Keywords

1 Introduction

In the most recent few decades, there has been a tremendous expansion of electric power networks. A nation's economy and rapid industrialization depend heavily on energy, which is available at power stations and then transmitted via HV transmission cables and substations to final users. Power plant operations have proven challenging due to the need for a new electrical system to respond to changing load demands and provide high-quality electricity. Because of the intense pressures placed on these systems, such as unexpected power flows of power and significant losses, these systems are constantly challenged, And the power grid has to deal with voltage instability so that all load buses get the voltage they need. When these conditions aren't met, the voltage drops and instability will ensue. Therefore, it is crucial to improve the voltage stability margin of power systems. In 1988, Hingorani [2] created FACTS in an effort to overcome the aforementioned difficulties and optimise the use of power system equipment. This means that transmission line capacities and power system security have both seen major boosts owing to FACTS devices and their associated circuits.

SVCs, familiar FACTS devices, consume or generate bus reactive power to stabilise voltage./ The SVC is a common FACTS device that helps keep voltage steady by consuming or producing bus reactive power depending on its location. When it comes to the planning and execution of electrical systems, OPF is an indispensable factor. OPF is essential from the perspective of the design and operation of electrical systems. In general, OPF is a multidimensional, nonlinear, and large numerical problem that is dependent on system data. It becomes even more difficult when variable restrictions and FACTS device settings are incorporated. To manage convergence to the optimal solution, a number of analytical and conventional optimization techniques have been employed, such as the extended reduced gradient technique, the Newton–Raphson technique, the interior point approach, the P– Q decomposition technique, the linear programming technique, and the quadratic programming method, among others. For traditional optimization methods to avoid being caught in local minima, it is necessary to start from a position that is relatively close to the solution. The standard of the responses becomes extremely dependent on the starting values as the number of problem parameters grows. These traditional approaches have their limitations, especially with the rise of computational technologies,/ As a result of the drawbacks of these traditional approaches and the advancement of computer technology, computer-based methods are now preferred. Over the past few decades, there has been a sharp increase in interest in applying heuristic optimization techniques to problems with the power system. The heuristic optimization techniques employ probabilistic transition rules as opposed to deterministic ones. avoid using information that is derived from other things, have the ability to avoid getting stuck in a local minimum, and be able to deal with large-scale nonlinear issues. In recent years, some of the most well-known heuristics, such as differential evolution [10], particle swarm optimization [11], and genetic algorithm [12], have been put to use in an effort to minimise the total cost of generation while also maintaining load bus voltages within the parameters of the constraints. This was accomplished by optimally determining the locations of SVC devices and the values of their parameters. In power systems, both heuristic optimization algorithms and conventional optimization techniques have been applied, each with its own advantages and disadvantages.

An optimization approach that is named the artificial bee colony (ABC) algorithm is given in this paper as a way to address the OPF problem. This algorithm was modelled after the way honey bee swarms search for food, which served as the inspiration for developing this method. Further, the power system is viewed as combined with SVC susceptance concept, and OPF based on ABC is used to address the resulting problem. Under the stated maximum power limitations, generation limits of active power values, and bus voltage limits, the total generating cost is minimised. In order to solve the OPF problem, the factors of FACTSwere found by applying ABC for both the IEEE 11-bus and IEEE 30-bus systems. In order to gauge how well this technique performs, it is compared to other optimization strategies. Like the second order gradient method, the reduced Hessian method, differential evolution.

The structure of this paper consists of the following sections: In Sect. 2, we see the OPF problem formulated and the modelling of the power system explained. In Sect. 3 we discuss the ABC algorithm and how it was applied to OPF. The results are displayed in Sect. 4, and the discussion and conclusions are presented in Sect. 5.

2 The Formulation of the OPF's Problems

2.1 SVC Model

The SVC susceptance model, which was built for the OPF that was employed in this work and is regarded to be a reactive power injection at the load buses [13], The SVC susceptance model shown in Fig. 1 was developed for the OPF employed in this research and is accounted for as MVAr injection at load buses [13]. When it comes to managing characteristics like load bus voltages, this shunt-connected SVC's output is made to switch between capacitive and inductive current.

Fig. 1
An illustration depicts the susceptance model of S V C. It depicts a bus represented by a vertical thick line connected to B subscript S V C.

