology

This section explains the optimization technique based on the Chernobyl disaster optimizer (CDO) for optimally integrating the HPVs and ONR approaches for ensuring reliable and quality power supply to NRRs.

Uncertain exploration of nuclei can emit three different forms of radiation, called Alpha, Beta, and Gamma, which can travel fast and far and be hazardous to people in various ways. The Chernobyl catastrophe, dubbed the worst civil nuclear incident in history, was a nuclear accident that happened at the Chernobyl Nuclear Power Plant’s No. 4 reactor on 26 April 1986, close to the city of Pripyat in northern Ukraine SSR in the Soviet Union. By modeling the consequences of this catastrophe and the outdoor walking speed of adults, Shehadeh HA introduced the Chernobyl disaster optimizer (CDO) in 2023 [35]. This section explains the CDO mathematically.

4.1. Gamma Rays

Gamma (λ) rays are electromagnetic radiation waves with a high frequency and small wavelength that have a maximum speed of 300,000 km/s. A thick layer of lead may readily absorb gamma radiation. These particles are extremely dangerous to people and only mildly ionizing. The gradient descent factor of λ rays while assaulting humans is given by

$$v_{\gamma }=X_{\gamma \left(k\right)}-{\varsigma }_{\gamma }{\left|d\right|}_{h\lambda }$$

(27)

$${\varsigma }_{\gamma }={\displaystyle \frac{A_h}{N_{\lambda }}}-\left(N_{aws}\times r_1\right)$$

(28)

$${\left|d\right|}_{h\lambda }=\left|A_{\lambda }{X}_{\gamma \left(k\right)}-x_{T\left(k\right)}\right|$$

(29)

$$A_h=\pi r_2^2$$

(30)

$$N_{\lambda }=\log \left(rand\left(1:\mathrm{300,000}\right)\right)$$

(31)

$$N_{aws}=3\times \left(1-k_{\max }^{-1}\right)$$

(32)

$$A_{\lambda }=\pi r_3^2$$

(33)

$$x_{T\left(k\right)}=\left(v_{\alpha }+v_{\beta }+v_{\gamma }\right)/3$$

(34)

where $v_{\gamma }$, ${\varsigma }_{\gamma }$, and $X_{\gamma \left(k\right)}$ are the gradient descent factor, propagation, and position of λ rays at iteration k, respectively; ${\left|d\right|}_{h\lambda }$ is the physical distance between the human and λ rays; $x_{T\left(k\right)}$ is the average of all positions; $A_h$ and $A_{\lambda }$ are the adults’ waking area and λ rays penetration area, respectively; $N_{\lambda }$ is the random speed of λ rays in the range of 1 to 300,000 km/s; $N_{aws}$ is the adults’ outdoor walking speed, which is assumed maximum of 3 miles/h; and $r_1$, $r_2$, and $r_3$ are uniformly distributed random numbers in the range (0, 1).

4.2. Beta Rays

Beta (β) rays may pierce humans/substances at a high speed of 270,000 km/s due to their great energy and comparatively low mass with a negative charge. These particles are dangerous to humans and only modestly ionize. When attacking a human, the gradient descent factor of the β rays is given by

$$v_{\beta }=0.5\left(X_{\beta \left(k\right)}-{\varsigma }_{\beta }{\left|d\right|}_{h\beta }\right)$$

(35)

$${\varsigma }_{\beta }=A_h{\left(0.5\times N_{\beta }\right)}^{-1}-\left(N_{aws}\times r_4\right)$$

(36)

$${\left|d\right|}_{h\beta }=\left|A_{\beta }{X}_{\beta \left(k\right)}-x_{T\left(k\right)}\right|$$

(37)

$$A_h=\pi r_5^2$$

(38)

$$N_{\beta }=\log \left(rand\left(1:\mathrm{270,000}\right)\right)$$

(39)

$$A_{\beta }=\pi r_6^2$$

(40)

where $v_{\beta }$, ${\varsigma }_{\beta }$, and $X_{\beta \left(k\right)}$ are the gradient descent factor, propagation, and position of β rays at iteration k, respectively; ${\left|d\right|}_{h\beta }$ is the physical distance between the human and β rays; $A_{\beta }$ is the penetration area of β rays; $N_{\beta }$ is the random speed of β rays in the range of 1 to 270,000 km/s; and $r_4$, $r_5$, and $r_6$ are uniformly distributed random numbers in the range (0, 1).

