1 Introduction

1.1 Background and motivation

On a large scale, the utilization of fossil fuel-based power plants addresses pressing problems like the depletion of fossil fuel resources and the adverse effects of environmental pollution (Chen et al. 2006). Wind energy (WE) and photovoltaic energy (PVE) have made significant contributions to electric power generation among various renewable energy sources (RESs) in recent decades (Alasali et al. 2021). Two enticing characteristics of WE and PVE are their wide availability and minimal environmental impact, whereas WE and PVE are highly intermittent due to their uncertain existence. The use of plug-in electric vehicles (PEV) has grown globally due to the rising threat posed by greenhouse gas emissions and the deleterious effects of sound-related pollution (Zhao et al. 2012). It is known from the statistical results that most of the privately owned PEVs are in use for one hour every day. As a result, the unused energy of that PEV is discharged back to the grid to support peak load hours using the vehicle-to-grid (V2G) technology. However, the unpredictable driving pattern of vehicle owners increases the uncertainty level in the power system’s operation (Zhao et al. 2012). Exploiting tidal energy (TDE) as a renewable energy source is a current trend in power system engineering. However, it poses operational hazards to the power system due to its intermittent nature (Duman et al. 2021). Another prominent example of RES is small hydro (SH), which power system engineers highly prefer due to its environmentally friendly power generation characteristics; yet, SH, like other RESs, exhibits considerable volatile features (Sarda et al. 2021). Due to the intermittent nature of the RESs, it is sometimes difficult to get the required renewable energy output, leading to an unreliable power system operating state. To mitigate this issue, one prominent option that draws the attention of the power industry is the BES system (Meyyappan 2020). Due to the minimum operating cost, the BES system is preferred by the power system researcher to support and compensate for the power production of the power system based on volatile RESs that caused generation uncertainty.

The optimal power flow (OPF) technique plays a significant role in ensuring the economic and security concerns of the power system (Attia et al. 2018). OPF technique minimizes different power system objectives when meeting various system constraints. The key hindrances faced by power system engineers to process OPF techniques are producing efficient solution sets, smoothly handling various system constraints, and acquiring less processing time (Attia et al. 2018; Mohamed et al. 2017). Optimization of fuel cost, transmission loss, and maintaining the profile of bus voltage are objectives that the power system engineer prioritizes to ensure the economical and safe operation of the power system (Mohamed et al. 2017). The emission objective has also recently been given the utmost importance to address the overwhelming effects of environmental pollution from various pollutants (Mohamed et al. 2017). Recently, implementing the OPF technique considering RESs and PEV-equipped V2G systems has drawn the attention of the power system researchers to a significant extent (Morshed et al. 2018). In Morshed et al. (2018), the researchers studied the economic and environmental benefits of the hybrid power system, considering thermal generators, WE, PVE and PEV. Zhao et al. (2012) proposed a probabilistic approach to solve an economic load dispatch paradigm that includes an uncertain wind power generation system and V2G-enabled PEV. The researchers conducted optimization of WE and PEV-based power systems to reduce the cost of power generation, which is highly affected by uncertainties involved with WE and PEV (Jadhav and Roy 2015). The researchers addressed the cost objective of power generation and emission objective for renewable energy-integrated power systems considering the random pattern of WE and PVE in Biswas et al. (2017). An OPF approach was implemented by Panda et al. (2017) for a thermal, wind and hydroelectric energy-incorporated power system to address economic and voltage issues considering the normal and stressed operating states. The researchers formulated and solved an OPF paradigm including significant uncertainties penetrated by a doubly fed induction generator (DFIG)-coupled WE production system (Mishra et al. 2011). To streamline the economic aspects of the hybrid power system, the researchers applied a probabilistic OPF method, which is challenged by severe forecasting issues caused by intermittent WE and TDE (Fernandes et al. 2019). Furthermore, for the power system to operate as economically as possible, the researchers minimized a complex OPF problem integrating multiple RESs, such as WE, PVE, and SH (Sarda et al. 2021). Hence, various kinds of the literature suggest that intermittent characteristics of RESs such as WE, PVE, TDE, and small hydro result in forecast error, further rendering the power system’s uncertain operation. As a result, a power system with RESs brings various uncertainties into the OPF approach, necessitating the development of appropriate optimization algorithms to solve the concerns. However, the mentioned OPF problems, as discussed above, focus on power system optimization for a single hour of a day. Hence, this kind of OPF problem is referred to as a static OPF problem. The static OPF problems are not eligible to produce stable solutions set in the multi-period time horizon. As a matter of fact, the researchers prefer MOPF problems for day-ahead scheduling, where load demands vary at each hour of the day. The researchers solved static and MOPF problems for the reliable operation of the power system using an enhanced charge system search algorithm (ECSSA) (Niknam et al. 2012). Furthermore, the researchers used the MOPF problem considering different RESs for the optimization of different test power systems (Maheshwari et al. 2023). The researchers optimized the smart grid considering RESs and BES to enhance the power system’s performance and security (Alzahrani et al. 2023). Another study revealed the optimization framework of a microgrid system that considers RESs and BES (Levron et al. 2013).

1.2 Literature review

The researchers implement evolutionary optimization algorithms (EOA) to solve the OPF technique, considering different challenging objectives (Mohamed et al. 2017). For the benefit of tracking the global optimal solution point, achieving excellent computational efficacy and speeding up the convergence ability, the phasor particle swarm optimization (PPSO) introduces phasor angle theta (\(\theta \)) based control parameters and was applied for the solution of economic load dispatch problem (Gholamghasemi et al. 2019). A modified crow search algorithm (MCSA) that includes a method of priority search and efficient flight-length of crows to boost the local searchability was implemented to solve non-convex objectives of ED problem (Mohammadi and Abdi 2018). A cooperative artificial bee colony (ABC) algorithm known as (CABC) was brilliantly applied by the researchers for solving various challenging OPF objectives considering the IEEE 30-bus power system to achieve a computationally robust and efficient set of solutions (Zhou et al. 2019). The researchers proposed a hybrid form of modified imperialist competitive algorithm and sequential quadratic programming, HMICA-SQP, to gain effective local search capability and greater precision of the solution in handling uncertainties caused by WE and PVE in the OPF model (Hmida et al. 2019). Biswas et al. (2018) recommended a differential evolution algorithm (DE), which exploits the features of a feasible solution (SF) superiority, self-adaptive penalty (SP) and an ensemble of these two constraint handling techniques (ECHT) to produce effective OPF solutions for a range of practical objectives of the power system. The conventional particle swarm optimization (PSO) was integrated with the interior point method in an enhanced particle swarm optimization (EPSO) with an efficient solution searching strategy to solve the economic dispatch model, which includes the probabilistic paradigm of WE and PEV (Zhao et al. 2012). Further, another research study shows the solution to the intermittent wind power incorporated chance-constrained economic emission dispatch problem using a chaotic sine-cosine algorithm (CSCA) (Guesmi et al. 2020). The moth swarm algorithm (MSA) modelled the behavioural pattern of moths to get the efficient output of the OPF model that minimized several objectives based on the economical, secure and eco-friendly operation of the power system (Mohamed et al. 2017). To reach an efficient solution, the researchers simulated a multi-objective adaptive guided differential evolution (MOAGDE) based OPF technique to deal with uncertainties in power generation by WE, PVE, and TDE (Duman et al. 2021). In the case of two different benchmark power systems, an equilibrium optimization (EO) was utilized to solve various test objectives of the OPF problem, taking WE and thermal generator into consideration (Amroune 2022). Hassan et al. (2023) integrated the elite evolutionary strategy (EES) with the wild horse optimizer (WHO) technique for better search efficacy and proposed a modified version of the WHO called EESWHO. The EESWHO was applied to optimize the power generation cost objective of the IEEE 30-bus system, considering intermittent RESs such as WE and PVE. Further, an enhanced coati optimization algorithm (ECOA) was devised to solve the probabilistic OPF (POPF) issue by considering the thermal power and the intermittent source of WE, PVE, and electric vehicle (EV) (Hasanien et al. 2023). To address the OPF problems, including CTU, WE, PVE, and SH generation system, the researchers used a flow direction algorithm (FDA) that mimics the movement of the flow directed to the drainage basin outlet (Maheshwari et al. 2023). Furthermore, the FDA-based OPF approach is validated using three different IEEE standard test systems. The researchers introduced a modified jellyfish search optimizer (MJSO) to solve the optimal reactive power dispatch (ORPD) problem, which incorporates intermittent WE and PVE (Gami et al. 2022). The proposed MJSO was integrated with a chaotic mutation mechanism for better exploration, whereas the spiral orientation motion-based sorted jellyfish population was used to enhance exploitation ability. Further study demonstrates the application of a multi-objective multi-verse optimization (MOMVO) for the solution of dynamic ELD problems with WE and PVE (Acharya et al. 2023). Moreover, the MOMVO successfully solved the cost and emission objective for the said ELD problem for enhancing power system reliability. A novel metaheuristic algorithm called weighted mean of vectors (INFO) was proposed for solving the OPF problem that considers various RESs, e.g., WE, PVE, and a combination of PVE and SH (Farhat et al. 2023). The researchers devised an adaptive lightning attachment procedure optimizer (ALAPO) to solve different power-related OPF objectives, considering stochastic operational scenarios in the presence of WE and PVE (Adhikari et al. 2023).

