Introduction

Optimal reactive power dispatch (ORPD) has an significant role for proper planning & operation of existing power networks. In order to keep the voltages at all system buses within acceptable ranges and to minimize network APL, reactive power needs to be controlled and managed in the system. The majority of system loads are inductive, and since reactive power is used by components like transformers and transmission lines, it cannot be avoided in the system. As reactive power flow results in APL, ORPD sets the minimizing of system APL as its major goal. A typical network will alter the passive tool settings like transformers and shunt VAR compensators to get the desired result. Power researchers are therefore making constant efforts to reduce predetermined OFs without violating a range of system restraints in order to address ORPD difficulties1,2,3..

To achieve the goals within the allowed system limitations (both equality and inequality), the most advantageous adjustment of specific control variables are the generating buses’ voltages, the tap settings of the transformer, the distribution of the VAR shunt compensator etc.4,5.

Presently integrating RESs with the conventional power grid is becoming gradually popular for the reasons of sustainability. However, the character of RESs is not deterministic rather stochastic. As result, introduction of RESs with traditional configurations enhances system complexity and makes achieving the ORPD solution is a more difficult task6. The modern advancement of power electronic technologies enhances the utilization of flexible AC transmission system (FACTS) devices (like flexible, stable & dependable VAR compensators) to effectively address the ORPD issues7.

The reduction of AVD, the diminution of APL etc are very often chosen as the OFs in the area of ORPD studies. Conventional optimization techniques, such as dynamic programming, linear programming, and others, have been discussed in the literature. However they are unable to solve non-differentiable functions. These methods took more iterations to generate results, therefore they were more time consuming. More often they were producing a local optimal solution instead of global solutions. Moreover, these traditional approaches of optimizations were inefficient to handle complex, non-linear optimization problem. Thus, more advanced optimization techniques have been incessantly created, and being applied on different power system applications, like ORPD problem which addresses the shortcomings of earlier approaches in solving ORPD problem8,9. Presently, development and utilization of meta heuristic methods have demonstrated successful outcomes in accomplishing ORPD difficulties10.

Table 1 is represented here to show a brief contemporary situations of ORPD research where different optimization algorithms has been applied to achieve several single OFs like declining APL, reduction in AVD, VSI enhancement, fuel cost reduction, reduction in emission, operational cost minimization, and uniting more OFs together as a multi-OFs11.

Laouafi12 proposed improved grey wolf optimizer (IGWO) to get the solution of the problem of optimal reactive power dispatch (ORPD). The effectiveness of the method was tested on the IEEE 30 bus test system with and without solar and wind energy as renewable energy resources (RESs). Megantoro et al13. used meta-heuristic algorithm to sort out the ORPD problem. Here, Wind and solar power are used as RESs. This technique was tested on IEEE 57 Bus system for identifying the robustness. In the aforesaid paper, objective functions of power loss, minimize voltage deviation, and improvement the voltage stability index (VSI) were minimize. Das et al14. used the rock hyraxes swarm optimization (RHSO) algorithm to find the solutions of ORPD problem. Paul et al15. implemented chaotic-oppositional (CO) based DTBO approach in CHPED based OPF problem by considering wind-solar-EV to minimize the generation cost and emission and to improve the voltage deviation. This proposed technique was tested on the IEEE 33 and IEEE 141 bus systems with and without PV-Wind power. Tu et al16. suggested an improved multi-objective equilibrium optimizer (IMOEO) for fixing the OARPD issue with renewable sources. The procedure is used to an modified IEEE-33 distribution network to check its performance. Hasanien et al17. proposed hybrid particle swarm Optimization/sea horse optimization (PSOSHO) algorithm to handle ORPD for electric vehicle integrated system. It was tested on IEEE 30-bus and IEEE 57-bus networks to verify its efficacy. Paul et al18. applied COWOA optimization to analyze hydro-thermal scheduling problem integrated with wind and solar for optimal solution of cost and emission. Nagrajan19 et al. focused on the enhanced wombat optimization algorithm (EWOA) for solving the optimal power flow (OPF), taking into account the RES-solar photovoltaic (PV) system, Wind energy (WE), Electric vehicles (EVs). The potential of this optimization technique was checked by applying it over IEEE 30-, IEEE 57-, & IEEE 118-bus networks. Ahmed et al20. applied gradient jellyfish search optimizer (GJSO) to accomplish the ORPD issue in electric networks. It was conducted on typical IEEE-30 & IEEE-57 bus systems to measure the effectiveness of the GJSO methodology. Chandra et al21. suggested an approach to analysis the voltage stability in the grid for ORPD problem. Elkholoy et al22.proposed an approach to improve the power quality in the distribution network (IEEE 13 bus) for unbalance load with utilizing different FACTS devices. Chandra et al23.applied competitive swarm technique integrated with oppositional-based learning to find the optimal ___location of the solar charging station for EV in the radial distribution. Split bregman approach applied by Rong et al24. for ORPD in induction generator in a wind power plant.

It is clear from Table 1 that as of 2023, there are still researchers striving to enhance ORPD solutions, hence ORPD research has not arrived its zenith yet. When the Table 1 which presents an overview of the nearly five years of ORPD study, is quantitatively evaluated, it is discovered that, out of 37 investigations, the IEEE 30-bus network was selected in 30 times, giving it a higher preference than other IEEE bus systems, as seen in Table 1. When the various OFs that were considered in those studies are totaled, it is found that every study has chosen lowering APL as one of the OFs, and 26 investigations have also considered decreasing AVD. This suggests that declining APL and AVD are the most common OFs in ORPD. The current work, which uses the IEEE 30-bus test system to achieve the bare minimum APL and AVD, is motivated by these quantitative studies.

Table 1 Different ORPD research from literature.
Full size table

Here, based on the selection of OFs, kind of load demand and test settings we have developed 30 distinct cases which are covered in five test modules. Trials are being conducted in module one under fixed load with out considering RESs. However, only half of the cases in this module use STATCOM; as FACT device. In both modules one and two, the same strategy has been applied with regard to the use of STATCOM. The second module, in contrast to the first, adds RESs with test setup and runs tests under various load scenarios. Test module three to five are also conducted by considering RESs & STATCOM tools. In these cases, the RESs and changeable load requirements are being modeled using best matched PDFs to capture their uncertainties. Furthermore, utilizing MCS and BRA, 25 plausible situations are generated, over which the testing in the second module are conducted. The goal in creating the scenarios is to replicate the events that take place in actual power networks as accurately as possible. In this study, DTBO algorithm58 has been proposed to resolve the ORPD issue. Few graceful members from DTBO population are chosen as driving instructors, while the remaining members are categorized as trainee drivers.

Model: STATCOM and RESs

Modeling of STATCOM

To control the power flow, the static synchronous compensator (STATCOM)59 device is considered in this experiment. The following is an explanation of this FACTS device’s static model. The main goal of STATCOM is reactive power compensation, which is achieved by varying the power network’s reactive power and voltage magnitude. The components of this device are a transformer, a voltage source converter (VSC), and a capacitor. STATCOM is used parallel with the power system network. A controllable voltage source (\({E}_{p}\)) in series with an impedance will be used to model the STATCOM. The STATCOM circuit model is shown in Fig. 1, attached to the power system’s \({i}^{th}\) bus.

