Optimal Reactive Power Dispatch Using a Chaotic Turbulent Flow of Water-Based Optimization Algorithm

: In this study, an optimization algorithm called chaotic turbulent ﬂow of water-based optimization (CTFWO) algorithm is proposed to ﬁnd the optimal solution for the optimal reactive power dispatch (ORPD) problem. The ORPD is formulated as a complicated, mixed-integer nonlinear optimization problem, comprising control variables which are discrete and continuous. The CTFWO algorithm is used to minimize voltage deviation (VD) and real power loss (P_loss) for IEEE 30-bus and IEEE 57-bus power systems. These goals can be achieved by obtaining the optimized voltage values of the generator, the transformer tap changing positions, and the reactive compensation. In order to evaluate the ability of the proposed algorithm to obtain ORPD problem solutions, the results of the proposed CTFWO algorithm are compared with different algorithms, including artiﬁcial ecosystem-based optimization (AEO), the equilibrium optimizer (EO), the gradient-based optimizer (GBO), and the original turbulent ﬂow of water-based optimization (TFWO) algorithm. These are also compared with the results of the evaluated performance of various methods that are used in many recent papers. The experimental results show that the proposed CTFWO algorithm has superior performance, and is competitive with many state-of-the-art algorithms outlined in some of the recent studies in terms of solution accuracy, convergence rate, and stability.


Introduction
The optimal reactive power dispatch (ORPD) problem plays a very important role in the optimal operation of electric power systems. It is a subclass of the optimal power flow (OPF) problem [1]. The power system must be operating with high reliability, and finding a safe way to achieve this should obtain the optimal operating state and the control variable values (such as the generator voltage ratings, the tap ratios for the tap setting transformers, and the reactive power of the shunt capacitors/reactors) [2]. There are three main objectives of ORPD, which include reducing and minimizing the active power losses, the voltage deviation values, and the stability index. Researchers have studied several problems related to the power systems, including the security assessment of online power systems [3], a two-stage active and reactive power coordinated optimal dispatch for an active distribution network, considering load flexibility [4], the early detection and prevention of blackouts in power interconnections [5], OPF [6], and economic emissions dispatch [7].
Recently, different optimization methods have been studied to solve the ORPD problem; various optimization methodologies are recommended, such as deterministic and metaheuristic algorithms [8]. These algorithms include original, modified deterministic, 1.
Applying four different algorithms as search algorithms, including artificial ecosystembased optimization (AEO), the equilibrium optimizer (EO), the gradient-based optimizer (GBO), and turbulent flow of water-based optimization (TFWO), on IEEE 30-bus and IEEE 57-bus power systems to solve ORPD problem.

2.
The TFWO algorithm gives the best results for different single-objective functions, namely, the minimization of power losses and voltage deviation in both tested power systems. 3.
Proposing a new chaotic TFWO algorithm (CTFWO), which based on applying the chaotic approach to improve the performance of the original TFWO 4.
The proposed CTFWO algorithm solves the ORPD problem and gives better results than all other compared algorithms on the tested power systems, the 30-bus and the 57-bus systems, for all studied cases.
The rest of the paper is organized as follows: The ORPD problem is formulated in Section 2. In Section 3.1 the conventional TFWO algorithm is described and in Section 3.2 the proposed CTFWO algorithm is explained. In Section 4, the main achieved results and discussion are given. In Section 5, the conclusion drawn from this research is illustrated.

Materials and Methods
The ORPD has three main objectives: first, minimize and reduce the active power losses (P loss ); second, reduce the voltage deviation (VD), which is the difference between load voltage (which changes continually) and the reference voltage (with a value of 1.0 pu); finally, minimize the stability index (L-index), which takes values from 0 to 1, with 0 meaning that the system is stable and 1 meaning that there is a system disturbance.

Objective Functions
The two key objectives of this paper are as follows:

Minimization of the Active Power Loss
When operating any power systems, we can consider that the total active power loss is the main objective of the ORPD: where: P loss is the active power loss. G k is the conductance of the kth branch connected between the ith and the jth bus. α ij is the admittance angle of the transmission line connected between the ith and the jth bus. N TL is the number of transmission lines (branches). V i and V j are the voltage magnitudes of the ith and the jth bus, respectively.

Improvement of the Voltage Profile
The difference between the voltage magnitude at each load bus and what the specified reference value of the voltage ought to be is outlined in the following equation: where: V li is the voltage at the ith load bus. V sp li is the desired voltage at the ith load bus, which is usually set to (1.0 p.u). This constraint ensures that there is load balance (i.e., the generation of real and reactive power is balanced against consuming): where: P i = (P Gi − P Di ) and Q i = (Q Gi − Q Di ) represent the real and reactive power injection at bus i. P Gi and Q Gi are the active and reactive power generation of the ith bus. P Di and Q Di are the active and reactive load demand of the ith bus. G ij is the real part of the bus admittance matrix of the (i, j)th entry. B ij is the imaginary part of the bus admittance matrix of the (i, j)th entry. N B is numbers of buses.