From the literature reviewed, the following research gaps are identified as stated below.

  • Due to economic, environmental, operational, and planning concerns, the researchers focus on optimizing power systems incorporating one or more RESs. As a result, various research publications based on RESs integrated power system optimization have been developed; however, the majority of these research articles dealt with the OPF technique, which includes WE (Amroune 2022; Chen et al. 2006; Mishra et al. 2011; Guesmi et al. 2020), WE-PVE (Adhikari et al. 2023; Alasali et al. 2021; Biswas et al. 2017; Gami et al. 2022; Hassan et al. 2023; Hmida et al. 2019), WE-PEV (Zhao et al. 2012; Jadhav and Roy 2015), WE-PVE-SH (Farhat et al. 2023; Maheshwari et al. 2023; Sarda et al. 2021), WE-PVE-PEV (Hasanien et al. 2023; Morshed et al. 2018), WE-hydro (Panda et al. 2017), WE-PVE-TDE (Duman et al. 2021. Therefore, it is deeply apparent from the literature review that no research has been carried out integrating WE, PVE, TDE, SH and PEV simultaneously in the OPF method.

  • According to the literature review, the greater the number of RESs in an OPF model, the greater the uncertainty; as a result, it has always been challenging to design an EOA that can effectively address the significant number of uncertainties imposed by WE, PVE, TDE, SH and PEV as described in the article.

1.3 Article contribution

The differential search algorithm (DSA) is an EOA that conceptualizes random-walk movement, resembling Brownian-like motion, done by an organism (Civicioglu 2012). The researchers implemented DSA to solve the OPF problem while addressing different concerns related to the optimal operational balance of the power systems with significant efficacy (Bouchekara and Abido 2014; Abaci and Yamaçli 2017; Abaci and Yamacli 2016). According to the “No Free Lunch” theorem, no particular EOA best suites every optimization problem (Xiong et al. 2013). On that note, the researchers are eager to develop new EOAs or improve existing ones.

As per the preceding discussion, it can be justified that including RESs in the OPF problem leads to economically efficient, environment-friendly power system operation and planning. As a matter of fact, in the present article, we develop a hybrid static OPF model incorporating conventional thermal units (CTUs) and four diverse RESs for efficient power system operation and planning. Due to their- a) economic and pollution-free power generation capacity and b) inexhaustible presence in nature, WE, PVE, TDE, and SH are integrated into our formulated hybrid OPF model. In this context, it is worth noting that, unlike large-scale hydropower generation systems, SH generation has no detrimental environmental impact (Paish 2002). Also, following the recent global research trend, V2G-enabled PEV as a gridable energy source is incorporated in the devised hybrid OPF model due to its pollution-free and cheap power generation ability. Additionally, we devise the MOPF problem by considering CTU, WE, PVE, and BES. Further, to solve the formulated hybrid OPF paradigm, a novel enhanced performance-based DSA (EPDSA) is developed with the implementation of three techniques, namely, A. a dynamic population reduction scheme to keep good exploration in the preliminary iteration phases while rendering substantial exploitation to reach near the global optimal solution point at the latter iteration phase, B. a quasi-oppositional-based learning (QOBL) to enhance exploration capacity, as well as to avoid the risk of getting caught in local minima, and C. a highly sophisticated, best artificial-organism-guided stopover site discovering technique to benefit the exploitation of the proposed EOA. The proposed EPDSA is applied to minimize four significant OPF objectives, such as cost minimization of active power production considering valve-point loading phenomenon, emission level, active power loss minimization, and voltage-stability index (L-index) for modified IEEE 30 and 118-bus test power system incorporating CTU and various intermittent RESs such as WE, PVE, combined wind-tidal (WTD) and PEV with V2G facility. Additionally, the proposed algorithm is subjected to minimizing power production cost with valve-point loading effect and emission level for the modified IEEE 30 and 118-bus system with CTU, mentioned RESs, and combined photovoltaic-small hydro (PVSH), respectively. Furthermore, the MOPF problem is also formulated and solved to minimize the generation cost of power considering the modified IEEE 30-bus system with the incorporation of CTU, WE, PVE, and BES by using the proposed algorithm. It should be noted that the current article concentrated on the modelling techniques of a TDE-based generation system based on a tidal range type energy system (Duman et al. 2021). The OPF-simulation outcomes are compared with eight state-of-the-art EOAs i.e., the DSA, political optimizer (PO) (Askari et al. 2020), marine predators algorithm (MPA) (Faramarzi et al. 2020), enhanced bat algorithm (EBA) (Yılmaz and Küçüksille 2015) and yellow saddle goatfish algorithm (YSGA) (Zaldivar et al. 2018), MSA, quantum firefly algorithm (QFA) (Zitouni et al. 2021), and biogeography-based optimization (BBO) (Simon 2008) to substantiate the power of the proposed EPDSA.

The significant contributions of the article are listed below.

  1. (1)

    A novel EPDSA is developed for solving the uncertainty-based hybrid OPF model.

  2. (2)

    Three alteration techniques (ATs), namely, dynamic population reduction scheme, QOBL, and best artificial-organism-guided stopover site discovering method, are integrated into the proposed EPDSA for better exploration, exploitation, and speedy convergence profile.

  3. (3)

    A hybrid OPF model comprising CTU and four highly intermittent RESs, namely, WE, PVE, combined WTD, combined PVSH, and V2G-enabled PEV, is designed for economical, eco-friendly, and secure power system operations. To the best of the authors’ knowledge, no single research article has been published to date that addresses a hybrid OPF model incorporating a large number of RESs all at once, as considered in this article.

  4. (4)

    The unpredictability of the investigated RESs is successfully integrated with the established hybrid model of OPF, employing the interpretable mathematical models that use Weibull, Lognormal, Gumbel, and Normal probability density functions (PDF).

  5. (5)

    Exhaustive power-based simulation tests are performed to validate the proposed EPDSA’s solution accuracy for solving the hybrid OPF model with significant intermittency, and the results are compared with eight highly competitive EOAs.

  6. (6)

    To validate the proposed EPDSA’s solution accuracy for solving the hybrid OPF model in the presence of significant intermittency, exhaustive power-based simulation tests are performed, and the results are compared with eight effective EOAs.

  7. (7)

    Based on four practical power system objectives, the simulation experiments correspond to the small and the large-scale modified RES-integrated IEEE test power systems.

  8. (8)

    Moreover, this article includes a practical test instance that considers the solution to the MOPF problem by considering intermittent RESs such as WE, PVE, and BES with charging/discharging patterns.

The remainder of the article is organized into different sections: Sect. 2 details the probabilistic modelling of RESs and PEV, section 3 shows the formulation of the OPF-objectives and different constraints, section 4 states solution methodology, simulation outcomes are discussed in section 5, and section 6 includes the concluding discussion.

2 Modelling of RESs, PEV, and BES

2.1 Probabilistic modeling: WE

2.1.1 Assumptions made for the modeling of wind power output

The following points are considered to model wind power output in the present article.