Fig. 1
figure 1

Schematic static model of STATCOM.

Full size image

STATCOM takes in the right extent of reactive electricity through the grid to maintain voltage stability over the power system loads under acceptable limits. The injected active & reactive power flow equations of the \({i}^{th}\) bus are shown below:

$$\begin{aligned} & \begin{array}{cc} P_i& =G_p\left| V_i \right| ^{2}-\left| V_i\right| \left| E_p\right| \left| Y_p \right| \cos (\delta _i-\delta _p-\theta _p) \\ & +\sum \limits _{j=1}^{N}\left| V_i\right| \left| V_j \right| \left| Y_{ij}\right| \cos (\delta _i-\delta _j-\theta _{ij}) \\ \end{array} \end{aligned}$$
(1)
$$\begin{aligned} & \begin{array}{ll} {{Q}_{i}}& =-{{B}_{p}}{{\left| {{V}_{i}} \right| }^{2}}-\left| {{V}_{i}} \right| \left| {{E}_{p}} \right| \left| {{Y}_{p}} \right| \sin ({{\delta }_{i}}-{{\delta }_{p}}-{{\theta }_{p}}) \\ & +\sum \limits _{j=1}^{N}{\left| {{V}_{i}} \right| }\left| {{V}_{j}} \right| \left| {{Y}_{ij}} \right| \sin ({{\delta }_{i}}-{{\delta }_{j}}-{{\theta }_{ij}}) \\ \end{array} \end{aligned}$$
(2)

STATCOM brings in two state variables (\(\left| {{E}_{p}} \right|\) and \({{\delta }_{p}}\)) into the transmission network. In a steady state, it is confirmed that the power used by the source should be zero and represented as

$$\begin{aligned} {\begin{matrix} & {{P}_{Ep}}=\operatorname {Re}al[{{E}_{p}}I_{p}^{*}] \\ & \quad \ \,=-{{G}_{p}}{{\left| {{E}_{p}} \right| }^{2}}+\left| {{E}_{p}} \right| \left| {{V}_{i}} \right| \left| {{Y}_{p}} \right| \cos ({{\delta }_{i}}-{{\delta }_{p}}+{{\theta }_{p}}) \\ & \quad \ =0 \\ \end{matrix}} \end{aligned}$$
(3)

where \({{V}_{i}}\) is the magnitudes of the voltage at the \({{i}^{th}}\) bus; \({{Y}_{p}}\) is the parallel component’s admittance; \({{B}_{p}}\) and \({{G}_{p}}\) are the susceptance and conductance of STATCOM ‘s parallel components, respectively; \({{\theta }_{ij}}\) is the transmission line’s (placed within \({{i}^{th}}\) and \({{j}^{th}}\) bus) angle of admittance; \(\theta _p\) is the angle of STATCOM’s voltage source; \(E_p\) is of STATCOM’s voltage sources.

WP model

Two parameters namely, scale (\(\iota\)) & shape (\(\varkappa\)) parameter, provide a good illustration of the WS variation (v m/s)60,61 using the Weibull PDF as:

$$\begin{aligned} f\left( v \right) =\left( \frac{\varkappa }{\iota } \right) \times {{\left( \frac{v}{\iota } \right) }^{\varkappa -1}}\times \left( {{e}^{-\left( \frac{v}{\iota } \right) }}^{\varkappa } \right) \quad \quad 0<v<\infty \end{aligned}$$
(4)

According to the cut-in pace \({{v}_{in}}\), rated pace \({{v}_{r}}\), cut-out pace \({{v}_{out}}\), and output ratting of the wind turbine (WT) \({{P}_{wr}}\), the output power from a WT is given as follows:

$$\begin{aligned} {{P}_{w}}(v)=\left\{ \begin{array}{cc}0\\ {{P}_{wr}}\left( \frac{v-{{v}_{in}}}{{{v}_{r}}-{{v}_{in}}} \right) \quad \quad \\ {{P}_{wr}} \\ \end{array}\begin{array}{cc} for\,{{v}_{in}}>v\ \& \ v>{{v}_{out}} \\ for\ {{v}_{in}}\le v\le {{v}_{r}} \\ for\ {{v}_{r}}<v\le {{v}_{out}} \\ \end{array} \right. \end{aligned}$$
(5)

Now, the likelihood of WP in various WS zones may be explained by:

$$\begin{aligned} & f{{\left( {{P}_{w}} \right) }_{\left| {{P}_{w}}=0 \right. }}=1-\exp \left[ -{{\left( \frac{{{v}_{in}}}{\iota } \right) }^{\varkappa }} \right] +\exp \left[ -{{\left( \frac{{{v}_{out}}}{\iota } \right) }^{ \varkappa }} \right] \end{aligned}$$
(6)
$$\begin{aligned} & f{{\left( {{P}_{w}} \right) }_{\left| {{P}_{w}}={{P}_{wr}} \right. }}=\exp \left[ -{{\left( \frac{{{v}_{r}}}{\iota } \right) }^{\varkappa }} \right] -\exp \left[ -{{\left( \frac{{{v}_{out}}}{\iota } \right) }^{\varkappa }} \right] \end{aligned}$$
(7)
$$\begin{aligned} & {\begin{matrix} & f{{\left( {{P}_{w}} \right) }_{\left| 0<{{P}_{w}}<{{P}_{wr}} \right. }}=\left[ \frac{\varkappa \times \left( {{v}_{r}}-{{v}_{in}} \right) }{{{\iota }^{\varkappa }}\times {{P}_{wr}}} \right] \times {{\left[ {{v}_{in}}+\left( \frac{{{P}_{w}}}{{{P}_{wr}}} \right) \left( {{v}_{r}}-{{v}_{in}} \right) \right] }^{\varkappa -1}} \\ & \quad \quad \quad \quad \quad \quad \times \exp \left[ -{{\left( \frac{{{v}_{in}}+\left( \frac{{{P}_{w}}}{{{P}_{wr}}} \right) \times \left( {{v}_{r}}-{{v}_{in}} \right) }{\iota } \right) }^{\varkappa }} \right] \\ \end{matrix}} \end{aligned}$$
(8)

In this study, \(\xi =10\), \(\kappa =2\) & \({P}_{wr}\)=80MW have been considered.

PV model

The function of PV unit is to transform the solar energy into electrical energy. The amount of SI and other environmental factors can affect power output. Since the lognormal PDF L(I) is very much closed with respect to the probability distribution of SI (I : denotes SI)60,62, it is frequently used to estimate SI and is expressed as:

$$\begin{aligned} L\left( I \right) =\frac{1}{I\lambda \sqrt{2\pi }}\exp \left( \frac{-{{\left( \ln I-\varepsilon \right) }^{2}}}{2{{\lambda }^{2}}} \right) \quad ,\ \ I>0 \end{aligned}$$
(9)

\(\varepsilon\) & \(\lambda\), respectively, represent the mean and standard deviation of the I distribution. \(\varepsilon =6\) & \(\lambda =0.6\) are being chosen here.