4.3. Alpha Rays

Although alpha (α) ray radiation is highly strong and positive, it may not travel far. The radiation can move at a top speed of 16,000 km/s and is easily absorbed by human skin. These particles are only strongly ionizing and extremely harmful to humans. The gradient descent factor of the beams used to assault a human is determined by

$$v_{\alpha }=0.25\left(X_{\alpha \left(k\right)}-{\varsigma }_{\alpha }{\left|d\right|}_{h\alpha }\right)$$

(41)

$${\varsigma }_{\alpha }=A_h{\left(0.25\times N_{\alpha }\right)}^{-1}-\left(N_{aws}\times r_7\right)$$

(42)

$${\left|d\right|}_{h\alpha }=\left|A_{\alpha }{X}_{\alpha \left(k\right)}-x_{T\left(k\right)}\right|$$

(43)

$$A_h=\pi r_8^2$$

(44)

$$N_{\alpha }=\log \left(rand\left(1:\mathrm{16,000}\right)\right)$$

(45)

$$A_{\alpha }=\pi r_9^2$$

(46)

where $v_{\alpha }$, ${\varsigma }_{\alpha }$, and $X_{\alpha \left(k\right)}$ are the gradient descent factor, propagation, and position of α rays at iteration k, respectively; ${\left|d\right|}_{h\alpha }$ is the physical distance between the human and α rays; $A_{\alpha }$ is the penetration area of α rays; $N_{\alpha }$ is the random speed of α rays in the range of 1 to 16,000 km/s; and $r_7$, $r_8$, and $r_9$ are uniformly distributed random numbers in the range (0, 1).

By utilizing these penetration behaviors of gamma, beta, and alpha rays and treating gamma ray fitness as the target function value, CDO is presented as a simple, efficient, and competitive meta-heuristic algorithm for real-time optimization problems [35].

4.4. Distribution System Load Flow Study

As described in Section 4.4, the evaluation of objective functions is interdependent on the load flow analysis of the network. In view of the convergence issues associated with using conventional power flow techniques, distribution systems were solved using different derivative-free approaches considering their high X/R ratio branches and radial configuration [39]. Thus, in this paper, one such derivative-free and efficient backward/forward (BW/FW) load flow method [40] is adapted and explained in this section.

In the backward sweep, the net-effective loadings of bus i are determined by summation of local load at bus j and branch losses that are incident to bus i. This procedure continues for all buses from the ending bus to the starting bus on the feeder.

$$P_{i\left(eff\right)}=P_i+\left(P_j+P_{loss\left(ij\right)}\right)\mathrm{and}{Q}_{i\left(eff\right)}=Q_i+\left(Q_j+Q_{loss\left(ij\right)}\right)$$

(47)

$$P_{loss\left(ij\right)}=\left[\left(P_j^2+Q_j^2\right)/V_j^2\right]r_{ij}\mathrm{and}{Q}_{loss\left(ij\right)}=\left[\left(P_j^2+Q_j^2\right)/V_j^2\right]x_{ij}$$

(48)

On the other side, in the forward sweep, the voltage of bus j is determined by subtracting the voltage drops from bus i due to real and reactive power flows in branch ij. This procedure continues for all buses from the starting bus to the ending bus on the feeder.

$$V_j=\sqrt{{\left(V_i-∆v_1\right)}^2+{∆v_2}^2}$$

(49)

$${\delta }_j={\delta }_i-{\tan }^{-1}\left[∆v_2{\left(V_i-∆v_2\right)}^{-1}\right]$$

(50)

where $∆v_1=\left(P_j{r}_{ij}+Q_j{x}_{ij}\right)/V_i$ and $∆v_2=\left(P_j{x}_{ij}-Q_j{r}_{ij}\right)/V_i$ are the longitudinal and transversal voltage drops, respectively.

To generalize this procedure for a distribution system, the real and reactive power losses of all branches are set to zero at the initial stage. Further, by assuming the initial voltage magnitudes and load angles for all buses are equal to 1.00 p.u and 0 radians, respectively, the losses in all branches are determined. At this stage, the computational procedure completes one iteration. In the next iteration, by updating the net effective loadings of each bus, as given in Equation (47), new voltages and angles, and consequently, branch and total losses, are updated.