  1. (1)

    In the wind power unit or WPU, a wind turbine is connected to the generator bus of the considered system via a DFIG (Panda et al. 2017).

  2. (2)

    In the proposed study, the intermittency-based wind power is considered to be a function of wind speed, which is a random variable (Jadhav and Roy 2015).

  3. (3)

    Due to its simplistic mathematical formulation and significant efficiency, the linear model of the wind power curve is used in our study to predict the wind power output of WPU subjected to intermittent wind speed (Jadhav and Roy 2015).

  4. (4)

    The wind speed is modelled in our approach using the Weibull PDF because of the advantages such as a) its generalized mathematical structure because it follows a two-parameter distribution and b) its higher accuracy level to predict variable wind speed.

As stated above, the wind speed follows the Weibull PDF (Chen et al. 2006), as shown below.

$$\begin{aligned} {{f}_{wind}}(v)=\left( \frac{k}{c} \right) {{\left( \frac{v}{c} \right) }^{(k-1)}}{{e}^{-{{({\scriptstyle {}^{v}/{}_{c}})}^{k}}}}\text { }0<v<\infty \end{aligned}$$
(1)

where v refers to the average wind speed, c refers to the scale parameter, and k represents the shape parameter (Chen et al. 2006).

The cumulative distribution function (CDF) of the wind speed is as follows.

$$\begin{aligned} {{F}_{wind}}(v)=1-{{e}^{-{{({\scriptstyle {}^{v}/{}_{c}})}^{k}}}} \end{aligned}$$
(2)

In the present article, the linear model of the wind power curve (Chen et al. 2006) is considered for expressing the relationship between wind power (w) and v, as stated below.

$$\begin{aligned} w=\left\{ \begin{aligned}&0\text {}v<{{v}_{cin}},v>{{v}_{cout}} \\&{{P}_{w,r}}\frac{(v-{{v}_{cin}})}{({{v}_{r}}-{{v}_{cin}})}\text {}{{v}_{cin}}\le v\le {{v}_{cout}} \\&{{P}_{w,r}}\text {}{{v}_{r}}\le v\le {{v}_{cout}} \\ \end{aligned} \right. \end{aligned}$$
(3)

In the above Eq. (3), \({{P}_{w,r}}\) is the output power rating of the WPU, \({{v}_{cin}}\), \({{v}_{r}}\), and \({{v}_{cout}}\) refer to the cut-in velocity, rated velocity and cut-out velocity (m/second) of wind. It appears from Eq. (3) that the output of the WPU varies continuously between zero and rated levels. Wind power following Weibull PDF may be formulated in the continuous domain by linear transformation (Chen et al. 2006), as shown below.

$$\begin{aligned} {{f}_{wind}}(w)=\frac{kl{{v}_{in}}}{c}{{\left( \frac{\left( 1+\rho l \right) {{v}_{in}}}{c} \right) }^{k-1}}\exp \left( -{{\left( \frac{\left( 1+\rho l \right) {{v}_{in}}}{c} \right) }^{k}} \right) \end{aligned}$$
(4)

where \(\rho =\frac{w}{{{P}_{w,r}}}\) and \(l=\left( \frac{{{v}_{r}}-{{v}_{in}}}{{{v}_{in}}} \right) \)

According to Eq. (3), w has the following discontinuous probability (Chen et al. 2006): When the probability event is zero \((w=0)\)

$$\begin{aligned} P1=1-\exp \left( -{{\left( \frac{{{v}_{in}}}{c} \right) }^{k}} \right) +\exp \left( -{{\left( \frac{{{v}_{out}}}{c} \right) }^{k}} \right) \end{aligned}$$
(5)

When the probability of event \(w={{P}_{w,r}}\)

$$\begin{aligned} P2=\exp \left( -{{\left( \frac{{{v}_{r}}}{c} \right) }^{k}} \right) -\exp \left( -{{\left( \frac{{{v}_{out}}}{c} \right) }^{k}} \right) \end{aligned}$$
(6)

2.2 Probabilistic modeling: PVE

2.2.1 Assumptions made for the modeling of PV power output

To model uncertain PV power, the following assumptions are made in the approach followed in the present article.

  1. (1)

    In the present study, the PVE source is linked with the respective generator bus by a DC/AC inverter (Biswas et al. 2017).

  2. (2)

    As widely adopted in different research works (Biswas et al. 2017; Alasali et al. 2021), the output power of the photo voltaic unit (PVU) is considered as a function of solar irradiance \((I_R)\) in the present article.

  3. (3)

    For accurate and realistic prediction of \(I_R\) in the present study, a Lognormal PDF is utilized for modelling \(I_R\) (Biswas et al. 2017).

As stated above, \(I_R\) varies with daytime and follows a Lognormal PDF, given below (Biswas et al. 2017).

$$\begin{aligned} {{f}_{PV}}({{I}_{R}})=\frac{1}{({{I}_{R}}\sigma \sqrt{2\pi })}\exp \left( \frac{-{{(\ln {{I}_{R}}-\mu )}^{2}}}{{{(2\sigma )}^{2}}} \right) \text {; }{{I}_{R}}>0 \end{aligned}$$
(7)

where \({\mu (=5.2)}\) is the mean and \(\delta (=0.6)\) is the standard deviation (SD) of Lognormal PDF.

The output of the PVU is a function of \((I_R)\) and is expressed below (Biswas et al. 2017).

$$\begin{aligned} s=\left\{ \begin{aligned}&{{P}_{PV,r}}\left( \frac{I_{R}^{2}}{{{I}_{R,std}}{{R}_{c}}} \right) ;\text { }for\text { }0<{{I}_{R}}<{{R}_{c}} \\&\\&{{P}_{PV,r}}\left( \frac{{{I}_{R}}}{{{I}_{R,std}}} \right) ;\text { }for\text { }{{I}_{R}}>{{R}_{c}}\text { } \\ \end{aligned} \right. \end{aligned}$$
(8)

where \({{P}_{PV,r}}\) express the rated power output of the PVU, \({{I}_{R,std}}=(1000w/m^2)\) represents the standard irradiance point of the PVE, \({{R}_c}=(150w/m^2)\) is called the specific irradiance point (Biswas et al. 2017).

2.3 Probabilistic modeling: TDE

2.3.1 Assumptions made for the modeling of tidal power output

Tidal power is modelled in the present study by considering the following key assumptions.

  1. (1)

    Our study assumes that the tidal source is connected to the generator bus through an AC generator.

  2. (2)

    Tidal range (Duman et al. 2021), a very common tidal energy harvesting scheme, is used to formulate tidal power output.

  3. (3)

    The tidal power output is expressed as a function of the tidal discharge rate (Duman et al. 2021).

  4. (4)

    Gumbell PDF is used to predict the uncertainty of the intermittent tidal discharge rate \(({{Q}_{TD}})\) (Duman et al. 2021).

Tidal discharge rate \({{Q}_{TD}}\) and tidal power output are random and interrelated variables (Duman et al. 2021). \({{Q}_{TD}}\) is characterized by Gumbel PDF (Duman et al. 2021), as stated below.

$$\begin{aligned} {{f}_{TDE}}({{Q}_{TD}})=\frac{1}{m}\exp \left( \frac{{{Q}_{TD}}-r}{m} \right) \exp \left[ -\exp \left( \frac{{{Q}_{TD}}-r}{m} \right) \right] \end{aligned}$$
(9)

where \(m(=24.52)\) is the scale parameter, and \(r(=220)\) is the location parameter, as taken from Duman et al. (2021). The output power \({{P}_{TE}}\) delivered by the tidal power unit (TDPU) is a function of \({{Q}_{TD}}\) and is given below (Duman et al. 2021).

$$\begin{aligned} {{P}_{TDE}}=\eta {{Q}_{TD}}\rho g{{H}_{TD}} \end{aligned}$$
(10)

where \({\eta (=0.85}\) is the efficiency of the TDPU, \({{Q}_{TD}}\) is in (\(m^3/sec\)), \(\rho (=1025 kg/m^2)\) refers to the density of water, g\((=9.81 m/s^2)\) is the acceleration of gravity, \({{H}_{TD}}(=3.2 m)\) indicates the pressure head due to the level difference between high and low tide.