The formula for the relationship between SI and the electrical output power of a PV unit is depicted as:

$$\begin{aligned} P\left( I \right) =\left\{ \begin{array}{cc} {{P}_{nm}}\frac{{{I}^{2}}}{{{I}_{st}}{{I}_{c}}},\quad for\ 0<I<{{I}_{c}} \\ \\ {{P}_{nm}}\frac{I}{{{I}_{st}}},\quad for\,I\ge {{I}_{c}} \\ \end{array} \right. \end{aligned}$$
(10)

The output power nominal of a PV unit, SI standard, & point of critical irradiance are denoted by \({{P}_{nm}}\), \({{I}_{st}}\) and \({{I}_{c}}\) respectively.

HP model

The behaviour of the fluctuations of WFR is usually modeled using Gumbel PDF6, which is expressed as follows:

$$\begin{aligned} f\left( {{Q}_{h}} \right) =\left( \frac{1}{\gamma } \right) \times {{e}^{\left( \frac{{{Q}_{h}}-\tau }{\gamma } \right) }}\times {{e}^{-{{e}^{\left( \frac{{{Q}_{h}}-\tau }{\gamma } \right) }}}} \end{aligned}$$
(11)

where, with values of 15 and 1.2, respectively, \(\tau\) & \(\gamma\) denote the ___location and scale factors of the WFR under consideration. The WFR is \(Q_{h}\). The following formula is used to determine the power from the HP-unit.

$$\begin{aligned} {{P}_{h}}\left( {{Q}_{h}} \right) =0.85\times \sigma \times \delta \times {{Q}_{h}}\times {{H}_{h}} \end{aligned}$$
(12)

The water density is denoted by \(\sigma\) which is approximately \(1000 kg/m^2\). The gravitational acceleration is represented by \(\delta\). 0.85 is the hydro turbine’s efficiency. \(H_h\) stands for the water’s head across the turbine.

Formulating problem

Objective function

Formulation of single objectives describes63 the reduction of APL, AVD, VSI,Reactive power loss & STATCOM installation cost. Below are the explanations of the previously mentioned objectives:

APL

Within transmission lines, inherent resistance results in APL. A representation of APL that must be minimized is as follows:

$$\begin{aligned} Min{{\mathcal {F}}_{1}}=\sum \limits _{n=1}^{{{N}_{L}}}{{{G}_{n\left( pq\right) }}\left( V_{p}^{2}+V_{q}^{2}-2{{V}_{p}}{{V}_{q}}\cos {{\varphi }_{pq}} \right) } \end{aligned}$$
(13)

The \({{n}^{th}}\) line’s transfer conductance, which connects between buses p and q, is \({{G}_{n(pq)}}\). There are \({{N}_{L}}\) total transmission lines. Between buses p and q, there is a voltage angle \({{\phi }_{pq}}\)

AVD

AVD over the load buses should be reserved to a smallest to maintain a decent voltage profile, and it is determined by:

$$\begin{aligned} Min{{\mathcal {F}}_{2}}=\sum \limits _{l=1}^{{{N}_{B}}}{\left| {{V}_{l}}-1 \right| } \end{aligned}$$
(14)

\({{V}_{l}}\) : Voltage at load bus l. No. of load buses is \({{N}_{B}}\)

VSI

The third goal function is to improve voltage stability. Voltage variations result in voltage instability, which can harm power networks or even induce voltage collapse, either suddenly or gradually. VSI needs to be enhanced in order to keep the voltage from dropping. The VSI is presented as per following equations:

$$\begin{aligned} & {{F}_{3}}=\min \left( {{L}_{\max }} \right) =\min \left( \max \left( {{L}_{j}} \right) \right) \ \forall \ j=1,2....,{{N}_{b}} \end{aligned}$$
(15)
$$\begin{aligned} & {{L}_{j}}=\left| 1-\sum \limits _{i=1}^{{{N}_{G}}}{{{F}_{ji}}\frac{{{V}_{i}}}{{{V}_{j}}}} \right| \,\forall \,j=1,2......{{N}_{L}} \end{aligned}$$
(16)

Here, \(L_j\) represents stability index of \(j^{th}\) bus; \({{F}_{ji}}=-{{\left[ {{Y}_{1}} \right] }^{-1}}\left[ {{Y}_{2}} \right]\); \(Y_1\) & \(Y_2\) are network’s \(Y_{BUS}\) sub-matrices.

Multi-objective

Multi-objective function25 has been formed by taking the linear combination of APL and AVD as:

$$\begin{aligned} Min{{\mathcal {F}}_{4}}=APL + \lambda .(AVD) \end{aligned}$$
(17)

where \(\lambda (=10\)), is known as weight factor.

$$\begin{aligned} MinF=APL+{{\lambda }_{1}}(AVD)+{{\lambda }_{2}}(VSI) \end{aligned}$$
(18)

where \(\lambda _1(=10\)) and \(\lambda _2(=10\)) are known as weight factors.

Reactive power loss (RPL)

$$\begin{aligned} Min{{F}_{s}}=\sum \limits _{n=1}^{{{N}_{L}}}{{{B}_{npq}}}(V_{p}^{2}+V_{q}^{2}-2{{V}_{p}}{{V}_{q}}\operatorname {Sin}{{\phi }_{pq}}) \end{aligned}$$
(19)

The \({{n}^{th}}\) line’s transfer susceptance, which is connected between buses p and q, is \({{B}_{n(pq)}}\). There are \({{N}_{L}}\) number of transmission lines. Between buses p and q, there is a voltage angle \({{\phi }_{pq}}\).

STATCOM installation cost in ($/hr)

The installation cost of STATCOM64 is expressed in terms of operating range of the STATCOM in MVA, number of FACTS and capital recovery factors and is given by:

$$\begin{aligned} & {{C}_{STATCOM}}=\sum \limits _{j=1}^{D}{({{F}_{STATCOMj}}\times {{S}_{j}}}\times 1000\times \beta )/8760 \end{aligned}$$
(20)
$$\begin{aligned} & {{F}_{STATCOMj}}=0.0003S_{j}^{2}-0.2691{{S}_{j}}+188.22 \end{aligned}$$
(21)
$$\begin{aligned} & \beta =\frac{r{{(1+r)}^{n}}}{{{(1+r)}^{n}}-1} \end{aligned}$$
(22)

where \({S}_{j}\): Rating of the \({(j)}^{th}\) STATCOM in MVAr; r:interest rate=0.05; n: capital recovery plan for 10 years; \(\beta\)=0.1295;

Constraints

The following limitations are applied to the ORPD with STATCOM devices:

