A Hybrid Firefly–JAYA Algorithm for the Optimal Power Flow Problem Considering Wind and Solar Power Generations

Abstract

Optimal power flow (OPF) is widely used in power systems. This problem involves adjusting variables such as online capacity, generator output, power stability, and bus voltage to reduce production costs. This paper presents HFAJAYA, a combined evolution method using the Firefly and JAYA algorithms to solve the OPF problem effectively and efficiently. While considering renewable energy, including solar energy and wind energy systems, the problem is regarded as a single-objective and multi-objective function. It considers power losses, emissions, emissions taxes, the total cost of fuel, and voltage deviation as objective functions of the problem. I have successfully implemented all simulations with different scenarios on a standard 30-bus IEEE network. A comparison of the results obtained from the HFAJAYA simulation with results from other well-known works has been undertaken to confirm the efficiency of the recommended HFAJAYA method.

1. Introduction

There is no standard and comprehensive method for solving all complex optimization problems in various fields in the engineering world. Therefore, hundreds of different approaches have been introduced in recent years to address some optimization problems effectively. One of the complicated problems in the planning of power networks is multi-constraint optimal power flow (OPF). The classical OPF problem for over 50 years has attracted the attention of researchers in this field. Over the years, the goals of optimal load dispatch have changed dramatically. These changes can be attributed to the flexible structure of electrical networks due to new equipment such as FACTS devices, storage elements, and renewable resources. The classical optimal load dispatch problem involves the distribution of active generating power of various sources (mainly thermal power plants). The cost of producing resources is minimized by considering the technical and safety constraints [1]. The complexity of the real load dispatch (due to its nonlinear, convex, and derivative nature) [1,2,3] has led researchers to use efficient algorithms to optimize the problem in the past years. As a result, numerous meta-heuristic optimization algorithms have been offered that are often inspired by common natural phenomena in the evolution of creatures and natural phenomena such as gravity and the universe, briefly reviewed as follows.

Symbiotic Organisms Search (SOS) and Moth Swarm Algorithm (MSA) for Alternating Current OPF (ACOPF) problems for different energy systems are two suggested nature-inspired techniques that have been utilized to solve OPF issues [4,5,6,7,8,9]. The following are some related works regarding the solving the OPF problems. An improved colliding bodies optimization (ICBO) [10], OPF solution incorporating wind power [11], a new parallel genetic algorithm (GA) (EPGA) [12], Jaya and modified Jaya algorithm [13,14], artificial bee colony (ABC) [15], tabu search (TS) [16], differential search algorithm (DSA) [17], TLBO algorithm enhanced with Lévy mutation (LTLBO) [18], adaptive group search optimization (AGSO) [19], a multi-objective particle swarm optimization (PSO) [20], interior search algorithm (ISA) [21], a modified bacteria foraging [22], differential evolution (DE) [23], chaotic invasive weed optimization (CIWO) [24], DE for OPF solutions incorporating stochastic wind and solar power [25], a new improved adaptive DE [26], an enhanced self-adaptive DE [27], multi-objective DE algorithm [28], the security constrained OPF with wind and thermal generators by a new hybrid algorithm [29], BAT search algorithm [30], multi-objective dynamic OPF (MODOPF) [31], an improved artificial bee colony (IABC) [32], a modified ABC algorithm [33], a chaotic ABC algorithm [34], solving OPF by means of multi-objective glowworm swarm optimization (MOGSO) which is applicable in a wind energy integrated system [35], a powerful optimizer using DE and PSO (DEPSO) algorithms [36], a sine–cosine algorithm (SCA) [37] and SCA for OPF dependent on a hydro-thermal wind arrangement of mixture power scheme [38], social spider optimization (SSO) [39], uncertainty modeling [40], ant lion algorithm (ALA) [41], grey wolf optimizer (GWO) [42], modified honey bee mating optimization (MHBMO) [43], a new hybrid algorithm using PSO [44], an improved strength Pareto evolutionary algorithm [45], a hybrid optimizer via genetic GA and TLBO algorithms (G-TLBO) [46], multi-objective electromagnetism-like algorithm (MOELA) [47], a hybrid PSO and SFLA algorithm [48], considering optimal FACTS devices in OPF by PSOGSA [49], etc.

The firefly insect algorithm [50] is one of the effective intelligent methods used in nonlinear optimizations. Xin-She Yang introduced this method in 2008, and it has achieved many successes in solving optimization problems, including autonomous mobile robots with optimized parameters [51], privacy preservation in social networks [52], clustering on different functions [53], manufacturing cell formation via a new discrete FA (DFA) to [54], etc. However, it has been observed that this algorithm for complex nonlinear problems sometimes becomes stuck in the local solution and loses its optimization power [51,52,53,55].

Alternatively, evolutionary algorithms can escape local optima but are slow to converge. I developed HFAJAYA, a hybrid evolutionary algorithm to optimize power flow problems to resolve this problem. According to the simulation results, the current proposed hybridization approach significantly improves the ability of the JAYA and FA algorithms to escape the local optimum due to their rapid convergence speed.

This study is continued in four sections as the OPF problems formulation is introduced in Section 2. The concepts and structure of FA, JAYA, and the proposed HFAJAYA algorithm are introduced in Section 3. The outcomes of the simulations and a discussion of the findings are reported in Section 4. Lastly, Section 5 details the outcomes of the paper.