2.4 Probabilistic modeling: SH

2.4.1 Assumptions made for the modeling of small hydropower output

Considering the following key assumptions, the small hydropower is modelled in the present study.

  1. (1)

    The current research assumes that the SH source is connected to the generator bus via an AC generator.

  2. (2)

    The SH source under consideration is connected to run-of-river installations, indicating no dam is needed (Paish 2002).

  3. (3)

    The output power of SH generation depends on the river flow rate (Farhat et al. 2023).

  4. (4)

    A Gumbell PDF is used to estimate the uncertainty associated with the river flow rate \(({{Q}_{SH}})\) of the SH unit or SHU (Farhat et al. 2023).

\({{Q}_{SH}}\) and power output of SHPU are random and interrelated variables (Farhat et al. 2023). \({{Q}_{SH}}\) is expressed by a Gumbel PDF (Farhat et al. 2023), as shown below.

$$\begin{aligned} {{f}_{SH}}({{Q}_{SH}})=\frac{1}{\zeta }\exp \left( \frac{{{Q}_{SH}}-{k}'}{\zeta } \right) \exp \left[ -\exp \left( \frac{{{Q}_{SH}}-{k}'}{\zeta } \right) \right] \end{aligned}$$
(11)

where \(\zeta (=1.2)\) indicates the scale parameter and \({k}'(=15)\) refers to the location parameter as taken from Farhat et al. (2023). The output power \({{P}_{SH}}\) of the SHU is a function of \({{Q}_{SH}}\) and is given below (Farhat et al. 2023).

$$\begin{aligned} {{P}_{SH}}={\eta }'{{Q}_{SH}}{\rho }'g{{H}_{SH}} \end{aligned}$$
(12)

where \({{\eta }'(=0.85}\) is the efficiency of the SHU, \({{Q}_{SH}}\) is in (\(m^3/sec\)), \({\rho }'(=1000 kg/m^2)\) is the density of water, g\((=9.81 m/s^2)\) is the acceleration of gravity, \({{H}_{SH}}(=25 m)\) refers to the effective pressure head of the SHU.

2.5 Probabilistic modeling: PEV

2.5.1 Assumptions made for the modeling of PEV output power

Some assumptions in this article are taken into account for modelling the PEV as a gridable source for the V2G technique and are listed as follows.

  1. (1)

    Private PEVs with more flexible daily driving schedules are used as a gridable source for V2G facilities (Zhao et al. 2012).

  2. (2)

    The owners of the vehicles must undergo a voluntary registration process to register their vehicles to meet the peak load requirement (Zhao et al. 2012).

  3. (3)

    As a necessary condition of the registration, the owners ensure that their PEVs are charged during the off-peak load hour. Therefore, recharging the PEV battery would not conflict with its hours of discharge (Zhao et al. 2012).

  4. (4)

    The PEVs are connected to the grid during peak load hours, and the grid operator can use the PEVs as a gridable source to support the peak load demand (Zhao et al. 2012).

  5. (5)

    A DC/AC inverter controls the discharging rate of PEV, which is operated as a V2G source (Zhao et al. 2012).

  6. (6)

    The system operator must remotely control the discharge operation of the PEV (Zhao et al. 2012).

  7. (7)

    Although the uncertain V2G output power (\(\rho \)) of PEV is dependent on a variety of parameters, and it is firmly established by (Zhao et al. 2012), that, according to the Jarque-Bera (JB) test for normality and derived P-values, the PEV output (p) can be appropriately modelled using a Normal PDF, as shown below.

$$\begin{aligned} {{f}_{PEV}}(p)=\frac{1}{\sqrt{2\pi {{\phi }^{2}}}}{{e}^{-\frac{{{(p-\mu )}^{2}}}{2{{\phi }^{2}}}}} \end{aligned}$$
(13)

2.6 Modelling of BES system

2.6.1 Assumptions made for the modelling of BES dynamics

The following points are considered to model BES in the present article.

  1. (1)

    In this study, it is taken into consideration that the BES is connected via a DC/AC converter with the individual bus bar.

  2. (2)

    For our research, we assume that BES has a charging or discharging efficiency of 90

  3. (3)

    It is also assumed that the lowest and highest charging/discharging power of the considered BES is 0 and 15 MW, respectively.

  4. (4)

    The battery’s state of charge (SOC) in each time interval is computed by the equation below.

$$\begin{aligned} SOC_{k}^{t+1}=SOC_{k}^{t}+\left( P_{ch,k}^{t}\times {{\eta }_{ch,i}}-\frac{P_{dch,k}^{t}}{{{\eta }_{ch,i}}} \right) \times \Delta t \end{aligned}$$
(14)

where \(k=1,2,..,{{n}_{BES}}\).

3 Formulation of OPF-Objective

3.1 Formulation of objective function for the static OPF problem

As stated below, four distinct single objective functions (SOFs) are formulated considering the uncertain behaviour of WE, PVE, TDE, SH, and PEV.

(1) Total cost The total fuel cost for power generation is formulated using the following equation.

$$\begin{aligned} \begin{aligned} SOF1=&\sum \limits _{k=1}^{{{n}_{GB}}}{T{{C}_{k}}}+\sum \limits _{k=1}^{{{n}_{WPU}}}{W{{C}_{k}}+}\sum \limits _{k=1}^{{{n}_{PVU}}}{PV{{C}_{k}}+}\sum \limits _{k=1}^{{{n}_{WTDPU}}}{WTD{{C}_{k}}}\\&+\sum \limits _{k=1}^{{{n}_{PVSHU}}}{PVSH{{C}_{k}}}+\sum \limits _{k=1}^{{{n}_{PEV}}}{PEV{{C}_{k}}} \end{aligned} \end{aligned}$$
(15)

Due to space limitations, the detailed formulation of the cost function for the RESs is described in the supplementary file.

The cost of fossil fuel for the \({{k}^{th}}\) CTU is represented and formulated below, taking into account the effect of valve-point loading, in order to increase the viability of the research work in view of the real-world power system (Mohamed et al. 2017).

$$\begin{aligned} \begin{aligned} T{{C}_{k}}&=\left( \left( \sum \limits _{k=1}^{{{n}_{GB}}}{{{\alpha }_{k}}+{{\beta }_{k}}{{P}_{g,k}}+{{\gamma }_{k}}P_{g,k}^{2}}\right. \right. \\&\quad \left. \left. +\left| {{d}_{k}}\times \sin ({{e}_{k}}\times \left( P_{g,k}^{\min }-{{P}_{g,k}}\right) \right) \right| \right) (\$/Hr) \end{aligned} \end{aligned}$$
(16)

(2) Level of emission From an environmental standpoint, the outflow of pollutants from the electrical power plant is a significant challenge. \({CO}_2\), \({SO}_2\), and \({NO}_x\) are pollutants that have a negative impact on the environment’s balance. The following function, which includes polynomial and exponential terms, is used to calculate the level of emissions from these pollutants.

$$SOF2 = \left( {\sum\limits_{{k = 1}}^{{n_{{GB}} }} {c_{k} P_{{g,k}}^{2} + b_{k} P_{{g,k}} + a_{k} + \varepsilon _{k} \exp ^{{(\lambda _{k} P_{{g,k}} )}} } } \right)(ton/Hr)$$
(17)

The minimization of the emission level objective function for the OPF problem is represented as the sum of every sort of emissions sourced from \(C{{O}_{2}}\), \(S{{O}_{2}}\), and \(N{{O}_{x}}\), using the appropriate pricing or weighting for each pollutant released from the CTU during electric power generation. The above equation shows that the emission minimization objective is a function of active power, i.e. \({{P}_{g,k}}\) produced by the \({{k}^{th}}\) CTU. Using Eq. (17), the amount of emitted pollutants is computed in ton/Hr in our proposed approach. Further, using the proposed EOA, the optimal settings of \({{P}_{g,k}}\) is found iteratively for which the level of emission is minimum.