Equality constraints

Constraint (23) provides a power flow equation which is shown below:

$$\begin{aligned} \left\{ \begin{array}{cc} \sum \limits _{c=1}^{{{N}_{s}}}{\left( {{P}_{Gc}}-{{P}_{Lc}} \right) }=\sum \limits _{c=1}^{{{N}_{s}}}{\sum \limits _{d=1}^{{{N}_{s}}}{{{V}_{c}} }{{V}_{d}}}\left( {{g}_{cd}}\cos {{\varphi }_{cd}}-{{h}_{cd}}\sin {{\varphi }_{cd}} \right) \\ \sum \limits _{c=1}^{N _{s}}{\left( {{Q}_{Gc}}-{{Q}_{Lc}} \right) {=-\sum \limits _{c=1}^{{{N}_{s}}}{\sum \limits _{d=1}^{{{N}_{s}}}{{{V}_{c}} }{{V}_{d}}}\left( {{g}_{cd}}\sin {{\varphi }_{cd}}-{{h}_{cd}}\cos {{\varphi }_{cd}} \right) }} \\ \end{array} \right. \end{aligned}$$
(23)

Where \({{P}_{Lc}}\), \({{Q}_{Lc}}\): real and reactive load of \({{c}^{th}}\) node (i.e. bus); \({{P}_{Gc}}\), \({{Q}_{Gc}}\): real and reactive generation of \({{c}^{th}}\) node; \({{g}_{cd}}\), \({{h}_{cd}}\) are conductance and susceptance of the \(c-d\) branch; \({{\varphi }_{cd}}\) is the admittance angle of the transmission line between \(c-d\) nodes.

Inequality constraints

  1. (i)

    Generator constraints:

    $$\begin{aligned} \left\{ \begin{array}{cc} V_{Gb}^{\min }\le {{V}_{Gb}}\le V_{Gb}^{\max } \\ \begin{array}{cc} P_{Gb}^{\min }\le {{P}_{Gb}}\le P_{Gb}^{\max } \\ \end{array} \\ Q_{Gb}^{\min }\le {{Q}_{Gb}}\le Q_{Gb}^{\max } \\ \end{array} \right. \begin{array}{ccc} & b\in {{N}_{P}} & \\ \end{array} \end{aligned}$$
    (24)
  2. (ii)

    Load bus constraints:

    $$\begin{aligned} V_{Lb}^{\min }\le {{V}_{Lb}}\le V_{Lb}^{\max }\begin{array}{cc} & b\in {{N}_{BL}} \\ \end{array} \end{aligned}$$
    (25)
  3. (iii)

    Transmission line constraints:

    $$\begin{aligned} {{S}_{L}}_{b}\le S_{Lb}^{\max }\begin{array}{cc} & b\in {{N}_{LT}} \\ \end{array} \end{aligned}$$
    (26)
  4. (iv)

    Transformer tap constraints:

    $$\begin{aligned} T_{b}^{\min }\le {{T}_{b}}\le T_{b}^{\max }\begin{array}{cc} & b\in {{N}_{T}} \\ \end{array} \end{aligned}$$
    (27)
  5. (v)

    Shunt compensator constraints:

    $$\begin{aligned} Q_{Cb}^{\min }\le {{Q}_{Cb}}\le Q_{Cb}^{\max }\begin{array}{cc} & b\in {{N}_{sc}} \\ \end{array} \end{aligned}$$
    (28)
  6. (vi)

    STATCOM voltage and phase angle constraints are respectively depicted in (29) and (30):

    $$\begin{aligned} & E_{Sb}^{\min }\le {{E}_{Sb}}\le E_{Sb}^{\max }\begin{array}{cc} & b\in \\ \end{array}{{N}_{STATCOM}} \end{aligned}$$
    (29)
    $$\begin{aligned} & \delta _{Sb}^{\min }\le {{\delta }_{Sb}}\le \delta _{Sb}^{\max }\begin{array}{cc} & b\in \\ \end{array}{{N}_{STATCOM}} \end{aligned}$$
    (30)

Here \(V_{Gb}^{min },V_{Gb}^{max }\) indicate voltage operating range; \(P_{Gb}^{min }, ~P_{Gb}^{max }\) represent real power generation operating range; \(Q_{Gb}^{min }, Q_{Gb}^{max }\) depict reactive power generation operating range; \(V_{Lb}^{min}, V_{Lb}^{max }\) indicate load voltage range; \({{S}_{Lb}}^{min }, S_{Lb}^{max }\) power flow limits of transmission line; \(T_{b}^{min }, T_{b}^{max }\) shows tap setting limits; \(Q_{Cb}^{min }, Q_{Cb}^{max }\) represent VAr compensation range; \(E_{sb}^{\max },~ E_{sb}^{\min }\) indicate voltage range of the STATCOM; \(\delta _{sb}^{\max },\delta _{sb}^{\min }\) are phase angle range of STATCOM; \({{N}_{P}}\) depicts generating buses; \({{N}_{BL}}\) represents load buses; \({{N}_{LT}}\) represents transmission line; \({{N}_{T}}\) is the number of regulating transformers; \({{N}_{sc}}\) is the number of shunt compensators and \({{N}_{STATCOM}}\) is count of STATCOM.

Algorithm for optimization

Driving training based optimization(DTBO)

DTBO approach is based on driving behaviors and training models. The idea of optimizing driving training-based is to improve driving efficiency, safety, and experience. Driving performance is intended to be optimized through the use of data, driving habits, and continuous input. This involves gathering information on a driver’s driving behaviors, including acceleration, braking, cornering, speed, and fuel consumption. Drivers can receive individual training programs that focus on their particular areas for improvement using the data collected. Reducing pollutants, fuel consumption, and expenses can all be achieved by optimizing driving behavior. This driving behavior helps to provide an optimal solution in optimizing technique with dynamic adaptability.

Dehghani et al. introduced DTBO at first58. DTBO is a population-based meta-heuristic technique. The DTBO program emulates the manner in which a driving instructor instructs trainees in a driving school. The mathematical framework of DTBO contains three phases: (1) training by the driving instructor, (2) patterning of students from instructor skills, and (3) practice. The ability of novice drivers to learn and master the skill of driving depends on their level of intelligence. A seasoned driver can learn from a variety of instructors in driving school. Driving skills are developed by new drivers through practicing on their own and by according to their instructor’s instructions. The foundation of the mathematical modeling of DTBO is these learner-teacher interactions and self-practice for improving driving skills. The following represents the DTBO population matrix, where each row member is one of the possible solutions to the given problem:

$$\begin{aligned} Z={{\left[ {\left\{ \begin{array}{ll} & {{Z}_{1}} \\ & . \\ & . \\ & {{Z}_{p}} \\ & . \\ & . \\ & {{Z}_{N}} \\ \end{array}\right. } \right] }_{N\times m}}={{\left[ {\left\{ \begin{array}{ll} & {{z}_{11}}\quad .\quad .\quad {{z}_{1q}}\quad .\quad {{z}_{1m}} \\ & .\quad \quad .\quad .\quad .\quad \quad .\quad \ . \\ & .\quad \quad .\quad .\quad .\quad \quad .\quad \ . \\ & {{z}_{p1}}\quad .\quad .\quad {{z}_{pq}}\quad .\quad {{z}_{pm}}\ \\ & .\quad \quad .\quad .\quad .\quad \quad .\quad \ . \\ & .\quad \quad .\quad .\quad .\quad \quad .\quad \ . \\ & {{z}_{N1.}}\ \ \ .\quad .\quad {{z}_{Nq}}\quad .\quad {{z}_{Nm}} \\ \end{array}\right. } \right] }_{N\times m}} \end{aligned}$$
(31)

The DTBO population is indicated by Z; \({{p}^{th}}\) member of Z is \({{Z}_{p}}\) i.e. \({{p}^{th}}\) candidate solution of the problem; \({{z}_{pq}}\) is the \({{q}^{th}}\) variable of the \({{p}^{th}}\) solution of the problem;The population size is N; No of problem variables is indicated by m.