Susceptance model of SVC

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For the bus k(Qk), the reactive power consumed by the SVC (Qsvc) can be represented as:

$$ {\text{Q}}_{{{\text{SVC}}}} = {\text{Q}}_{{\text{K}}} = - {\text{V}}_{{{\text{K}}^{2} }} {\text{B}}_{{{\text{SVC}}}} $$

where Vk—voltage on bus k

BSVC—bus-to-source voltage converter's equivalent susceptance.

2.2 Optimal Power Flow

The OPF is a problem that involves the optimization (minimization) of nonlinear constraints and a nonlinear objective function. The primary goal of this research is to optimise the total generating cost by identifying the appropriate power values of generation units and the specifications of SVC devices in the OPF problem considered here.

A mathematical description of the OPF is as follows:

$$ \begin{aligned} & {\text{Min}}({\text{overall}}\,{\text{generation}}\,{\text{cost}}\,{\text{f}}\left( {{\text{x}},{\text{u}}} \right)) = \mathop \sum \limits_{{{\text{i}} = 1}}^{{{\text{N}}_{{\text{g}}} }} {\text{a}}_{{\text{i}}} + {\text{b}}_{{\text{i}}} {\text{P}}_{{{\text{Gi}}}} + {\text{C}}_{{\text{i}}} {\text{P}}_{{{\text{GI}}}}^{2} \\ & {\text{subject}}\,{\text{to}}\left\{ {\begin{array}{*{20}l} {{\text{g}}\left( {{\text{x}},\,{\text{u}}} \right) = 0} \hfill \\ {{\text{h}}\left( {{\text{x}},\,{\text{u}}} \right) \le 0} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

Here

$$ X = \left[ {P_{GSlack} V_{L} Q_{G} } \right]{ } = State\,variables{ } $$
$$ U = \left[ {P_{G} VQ_{G} Q_{SVC} } \right] = {\text{Control }}\,{\text{variables }} $$

PGslack = The slack bus’s true power value.

VL = load bus voltage.

QG0 = reactive power generation.

PG = real power.

VG = the generator voltage.

QSVC = SVC reactive power.

In order to achieve an optimal active power dispatch, it is necessary to lower the overall generating cost, which is denoted by the expression f (x, u).

$$ f = \mathop \sum \limits_{i = 1}^{{N_{g} }} a_{i} + b_{i} P_{Gi} + C_{i} P_{GI}^{2} $$

where Ng—number of the generator

PGi—real power generation at bus i;

\({\text{P}}_{{{\text{Gi}}}} - {\text{P}}_{{{\text{Di}}}}\) = losses

$$ P_{Gi} - P_{Di} - \mathop \sum \limits_{j = 1}^{Nb} \left| {V_{i} } \right|\left| {V_{j} } \right|\left| {Y_{ij} } \right|\cos \left( {\theta_{ij} + \delta_{i} + \delta_{j} } \right) + P_{injeSv Ci} = 0 $$
(3a)
$$ Q_{Gi} - Q_{Di} - \mathop \sum \limits_{j = 1}^{Nb} \left| {V_{i} } \right|\left| {V_{j} } \right|\left| {Y_{ij} } \right|\cos \left( {\theta_{ij} + \delta_{i} + \delta_{j} } \right) + Q_{injeSv Ci} = 0, 3 $$
(3b)

where

Equations 3a and 3b gives th load flow equations g(x,u).

QGi = the generation reactive power.

PDi and QDi = Ith bus active and reactive load demand,

Yij = the bus admittance value;

Pinjesv ci and Qinjesv Ci = Ith bus injected active and reactive power respectively.

the constraints are:

$$ {\text{V}}_{{{\text{Gi}}}}^{{{\text{min}}}} \le {\text{V}}_{{{\text{Gi}}}} \le {\text{V}}_{{{\text{Gi}}}}^{{{\text{maxi}}}} = 1, \ldots ,{\text{N}}_{{\text{a}}} $$
(4a)
$$ P_{Gi}^{min} \le P_{Gi} \le P_{Gi}^{maxi} = 1, \ldots ,N_{a} $$
(4b)
$$ Q_{Gi}^{min} \le Q_{Gi} \le Q_{Gi}^{maxi} = 1, \ldots ,N_{a} $$
(4c)
$$ Q_{GiSV Ci}^{min} \le Q_{SV CiGi} \le Q_{GiSV Ci}^{max } = 1, \ldots ,N_{SV C} $$
(4d)

where NSVC—SVC devices number.