Inequality Constraints
The inequality constraints should be within limited values, as follow: For i = 1, . . . . . . . . . , N T where: V min Gi and V max Gi are the minimum and maximum generator voltage values of the ith bus, respectively. Q min Ci and Q max Ci are the minimum and maximum values of the reactive power injection of the ith shunt compensator, respectively.
T min i and T max i are the minimum and maximum tap setting values of the ith transmission line, respectively. N C , N G , and N T are the numbers of shunt compensators, generators, and tap changing transformers, respectively.
The inequality constraints on the dependent variable are given by:

The Conventional TFWO
In this subsection, we briefly explain the concept of the original turbulent flow of water-based optimization (TFWO) algorithm. It is inspired by the whirlpools created in the turbulent flow of water. The whirlpool (Whj) is a random behavior of nature that happens in seas, rivers, and oceans. Its rotation and flow are affected by the force of gravity. The center of the whirlpool (Whj) functions as a sucking hole that attracts the objects and particles nearby towards its middle via internal forces. Though the centripetal force attracts the moving objects towards the whirlpool, the centrifugal force takes the object away from the corresponding center. The effects of the Whj on the object's particles are displayed in Figure 1. As can be seen from Figure 2, the objects (X) move with their special angle (δ) around the Whj's center. Therefore, this angle at each moment is changing as follows: Mathematics 2022, 10, x FOR PEER REVIEW 6 of 28

The Proposed CTFWO
The proposed CTFWO technique is the combination of the conventional TFWO algorithm with chaotic maps. Chaotic systems are deterministic systems that present unpredictable conduct, whose action is complex and similar to randomness [67]. In [67], a chaos-based exploration rate was proposed to enhance the performance of three wellknown optimization algorithms. Based on this proposed, the real random numbers (rand 1 , rand 2 ) in Equation (11) are replaced by a chaotic number. Figure

Simulation Results and Discussion
The algorithms proposed in our study are applied to two different standard power systems (IEEE 30-bus and IEEE 57-bus test systems). Figure 3 displays the IEEE 30-bus system, while Table 1 presents the description of the two test power systems. The proposed technique uses MATLAB 2018a programming, and all sections of the simulations have been executed on a PC with a 2.40 GHZ frequency CPU, and the installed memory (RAM) is 4.0 GB.

Simulation Results and Discussion
The algorithms proposed in our study are applied to two different standard power systems (IEEE 30-bus and IEEE 57-bus test systems). Figure 3 displays the IEEE 30-bus system, while Table 1 presents the description of the two test power systems. The proposed technique uses MATLAB 2018a programming, and all sections of the simulations have been executed on a PC with a 2.40 GHZ frequency CPU, and the installed memory (RAM) is 4.0 GB.    The software used is MATLAB 2018, and our computer has a 2.67 GHz Intel Core i5 processor and 4 GB RAM. The results relating to the performance for all our algorithms are taken after many trials. In our study, we have taken the population size of 30, while the number of iterations is 500 in both tested systems. In Table 1, we show that the values produced by the CTFWO algorithm, in the case of power losses, are better and more optimal values compared with the other four algorithms for the IEEE 30-bus system. In Table 2, we show the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values for case one, which simulates power losses in the 30-bus system. The best values obtained are in bold.
In Table 3, we show that the values for the CTFWO algorithm are better and more optimal compared with the other algorithms in the case of power losses in the IEEE 30-bus system. In Figure 4, the CTFWO algorithm gives the minimal values in the case of power losses compared to the other algorithms. The best values obtained are in bold.
The voltage profiles of all the algorithms for the 30 buses in this system are illustrated in Figure 5. The figure shows that the voltages magnitudes for all buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the other algorithms. Figure 6 shows the reactive power values of the six generators for the 30-bus power system in case one, which simulates power losses, for all algorithms. The best values obtained are in bold. The voltage profiles of all the algorithms for the 30 buses in this system are illustrated in Figure 5. The figure shows that the voltages magnitudes for all buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the other algorithms. Figure 6 shows the reactive power values of the six generators for the 30-bus power system in case one, which simulates power losses, for all algorithms.   In Table 4, the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values are shown for the voltage deviation simulation with the 30-bus system. Table 5 shows that the values obtained by the CTFWO algorithm are better and more optimal than those obtained by the others in the case of voltage deviation for the IEEE 30-bus system.  In Table 4, the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values are shown for the voltage deviation simulation with the 30-bus system. Table 5 shows that the values obtained by the CTFWO algorithm are better and more optimal than those obtained by the others in the case of voltage deviation for the IEEE 30-bus system.  The best values obtained are in bold. The best values obtained are in bold.
In Figure 7, the CTFWO algorithm gives the lowest values in the case of voltage deviation compared to the other algorithms in the 30-bus power system. The voltage profiles in p.u. for all algorithms with the 30 buses in this system are illustrated in Figure 8. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than other algorithms. Figure 9 shows the reactive power values of the six generators for the 30-bus power system in case two, which simulates the voltage deviation, for all the algorithms. The voltage profiles in p.u. for all algorithms with the 30 buses in this system are illustrated in Figure 8. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than other algorithms. Figure 9 shows the reactive power values of the six generators for the 30-bus power system in case two, which simulates the voltage deviation, for all the algorithms.   Table 6 shows the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values for the power losses in the 57-bus power system.   Table 6 shows the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values for the power losses in the 57-bus power system.  Table 6 shows the generator voltage, transformer tap ratio, capacitor bank, and generator reactive power values for the power losses in the 57-bus power system. The best values obtained are in bold.
In Table 7, we observe that the CTFWO algorithm gives better, more optimal values in the case of power losses for the 57-bus system than those obtained from the other algorithms. In Figure 10, we see that the CTFWO algorithm gives the best values at all individual runs in the case of power losses compared to the other algorithms for the 57-bus power system. The best values obtained are in bold.