The starting positions of DTBO members (i.e., potential solutions) are initialized at random at the start of DTBO implementation in the following ways:

$$\begin{aligned} {{z}_{pq}}=z_{pq}^{\min }+r*\left( z_{pq}^{\max }-z_{pq}^{\min } \right) \quad for\,p=1\,to\,N\ \varsigma \ q=1\ to\ m \end{aligned}$$
(32)

where the upper and lower bounds, respectively, of the \({{q}^{th}}\) variable of the problem under consideration are denoted by \(z_{pq}^{\max }\) and \(z_{pq}^{\min }\); An unbiased random number between 0 and 1 is denoted by r.

The objective function’s value is calculated for each unique candidate solution and is shown as follows:

$$\begin{aligned} F={{\left[ {\left\{ \begin{array}{ll} & {{F}_{1}} \\ & . \\ & . \\ & {{F}_{p}} \\ & . \\ & . \\ & {{F}_{N}} \\ \end{array}\right. } \right] }_{N\times 1}}={{\left[ {\left\{ \begin{array}{ll} & F\left( {{Z}_{1}} \right) \\ & \\ & \\ & F\left( {{Z}_{p}} \right) \\ & \\ & \\ & F\left( {{Z}_{N}} \right) \\ \end{array}\right. } \right] }_{N\times 1}} \end{aligned}$$
(33)

The decisive criterion for evaluating the merits of the solutions under consideration is the computed values of the objective function. The best member is determined by selecting the candidate solution that yields the best value of the objective function. As the iteration moves forward, the top member gets updated. The following three processes make up the process of revising candidate solution in DTBO:

step 1.:

Training by the driving instructor (Exploration): Few graceful members from DTBO population are chosen to be driving instructors, while the remaining members are categorized as trainee drivers. The capacity to perform a global search to find the optimal solution area for the given problem is accomplished by the skillful selection of instructors and the attaining the instructor’s skill. L DTBO members are selected as instructors in each iteration based on a comparison of the objective function values. These members are represented as the driving matrix DI in the following manner:

$$\begin{aligned} DI={{\left[ {\left\{ \begin{array}{ll} & D{{I}_{1}} \\ & . \\ & . \\ & D{{I}_{p}} \\ & \\ & D{{I}_{L}} \\ \end{array}\right. } \right] }_{L\times m}}={{\left[ {\left\{ \begin{array}{ll} & D{{I}_{11}}\quad .\quad .\quad D{{I}_{1q}}\quad .\quad D{{I}_{1m}} \\ & .\quad \quad \,\,.\quad \quad \quad .\quad \ \ \ \ .\quad \ \,. \\ & \quad \quad \,\,.\quad \quad \quad .\quad \ \ \ \ .\quad \ \,. \\ & D{{I}_{p1}}\quad .\quad .\quad D{{I}_{pq}}\quad .\quad D{{I}_{pm}}\ \\ & .\quad \quad \,\,.\quad \quad \quad .\quad \ \ \ \ .\quad \ \,. \\ & .\quad \quad \,\,.\quad \quad \quad .\quad \ \ \ \ .\quad \ \,. \\ & D{{I}_{L1.}}\ \ \ .\quad .\quad D{{I}_{Lq}}\quad .\quad D{{I}_{Lm}} \\ \end{array}\right. } \right] }_{L\times m}} \end{aligned}$$
(34)

\(D{{I}_{p}}\) is \({p}^{th}\) driving instructor. \(D{{I}_{pq}}\) is \({{q}^{th}}\) variable of \({p}^{th}\) instructor.

$$\begin{aligned} L=\left\lfloor 0.1\times N\times \left( \frac{1-s}{S} \right) \right\rfloor \end{aligned}$$
(35)

S is the maximum iteration, while s represents the current iteration. The adjusted position of the DTBO population member is obtained as follows in this step:

$$\begin{aligned} z_{pq}^{st1}={\left\{ \begin{array}{ll} & {{z}_{pq}}+r.\left( D{{I}_{kpq}}-I.{{z}_{pq}} \right) ,\ \ {{F}_{DI{{k}_{p}}}}<{{F}_{p}} \\ & {{z}_{pq}}+r.\left( {{z}_{pq}}-D{{I}_{kpq}} \right) ,\ otherwise \\ \end{array}\right. } \end{aligned}$$
(36)

In the set \(\{1,2\}\), I represents a random number, and r represents a random value between 0 and 1. A random selection of k is made from the collection 1,2,...,L in \(D{{I}_{kpq}}\) i.e. \({{k}^{th}}\) driving instructor whose objective function value is \({{F}_{DI{{k}_{p}}}}\), p denotes \({{p}^{th}}\) trainee member of the population which is under the training of \({{k}^{th}}\) instructor. When new position provides fitter solution than earlier position then the position is updated by (37).

$$\begin{aligned} {{Z}_{p}}= {\left\{ \begin{array}{ll} & Z_{p}^{st1},\quad F_{p}^{st1}<{{F}_{p}} \\ & {{Z}_{p}},\quad otherwise \\ \end{array}\right. } \end{aligned}$$
(37)

The revised \({{p}^{th}}\) candidate solution at the \(1^{st}\) DTBO step is \(Z_{p}^{st1}\); \(z_{pq}^{st1}\) is its \({{q}^{th}}\) problem variable, The value of its objective function is \(F_{p}^{st1}\).

step 2.:

Patterning of the instructor skills of the student driver (Exploration): In the \(2^{nd}\) step, the trainee driver mimics the instructor’s techniques and actions to enhance the DTBO solutions. Members of the DTBO reach a new area of the search space through this procedure. It strengthens DTBO’s exploration power. The DTBO members and instructors combine linearly to form a modified position, which is mathematically represented by (38). If the value of the objective function is better at the new position than it was at the previous one, then (39) is used to replace the previous position.

$$\begin{aligned} & z_{pq}^{st2}=\xi .{{z}_{pq}}+\left( 1-\xi \right) .D{{I}_{kpq}} \end{aligned}$$
(38)
$$\begin{aligned} & {{Z}_{p}}={\left\{ \begin{array}{ll} & Z_{p}^{st2}\,\ F_{p}^{st2}<{{F}_{p}} \\ & {{Z}_{p}},\ \ otherwise \\ \end{array}\right. } \end{aligned}$$
(39)

The \(Z_{p}^{st2}\) is the updated \({{p}^{th}}\) candidate solution on the DTBO second stage, \(z_{pq}^{st2}\) is its \({{q}^{th}}\) variable, The related objective function value is \(F_{p}^{st2}\). The patterning index \(\xi\) is given by:

$$\begin{aligned} \xi =0.01+0.9\left( 1-\frac{s}{S} \right) \end{aligned}$$
(40)
step 3.:

Personal practice (Exploitation): Based on individual practice, the novice drivers’ driving abilities are improved in this phase. It is akin to exploiting DTBO’s local search capability. Every learner looks for a better position around their existing position. By (41), new positions are generated in close proximity to the existing position. The previous position is replaced by the new one using (42) while it upgrades the objective function value as follows:

$$\begin{aligned} & z_{p,q}^{st3}={{z}_{pq}}+\left( 1-2r \right) .R.\left( 1-\frac{s}{S} \right) .{{z}_{pq}} \end{aligned}$$
(41)
$$\begin{aligned} & {{Z}_{p}}={\left\{ \begin{array}{ll} & Z_{p}^{st3},\ F_{p}^{st3}<{{F}_{p}} \\ & {{Z}_{p}},\ otherwise \\ \end{array}\right. } \end{aligned}$$
(42)

\(Z_{p}^{st3}\) is modified \({{p}^{th}}\) possible solution at the \(3^{rd}\) DTBO phase; \(z_{p,q}^{st3}\) is its \({{q}^{th}}\) variable; the value of the related objective function is \(F_{p}^{st3}\); r is arbitrary quantity, ranging from 0 to 1.; R is 0.05, s is present iteration & S is the maximum iteration. Steps one through three update the DTBO population, completing one DTBO iteration. Then, with a freshly updated population, the subsequent iteration begins and this procedure is ongoing [through (34) to (42)] till the end of the last iteration. The best potential solution is noted as the problem’s solution at the conclusion of the last iteration. Flowchart of DTBO is shown in Fig. 2

Fig. 2
figure 2

Flowchart of DTBO.

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Simulation outcomes & key observations

This section presents the simulation findings for various ORPD case studies using the DTBO algorithm and compares them with the results given in6. The entire simulation is run within the MATLAB framework. The selection of test systems includes the conventional IEEE 30, 57,118, 300 bus networks and their modified architectures in modules: one, two, three, four & five. Table 2 provides a brief summary of the test systems under module one and two. Two test networks that are listed in Table 2 are base configuration and adapted configuration. There are five main test modules that comprise the current study. The module three comprises of IEEE 57 bus network. Module four includes IEEE −118 bus system and Module five considers IEEE-300 bus network. In order to provide an impartial comparison, the test systems are selected based on the system utilized in6.

Table 2 IEEE 30-bus description.
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Only thermal generation is taken into account in test module one, however RESs are added together with an earlier test system in test module two. A total of thirty cases are examined over these five test networks; these are compiled at Table 3. Fig. 3 displays the WS PDF (weibull based), SI PDF (lognormal based), and WFR PDF (Gumbel distribution based) according to the previously indicated parameter values. These are employed to estimate the uncertainty of the RESs.

Table 3 Proposed case studies.
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Cases 1 through 8 are examined in test module one, and cases 9 through 16 are investigated in module two. It is possible to split test modules one and two into two categories: those that use STATCOM as a FACTs tool integrated into the test network and those that do not. Cases 1–4 and 9–12 explicitly do not take STATCOM into account, while cases 5–8 and 13–16 are investigated taking STATCOM into account along with a test system.Cases 17 to 22 are taken in Module three. Cases 23 through 26 are considered in Module four & Cases 27 through 30 are under Module five. With the exception of the swing generator, the active power settings for generators in an optimization problem must be carefully selected within the generators’ specific operating parameters. Throughout the course of the study, these amounts are shown for cases 1–8, as well as for cases 9–16, in Table 2. There are different objectives with considered test setting. These are the following: lowering APL, reactive power loss minimization, cost minimization for STATCOM, minimizing AVD and reducing VSI as single objective cases, and concurrently lowering APL and AVD as multi-objective cases.

Fig. 3
figure 3

Weibull based WS PDF with \(\iota =10\) and \(\varkappa =2\), Lognormal based SI \((W/m^2)\) PDF with \(\varepsilon =6\) and \(\lambda =0.6\) & Gumbel distribution based WFR PDF with ___location factor 15 & scale factor 1.2.

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Module one

In the left column of Table 2, the test network for this module is provided under the “Base configuration” heading. For this test setup, cases 1 through 8 are run, and for cases 5–8, STATCOM is integrated with the test network. The module has been taken into consideration for a constant 100% network loading. Table 4 and Table 5, respectively, include the computed results for cases 1–4 and cases 5–8. The estimated magnitudes of the objective quantities are displayed in these tables together with the optimal and extreme border values of each variable.

Table 4 IEEE 30-bus simulation results for fixed loading (100%).
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Table 5 IEEE 30-bus simulation results for fixed loading (100%) with STATCOM.
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Table 6 Twenty five scenarios based on probable wind speed, solar irradiance and water flow rate and % of loading.
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Table 7 Output power of different generators for different loading.
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The modified artificial hummingbird algorithm (MAHA) used in6 is used as a comparable test setup. The DTBO algorithm is being used in this work to reduce APL, AVD, VSI as single OFs and to reduce APL and AVD at the same time.

From Table 4, crucial observations are:

  • APL is determined as 4.3101 (MW) in case 1 using DTBO, whereas it was 4.5086 (MW) in6. Therefore, DTBO lowers APL in relation to6 by 0.1985(MW).

  • Using DTBO, the AVD in case-2 is 0.0794 p.u., which is less than the AVD found in6 by 0.0085 p.u.

  • With DTBO, the VSI in case-3 is 0.1104, whereas in6, it was 0.1132. DTBO hence lowers VSI by 0.0028 as opposed to6.

  • The outcomes in case 4 are noteworthy when simultaneous aims, namely APL & AVD, are taken into account. Case 4’s APL and AVD are both higher than Case 1 and Case 2, respectively, but taken as a whole, APL & AVD are better than Case 1 or Case 2.

The final row of this result lists the computational duration for each case. It indicates that, when compared to6, using DTBO not only improves the optimization results for cases 1–3, but also helps to occupying better results in a shorter amount of time.

The results of the experiments that were carried out while taking into account STATCOM with base configuration are shown in Table 5, which reveals that:

  • The computed APL value in case 5 is 4.27(MW), which is less than 0.0401(MW) from the APL of case 1.

  • The calculated AVD in case 6 is 0.0731(p.u.). The AVD, found in case 2 is higher than the AVD of case 6 by 0.0063(p.u.).

  • The calculated VSI for case 7 is 0.1045(p.u.). As for case 7, the VSI is lower by 0.0059(p.u.) than the VSI obtained in case 3.

  • APL & AVD in the multi-objective situation (case 8) are 5.0701(MW) & 0.1221 (p.u.), respectively, which are better than those figures in case 4.

  • As previously stated, the test configuration that was used for cases 1–4, has been altered by adding STATCOM, and cases 5–8 have been resolved using this updated setup.

The aforementioned findings (cases 1 through case 8)make it abundantly evident that the success of the ORPD issue is greatly aided by the use of STATCOM in the power network. Fig. 4 provides the convergence characteristics of APL minimization, AVD minimization & VSI minimization with and without consideration of STATCOM. From the curves in Fig. 4, it is clear that adding STATCOM to the power network improved system performances.