2. Problem Formulation

In the case of optimal load dispatch, the values of some or all of the control variables need to be specified to optimize (minimize or maximize) a predetermined objective function. The quality of a solution depends on the model’s accuracy under study. In general, an OPF optimization problem can be defined as follows [9]:

$$\mathrm{Min} J\left(x, u\right)$$

(1)

$$\mathrm{Subject} \mathrm{to}: g\left(x, u\right)=0$$

(2)

$$h\left(x, u\right) \le 0$$

(3)

where x is the vector of the state variables (dependent variables). This vector includes the power of the reference generator, the voltage of the bus bar, the reactive power of the generators, and the load of the transmission lines. Therefore, x can be expressed as follows:

$$x^T=\left[P_{G_1},V_{L_1},\dots ,V_{L_{NPQ}},Q_{G_1},\dots ,Q_{G_{NG}},S_{l_1},\dots ,S_{l_{NTL}}\right]$$

(4)

where NPQ, NTL, NG, are the number of load bus bars, the number of transmission lines, and the number of generators, respectively.

In addition, u is the vector of the decision variables. The primary control variables commonly used for this purpose include the generators’ output voltage and active power, the transformers’ settings, and the capacity of the reactive power compensators, which can be expressed as Equation (5):

$$u^T=\left[P_{G_2},\dots ,P_{G_{NG}},V_{G_1},\dots ,V_{G_{NG}},Q_{C_1},\dots ,Q_{C_{NC}},T_1,\dots ,T_{NT}\right]$$

(5)

where NC and NT are the number of generators and the number of reactive power compensators.

2.1. OPF Constraints

This problem includes various inequality and equality constraints related to systems and units, which are as follows.

2.1.1. OPF Equality Constraints

Load dispatch equations present these constraints of grid powers according to Formulas (6) and (7). The real and reactive power supply constraints of network buses are defined as follows [9]:

$$P_{Gi}-P_{Di}=V_i{\displaystyle \sum }_{j=1}^{NB}{V}_j\left(G_{ij}\cos {\delta }_{ij}+B_{ij}\sin {\delta }_{ij}\right)$$

(6)

$$Q_{Gi}-Q_{Di}=V_i{\displaystyle \sum }_{j=1}^{NB}{V}_j\left(G_{ij}\sin {\delta }_{ij}-B_{ij}\cos {\delta }_{ij}\right)$$

(7)

In the above equations, i and j are the indices of bus numbers; V_i and V_j represent the voltage magnitudes; P_Gi and Q_Gi are the real and reactive power outputs of power generators, respectively; Q_Di and P_Di are the reactive and real load demands, respectively; B_ij, G_ij are the susceptance and conductance of the branch between buses i and j, respectively; δ_ij is the difference between the phase angles of voltages between these buses; and NB denotes the number of buses.

2.1.2. OPF Inequality Constraints

The inequality constraints, $h\left(x,u\right)$, represent the constraints on all the network variables:

• Constraints relating to the generation units: for all the generators, the voltage, as well as the reactive and active powers, should be defined in upper and lower limits as follows:

$$\begin{gathered} P_{Gi}^{\min }\le P_{Gi}^{}\le P_{Gi}^{\max } \\ {Q}_{Gi}^{\min }\le Q_{Gi}^{}\le Q_{Gi}^{\max } \\ {V}_{Gi}^{\min }\le V_{Gi}^{}\le V_{Gi}^{\max } \end{gathered}$$

(8)

where $V_{Gi}^{\min }$and $V_{Gi}^{\max }$ represent the allowable range limits of the voltage magnitude of the ith generator; $P_{Gi}^{\min }$ and $P_{Gi}^{\max }$ represent the acceptable range of actual power output of ith generator; and $Q_{Gi}^{\min }$and $Q_{Gi}^{\max }$ describe the permitted limit of the reactive power output of the ith generator.

• Transformers’ tap adjustments: transformers’ taps can be altered in their approved limitation as in [13]:

$$T_i^{\min }\le T_i\le T_i^{\max }; \forall i=1,\dots , NT$$

(9)

where $T_i^{\min }$ and $T_i^{\max }$ express the acceptable range for altering the ith transformer’s tap.

• Constraints of VAR compensating units: the productions of these units are constrained in the following way [13]:

$$Q_{Ci}^{\min }\le Q_{Ci}\le Q_{Ci}^{\max }; \forall i=1,\dots , NC$$

(10)

where $Q_{Ci}^{\min }$and $Q_{Ci}^{\max }$are the allowable range of VAR injection of ith compensating unit.

• Voltage limits: the acceptable range of load bus voltage magnitudes are known as a security constraint [13]:

$$V_{Li}^{\min }\le V_{Li}\le V_{Li}^{\max }$$

(11)

The maximum and minimum ranges of voltage magnitudes for the ith load bus signify by $V_{Li}^{\min }$and $V_{Li}^{\max }$, respectively.

• Transmission lines constraints: the apparent power flows across branches should be restricted by:

$$S_{li}\le S_{li}^{\max }$$

(12)

Moreover, the apparent power via an ith transmission line and its higher bound are represented by $S_{li}$ and $S_{li}^{\max }$, respectively.

It is worth noting that the optimization process considers the allowable ranges of the decision variables. The penalty factors approach, which transforms a constrained optimization problem into an unconstrained one by applying penalized values of violated constraints to the objective function, is one of the most common practices for dealing with constrained problems, making an enlarged objective function [54]. This process is shown here as [25]:

$$\mathrm{J}={\displaystyle \sum }_{i=1}^{NG}{F}_i\left(P_{Gi}\right)+{\lambda }_P{\left(P_{G1}-P_{G1}^{\lim }\right)}^2+{\lambda }_Q{\displaystyle \sum }_{i=1}^{NG}{\left(Q_{Gi}-Q_{Gi}^{\lim }\right)}^2+{\lambda }_V{\displaystyle \sum }_{i=1}^{NPQ}{\left(V_{Li}-V_{Li}^{\lim }\right)}^2+{\lambda }_S{\displaystyle \sum }_{i=1}^{NTL}{\left(S_{li}-S_{li}^{\lim }\right)}^2$$

(13)

where λ_P, λ_V, λ_Q, and λ_S are the penalty factors; and the supplementary variable u^lim is given as [25]:

$$u^{lim}=\left\{\begin{array}{c}u; u^{min}\le u\le u^{max}\\ u^{min}; u\le u^{min}\\ u^{max}; u\ge u^{max}\end{array}\right.$$

(14)

3. Proposed Mix Optimizer

In the first step, the two algorithms FA and JAYA, are reviewed. Then the proposed HFAJAYA algorithm is described and formulated.