(3) Active line loss The following objective is considered in order to minimize active line loss and increase the operational efficiency of the power system.

$$\begin{aligned} SOF3= & {} \sum \limits _{k=1}^{{{n}_{TL}}}{{{G}_{k}}}[~~{{\left| {{V}_{e}} \right| }^{2}}+{{\left| {{V}_{f}} \right| }^{2}}\nonumber \\{} & {} \quad -2\left| {{V}_{e}} \right| \left| {{V}_{f}} \right| \cos ({{\delta }_{e}}-{{\delta }_{f}})~]MW \end{aligned}$$
(18)

where the \({{k}^{th}}\) line connecting \({{e}^{th}}\) and \({{f}^{th}}\) the bus has a conductance of \({{G}_{k}}\), \({{n}_{TL}}\) is the number of transmission lines.

(4) Voltage stability index The L-index, or voltage stability index, allows the system operator to track whether the system’s bus voltages are within the prescribed limit under rated operating conditions. The following equation expresses the L-index for the \({{k}^{th}}\) bus.

$$\begin{aligned} {{L}_{k}}=\left| 1-\sum \limits _{i=1}^{{{n}_{GB}}}{{{F}_{ki}}\frac{{{V}_{i}}}{{{V}_{k}}}} \right| \end{aligned}$$
(19)

where \(k=1,2,..,{{n}_{LB}}\)

$$\begin{aligned} {{F}_{ki}}=-{{\left[ {{Y}_{1}} \right] }^{-1}}\left[ {{Y}_{2}} \right] \end{aligned}$$
(20)

where \({{Y}_{1}}\) and \({{Y}_{2}}\) are referred to as sub-matrices of \({{Y}_{bus}}\) of the considered system.

The objective L-index for the entire system is denoted as \({{L}_{max}}\) and is given below.

$$\begin{aligned} SOF4={{L}_{\max }}=\max ({{L}_{k}}) \end{aligned}$$
(21)

where \(k=1,2,..,{{n}_{LB}}\)

In this connection, it must be noted that a low value of \({{L}_{max}}\) signifies a higher degree of the voltage stability of the power system.

3.2 Formulation of objective function for multi-period OPF problem

The MOPF problem is conducted for a complete daytime horizon, i.e., on a 24-hour basis. Each period or interval for the MOPF problem is divided into one-hour time duration. The key objective of the power system operator is to find the power system’s optimal settings for each hour using the MOPF problem. The following SOF, which considers thermal cost with valve-point loading effect, uncertainty cost of WE, PVE, and the BES system’s charging/discharging cost, is used for the multi-period OPF and is expressed as the following equation.

$$\begin{aligned} SOF5= & {} \sum \limits _{t=1}^{T}\sum \limits _{k=1}^{{{n}_{GB}}}{TC_{k}^{t}}\nonumber \\{} & {} \quad +\sum \limits _{k=1}^{{{n}_{WPU}}}{WC_{k}^{t}+}\sum \limits _{k=1}^{{{n}_{PVU}}}{PVC_{k}^{t}+}\sum \limits _{k=1}^{{{n}_{BES}}}{BESC_{k}^{t}} \end{aligned}$$
(22)

The following equation is implemented for the cost computation of the BES system.

$$\begin{aligned} BESC_{k}^{t}={{c}_{dch,k}}P_{dch,k}^{t}-{{c}_{ch,k}}P_{ch,k}^{t} \end{aligned}$$
(23)

The supplementary file includes all the operational constraints considered in the present research work for the static OPF and MOPF problems.

4 Solution methodology

4.1 Differential search algorithm (DSA)

The DSA is a very popular EOA introduced by Civicioglu (2012). DSA strategy mimics the \(superorganism's\) migratory pattern in search of a more resourceful habitat. As per the DSA, Brownian-like random-walk movement is implemented to model the direction of the superorganism. The key steps of the DSA are as follows.

  • An artificial-organism is equivalent to a population consisting of randomly generated solutions.

  • The superorganism, which consists of the artificial-organism, continues to migrate until the global minimum point is found.

  • The superorganism checks for the temporary suitability of a position randomly selected while migrating.

  • If a position is suitable for a temporary stopover, then every member of the superorganism i.e., artificial-organism, immediately occupies that position and continues to migrate.

Different phases of the DSA are stated below.

Phase 1:

This phase initializes the \(Superorganis{{m}_{t}}\text { }\)(=\([{{X}_{i}}])\), t=1,2,...,\(\max \_ite\) (maximum iteration count) via initializing each of its artificial-organism (\({{X}_{i}}=[{{x}_{i,j}}]\)) in the following manner:

$$\begin{aligned} {{x}_{i,j}}=rand.(u{{p}_{j}}-lo{{w}_{j}})+lo{{w}_{j}} \end{aligned}$$
(24)

where \(i=1,2,...,{SP}\); \(j=1,2,...,d\); SP and d indicate the total population number and total control variable number of the concerned optimization model, up and low represent the upper and lower boundary of each control variable and randg is a random number with a range [0,1].

Phase 2:

In the migration operation of the DSA, a stopover site is generated in the search space that remains between the artificial-organism employing a mechanism called \(Brownian-like\) random walk as described below.

$$\begin{aligned} StopoverSit{{e}_{i}}={{X}_{i}}+scale.(donor-{{X}_{i}}) \end{aligned}$$
(25)

where

$$\begin{aligned}{} & {} scale=randg.\left[ 2.ran{{d}_{1}} \right] .\left( ran{{d}_{2}}-ran{{d}_{3}} \right) , \end{aligned}$$
(26)
$$\begin{aligned}{} & {} donor={{X}_{Random\_Shuffling(i)}} \end{aligned}$$
(27)

The scale parameter enables respective artificial-organisms to change their search direction radically. rand is a random number generator by a gamma probability distribution. \({{rand}_{1}}\), \({{rand}_{2}}\), and \({{rand}_{3}}\) are random numbers ranging [0,1]. Random_Shuffling function alters the position of different control variables of the different artificial-organisms in a random manner.

Phase 3:

The eligibility of the individuals of the artificial-organisms corresponding to a superorganism for taking part in the search process of the stopover site is determined by a random method as follows.

$$\begin{aligned} StopoverSite({{x}_{I,J}})=Superorganism({{x}_{I,J}}) \end{aligned}$$
(28)
$$\begin{aligned} {{x}_{I,J}}\leftarrow {{r}_{I,J}}>0\text {}\!\!|\!\!\text { }I\in i,J\in [1,d] \end{aligned}$$
(29)

where \({{r}_{I,J}}\) is randomly assigned with ‘0’ or ‘1’.

Phase 4:

Here, the stopover site is updated by the following logic.

$$\begin{aligned}{} & {} StopoverSite_{i,j}:=rand.(u{{p}_{j}}-lo{{w}_{j}})\nonumber \\{} & {} \quad +lo{{w}_{j}}\text {}if\left\{ \begin{aligned}&StopoverSite_{i,j}<lo{{w}_{j}} \\&StopoverSite_{i,j}>u{{p}_{j}} \\ \end{aligned} \right\} \end{aligned}$$
(30)
Phase 5:

The movement of the superorganism towards the stopover site is determined via two key factors, i.e., the fitness of the superorganism and the stopover site, by the below-mentioned method.

$$\begin{aligned} {{X}_{i+1}}=\left\{ \begin{aligned}&StopoverSit{{e}_{i}}\,\text {}if\,\text {}fi{{t}_{StopoverSit{{e}_{i}}}}<fi{{t}_{{{X}_{i}}}} \\&{{X}_{i}}\text {}else \\ \end{aligned} \right. \end{aligned}$$
(31)

where \({fi{{t}_{StopoverSit{{e}_{i}}}}}\) and \({fi{{t}_{{{X}_{i}}}}}\) are the fitness values of the stopover site and the superorganism, respectively.

4.2 Enhanced performance-based differential search algorithm

Three ATs modify the conventional DSA to obtain a high-quality solution with greater computational efficacy by addressing various uncertainties and complexities incorporated in the hybrid OPF model.