Fig. 4
figure 4

Using DTBO with and without STATCOM, the convergence characteristics of APL, AVD & VSI.

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Module two

As indicated in Table 3, eight examples (cases 9–16) are taken into consideration in this phase of the experiment, the first four cases (cases 9–12) are carried out using a test setup without the introduction of STATCOM, and the remaining cases (cases 13–16) are carried out over a test network that has STATCOM. Table 2’s right portion displays the test configuration that was used for this part of study. This type of adjustment, referred as “adapted configuration,” involves combining a conventional model with RESs (WP, PV & HP). Furthermore, in this experimental mode, the process of scenario generation and scenario downsizing has been utilized to address the volatility of RESs and the unpredictability of load25. For estimating variable load demands, an average PDF with mean=70 and standard deviation=10 has been considered25. Weibull, lognormal, and gumbel PDFs are used to model uncertain WS, SI, and WFR, respectively, during the scenario design process. During the construction of the scenarios, nil irradiance is assigned with 50% chance because the sun is present for just about half of a 24-hour day. The remaining 50% of the possibilities are allocated with non-zero PV power contribution to the scenarios. The load demand, WS, SI, and WFR are the elements of a single scenario.

To begin with, the 1000 Monte-Carlo options for load demand, WS, SI, and WFR are combined to create a set of 1000 scenarios. The 1000 situations have been reduced to 25 scenarios by BRA65 since handling 1000 possibilities is not manageable. Initially, \(N_{0}\) scenarios are considered where each of them having probabilities of (\({{\rho }_{0}}=\frac{1}{N_0}\)). After every BRA iteration, one scenario is removed in an effort to reduce the total number of possibilities. The following are the steps that the BRA takes to reduce scenarios:

1. Initialization

  • Create \(N_{0}\) scenarios (\(S_{i}\) for \(i= 1, 2, .~.~., N_{0}\)). Currently: \(N_{0} = 1000\).

  • At the beginning, the chance of every scenario is identical (\({{\rho }_{0}}=\frac{1}{N_0}\)). Determine the distance \(d_{ij}\) among each pair of scenarios. where \({{d}_{ij}}=\left\| {{S}_{i}}-{{S}_{j}} \right\|\).

  • With \(d_{ij}\), set up distance matrix D with starting dimension \({{N}_{0}}\times {{N}_{0}}\) and diagonal elements \(d_{ii}=0\).

  • Allot a running variable \(N_r = N_{0}\) & stopping criterion \(N_{ec}\), indicates the count of final preferred scenarios.

2. Looping events

step 1.:

Find least distance value (apart from self-distance \(d_{ii}=0\)) from D. Suppose \(d_{mn}\) is least in D (i.e. separation between \(m^{th}\) and \(n^{th}\) scenarios), and suppose scenarios \(S_{m} and S_{n}\) having likelihoods of \(\rho _{m}\) and \(\rho _{n}\) respectively.

step 2.:

If \({{\rho }_{m}}\ge {{\rho }_{n}}\), remove scenario n. Modify likelihood \({{\rho }_{m}}={{\rho }_{m}}+{{\rho }_{n}}\). Else, take away scenario m. Alter probability \({{\rho }_{n}}={{\rho }_{m}}+{{\rho }_{n}}\).

step 3.:

Allocate \({{N}_{r}}={{N}_{r}}-1\). reassess the matrix D, composed of distance between each pair of existing scenarios.

step 4.:

If \({{N}_{r}}>{{N}_{ec}}\), jump to STEP 1 of reiterating. Else, END.

These 25 scenarios, together with their associated possibilities, are displayed in Table 6 and are generated by applying BRA to 1000 initial scenarios. The load demand is presented in Table 6 as % loading. From the scenarios given in Table 6 and through equations (5), (10) and (12), respectively, the corresponding WP, PV & HP are evaluated and shown in Table 7.

The optimization algorithm is then executed over every scenario independently. The results of running those algorithms are the OFs, which are the multi-objective minimization of combined APL and AVD and the single-objective minimization of APL, AVD, and VSI. As the current study consists of 25 situations, the optimization technique is run 25 times to cover all the developed scenarios in order to thoroughly investigate any case.

For each constructed scenario in Table 6, Table 8 displays the minimal APL (corresponds to case 9), minimum AVD (corresponds to case 10), and minimum VSI (corresponds to case 11). An expected APL (EAPL) (for case 9), an expected reactive power loss (ERPL) (in case 9 A), an expected AVD (EAVD) (for case 10), and an expected VSI (EVSI) (for case 11) are computed and reported in Table 8 from these probable computed APL, AVD, and VSI for each scenario. These calculations are done as follows:

$$\begin{aligned} & EAPL=\sum \limits _{i=1}^{{{N}_{ec}}}{{{\rho }_{i}}\times APL_{i}} \end{aligned}$$
(43)
$$\begin{aligned} & EAVD=\sum \limits _{i=1}^{{{N}_{ec}}}{{{\rho }_{i}}\times AVD_{i}} \end{aligned}$$
(44)
$$\begin{aligned} & EVSI=\sum \limits _{i=1}^{{{N}_{ec}}}{{{\rho }_{i}}\times VSI_{i}} \end{aligned}$$
(45)
$$\begin{aligned} & ERPL=\sum \limits _{i=1}^{{{N}_{ec}}}{{{\rho }_{i}}\times RPL_{i}} \end{aligned}$$
(46)

where: i indicates scenario index; \(N_{ec}\) is the number of scenario; \(\rho _{i}\) indicates probability of \(i^{th}\) scenario.

Table 8 Single-objective ORPD solution of IEEE 30 bus with RESs.
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Table 9 Multi-objective ORPD solution of IEEE 30 bus with RESs.
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To display the outcomes of case 12, where the target is simultaneously minimize the APL and the AVD for 25 scenarios of Table 6, Table 9 is created.

Experiments for cases 13–16 are carried out on the test setup with STATCOM under the identical scenarios as listed in Table 10. Table 10 contains scenario-based experimental results for instances 13–15, whereas Table 11 has results for case 16. Applying DTBO to test systems with and without STATCOM, the obtained results indicate that:

  • Without STATCOM, EAPL was 2.6719 MW (case 9), but with STATCOM, it decreases to 2.5426 MW (case 13).

  • The EAVD in case 10 was 0.0627 p.u. without STATCOM, while in case 14 (which includes STATCOM), it decreases to 0.0596 p.u..

  • Without STATCOM, the EVSI in case 11 was 0.0858 p.u.; with STATCOM included, the EVSI drops to 0.0818 p.u. in case 15.

  • When case-12 and case-16 are observed simultaneously, it is discovered that, in contrast to case-12 (i.e. without STATCOM), connecting STATCOM (in case-16) lowers EAPL and EAVD by 0.1974 MW and 0.0035 p.u., respectively.

These findings imply that to minimize individually APL, AVD, VSI as well as to reduce jointly APL and AVD utilization of STATCOM devices offers positive impacts in the operations of power networks.

Table 10 Single-objective ORPD solution of IEEE 30 bus with RESs-STATCOM.
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Fig. 5
figure 5

Load bus voltage deviations for case-2, 6, 10 & 14.