3.1. Firefly Algorithm (FA)

FA [50] is one of the most powerful and creative algorithms inspired by firefly insect behavior and flashing light. There are roughly 2000 different kinds of fireflies, and most of them generate brief, repetitive flashes. Bioluminescence produces the flashing light, and the pattern of flashes is frequently unique to a species. However, two primary purposes of such flashes are to reach prospective prey and attract powerful mates. Furthermore, flashing can be used as a safety warning system. The signal system that draws both sexes together includes the rhythmic flash, the pace of flashing, and the length of time. Females react to a male’s distinctive flashing pattern within the same kind. In contrast, female fireflies of different kinds, such as photuris, can replicate the mating flashing pattern to attract and consume male fireflies who may misinterpret the flashes as an appropriate prospective partner.

The inverse square law governs the light intensity at a given distance r from the light source. In other words, as the distance r rises, the light intensity I drops in terms of I∝1/r².

The following three simplified principles are used to describe the Firefly Algorithm (FA) for clarity: (1) all populations are not distinguished based on sex, which means that one firefly will be captivated by another firefly irrespective of gender; (2) sexual attraction is proportionate to light, thus if two flashing fireflies are present, the less brilliant one will go towards the brighter one.

The change in light intensity and attraction construction are two significant concerns in the firefly algorithm. With the help of the above information, the FA rules are:

In the simplest form, the firefly’s brightness I of a firefly at a specific point x may be written as I(x) ∝ f(x). Nevertheless, attractiveness β is not absolute; it is better to be assessed by other fireflies or seen in the eyes of the beholder, as follows:

$$I\left(r\right)=\frac{I_s}{r^2}$$

(15)

The light intensity I alters with the distance r for an assumed medium with a permanent light absorption coefficient γ:

$$I=I_0{e}^{-\gamma r}$$

(16)

Therefore, a function that considers both the effects of absorption and distance can be approximated as follows:

$$I\left(r\right)=I_0{e}^{-\gamma r^2}$$

(17)

Furthermore, due to the direct relationship between the attractiveness and flashing of each firefly, we have:

$$\beta ={\beta }_0{e}^{-\gamma r^2}$$

(18)

In order to reduce the fluctuation rate in the attractiveness function and for faster calculations, this function can be approximated by:

$$\begin{gathered} e^{-\gamma r^2}\approx 1-\gamma r^2+\frac{1}{2}{\gamma }^2{r}^4+\dots , \\ \frac{1}{1+\gamma r^2}\approx 1-\gamma r^2+{\gamma }^2{r}^4+\dots , \\ \beta =\frac{{\beta }_0}{1+\gamma r^2} \end{gathered}$$

(19)

These two above equations have a characteristic distance $\Gamma =1/\sqrt{\gamma }$, in which the exponential function changes from ${\beta }_0$ to ${\beta }_0{e}^{-1}$and in the second function from ${\beta }_0$ to ${\beta }_0/2$. The actual form of the attractiveness function in the implementation can be any declining function depending on the type of problem.

In this case, the characteristic distance can be as follows:

$$\beta \left(r\right)={\beta }_0{e}^{-\gamma r^m},\left(m\ge 1\right)$$

(20)

$$\mathsf{\Gamma }={\gamma }^{-\frac{1}{m}}\to 1,\left(m\to \infty \right)$$

(21)

However, if this characteristic distance is known in an optimization problem, the attractiveness coefficient can be easily defined. Finally, the length between any two populations, in FA, i and j at x_i and x_j is:

$$r_{ij}=||x_i-x_j||=\sqrt{{\displaystyle \sum }_{k=1}^d{\left(x_{i,k}-x_{j,k}\right)}^2}$$

(22)

When an ith firefly is fascinated by a more brilliant one (jth), its movement is controlled by:

$$x_i=x_i+{\beta }_0{e}^{-\gamma r_{ij}^2}\left(x_j-x_i\right)+\alpha {\in }_i$$

(23)

Another rule is owing to attraction, whereas the third rule is randomness, with α as the randomized parameter, and ${\in }_i$ is a uniformly distributed random number in the range [0, 1]. In most applications, we can consider ${\beta }_0=1$ and α ∈ [0, 1] [50].

It should be noted that the third term actually refers to a random step towards a brighter firefly, so if it is equal to zero, the firefly takes a simple random step.