Later, the total real and reactive power losses of the system are determined by aggregating all branch losses, which are given by

$$P_{loss\left(T\right)}={\displaystyle \sum _{k\in ij}^{nbr}{P}_{loss\left(ij\right)}}\mathrm{and}{Q}_{loss\left(T\right)}={\displaystyle \sum _{k\in ij}^{nbr}{Q}_{loss\left(ij\right)}}$$

(51)

This procedure continues until the convergence criterion is satisfied, $\epsilon =\left(S_{loss(T)}^k-S_{loss(T)}^{k-1}\right)\le {10}^{-5}$ (i.e., the change in total absolute loss with respect to earlier iteration).

4.5. Solution Methodology for the Proposed Objective Functions

4.5.1. Objective Function 1

By integrating PV systems optimally in the distribution feeder, the real power burden on the main grid is proposed to be reduced in this function. The impact of PVs is assessed in terms of reduction in GHG and real power loss. Thus, the search space for CDO is dependent on the number of PVs, their location, and their sizes. For instance, to allocate three PVs, the CDO needs to determine three optimal locations and their sizes. Therefore, the total number of search variables is six. In each iteration of CDO, the search variables (i.e., alpha, beta, and gamma particles) are updated in the load flow study as given by

$$P_j=P_j-P_{pv\left(j\right)},\forall j=1:npv$$

(52)

4.5.2. Objective Function 2

By integrating PVs and D-STATCOMs simultaneously, the real and reactive power burden on the main grid is proposed to be reduced in this function. The impact of PVs and D-STATCOMs is assessed in terms of improvement in voltage profile and THDv. Thus, for the allocation of three PVs and three D-STACOMs, the number of search variables becomes twelve. Further, these will be updated in the load flow study as given by

$$P_j=P_j-P_{pv\left(j\right)}\mathrm{and}{Q}_j=Q_j-Q_{ds\left(j\right)},\forall j=1:npv$$

(53)

4.5.3. Objective Function 3

In this scenario, by optimally switching (on/off) the branches and tie-lines, network reconfiguration is aimed to improve the reliability metrics of the network. For a network with b branches and t tie-lines, the possible configurations are 2^(b+t). Identifying the best one among these many combinations is a typical task. For the CDO, the search space is equal to the number of tie-lines and number of branches. In the optimization, the total number of on-condition branches and tie-lines should be always equal to (nbus-1). By satisfying the radiality constraint as defined in Equation (25), load flow is computed with the revised branches at each iteration, and correspondingly, the objective function is updated.

4.5.4. Objective Function 4

By integrating adequate ESS at an optimal location, this scenario is aimed to ensure uninterrupted power supply even under the islanding mode of operation. The analysis is extended to design the ESS with and without PVs and D-STATCOMs in the network.

4.6. Computation Procedure of CDO Algorithm

In solving any optimization function using CDO, the following steps are involved.

Step 1.

Define/read the optimization function and its associated constraints.

Step 2.

Set the parameters of CDO, i.e., lower and upper limits of search variables, number of search variables ($N$), population, and maximum iterations.

Step 3.

Initialize the positions for particles α, β, and λ, $X_i\left(i=1:N\right)$.

Step 4.

WHILE (the end iteration is not achieved) DO

• For all particles α, β, and λ, calculate the fitness $F\left(X_i\right)$
• IF fitness $F\left(X_i\right)<$λ Score $f\left(X_{\gamma \left(k\right)}\right)$
• Set $F\left(X_i\right)=f\left(X_{\gamma \left(k\right)}\right)$ and update the position of λ particle, END
• IF fitness $F\left(X_i\right)<$ β Score $f\left(X_{\beta \left(k\right)}\right)$
• Set $F\left(X_i\right)=f\left(X_{\beta \left(k\right)}\right)$ and update the position of β particle, END
• IF fitness $F\left(X_i\right)<$ α Score $f\left(X_{\alpha \left(k\right)}\right)$
• Set $F\left(X_i\right)=f\left(X_{\alpha \left(k\right)}\right)$ and update the position of α particle, END

Step 1.

For all particles α, β, and λ, update the position on the Cartesian plane (x, y)

Calculate gradient descent factors for particles α, β, and λ

Step 2.

Update average of total positions, END

5. Simulation Results

To analyze the computational efficiency of the proposed CDO and its superiority over other meta-heuristics, simulations were performed on IEEE 33-bus LV-EDN using MATLAB 2022b on a PC (Intel i3, 1.7 Hz, 8 GB RAM).