4.2.1 AT-1: Dynamic population reduction scheme

Small population size may be an option for improving the orthodox DSA’s convergence speed. Still, it comes at the expense of poor exploration capability due to reduced diversity among the various population sets, potentially increasing the chances of being trapped in the local optimal solution point. As a result of the preceding reasons, improving the capability of the DSA in determining the population size, with a progressive reduction over iterative steps, is a highly viable option. The proposed EPDSA includes a dynamic population reduction scheme to take advantage of the stated concept (Ghambari and Rahati 2018). The proposed EOA exhibits substantially strong exploration capability to track the fruitful food source in the search domain during the initial iteration process. In contrast, during the latter iteration stage, the proposed EOA follows a highly exploitative nature to find a mostly fruitful food source close to the global optimal point.

Based on the preceding analysis, the proposed dynamic population reduction scheme is mathematically formulated, as given below.

$$\begin{aligned} S{{P}_{current}}=\left\{ \begin{aligned}&\max ((S{{P}_{current}}-S{{P}_{\min }}),S{{P}_{}}\text { )}if\text { }{{S}_{R}}>{{r}^{/}} \\&S{{P}_{current}}\text {}else\text {} \\ \end{aligned} \right. \text {} \end{aligned}$$
(32)

where \(S{{P}_{current}}\) refers to the size of the current population, \(S{{P}_{min}}\) is the least possible value to which the size of the population can be reduced; \(S{{P}_{}}\) is called the population threshold that determines the possibly smallest size of the population, while the population reduction is executed; lastly, \({{r}^{/}}\) signifies a randomly produced number in the range of 0–1.

Equation (32) is devised keeping in mind the following two advantages, which help our proposed EOA obtain high-quality solutions with stable system performance.

  1. (1)

    In determining the population size, an elitism-based survival scheme (ESS) is implemented to allow the elite populations with higher fitness values to enter into the subsequent iteration, removing the populations with inferior fitness values. The proposed EPDSA experiences a dynamic population reduction scheme over the iteration steps. With this scheme, the implemented ESS enforces the proposed EPDSA to zoom out on the fruitful sources of food by allowing the elite populations to survive in the upcoming iteration steps.

  2. (2)

    In Eq. (32), a survival rate factor \({{S}_{R}}\) with a constant value in the range of 0–1 is used to regulate the reduction rate of the population size in each iteration step. In this context, it must be mentioned that we avoid the large value for \({{S}_{R}}\) that results in the relatively smaller size of the population that forced our proposed EOA to converge prematurely. On the contrary, \({{S}_{R}}\) with a lower value substantially reduces the convergence speed. Thus, the computational efficacy of the proposed EPDSA is highly affected. Moreover, \({{S}_{R}}\) assists the size of the population in reducing linearly in a probabilistic way, thus causing an efficient and robust search pattern followed by the proposed EPDSA by maintaining an appropriate balance between exploration and exploitation.

4.2.2 AT-2: Quasi-oppositional-based learning

Based on the probability principle, it is checked that, in more than 50\(\%\) of cases, the solution generated by quasi-oppositional assumptions is closer to the global best point than the solution generated by random initialization. Hence, due to the capability to direct the solution search process in an alternative direction, QOBL is applied to enhance the exploration ability potentially, thus avoiding solution trapping in local minima (Rahnamayan et al. 2007). The basic principle of generating quasi-opposite numbers using QOBL is as follows.

  • 1st step- Generating opposite number Let’s assume X is a real number with a range [low, up]. If \({{x}^{O}}\) is an opposite number of X, then,

    $$\begin{aligned} {{x}^{O}}=low+up-x \end{aligned}$$
    (33)

  • 2nd step- Generating quasi-opposite number \({{x}^{QO}}\) is the quasi-opposite number of \({{x}^{O}}\). The following equations are used to obtain \({{x}^{QO}}\).

    $$\begin{aligned} {{x}^{QO}}=rand\left[ \left( \frac{low+up}{2} \right) ,{{x}^{O}} \right] \end{aligned}$$
    (34)

QOBL is applied to the proposed EPDSA in its two operational phases, as stated below.

  1. (1)

    The following equations apply the QOBL technique to generate a quasi-oppositional population at the initialization phase, simultaneously with the random population initialization.

    Generation of the oppositional population

    $$\begin{aligned} x_{i,j}^{O}=lo{{w}_{j}}+u{{p}_{j}}-{{x}_{i,j}} \end{aligned}$$
    (35)

    where \({{x}_{i,j}^{O}}\) is the \({{j}^{th}}\) variable of the \({{i}^{th}}\) oppositional population.

    Generation of the quasi-oppositional population

    $$\begin{aligned} {{x}_{i,j}^{QO}}=rand\left[ \left( \frac{low+up}{2} \right) ,{{x}_{i,j}^{O}} \right] \end{aligned}$$
    (36)

    where \({{x}_{i,j}^{QO}}\) is the \({{j}^{th}}\) variable of the \({{i}^{th}}\) quasi-oppositional population.

  2. (2)

    After updating based on boundary criterion, employing jumping probability rate \({{J}_{r}}\), Eqs. (35) and (36) generate a quasi-oppositional population and corresponding fitness is evaluated. By considering the fitness value, the SP fittest populations are selected from the quasi-oppositional population set and the artificial-organisms of the present generation.

Importance of QOBLThe significant QOBL characteristics listed below allow the proposed EOA to obtain solutions with a higher level of accuracy, quality and robustness.

  1. (1)

    During the initialization phase, QOBL is used to direct the proposed EOA in an alternative search direction, resulting in the proposed EOA visiting the promising area of the search space. This mechanism enhances the exploration of the proposed EPDSA, thus increasing the chances of our proposed algorithm reaching an accurate and robust solution point.

  2. (2)

    The efficacy of QOBL-based initialization aids the proposed EPDSA in approaching closer to the global best solution by following an opposite search path, resulting in a fast convergence profile.

  3. (3)

    Based on \({{J}_{r}}\), QOBL is further implemented in the proposed EOA’s later operational phase to generate a quasi-oppositional population. This strategy assists the proposed EOA in avoiding becoming trapped in the local optimal solution to achieve a high-quality solution. In addition, \({{J}_{r}}\) enables the proposed EPDSA to make a dynamic jump in the process of quasi-oppositional population creation, thus improving the proposed algorithm’s search and computational efficacy.

4.2.3 AT-3: Discovery of stopover site utilizing the best artificial-organism

The conventional strategy for producing a stopover site is utilising the information gathered in the donor vector, which is the randomly shuffled form of the superorganism. This process highly depends on the donor vector’s quality, which cannot always guarantee to direct the superorganism towards a fruitful food location due to its random generation process. Hence, in the process of stopover site generation, orthodox DSA may fail to provide adequate exploitation, thus leading to a slower convergence profile. To mitigate this flaw, in our proposed EOA, Eq. (25) is replaced by Eq. (37), which assists the existing superorganism in bringing towards the food location discovered by the best artificial-organism of the present iteration (Ghambari and Rahati 2018). A randomly selected differential vector is incorporated in Eq. (28) to ensure the substantial exploitation of the proposed EPDSA. The following equation represents the modified form of a stopover site production strategy.

$$\begin{aligned} StopoverSit{{e}_{i,j}}={{x}_{i,j}}+{{s}^{/}}.({{x}_{b,j}}-{{x}_{i,j}}+{{x}_{{{r}_{1}},j}}-{{x}_{{{r}_{2}},j}}) \end{aligned}$$
(37)

where \({{x}_{i,j}}\) is the \({{j}^{th}}\) dimension of the \({{i}^{th}}\) artificial-organism, \({{x}_{b,j}}\) refers to the \({{j}^{th}}\) dimension of the best artificial-organism of the superorganism for the present iteration, \({{x}_{{{r}_{1}},j}}\) and \({{x}_{{{r}_{2}},j}}\) are the \({{j}^{th}}\) dimension of two mutually independent randomly selected artificial-organism considering \({{r}_{1}}\ne {{r}_{2}}\ne i\). Lastly, \({{s}^{/}}\) is a random number generated by uniform distribution with a range of 0–1.

4.2.4 Functional steps of the proposed EPDSA for solving the hybrid OPF model

The functional steps considered for the proposed EPDSA for solving the hybrid-OPF model are stated in this subsection:

  1. Step 1:

    Initialization of input parameters for the OPF model: SP, \(S{{P}_{\min }}\), \(S{{P}_{\max }}\), \(S{{P}_{}}\), \({{S}_{R}}\), d, the upper and lower boundary of control variables: up and low, and maximum iteration count \(\max \_ite\).