3.2. JAYA Algorithm

The JAYA method introduced by Rao in 2016 [56] is a recently suggested algorithm for real-world optimization problems, and the optimization process of the original JAYA algorithm is shown in Figure 1. It is a robust, efficient, and free-parameters optimizer. Each member of the whole population (N) has its position in the tth iteration. $X_k^i$ (k = 1: N) is explained by the parameters of the problem in a d-dimensional solution exploration area, $X_k^i=\left[x_{1,k}^i,x_{2,k}^i,\dots ,x_{d,k}^i\right]$. The new location value for kth member, $X_k^{i+1}=\left[x_{1,k}^{i+1},x_{2,k}^{i+1},\dots ,x_{d,k}^{i+1}\right]$ (for $X_k^i$) is attained by updating the positions in repetitions; if $f\left(X_k^{i+1}\right)\le f\left(X_k^i\right)$, the new location value ($X_k^{i+1}$ ) is substituted by the new position ($X_k^i$) by means of below relation [56]:

$$X_k^{i+1}=X_k^i+{\mathrm{rand}}_1^i\left(X_{\mathrm{best}}^i-\left|X_k^i\right|\right)-{\mathrm{rand}}_2^i\left(X_{\mathrm{worst}}^i-\left|X_k^i\right|\right)$$

(24)

where$f\left(X_k^{i+1}\right)$ is the objective function value for $X_k^{i+1}$, $X_{\mathrm{best}}^i=\left[x_{1,\mathrm{best}}^i,x_{2,\mathrm{best}}^i,\dots ,x_{d,\mathrm{best}}^i\right]$ and $X_{\mathrm{worst}}^i=\left[x_{1,\mathrm{worst}}^i,x_{2,\mathrm{worst}}^i,\dots ,x_{d,\mathrm{worst}}^i\right]$ are the best solutions obtained until the ith iteration of the algorithm, respectively. ${\mathrm{rand}}_1^i=\left[{\mathrm{rand}}_{1,1}^i,{ \mathrm{rand}}_{1,2}^i,\dots ,{ \mathrm{rand}}_{1,d}^i\right]$and ${\mathrm{rand}}_2^i=\left[{\mathrm{rand}}_{2,1}^i,{ \mathrm{rand}}_{2,2}^i,\dots ,{ \mathrm{rand}}_{2,d}^i\right]$ are both the random values in the range [0, 1] in the ith iteration of the algorithm. Moreover, $\left|X_k^i\right|$is the absolute value of $X_k^i$.

Figure 1. The optimization process of the original JAYA algorithm.

3.3. The Proposed HFAJAYA Optimization Algorithm

As mentioned above, in engineering, we encounter a diversity of complex problems. One of the effective ways to reach a range of acceptable solutions for these problems is to use a combination of two algorithms with different structures and functions [23].

Therefore, in this paper, we used a combination of two powerful and well-known algorithms, FA and JAYA, to achieve an optimal solution to our desired engineering problem, such as the optimal power flow. This method is shown in Figure 2. In the first modification, only the members with a better fitness merit than the ith member (e.g., jth member) can affect its movement. This can speed up the algorithm convergence rate and provide more opportunities to search. In the second modification, two ranges are defined regarding the objective function of the ith member and other members (e.g., jth member). In other words, if $F\left(x_j^t\right)\le 0.9*F\left(x_i^t\right)$, then the jth member is directed to the FA algorithm; otherwise, the jth member is directed to the JAYA algorithm. When the objective function value of the jth member is very close to the objective function of the ith member, i.e., $\left(x_j-x_i\right)$ tends to zero in the FA algorithm. The optimization process is assigned to the JAYA algorithm, which can help avoid trapping in the local optimum.

Figure 2. The optimization process of the HFAJAYA optimizer.

In FA, three popular test functions were designated. The optimization process was performed in two modes: dimension 30 with a population of 25 and a maximum value of iteration 400 and dimension 100 with a population of 75 and a maximum value of iteration 1200. For each function, 30 independent runs were carried out by each algorithm. The simulation results for FA and HFAJAYA are given in Table 1, showing that HFAJAYA is a more robust and more suitable algorithm than the original one. The convergence characteristics of the algorithms for the test functions 1 and 2 are also given in Figure 3. The suggested approach had a reasonable and operative convergence speed.

Table 1. Optimization results for three selected popular functions by the algorithms.

Function Limits	Mathematical Model Property	D = 30		D = 100
		FA	HFAJAYA	FA	HFAJAYA
		Mean Best Std	Mean Best Std	Mean Best Std	Mean Best Std
Rosenbrock [−32, 32]	$f_1={\displaystyle \sum }_{i=1}^{D-1}(100{(x_i^2-x_{i+1})}^2+\left(x_i-1{)}^2\right)$ Unimodal	56.7494 23.2999 81.0251	45.5246 0.1357 29.1884	26.3190 23.8039 10.8648	17.8910 15.8155 1.1293
Sphere [−100, 100]	$f_2={\displaystyle \sum }_{i=1}^D{x}_i^2$ Unimodal	2.7196×10−6 1.9314×10−6 3.5034×10−7	4.3637 × 10−15 7.3028 × 10−17 8.5226 × 10−15	1.4702×10–19 1.3292×10−19 8.1340×10−21	2.6643 × 10−;20 5.9287 × 10−21 2.0183 × 10−20
Rastrign’s [−5.12, 512]	$f_3={\displaystyle \sum }_{i=1}^D\left(x_i^2-10\cos (2\pi x_i\right)+10)$ Multimodal	20.5293 6.9647 6.0745	43.5562 4.7033 13.5772	34.9728 19.8992 11.8223	30.8437 17.8841 8.2046

Figure 3. Convergence of some optimization test functions by the proposed algorithm.

4. HFAJAYA for the Multi-Constraint OPF Problems

In order to show the power of the proposed algorithm, various problems of optimal load dispatch were considered to obtain the optimal solutions for each problem. We selected a common and well-known power system, the IEEE 30-bus power system, and performed the simulation operation. The population size and iterations were selected at 50 and 500, respectively. The simulation study was performed in MATLAB software. The generating units’ boundaries, cost coefficients, and the emission cost coefficients are given in Table A1, Table A2 and Table A3 in the Appendix A. As per each metaheuristic algorithm, the initial solutions were randomly generated between the lower and upper limits.

Eleven instances were examined to determine the effectiveness of HFAJAYA.