The system has 33 buses, 32 branches, and 5 tie-lines; the line and bus data are taken from [39]. In our simulation, each load bus is modeled with 30% residential (${\kappa }_R=0.3$), 30% commercial (${\kappa }_C=0.3$), and 40% industrial loads (${\kappa }_I=0.4$). In addition, buses 13, 24, and 30 are treated as NRRs with static non-linear loads (SNLs) of 400 kVA, 450 kVA, and 500 kVA with an operating power factor of 0.95 lagging, respectively. For the load modeling of NRRs, the proportional factors of ${\kappa }_p$, ${\kappa }_i$, and ${\kappa }_z$ are taken as 0.4, 0.3, and 0.3, respectively. Thus, the total peak demand of the network becomes (4757.55 kW + j 2173.49 kVAr). By using BW/FW load flow [41], the network performance is determined. The total real and reactive power losses are 312.4885 kW and 207.3051 kVAr, respectively. The minimum voltage magnitude is registered at bus-18 as 0.8912 p.u and the total GHG emission is equal to 10,382 lb/hr.

5.1. Performance Improvement

To improve the feeder performance in terms of reduced distribution loss, improved voltage profile, and reduced GHG emission, this case study explores the optimal allocation of three PV systems in the network. The search space for PV locations is considered as all buses except sub-station buses, and for PV sizes, the maximum capacity is treated as 2 MW. There are three PV locations.

Case 5.1.1: In this case, only loss minimization is the objective function; CDO is implemented to optimize the locations and sizes of PVs in the network. The best performance is observed with CDO by integrating PVs at bus # 13 (1.092 MW), # 24 (1.46 MW), and # 30 (1.432 MW). By this, the distribution losses are reduced to 59.472 kW and 40.907 kVAr. The lowest voltage is registered as 0.9728 p.u at bus # 33. In addition, the total GHG emission from the main grid resource is reduced to 1704.71 lb/MWh.

Case 5.1.2: In this case, both distribution losses and grid dependency for real power are minimized simultaneously. The best performance is observed with CDO by integrating PVs at buses # 3 (2 MW), # 13 (1.066 MW), and # 30 (1.385 MW). By this, the distribution losses are reduced to 71.876 kW and 50.356 kVAr. The lowest voltage is registered as 0.9729 p.u at bus # 33. In addition, the total GHG emission from the main grid resource is reduced to 773.95 lb/MWh. The total PV capacity in Case 5.1.2 is 93.561%, whereas in Case 1, it is observed as only 83.746%; thus, the total GHG emissions are less in Case 5.1.2 than in Case 5.1.1. The voltage profiles of the network for the base case, Case 5.1.1, and Case 5.1.2 are given in Figure 2.

5.2. Power Quality Improvement

In the second stage, the power quality of the PVs integrated feeder is further optimized by reducing the total harmonic distortion (THD) and individual harmonic distortion (IHD) via optimally allocating three D-STATCOM units. The search space for locations is the same as the PV allocation problem, whereas the maximum size for D-STATCOMs is considered as 1.5 MVAr. To create non-linear behavior, NRRs are modeled as static loads in ETAP software, and the harmonic load flow analysis is performed. The typical IEEE 6 Pulse 1 model is considered as a current harmonic source for all static loads.

Case 5.2.1: In this case, the total voltage deviation (TVD) and total harmonic distortion (THD) are considered as objective functions, and CDO is implemented to optimize the locations and sizes of PVs and D-STATCOMs in the network. The best performance is observed with CDO by integrating PVs and D-STATCOMs at buses # 13 (1.084 MW, 0.376 MVAr), # 24 (1.443 MW, 0.578 MVAr), and # 30 (1.416 MW, 0.855 MVAr). By this, the distribution losses are reduced to 10.073 kW and 8.335 kVAr. The lowest voltage is registered as 0.9933 p.u at bus # 8. In addition, the total GHG emission from the main grid resource is reduced to 1689.27 lb/MWh.

Case 5.2.2: In this case, grid dependency for both real and reactive powers and power quality are optimized simultaneously. The best performance is observed with CDO by integrating PVs and D-STATCOMs at buses # 3 (1.998 MW, 0.828 MVAr), # 13 (1.057 MW, 0.363 MVAr), and # 30 (1.366 MW, 0.833 MVAr). By this, the distribution losses are reduced to 24.307 kW and 19.228 kVAr. The lowest voltage is registered as 0.9832 p.u at bus # 25. In addition, the total GHG emission from the main grid resource is reduced to 739.82 lb/MWh.