  2. Step 2:

    Within the specified search space, the random generation of the artificial-organism that resembles control variables corresponding to the OPF model is done using Eq. (24), and the fitness of the present population is computed.

  3. Step 3:

    Generation of the quasi-oppositional population and evaluation of corresponding fitness is done.

  4. Step 4:

    Selection of the fittest \(S{{P}_{current}}\) number of the candidate solution is made among the randomly generated population set and the quasi-oppositional population set considering the fitness values evaluated in steps 2 and 3.

  5. Step 5:

    Shuffling between the different artificial-organisms is carried out to generate a donor vector in a random way.

  6. Step 6:

    The Scale parameter is obtained using Eq. (26) and the stopover site is mathematically positioned using Eq. (37).

  7. Step 7:

    Control parameters of the proposed EPDSA are declared: \({{p}_{1}=0.3*{{rand}_{4}}}\) and \({{p}_{2}=0.3*{{rand}_{5}}}\).

  8. Step 8:

    A random selection of individuals of the artificial-organisms corresponding to a superorganism is carried out to move towards the stopover site.

  9. Step 9:

    In case any of the individuals of the stopover site, i.e., the control variables of the OPF model, violate the boundary criterion, then using Eq. (30), those infeasible control variables are corrected, and the stopover site is updated accordingly.

  10. Step 10:

    Generation of the quasi-oppositional population and evaluation of corresponding fitness is done using \({{J}_{r}}\).

  11. Step 11:

    Selection of the fittest \(S{{P}_{current}}\) number of the candidate solution is made among the randomly generated population set of the present generation and the quasi-oppositional population set, considering the fitness values.

  12. Step 12:

    Selection of the fittest \(S{{P}_{current}}\) number of the candidate solution is made among the fittest solution identified in the previous step and the same corresponding to the stopover site, considering the fitness values.

  13. Step 13:

    Sort \(S{{P}_{current}}\) candidate solution from best to worst.

  14. Step 14:

    Print the improved best candidate solution if the termination condition is satisfied or return to step 5.

The flowchart of the proposed EPDSA is shown in Fig. 1, and the pseudocode of the proposed EPDSA is shown in algorithm 1.

Fig. 1
figure 1

Flowchart of the proposed EPDSA

Full size image
Algorithm 1
figure a

Pseudocode for the proposed EPDSA

Full size image

5 Analytical discussion of the obtained results

As described in Table 1, nine test instances are simulated using MATLAB 2018b on a PC with an i3 processor, 8 GB RAM, and 2.60 GHz clock speed. The simulation outcomes are compared with eight sophisticated EOA to validate the power of the proposed EPDSA. The performance of the proposed EPDSA is evaluated using four different test power systems. The proposed EOA is simulated considering 30 independent trials with different input parameter combinations on the implemented test power system to obtain suitable parameter settings, taking SOF1 into account. The objective values obtained by the proposed EPDSA, considering different input parameter configurations, are listed in Table 2. Input parameters of the proposed EPDSA and the rest of the EOAs are shown in Table 3.

Table 1 Description of various test instances
Full size table
Table 2 SOF1 value obtained by the proposed EPDSA considering different parametric configurations
Full size table
Table 3 Input parameters of the considered EOAs
Full size table

5.1 Modified IEEE 30-bus system-I

The input data set for this test power system is collected from Mohamed et al. (2017). The generator bus’s upper and lower limiting voltages are 0.95 p.u. and 1.1 p.u., respectively. On the other hand, load bus voltages are between 0.95 and 1.05 p.u. Some modifications are implemented to include RESs. The two WPUs are installed in place of the CTU on bus number 5. By replacing the equivalent CTU, one WPU and TDPU are combined and connected to bus number 8. The two PVUs replace the CTU of bus no. 11. 10,000 PEVs are used to develop a V2G system connected to bus 13 by replacing the corresponding CTU. The coefficients for the CTUs fuel cost and valve-point loading effect are taken from Mohamed et al. (2017). The coefficients of cost for the WPU, PVU, TDPU, SHU, and PEVs are provided in Table 4. As discussed and analyzed below, four test instances are simulated using an IEEE 30-bus test power system.

Table 4 Cost-efficients for the considered renewable energy sources
Full size table

5.1.1 Test instance 1

The present instance is developed and simulated to reduce the total cost for active power generation with the valve-point loading effect under high renewable energy (RE) penetration. This instance’s control variables are shown in the supplementary file. Table 5 illustrates the objective value produced by the proposed EPDSA and other EOAs considered. Table 6 clearly shows that the proposed EPDSA outperforms the other EOAs, thus proving its higher level of solution accuracy than the other EOAs considered under the challenges of the hybrid-OPF model that offers significant uncertainties and mathematical complexities imposed by the four intermittent RESs. It is also verified from Table 5 that the proposed EPDSA requires less CPU time than the other considered EOAs, demonstrating the proposed EOA’s computational efficacy in solving an OPF model that is highly challenging and mathematically complex due to the inclusion of a nonlinear, probabilistic mathematical model as well as a large number of operational constraints. Figure 2 validates the proposed EOA’s rapid convergence over other considered EOAs. Furthermore, the dynamic population size reduction process of the proposed EPDSA is shown in Fig. 3 for the present instance. It is distinctly revealed from Fig. 3 that to support the exploration in the early phase of iteration and exploitation in the latter iteration steps, the population of the proposed EOA is linearly reduced in a probabilistic way using various stages.

Table 5 Comparative representation of the OPF-objective and CPU time for test instance 1 obtained by the considered EOAs (Modified IEEE 30-bus system-I)
Full size table
Fig. 2
figure 2

Convergence characteristic of test instance 1

Full size image
Fig. 3
figure 3

Stages of dynamic population reduction scheme over various iterative steps for test instance 1

Full size image

However, for the space limitation, the analysis of the results supported by the control variables, comparative representation of the OPF-objectives of the considered algorithms, and different graphical illustrations, for instance, 2–4, are presented in the supplementary file.

5.2 Modified IEEE 30-bus system-II

In the present test system, the input data, as well as the number and position of the RESs, remain unchanged from the previous test system, i.e., modified IEEE 30-bus system-I, with one exception: one WPU and one SHU jointly connected as a combined PVSHU to bus no. 7. The cost coefficients of the SHU are presented in Table 4. Test instance 5, discussed below, is simulated considering the modified IEEE 30-bus system-II.

5.2.1 Test instance 5

Considering the CTU and the substantial generation uncertainty of multiple RESs, this instance aims to minimize the total cost of active power generation with the valve-point loading effect. The optimal settings of the control variables obtained by the proposed algorithm are furnished in the supplementary material. Table 6 displays the minimized generation cost objective values attained by various algorithms. It is explicitly demonstrated in Table 8 that, out of all the algorithms taken into consideration, the proposed EPDSA performs best. In this context, it is worth mentioning that by observing the outcomes of Table 6, the total cost for each algorithm is reduced compared to instance-1 due to the PVSHU’s integration into the power system alongside the other RESs. In addition, the CPU time in Table 8 validates the proposed EOA’s computational efficacy. Furthermore, Fig. 4 illustrates the convergence profile of the proposed EPDSA and the considered algorithms, revealing that the proposed EPDSA has a faster convergence speed than the other advanced algorithms used in this article.

Fig. 4
figure 4

Convergence characteristic of test instance 5

Full size image
Table 6 Comparison of the OPF-objectives for test instance 5 (Modified IEEE 30-bus system-II)
Full size table

5.3 Modified IEEE 118-bus system-I

All input data for this large-scale test power system is taken from (Mohamed et al. 2017). The upper and lower ranges of bus voltages are 0.94 p.u. and 1.1 p.u., correspondingly. Shunt regulating transformers have limiting values of 0.9 p.u. and 1.1 p.u. Some modifications are made to connect RESs. CTU of bus number 12 is replaced by a wind power plant that houses the two WPUs. By replacing the equivalent CTU, one WPU and TDPU are connected to bus number 18 to form combined WTDPU. The CTU of bus number 27 is replaced by a PV plant that includes two PVUs. The V2G system with 10,000 PEVs is connected to bus number 40 by removing the corresponding CTU. The cost coefficients for CTUs are taken from (Mohamed et al. 2017), and those for WPUs, PVUs and EVs are provided in Table 4. Following are the test instances simulated on this large-scale test power system.