• The considered cases for the IEEE 30-bus system:

Case 1: Minimizing the fuel cost based on [9];

Case 2: Minimizing the piecewise quadratic fuel cost functions based on [9];

Case 3: Minimizing the fuel cost considering valve point effects (VPEs) based on [9];

Case 4: Minimizing the fuel cost and real power loss based on [9];

Case 5: Minimizing the fuel cost and voltage deviation based on [9];

Case 6: Minimizing the fuel cost, emissions, voltage deviation, and losses based on [9];

Case 7: Minimization of the generation cost, including haphazard solar and wind power based on [25];

Case 8: Minimization of generation cost incorporating stochastic wind and solar power with the carbon tax based on [25].

4.1. OPF without Stochastic Renewable Energy

The generator information, bus data, and bounded ranges of decision variables for the IEEE 30-bus network were the same as reported in [57,58]. As demonstrated in Figure 4, six generators were located on buses 1, 2, 5, 8, 11, and 13 [58]. In contrast, four transformers with off-nominal tap ratios were located on lines 6–9, 6–10, 4–12, and 28–27. At a base capacity of 100 MVA, the total network demand was 2.834 p.u. All the load buses’ highest and lowest voltages were adjusted to 1.05–0.95 in p.u., respectively.

Figure 4. Single line diagram of the IEEE 30-bus test system.

Table 2 in the additional material contains all HFAJAYA’s results for six situations of the 30-bus power system.

Table 2. The optimal values for the six cases by HFAJAYA.

Variables Optimal Values	Limits		The Cases
	Min	Max	1	2	3	4	5	6
PG1 (MW)	50	250	177.0297	139.9985	198.7328	102.6069	176.3325	126.7899
PG2 (MW)	20	80	48.7224	54.9972	44.8834	55.5590	48.8232	52.0824
PG5 (MW)	15	50	21.3825	24.1391	18.4734	38.1133	21.6338	30.2948
PG8 (MW)	10	35	21.2881	4.9826	10.0000	35.0000	22.2391	34.9995
PG11 (MW)	10	30	11.9890	18.6929	10.0001	30.0000	12.2210	25.0682
PG13 (MW)	12	40	12.0017	17.3272	12.0000	26.6501	12.0000	20.0045
VG1 (p.u.)	0.95	1.1	1.0836	1.0735	1.0816	1.0698	1.0423	1.0737
VG2 (p.u.)	0.95	1.1	1.0605	1.0565	1.0579	1.0576	1.0228	1.0574
VG5 (p.u.)	0.95	1.1	1.0337	1.0298	1.0302	1.0359	1.0145	1.0323
VG8 (p.u.)	0.95	1.1	1.0382	1.0405	1.0371	1.0438	1.0055	1.0406
VG11 (p.u.)	0.95	1.1	1.0991	1.0915	1.0995	1.0834	1.0733	1.0405
VG13 (p.u.)	0.95	1.1	1.0519	1.0650	1.0633	1.0573	0.9873	1.0242
T6–9 (p.u.)	0.9	1.1	1.0713	1.0616	1.0417	1.0653	1.0999	1.1000
T6–10 (p.u.)	0.9	1.1	0.9183	0.9112	0.9692	0.9205	0.9000	0.9503
T4–12 (p.u.)	0.9	1.1	0.9774	1.0028	0.9951	0.9901	0.9381	1.0334
T28–27(p.u.)	0.9	1.1	0.9743	0.9743	0.9776	0.9750	0.9710	1.0048
QC10 (MVAR)	0.0	5.0	2.6745	1.1882	4.7749	4.9569	4.9638	3.1583
QC12 (MVAR)	0.0	5.0	1.2842	1.2617	1.9178	0.1636	0.0080	0.0291
QC15 (MVAR)	0.0	5.0	4.2681	4.7107	3.7824	4.4512	5.0000	3.9138
QC17 (MVAR)	0.0	5.0	5.0000	4.2344	4.5810	5.0000	0	4.9986
QC20 (MVAR)	0.0	5.0	4.2732	4.2153	4.3853	4.2168	5.0000	4.9972
QC21 (MVAR)	0.0	5.0	4.9997	4.9164	4.9552	5.0000	5.0000	4.9994
QC23 (MVAR)	0.0	5.0	3.4064	3.0390	2.8829	3.2778	5.0000	4.3156
QC24 (MVAR)	0.0	5.0	4.9966	4.8430	4.9368	5.0000	5.0000	4.9969
QC29 (MVAR)	0.0	5.0	2.6155	2.5157	2.6963	2.5602	2.6384	2.6202
Cost (USD/h)	-	-	800.4800	646.5020	832.1798	859.0165	803.7036	825.0311
Emission (t/h)	-	-	0.3659	0.2835	0.4378	0.2289	0.3638	0.2597
Power losses (MW)	-	-	9.0134	6.7375	10.6897	4.5293	9.8496	5.8393
V.D. (p.u.)	-	-	0.9047	0.9041	0.8578	0.9327	0.0948	0.2956

4.1.1. Case 1

The first target function, which was solitary, was to minimize the power generation’s overall fuel cost, which may be expressed using a quadratic curve the below [18]:

$$J_1={\displaystyle \sum }_{i=1}^{NG}{F}_i\left(P_{Gi}\right)={\displaystyle \sum }_{i=1}^{NG}\left(a_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)$$

(25)

where P_Gi and F_i are the output and fuel cost of the ith generator. Moreover, $a_i$, $b_i$ , and $c_i$ are the cost coefficients of the ith unit, respectively. Finally, NG is the whole quantity of generators. In [59], the cost coefficient values were accessible. The results of this case, as given in Table 3, indicates that the proposed method can achieve better solution in compared to the state-of-the-art algorithms.