In Case 5.2.1, real power compensation is 82.866%, whereas in Case 5.2.2, it is equal to 92.91%. Thus, the total GHG emission was reduced significantly in Case 5.2.2. On the other hand, reactive power compensation in Case 5.2.1 is 83.2%, whereas in Case 5.2.2, it is equal to 93.1%. Thus, the TVD is observed as low in Case 5.2.2 compared to Case 5.2.1. The voltage profiles of the network for the base case, Case 5.2.1, and Case 5.2.2 are given in Figure 3. In addition, THDv for different scenarios are given in Figure 4. It is observed that the fundamental THDv at bus-13 has the greatest THDv of 5.21%. The IEEE 519 standards require less than 5%, while the proposed approach reduces it to 3.98% by integrating D-STATCOMs.

5.3. Reliability Improvement

In the third stage, the reliability indices, namely, system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), and customer average interruption duration index (CAIDI), are optimized by optimal network reconfiguration (ONR). The data on the failure rate and repair rate for each branch are taken from [38].

Case 5.3.1: In this case, the network is assumed to have all tie-lines as open conditions and there are no faulty conditions in the network. Thus, all load buses are interconnected and served by branches. For this radial configuration, the reliability indices SAIFI, SAIDI, and CAIDI are evaluated as 2.5741, 2.2791, and 0.8854, respectively.

Case 5.3.2: In this case, the network is assumed to have some faulty conditions on branches 9 and 14. Thus, the objective is to exclude those branches and simultaneously to determine ONR for ensuring connectivity to all the load buses served by other branches and tie-lines. The ONR-determined CDO is that branches 7, 9, 14, 32, and tie-line 37 are opened. For this new optimal radial configuration, the reliability indices SAIFI, SAIDI, and CAIDI are evaluated as 2.5719, 1.7114, and 0.6654, respectively. In comparison to Case 5.3.1, the reliability indices are decreased, which indicates an improvement in the reliability aspects of the network operation and control even under faulty conditions.

5.4. Resilience Enhancement

In the fourth stage, the resilience of the feeder is optimized in terms of average service unavailability index (ASUI), expected energy not served (ENS), and average energy not supplied (AENS) via optimally sizing the energy storage systems (ESSs).

Case 5.4.1: In this case, ENS and AENS are determined without considering PVs in the network. As per data from [38], the ASUI, ENS, and AENS are determined for the network as 0.00026, 6593.35 kWh, and 206.04 kWh, respectively.

Case 5.4.2: In this case, the load profile for a typical day and annual PV generation are taken as given in

Table 1. Daily load profile and PV generation.

Daily Load Profile				PV Generation
Time	Load (kW)	Time	Load (kW)	Month	Radiation (kWh/m2 day)	AC Energy (kWh)
12–1 AM	3725.36	Noon–1 PM	4227.37	January	6.73	615,473
1–2 AM	3510.21	1–2 PM	4356.71	February	7.09	573,535
2–3 AM	3440.55	2–3 PM	4227.37	2–3 AM	7.24	633,902
3–4 AM	3368.06	3–4 PM	4227.37	3–4 AM	6.66	570,568
4–5 AM	3324.52	4–5 PM	4183.83	4–5 AM	5.61	499,689
5–6 AM	3224.63	5–6 PM	4012.22	5–6 AM	4.56	407,270
6–7 AM	3152.92	6–7 PM	4112.11	6–7 AM	4.55	420,146
7–8 AM	3368.06	7–8 PM	4642.29	7–8 AM	4.49	412,509
8–9 AM	3811.16	8–9 PM	4757.55	8–9 AM	4.47	399,811
9–10 AM	3811.16	9–10 PM	4642.29	9–10 AM	4.99	459,289
10–11 AM	4012.22	10–11 PM	4585.95	10–11 AM	5.37	485,851
11–Noon	4155.65	11–12 PM	4227.37	11–Noon	5.27	487,862
Total			95,106.95 kWh	Annual	5.59	5,965,905

. From this, the daily load factor of the network is determined as 0.833. Further, the daily generation from 3.942 MW PV plants (i.e., as determined in Case 5.2.1) is estimated as 16,345 kWh using the PVWatts Calculator [42] considering weather conditions of Bangalore (Lat, Lng: 12.97, 77.58). From [27], it is considered that the number of days of autonomy = 1, temperature correction = 0.964, aging factor = 0.85, depth of discharge (DoD) = 0.8, battery nominal voltage = 48 V, and battery efficiency = 0.95, the required ESS is determined as 2635 kWh. Thus, by having PVs and ESSs, the ENS and AENS can be neutralized, so that the resilience aspects of the feeder can be maintained effectively.