2.3 Sensitivity Analysis

To deploy SVC in the proper locations, a power system sensitivity analysis was conducted on the power system to identify the bus number most sensitive to variations in reactive power. Voltage stability can be significantly improved by the use of shunt compensation and In order to attain the highest possible level of efficiency, conducting a V–Q sensitivity analysis is necessary before deciding where to place SVC devices.

The Jacobian matrix for the power system was utilised for carrying out the sensitivity analysis. In this case, the diagonal elements of the matrix represent the stability indices at the steady state. Both the bus voltage sensitivity and the steady-state stability indices are represented by the diagonal elements of the matrix and the inverse reduced Jacobian matrix, respectively. In the sensitivity analysis, we only consider load buses and When the sensitivity index is positive, the margin of stability is reduced, and when it is negative, instability prevails.

Variation in J-matrix elements and reactive power equals a differential in voltage value:

$$ \Delta {\text{V}} = {\text{J}}_{{\text{R}}}^{ - 1} \nabla {\text{Q}} $$

\({\text{where J}}_{{\text{R}}} {\text{J}}_{4} - {\text{J}}_{3} .{\text{J}}^{ - 1} .{\text{J}}_{2} =\) system’s reduced jocbian matrix.

3 The Artificial Bee Colony Algorithm Description

Nonlinear and multidimensional optimization problems have attracted a plenty of focus and consideration in science and engineering in couple years, and swarm intelligence-based optimization techniques have been at the centre of this attention. An innovative heuristic optimization strategy, the artificial bee colony [16,17,18,19,20] is used to identify optimal convergent solutions to optimization issues by mimicking swarms of foraging honey bees. In 2005, Karaboga developed the artificial bee colony (ABC) algorithm. The ABC algorithm was first seen while studying the behaviour of real bees as they foraged for and distributed honey. i.e. ABC adapted its foraging strategy from those of swarms of honey bees.

Both self-organization and the division of work are cornerstone concepts in the field of swarm intelligence. The labor(bees) of the ABC algorithm is distributed across three classes: workers(employers), observers(onlookers), and scouts. Each bee performs a unique function throughout the optimization procedure, like workers keep their attention fixed on a nearby source of nectar so that they don’t forget where it came from.

As employed bees do, workers keep their attention fixed on a nearby source of nectar so that they don’t forget where it came from. While scout bees are subjected to a rigorous degree of calculation, onlooker bees are responsible for compiling information from worker bees and making a resource decision regarding nectar harvesting.

Below is a detailed description of the algorithm applied in this study:

  1. 1.

    Firstly, prepare the ABC food supplies.

  2. 2.

    Calculate the power flow using the Newton–Raphson method in a campout starting with the given data.

  3. 3.

    Third, get the fitness function from the power flow solution. Choose the most viable option for initial sustenance. For k = 1, set the iteration counter.

  4. 4.

    Locate positions for currently employed bees

  5. 5.

    Utilizing the Newton–Raphson method, provide a solution to the power flow problem while taking into account the positions of the workers bees.

  6. 6.

    The sixth step is to compute the new fitness function based on the relocated workers.

  7. 7.

    By contrasting the modified fitness function with the initialised one, we can learn which value is best.

  8. 8.

    Update the position of the observers to reflect their newfound proximity to food sources and proceed to Step 5.

  9. 9.

    Identify unimproved food sources when the “limit” is reached.

  10. 10.

    Assign the unaltered food sources found to the scout bees, solve the power flow problem, and then compute the fitness function.

  11. 11.

    Enhance the values of the fitness function that correspond to the best possible position for the food source.

  12. 12.

    K = K + 1 and repeat from step 4 if K is less than the maximum number of repetitions.

ABC Implementation to Address the OPF Problem

  1. 1.

    Specify the parameters of the ABC method and the limitations placed on the parent vector. Details such as voltages of line and bus, real and reactive power, and SVC susceptance data are included.

  2. 2.

    Arbitrary selection of the first parent vectors

    $$ {\text{P}}_{mn} = {\text{P}}_{{\text{n min}}} + {\text{rand X }}\left( {{\text{P}}_{{\text{n max}}} - {\text{P}}_{{\text{n min}}} } \right) $$
    $$ {\text{m}} = {\text{index number}},{\text{ n}} = {\text{dimension number}} $$
  1. 3.