Table 3. The obtained optimal results in the current works for Case 1.

Optimizer	Fuel Cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
PSOGSA [58]	800.49859	-	9.0339	0.12674
JAYA [13]	800.4794	-	9.06481	0.1273
MSA [9]	800.5099	0.36645	9.0345	0.90357
ARCBBO [59]	800.5159	0.3663	9.0255	0.8867
FPA [9]	802.7983	0.35959	9.5406	0.36788
GWO [42]	801.41	-	9.30	-
IEP [60]	802.46	-	-	-
TS [16]	802.29	-	-	-
MICA-TLA [61]	801.0488	-	9.1895	-
MGBICA [62]	801.1409	0.3296	-	-
ABC [17]	800.660	0.365141	9.0328	0.9209
SFLA-SA [63]	801.79	-	-	-
MFO [9]	800.6863	0.36849	9.1492	0.75768
MHBMO [42]	801.985	-	9.49	-
AGSO [20]	801.75	0.3703	-	-
DE [64]	802.39	-	9.466	-
SKH [65]	800.5141	0.3662	9.0282	-
EP [66]	803.57	-	-	-
MPSO-SFLA [48]	801.75	-	9.54	-
FA	800.7502	0.36532	9.0219	0.9205
HFAJAYA	800.4800	0.3659	9.0134	0.9047

4.1.2. Case 2

Thermal generators may use various fuel sources, including oil, natural gas, and coal, depending on the power system’s operation circumstances. As a result, the theoretical analysis for these generators’ fuel cost curves (units 1 and 2) may be considered a collection of restrictions [18].

$$F\left(P_{Gi}\right)=\left\{\begin{array}{l}{a}_{i1}+b_{i1}{P}_{Gi}+c_{i1}{P}_{Gi}^2; P_{Gi}^{min}\le P_{Gi}\le P_{Gi1}\\ a_{i2}+b_{i2}{P}_{Gi}+c_{i2}{P}_{Gi}^2; P_{Gi1}\le P_{Gi}\le P_{Gi2}\\ \dots \\ a_{ik}+b_{ik}{P}_{Gi}+c_{ik}{P}_{Gi}^2; P_{Gik-1}\le P_{Gi}\le P_{Gi}^{max}\end{array}\right.$$

(26)

The ith generator’s cost coefficients are characterized by a_ik, b_ik, and c_ik for the k-th fuel type. The other single fuel source units’ cost coefficients are unchanged and have similar values as scenario 1. As a result, the goal function for modeling the gasoline cost features can be depicted as:

$$J_2=\left({\displaystyle \sum }_{i=1}^2{a}_{ik}+b_{ik}{P}_{Gi}+c_{ik}{P}_{Gi}^2\right)+\left({\displaystyle \sum }_{i=3}^{NG}{a}_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)$$

(27)

It is important to note that the optimization technique was chosen. The unit data were linked to the type of objective function and the restrictions [41]. Table 4 compares the minimal fuel cost (USD/h), emission (t/h), power losses (MW), and V.D. (p.u.) of the FA and HFAJAYA algorithms and gives a comparison study between these results and the outcomes reported in the recent literature. It can be observed that HFAJAYA had a suitable optimization influence compared to the other algorithms.

Table 4. The obtained optimal results in the current works for Case 2.

Optimizer	Fuel Cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
FA	647.2612	0.2835	6.7302	0.8915
HFAJAYA	646.5020	0.2835	6.7375	0.9041
LTLBO [18]	647.4315	0.2835	6.9347	0.8896
MSA [9]	646.8364	0.28352	6.8001	0.84479
MICA-TLA [61]	647.1002	-	6.8945	-
MPSO-SFLA [48]	647.55	-	-	-
MDE [64]	647.846	-	7.095	-
GABC [67]	647.03	-	6.8160	0.8010
FPA [9]	651.3768	0.28083	7.2355	0.31259
SSO [39]	663.3518	-	-	-
MFO [9]	649.2727	0.28336	7.2293	0.47024
IEP [60]	649.312	-	-	-

4.1.3. Case 3

The VPEs were represented as an entire sinusoidal function applied to the cost features of the units under consideration in this example, as follows [42]:

$$J_3={\displaystyle \sum }_{i=1}^{NG\sum }{a}_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2+{\displaystyle \sum }_{i=1}^{NG}\left|d_i\sin \left[e_i\left(P_{Gi}^{\min }-P_{Gi}\right)\right]\right|$$

(28)

where $d_i$ is the VPE cost coefficient and $P_{Gi}^{\min }$ is the lower limit of the generating unit i. Table 5 displays the results produced by the HFAJAYA algorithm compared to the solutions described in various prior works, demonstrating the efficacy of the proposed HFAJAYA method.

Table 5. The obtained optimal results in the current works for Case 3.

Optimizer	Fuel Cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
FA	832.5596	0.4372	10.6823	0.8539
HFAJAYA	832.1798	0.4378	10.6897	0.8578
SP-DE [57]	832.4813	0.43651	10.6762	0.75042
PSO [10]	832.6871	-	-	-

4.1.4. Case 4

This case aimed to reduce both fuel costs and network losses. The objective function may be written like this:

$$J_4={\displaystyle \sum }_{i=1}^{NG}{a}_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2+\lambda p*P_{Loss}$$

(29)

where the value of 𝜆p is selected 40, similar to [9]. The power losses, $P_{Loss},$ are defined as:

$$P_{Loss}={\displaystyle \sum }_{\begin{array}{c}k=1\\ k=\left(i,j\right)\end{array}}^{NTL}{g}_k(V_i^2+V_j^2-2V_i{V}_j\cos {\delta }_{ij})$$

(30)

In (30), the overall transmission network active losses is P_Loss, the conductance of branch k is g_k, $V_i$ and $V_j$ express the voltages of ith and jth buses, respectively, NTL represents the value of transmission lines, NPQ shows the value of PQ buses, and δ_ij phase shift of voltages between the ith and jth buses.