5.5. Limitations and Future Scope

Although this work satisfies the requirement of reliable and quality power supply to the NRRs by integrating PVs, D-STATCOMs, ONR, and ESSs in a sequential and an individual optimization problem, there is scope for reformulating as a simultaneous optimization problem. Although the present work is solved for discrete operating states by modeling all the network components (including, loads, NRRs, PV, and ESS) as static conditions, there is scope for consideration of a probabilistic approach [43]. Further, to manage the uncertainty due to RE sources, the present work does not address the resilience feature of EES considering non-anticipative constraints [44]. These aspects can be treated as the future scope of this research.

5.6. Comparison of CDO and Other Meta-Heuristics

This section compares the computational efficacy of CDO with other meta-heuristics, specifically the coyote optimization algorithm (COA) [45], future search algorithm (FSA) [46], Archimedes optimization algorithm (AOA) [47], and pathfinder algorithm (PFA) [48]. Simulations are performed using standard IEEE 33-bus test system data [39]. It is considered that the maximum iterations and population for all algorithms are 50 and 30, respectively. Further, statistical analysis is performed considering 50 individual trials of each algorithm.

Three PVs are proposed for optimal integration in the network, comparing them with the existing literature.

Table 2. Comparison of different algorithms.

Method	Locations	Sizes (kW)	Ploss (kW)
			Best	Worst	Mean	Median	S.D.	Time (s)
COA	24, 30, 14	1071, 754, 1100	71.457	105.683	75.237	71.459	7.611	12.021
FSA	24, 30, 14	1071, 754, 1100	71.457	109.225	74.180	71.533	6.539	12.105
AOA	13, 30, 24	788, 1058, 1094	71.498	120.935	73.677	72.105	7.068	12.658
PFA	30, 14, 24	1071, 754, 1100	71.457	90.756	73.844	71.458	4.777	11.592
CDO	14, 30, 24	754, 1071, 1100	71.457	90.917	72.773	71.457	3.441	11.482

presents the results of all the algorithms. Upon comparison of all algorithms, CDO emerges as the superior choice. It constantly exhibits the lowest mean and median power loss, pointing to superior overall power-loss minimization performance. It also has the lowest standard deviation, which indicates that the results are the most consistent across many settings. Additionally, CDO executes the fastest, which increases its efficiency even more. While each method’s specific locations and sizes may affect its practicality, CDO stands out for its overall dependability and efficacy in terms of execution time and power-loss reduction. Figure 5 provides the convergence characteristics of these algorithms.

Further, in

Table 3. Comparison of CDO with results from the literature.

Method	Locations	Sizes (kW)	Ploss (kW)	Method	Locations	Sizes (kW)	Ploss (kW)
WCA [49]	14, 24, 29	855, 1012, 1181	71.052	EGWO-PSO [50]	14, 24, 30	754, 1099, 1071	71.457
MRFO [50]	24, 13, 30	1017, 788, 1035	72.876	CDO	14, 30, 24	754, 1071, 1100	71.457

, the results of CDO are compared with the water cycle algorithm (WCA) [49], the Mantra ray foraging optimization (MRFO) [50], and the enhanced grey wolf optimization–particle swarm optimization (EGWO-PSO) [50]. According to the findings, WCA has the lowest losses, whereas CDO has the best results compared to MRFO and EGWO-PSO. This comparison demonstrates the competitiveness of CDO in solving real-time optimization problems.

6. Conclusions

Due to the complexity and importance of nuclear research reactors (NRRs), their electricity must be reliable and high-quality to support ongoing exploration, progression, instruction, and teaching in nuclear science and engineering. For reliable power delivery to NRRs under adverse conditions, this research proposes the efficient meta-heuristic Chernobyl disaster optimizer (CDO). Finding the best photovoltaic system placement and size improves feeder performance by reducing distribution loss, voltage profile, and GHG emissions. In the second stage, optimally arranging D-STATCOM units reduced total harmonic distortion (THD) to optimize feeder power quality. Optimized network reconfiguration (ONR) and energy storage system integration improve feeder reliability and resilience in the third and fourth stages. NRR non-linearity, feeder loading profile variability, and PV variability are considered in simulations of a modified IEEE 33-bus feeder. According to the study, the proposed strategy can considerably improve NRR service requirements to better comprehend global nuclear power needs.