    Using the equation, determine the numbers for the fuel cost, as well as the FV.\({\text{FV}}\left( {\text{ i}} \right) = \left\{ {\begin{array}{*{20}c} {1/\left( {1 + {\text{FV}}_{{\text{i}}} } \right){ }0 \le {\text{f}}_{{\text{i}}} } \\ {1 + {\text{abs}}\left( {{\text{FVi}}} \right){ }0 > {\text{ f}}_{{\text{i}}} } \\ \end{array} } \right\}\)

  2. 4.

    Establish a fresh objective that is connected to Eq. (8). Through the utilisation of optimal power flow, one must determine the FV and fuel cost associated with each target vector. Determine which vector has the highest fitness rating and save that one. \({\text{R}}_{{{\text{ijnew}}}} = {\text{R}}_{{{\text{ij}}}} + \emptyset \times \left( {{\text{R}}_{{{\text{ij}}}} - {\text{R}}_{{{\text{kj}}}} } \right)\left( {{\text{i k}},\emptyset = {\text{rand}}\left[ { - 1,1} \right]} \right)\)

  3. 5.

    Estimate the possibility of the fuel cost for every food source connected to a certain rule. To meet Eq. (9), develop a different vector; check FV and pick the best (top) possibilities for each food source.

    $$ {\text{P}}_{{\text{i}}} = \frac{{{\text{fitness}}_{{\text{i}}} }}{{\mathop \sum \nolimits_{{{\text{i}} = 1}}^{{{\text{SN}}}} {\text{fitness}}_{{\text{i}}} }} $$
  1. 6.

    Drop the current target, choose a new one, and move on to Step 2 if the target has not been improved after the user-specified maximum number of tries.

  2. 7.

    The best answer that has been identified up to this point will be updated, and the iteration counter will advance by one.

  3. 8.

    When all of the end criteria have been met, the iteration process should be finished. In any other case, move on to Step 4 and carry out the iteration method as many times as necessary until the end requirements are met or the iteration counter reaches its maximum number.

  4. 9.

    Parameters of ABC method

No. of hired bees = 20.

source no. = 20.

max. iterations = 500.

4 Test Results and Discussion

4.1 Case.1 IEEE 11-Bus Test Systems

The table below provides information about the IEEE 11 bus system (Table 1).

Table 1 IEEE 11 bus system data
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In addition to further assess the efficacy of the ABC technique, we apply differential evolution to the same problem. The optimised values for Active power, Reactive power, voltage angle, and total generating cost are shown in Table 2 beneath. The RHM results reported in the published literature are also included in the table. ABC and DE are more viable than RHM. The results obtained by ABC are noticeably superior to those obtained by DE. The cost of generating is brought down to 1253.66 rupees per hour owing to the ABC solution for the OPF problem (Fig. 2).

Table 2 Analyses of three different methods for case 1
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Fig. 2
A multiple-line graph of performance evaluation of the A B C and D E methods. Both the legends depict a step-wise decreasing trend.

Performance evaluation of the ABC and DE methods using the IEEE 11-bus system

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4.2 Case.2 IEEE 30-Bus Test Systems

In addition, the convergence qualities of the ABC and DE-based OPF solutions for the IEEE 11-bus system were investigated. Figure 3 demonstrates that ABC needs only 24 iterations to arrive at the best solution, but DE requires 56 iterations.

Fig. 3
A line graph plot of convergence of A B C optimization on the I E E E 30-bus system. It depicts a decreasing line decreasing from approximately (0, 815) to (50, 798) approximately and then remains constant.

Convergence plot of ABC method on IEEE 30 bus system

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The efficacy of the ABC algorithm was demonstrated through the utilisation of the IEEE 30-bus system [10] (Table 3).

Table 3 IEEE 30 bus system data
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Figure 3 displays the algorithm’s convergence plot in its entirety. Values in Table 4.

Table 4 Analyses of three different methods for case-2
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The system was also put through the ongoing power flow approach, which involved calculating the MLP through the application of the ABC-based optimal power flow algorithm.

5 Conclusions

This paper examines the ABC metaheuristic approach to OPF control of a power system using SVC devices in great detail in an attempt to enhance voltage stability while also reducing generator fuel cost. Maximum loading point was discovered through CPF analysis and the recommended ABC algorithm was validated on IEEE 11-bus and 30-bus systems. Here, the ideal locations for the SVCs were determined with the aid of a sensitivity analysis. Lastly, ABC may be used to successfully handle the complicated nonlinear difficulties of power systems by virtue of its high qualities and quick convergence in a short runtime.