Based on the findings in Table 6, it can be concluded that the suggested HFAJAYA outperformed the original FA and the two other approaches.

Table 6. The obtained optimal results in the current works for Case 4.

Optimizer	Fuel cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
FA	859.8325	0.2289	4.5398	0.9330
HFAJAYA	859.0165	0.2289	4.5293	0.9327
SF-DE [57]	859.1458	0.2289	4.5245	0.92731
MSA [9]	859.1915	0.2289	4.5404	0.92852

4.1.5. Case 5

The voltage profile is the essential metric for determining the quality of a system’s service. The voltage profile can be improved by lowering the load bus voltage’s divergence from unity. A viable solution with undesired voltage deviations is obtained when only a cost-based goal function is considered. As a result, a multiple objective OPF problem to concurrently minimize voltage deviations (VD) and fuel cost is presented, which may be described as follows.

$$J_5={\displaystyle \sum }_{i=1}^{NG}\left(a_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)+\lambda v*{\displaystyle \sum }_{i=1}^{NPQ}\left|V_i-1.0\right|$$

(31)

where the value of factor 𝜆v was selected as 100 [57].

In Table 7, the minimal fuel cost (USD/h) and V.D. (p.u.) of the FA and HFAJAYA algorithms are examined, and the findings are compared to those described in the current works. The table’s findings demonstrate the strength of HFAJAYA, which successfully found appropriate solutions for this situation.

Table 7. The obtained optimal results in the current works for Case 5.

Optimizer	Fuel Cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
FA	804.9733	0.3637	9.8837	0.0950
HFAJAYA	803.7036	0.3638	9.8496	0.0948
MOMICA [68]	804.9611	0.3552	9.8212	0.0952
ECHT-DE [57]	803.7198	0.36384	9.8414	0.09454
BB-MOPSO [68]	804.9639	-	-	0.1021
MFO [9]	803.7911	0.36355	9.8685	0.10563
MNSGA-II [68]	805.0076	-	-	0.0989
MPSO [9]	803.9787	0.3636	9.9242	0.1202

4.1.6. Case 6

This case, similar to MSA [9] and MOMICA [68], included targets to minimize four conflicting objectives at the same time: emission, losses, fuel cost and voltage deviations.

$$J_6={\displaystyle \sum }_{i=1}^{NG}\left(a_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2\right)+\lambda v*{\displaystyle \sum }_{i=1}^{NPQ}\left|V_i-1.0\right|+\lambda p*P_{Loss}+\lambda e*{\displaystyle \sum }_{i=1}^{NG}{F}_{Ei}\left(P_{Gi}\right)$$

(32)

To balance the problem’s many objectives, the weight factors were used as in [57] with 𝜆v = 21, 𝜆p = 22 and 𝜆e = 19. The emission function is examined in this article as follows:

$$F_E={\displaystyle \sum }_{i=1}^{NG}{F}_{Ei}\left(P_{Gi}\right)={\displaystyle \sum }_{i=1}^{NG}\left({\alpha }_i+{\beta }_i{P}_{Gi}+{\gamma }_i{P}_{Gi}^2+{\xi }_i\exp ({\lambda }_i{P}_{Gi})\right)$$

(33)

where F_Ci characterizes the fuel cost. In addition, the F_Ei signifies the emission,${\gamma }_i$, ${\beta }_i$, ${\xi }_i$ and ${\lambda }_i$ show the emission coefficients of the ith generator, while ${\gamma }_i$ (ton/h MW²), ${\beta }_i$ (ton/h MW), ${\alpha }_i$ (ton/h) are related to SOX, and ${\xi }_i$ (ton/h), ${\lambda }_i$ (1/MW) are related to NOX, respectively.

The suggested method is an efficient and dependable approach for the multiple objective OPF in power systems, based on the findings compared in Table 8.

Table 8. The obtained optimal results in the current works for Case 6.

Algorithm	Fuel Cost (USD/h)	Emission (t/h)	Power Losses (MW)	V.D. (p.u.)
FA	828.1568	0.2602	5.9015	0.3137
HFAJAYA	825.0311	0.2597	5.8393	0.2956
MOMICA [68]	830.1884	0.2523	5.5851	0.2978
MFO [9]	830.9135	0.25231	5.5971	0.33164
MNSGA-II [68]	834.5616	0.2527	5.6606	0.4308
MSA [9]	830.639	0.25258	5.6219	0.29385
BB-MOPSO [68]	833.0345	0.2479	5.6504	0.3945

4.2. OPF Solutions, Including Stochastic Solar and Wind Power

4.2.1. The Mathematical Modeling of Generating Units, Including Solar and Wind Power

Renewable Energy Sources (RES) have been recognized for improving the steadiness and quality of the power grid. They have had a direct influence on the energy market. To identify the best power generation flow for a power grid using renewable energy sources while meeting a cost function, such as reducing greenhouse gas discharges, it is critical to boost the injected power from RES as much as possible. In a power grid with renewable energy sources, an OPF problem can be expressed as an optimization task with single or numerous objective functions. Nevertheless, renewable energy sources are intrinsically stochastic due to the extremely variable nature of meteorological conditions. As a result, the probability model is necessary to effectively address the OPF subject while coping with the significant reservations associated with RES. This study connected renewable energy sources to an IEEE 30-bus test system in several geographical situations. Solar and wind energy generation play an important part in OPF.