AEO
In Figure 10, we see that the CTFWO algorithm gives the best values at all individual runs in the case of power losses compared to the other algorithms for the 57-bus power system. The voltage profiles in p.u. for all the algorithms for the 57 buses in this system are illustrated in Figure 11. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the other algorithms. Figure 12 shows the reactive power values in the 57-bus power system in case three, which simulates voltage deviation, for all the algorithms. The voltage profiles in p.u. for all the algorithms for the 57 buses in this system are illustrated in Figure 11. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the other algorithms. Figure 12 shows the reactive power values in the 57-bus power system in case three, which simulates voltage deviation, for all the algorithms.    Table 8 illustrates the generator voltage, transformer tap ratio, capacitor bank and generator reactive power values for the case of voltage deviation in the 57-bus system.  Table 8 illustrates the generator voltage, transformer tap ratio, capacitor bank and generator reactive power values for the case of voltage deviation in the 57-bus system.  The best values obtained are in bold. Table 9 shows that the CTFWO algorithm gives better and more optimal values for the 57-bus system in the case of voltage deviation compared with the other algorithms. The best values obtained are in bold.
In Figure 13, the CTFWO algorithm gives the best values at 30 individual runs in the case of voltage deviation compared to the other algorithms in the 57-bus power system.  The best values obtained are in bold.
In Figure 13, the CTFWO algorithm gives the best values at 30 individual runs in the case of voltage deviation compared to the other algorithms in the 57-bus power system. Figure 13. Boxplots for all algorithms for the 57-bus system in case 4.
The voltage profiles in p.u. for all the algorithms for the 57 buses in this system are illustrated in Figure 14. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the The voltage profiles in p.u. for all the algorithms for the 57 buses in this system are illustrated in Figure 14. The figure shows that the voltages magnitudes for all the buses are within the specified limits. However, the voltage profile in the case of using the proposed CTFWO technique has the better profile for most buses in the system than the other algorithms. Figure 15 shows the reactive power values in the 57-bus power system in case four, which simulates voltage deviation, for all the algorithms.   In the case of the 30-bus power system and the 57-bus power system, we performed 30 different trials for each algorithm under study and recorded the best trial for each one and plotted them as shown in Figures 16-19.         Figures 16 and 17 illustrate the curves in the case of power loss and voltage deviation for the 30-bus power system, and from these we can see that the CTFWO algorithm achieves the best, most minimal, smoothest, lowest curve compared with the other algorithms. Figures 18 and 19 illustrate the curves in the case of power loss and voltage deviation for the 57-bus power system and from these we can see that the CTFWO algorithm achieves the best, most minimal, smoothest, lowest curve compared with the other algorithms.

Conclusions
In this paper, several optimization algorithms; artificial ecosystem-based optimization, the equilibrium optimizer, the gradient-based optimizer, turbulent flow of water-based optimization, and proposed CTFWO are applied as tools to solve the ORPD problem by minimizing the voltage deviation (VD) and total transmission power loss (p loss ) in two standard power systems, a 30-bus system and a 57-bus system. For example, the values of power loss for the 30-bus system varied from 4.945 (in GBO) to 4.9449 (in TFWO), but after using our algorithm (CTFWO), it became 4.94480. Additionally, for the 57-bus system, there was variation from 23.68991 (in EO) to 23.3654 (in TFWO), but after using the proposed algorithm (CTFWO), it became 23.3235. Moreover, the values for voltage deviation in the 30-bus system varied from 0.12308 (in AEO) to 0.12206 (in TFWO); by using the proposed algorithm (CTFWO), it became 0.12127. For the 57-bus system variation, these values ranged from 0.60495 (in AEO) to 0.58588 (in TFWO); after using the proposed algorithm (CTFWO), it became 0.58553. From the all above results and discussions, we find that the CTFWO algorithm gives better voltage deviation and transmission power loss values than other algorithms, and that these results are also better than the results of other recently developed algorithms, such as the many modifications of the DE algorithm, PGSWT-PSO, OGSA, WCA, and GBWCA. The results that we obtained by using the proposed CTFWO algorithm are encouraging for future research. In the future, the proposed CTFWO can be used to solve ORPD problems for systems with a large number of buses, and also to study multi-objective ORPD problems.

Conflicts of Interest:
The authors declare no conflict of interest.