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Fig. 5 depicts the deviation of voltage on different load buses for case 2 (on only traditional IEEE 30 bus set up), case 6(traditional network with STATCOM), case 10 (RESs included in conventional IEEE 30 bus network) and case 14 (Both RESs and STATCOM included). In these four cases objective was reduction on AVD. It can be observed from the Fig. 5 that the spread of voltage deviation is reducing when RESs & STATCOM devices are being introduced with conventional standard 30 bus network.

Table 11 Multi-objective ORPD solution of IEEE 30 bus with RESs-STATCOM.
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Module three

In this test module, the summary of the test system is given in Table 12 where IEEE 57 bus network is being considered. The scenarios, the % of loads, shearing of wind power, solar power and hydro power with scenario probabilities are furnished at Table 14. There are 6 cases (five single and one multi-objective) that have been taken care in this module from Case 17 to 22 and are mentioned earlier in Table 3. The outcomes of Case 17 to Case 21 (single objective) depending on scenarios are presented on Table 15 while the outcomes for Case 22 (multi-objective) are provided in Table 16. The variation of bus voltages for Case 20 under scenario 5 and 10 is given in Fig. 6. The relative hikes in EAPL, EAVD & EVSI in Case 22 is noticeable with respect to Case 17,20,21 respectively is due to the consideration of Multi objective in Case 22 while Case 17,20 &21 were single objective cases. Fig. 7 displays the fluctuation of voltages over buses for Case 22 on scenario 5 and 10.

From the overall simulation study, the superiority of DTBO to compute the efficacy of ORPD solutions has become starkly apparent over other recent optimization algorithms under all these cases, when the obtained test results are compared with the results which were presented in the literature on the same experimental platform. It is also evident from the simulation study that inclusion of STATCOM can significantly improve system’s performance. A brief overview of the IEEE 57-bus system are listed in Table 13.

Table 12 IEEE 57-bus details.
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Table 13 An overview of IEEE 57- bus System.
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Table 14 Different loading of different sources for RESs based IEEE 57 bus.
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Table 15 Results of Single-objective functions for IEEE 57 bus with RESs for ORPD.
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Table 16 Multi-objective Results for IEEE 57 bus with RESs for ORPD.
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Figure 6
figure 6

Load bus voltage deviations for case-20 at Scenario no 5 & Scenario no 10.

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Figure 7
figure 7

Load bus voltage deviations for case-22 at Scenario no 5 & Scenario no 10.

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Module Four

In test Module four, IEEE 118 bus network has been chosen where RESs and STATCOM devices are also being added with the system. Here, a single wind farm is placed in bus 25, a PV unit is kept at bus 40, a combination of wind unit and hydro unit is connected to bus 70. The 25 scenarios, % loading, contribution of sole wind farm, PV unit, combined wind-hydro unit, and probabilities of scenarios considered in this module are referred to Table 14 which was also used in Module 3. As mentioned in Table 3, cases 23 to 25 (as single objective) & case 26 (as a multi-objective) are examined. The results obtained for Cases 23 to 25 & case 26 are placed in Table17 & 18 respectively.

Table 17 Single-objective ORPD evaluted cases with time-varying demand and uncertain renewable power with STATCOM (IEEE 118 Bus system).
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Table 18 Multi-objective ORPD evaluted cases with time-varying demand and uncertain renewable power with STATCOM (IEEE 118 bus system).
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Module Five

Table 19 Single-objective ORPD evaluated cases with time-varying demand and uncertain renewable power with STATCOM (IEEE 300 Bus system).
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Table 20 Multi-objective ORPD evaluted cases with time-varying demand and uncertain renewable power with STATCOM (IEEE 300 bus system).
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The IEEE 300 bus system has been selected for test Module 5, and the system will also include RESs and STATCOM devices. In this instance, bus 84 has a single wind farm, bus 108 carries a photovoltaic unit, and bus 152 keeps a wind- hydro unit combo. Referring to Table 14, which was also utilized in the previous two modules, are the 25 scenarios, % loading, contribution of a single wind farm, PV unit, combined wind-hydro unit, and probability of scenarios taken into consideration in this module. Cases 27 to 29 (as single objective) and case 30 (as a multi-objective) are analyzed here, as indicated in Table 3. Table19 and Table20 include the results for Cases 27 to 29 and Case 30 respectively.

Conclusions

Using DTBO across five test setups, as demonstrated in five study modules, the ORPD problem has been tackled in the current work. The first one looks at a typical IEEE 30-bus network, and the second one looks at a traditional network that has been reconfigured with RESs connected. In the third, forth & the fifth modules of the study respectively IEEE 57 bus, 118 bus & 300 bus network have been used as test setup. A deterministic environment is used for the study’s earlier phases, and in the latter sections, the approach of scenario development and reduction procedure is used to address the stochasticity of load demand and RESs. Scenarios are created using MCS, and they are then condensed into a manageable number utilizing BRA. In this regard, appropriate PDFs of load demand and RESs are also being taken into account. The study has two objectives: first, it aims to minimize APL, AVD, and VSI individually as a single target; second, it aims to minimize APL and AVD jointly as a multi-objective. Experiments are run in both test configurations, once with STATCOM taken into account and once without. The results of the experiments show that DTBO is more effective than modern optimization algorithms in both deterministic experimental setups and test scenarios where volatility is prevalent. Additionally, it has been verified that all network constraints are currently kept within predetermined bounds. One intriguing result of this work is that, with regard to the ORPD issue, using STATCOM is highly beneficial due to the decrease in the system’s APL, AVD, and VSI. STATCOM continues to be incredibly effective in both fixed and uncertain loading scenarios. Despite the presence of unpredictable RESs in the power network, STATCOM nevertheless improves system performance.Further experiments can be conducted using higher ordered IEEE standard networks.

From simulation results it has been observed that for IEEE 30 bus, the average power loss (APL) is 4.5086 MW. However, after utilizing STATCOM, the APL is reduced by 5.3% and with the integration f renewable sources, the APL is reduced by 41%, and for both STATCOM and RESs system, it decreases to 43.6%. Hence, STATCOM and RES help to reduce the power losses using DTBO approach. Furthermore, the average voltage deviation (AVD) is improved by 97.4 % with incorporating STATCOM-RESs. Voltage stability index (VSI) is improved by 26.9% with scheduling STATCOM and renewable sources (RESs). For the multi-objective situation, APL & AVD both are simultaneously improved to 5.0701(MW) & 0.1221 (p.u.), respectively, after incorporating STATCOM and RESs using DTBO. Voltage deviation converges at 40 iterations for simulation study having STATCOM but for without STATCOM, it takes 80 iterations to converge. Similarly for voltage stability index with STATCOM converge 4 iterations earlier as compared to that of without STATCOM system. Again, for large scale IEEE 57 bus system, The DTBO approach incorporating STATCOM and RESs provides optimal results. So, for both IEEE 30 and IEEE 57 bus systems, DTBO proof its superiority and robustness satisfactorily. Furthermore, the study also covers the experiments on IEEE 118 & 300 bus network in Module four & five respectively where the outcomes are also remarkably well.