The current test system is used to execute test instances 6 and 7. In view of the space issue, the supplementary file contains a detailed discussion of results along with the control variables, a comparative representation of the OPF objectives, and a visual demonstration of the convergence properties as attained by the optimization techniques under consideration.

5.4 Modified IEEE 118-bus system-II

All of the system data in the current test system and the number and bus positions of the RESs are the same as those in the previous test system. Compared to the previous test power system, only one modification is made, i.e., one WPU and one SHU jointly coupled with bus no. 70. Table 4 provides the cost coefficients for the SH generator. Table 4 provides the cost coefficients of the SH generator. Considering the present test system, test instance 8 is carried out. The supplementary file includes the findings and convergence curves that the optimization techniques under study produce for test instance 8.

5.5 Multi-period optimal power flow: Modified IEEE 30-bus system-III

The input data of this test system is collected from (Mohamed et al. 2017). Except for the CTU, the mentioned test system houses two WPUs at bus no. 5 and two PVUs at bus no. 11. However, bus no. 16 is linked with the BES system. The charging and discharging cost coefficients and the other specifications of the BES system are given in Table 7.

5.5.1 Test Instance-9

In this instance, the power generation cost, considering the valve-point loading effect, is minimized for different time intervals in the case of the MOPF problem. The present instance included the RESs such as WE and PVE. However, in order to support the power generation from the RESs and CTUs in peak load situations, the BES system is used. Table 8 displays the hourly generations achieved from different generators. The charging and discharging power of the BES system are also included in Table 8. Moreover, the generation cost for the whole day, as obtained by different EOAs, is produced in Table 9. It is proved from the results of Table 9 that the proposed EOA exhibits superior performance than the considered algorithms. The simulation time, as given in Table 9, establishes the significant computational efficacy of our proposed algorithm. The total hourly cost of power generation over a 24-hour time duration is illustrated in Fig. 5, whereas Fig. 6 shows wind velocity and solar irradiance variation throughout the day. From Fig. 5, it is shown that the hourly cost is comparatively low at 5–18 hr due to the significant availability of WE and PVE. The SOC, charging, and discharging power of the BES system is represented in Fig. 7. Moreover, Fig. 8 shows the varying load demand in the day ahead. From Fig. 7, it appears that the BES system discharges at the peak load hour to support the existing generators connected to the system.

Table 7 Specifications of the BES system
Full size table
Table 8 Active power generations at different time intervals for the MOPF problem
Full size table
Table 9 Comparison of the OPF-objectives for test instance 9 (Modified IEEE 30-bus system-III)
Full size table
Fig. 5
figure 5

Hourly generation cost considering CTU, WE, and PVE for 24 Hr horizon

Full size image
Fig. 6
figure 6

A Variation of wind velocity throughout the day, B Variation of solar irradiance throughout the day

Full size image
Fig. 7
figure 7

SOC, charging and discharging power of the BES system over the 24 Hr horizon

Full size image
Fig. 8
figure 8

Hourly load demand for the MOPF problem over 24 Hr time frame

Full size image

5.6 Analysis of the load bus voltage profile

Considering the current paper’s length, the load bus voltage profiles and corresponding analysis for two representative test instances, namely 4 and 6, are provided in the supplementary file.

5.7 Statistical analysis

Conducting a statistical performance investigation is one of the most fundamental approaches to assessing an EOA’s performance (Derrac et al. 2011). The statistical robustness of the proposed EPDSA is validated in this subsection compared to the other techniques studied in this article. However, because of the limitations of space, the statistical results for each instance-aside from test instances 1 and 5-are discussed in the supplementary file. The SD values in Table 10 show that the proposed EPDSA are statistically far more robust than the other EOAs studied. For confirming the statistical effectiveness of the proposed EPDSA, a two-sided comparison based on the best mean value is performed using the Wilcoxon signed-rank test (0.05 significance level) between the proposed EPDSA and each of the individual EOAs considered (Derrac et al. 2011). Table 10 shows that the p-value achieved for each of the EOAs investigated is less than 0.05 for test instances 1 and 5, thus demonstrating the statistical robustness of the proposed EPDSA. Furthermore, Box plots in Fig. 9, for instance, 1 and 5, illustrate the statistical superiority of the proposed EOA over the other considered state-of-the-art EOAs.

Table 10 Statistical outcomes for test instances 1 and 5
Full size table
Fig. 9
figure 9

A Box plots for test instance 1, B Box plots for test instance 5

Full size image

6 Conclusion

This research work proposes a novel EPDSA to address the hybrid static OPF and MOPF model with non-linearity and uncertainty, driven WE, PVE, TDE, SH and V2G-enabled PEV. The hybrid static OPF model is integrated with four RESs utilising an interpretable probabilistic technique to model the RESs’ intermittent parameters. However, the MOPF model devised in this article is integrated with WE, PVE, and BES system. The shortcomings of the orthodox DSA are mitigated with the improvement of exploration, exploitation, and convergence profile of the proposed EOA using three intelligent ATs- dynamic population reduction scheme, QOBL, and the best artificial-organism guided stopover site discovering-technique to solve the uncertainty-induced, non-linear, and mathematically complex RESs-based OPF model effectively and stably. The dynamic population reduction scheme effectively boosts the exploration capability in the earlier iteration phase while improving the exploitation in the latter iteration steps. In addition, the mentioned population reduction scheme allows the elite populations to get into the next step of iteration using its intelligent ESS, thus enhancing the solution quality of the proposed EPDSA. A two-stage QOBL implementation in the proposed EPDSA enhances the overall exploration and exploitation of the proposed method for improving solution accuracy. Implementing the best artificial-organism-guided stopover site discovery technique enables the proposed EOA to direct the artificial-organisms towards a stable and fruitful stopover site in a highly efficient way, thus causing potentially effective local search ability to attain a speedy convergence profile.

The performance of the proposed EPDSA is examined by minimizing four practical OPF objectives on modified IEEE 30-bus system-I and 118-bus system-I, considering four intermittent RESs, i.e., WE, PVE, combined WTD, and PEV. Furthermore, two complex test instances integrating WE, PVE, combined WTD, combined PVSH, and PEV on modified IEEE 30-bus system-II and 118-bus system-II are used to verify the efficacy of our proposed algorithm. Additionally, the proposed EOA is tested to solve a highly challenging MOPF problem considering significant non-linearity due to the presence of intermittent WE, PVE, and BES systems along with charging/discharging dynamics. To confirm the strength of the proposed EPDSA, the test results are compared with eight advanced optimization algorithms. From the results analysis, it can be inferred that our proposed method exhibits more accuracy, solution quality, and computational efficacy to solve uncertainty-dependent, complex and non-linear hybrid OPF models than the eight other highly efficient EOAs. Moreover, from the convergence characteristics, it is verified that the proposed EPDSA converges to the optimal solution faster than the rest of the considered optimization algorithms. With the help of the load bus voltage profile, we confirm that our proposed method successfully keeps all voltage values of the load bus within the prescribed limit, which indicates the high ability to deal with constrained optimization. Further, from the statistical results, it is strongly verified that the proposed method is statistically more robust and precise in terms of generated solutions than the other techniques considered. Hence, the proposed alterations successfully improve the proposed EPDSA’s exploration, exploitation, and convergence speed adequately to address the uncertainty-based, non-linear, hybrid OPF model with higher accuracy, efficacy, convergence speed, and robustness than the other eligible competitors employed in this article.

However, in the present research study, the proposed optimization technique successfully deals with only the single objective-based OPF problem considering various RESs under steady system load conditions. Addressing the multi-objective-based OPF issue is out of the scope of the present article. Additionally, the load uncertainty is not taken into account in the present paper. Thus, from a future research perspective, the authors aim to develop an extended version of the proposed EOA for addressing a multiple conflicting power system objective-based OPF problem in an uncertain load scenario.