Furthermore, the stochastic character of RES outputs is exacerbated by the unreliability of weather conditions, posing several energy issues such as risk operation management. We used probabilistic estimate techniques to predict RES power generation in this study to accommodate this and to reduce the influence of stochastic power-generating behavior. When RESs delivered real power, the stochastic software design approach presented here was used to construct RES power profiles. To bring the RES into the OPF, power profiles from RES are used as a negative load. This means that the PV and wind power generators will deliver electricity to loads first. The thermal generators cover the remaining loads and network losses.

4.2.2. The Units of Wind Power

An upcoming wind energy profile must be anticipated to create a working optimization approach for addressing OPF issues. The predictors in this work were generated using a Weibull probability distribution function [6,7,8]. Before establishing the optimal solution technique, the wind energy estimating work can be completed independently. The wind power generation model is often built using wind speed variables. In this part, the wind speed f_v(v) is provided and simulated mathematically using the Weibull probability distribution function basis of dimensionless form factor (K) and step size (c) [25]:

$$f_{\nu }\left(\nu \right)=\frac{k}{c}{\left(\frac{\nu }{c}\right)}^{k-1}\times e^{-{\left(\frac{\nu }{c}\right)}^k}$$

(34)

As expressed in (35), Γ(x) is mainly accountable for the average of the Weibull probability distribution (M_wbl), as illustrated in Formula (36):

$$\Gamma \left(x\right)={{\displaystyle \int }}_0^{\infty }{e}^{-t}{t}^{x-1}dt$$

(35)

$$M_{wbl}=c*\Gamma \left(1+K^{-1}\right)$$

(36)

Generally, the wind turbine may transform the moving energy of the wind into electrical energy. Equation (37) shows the real production power of a wind turbine, P_w(v), as a function of wind speed:

$$P_w\left(\nu \right)=\left\{\begin{array}{l}0; \nu \le {\nu }_{in} \mathrm{and} \nu >{\nu }_{out}\\ P_{wr}\left(\frac{\nu -{\nu }_{in}}{{\nu }_r-{\nu }_{in}}\right); {\nu }_{in}<\nu \le {\nu }_r\\ P_{wr}; {\nu }_r<\nu \le {\nu }_{out}\end{array}\right.$$

(37)

where $P_{wr}$is the wind turbine’s rated power, the wind turbine’s cut-in wind rate is v_in, v_out is the cut-out wind rate, and v_r is the valued wind speed. The Weibull probability distribution was used in this study to include a stochastic procedure in the optimization approach. The Weibull probability distribution estimate offers the doubt findings for wind energy generation. The cost of wind power production units must be determined to reduce the power generation whole cost. The entire cost of a wind power generator is evaluated based on flow rate and real power (produced by wind turbines) to decrease the impact of unpredictability in wind power patterns. Equations (38)–(40) compute the direct, reserve, and penalty expenses in (USD/h), respectively. Equation (41) describes the overall cost of wind power generation in (USD/h), $C_W^T$, which comprises three primary items: straight wind turbine cost, reserve or overestimation, and penalty costs [25,44]

$$C_{w,j}\left(P_{ws,j}\right)=g_j{P}_{ws,j}$$

(38)

$$\begin{gathered} C_{Rw,j}\left(P_{ws,j}-P_{wav,j}\right)=K_{Rw,j}\left(P_{ws,j}-P_{wav,j}\right) \\ =K_{Rw,j}{{\displaystyle \int }}_0^{P_{ws,j}}\left(P_{ws,j}-P_{w,j}\right)f_w\left(P_{w,j}\right)d{p}_{w,j} \end{gathered}$$

(39)

$$\begin{gathered} C_{Pw,j}\left(P_{wav,j}-P_{ws,j}\right)=K_{Pw,j}\left(P_{wav,j}-P_{ws,j}\right) \\ =K_{Pw,j}{{\displaystyle \int }}_{P_{ws,j}}^{P_{wr,j}}\left(P_{w,j}-P_{ws,j}\right)f_w\left(P_{w,j}\right)d{p}_{w,j} \end{gathered}$$

(40)

$$C_W^T={\displaystyle \sum }_{j=1}^{N_W}\left[C_{w,j}\left(P_{ws,j}\right)+C_{Rw,j}\left(P_{ws,j}-P_{wav,j}\right)+C_{Pw,j}\left(P_{wav,j}-P_{ws,j}\right)\right]$$

(41)

where $g_j$ is the direct cost coefficient associated with jth wind power plant, $P_{ws,j}$ is the scheduled power from the same plant, $K_{Rw,j}$ indicates the reserve cost coefficient pertaining to jth wind power plant, $P_{wav,j}$ defines the actual available power from the same plant, $K_{Pw,j}$ is the penalty cost coefficient for the jth wind power plant, and $P_{wr,j}$ is rated output power from the same windfarm.

Wind energy providers generally supply power network operators with an expected power generation profile. The network operator creates an operation plan for all producing units based on the expected wind power output to fulfill the demand. Suppose the actual wind power output is less than the predicted value. In that case, the reserve cost is applied to offset the anticipated value. A penalty fee is applied if the actual wind power consumption surpasses the predicted value. As a result, having an exact estimation technique for the wind power profile is critical. The costs are computed in USD/h as described in [49].

4.2.3. Solar Power Units

Sun power is stochastic and fluctuating due to meteorological circumstances such as clouds and solar irradiation. As a result, solar systems’ output power is calculated using the sun irradiance variable (G). The solar irradiance (G) is given and probabilistically represented in this section using the lognormal probability distribution function, in which in [25], the function f_G(G), is defined:

$$f_G\left(G\right)=\frac{k}{G\sigma \sqrt{2\pi }}\times e^{-\left(\frac{\left(\ln x-\mu \right)}{2{\sigma }^2}\right)} \mathrm{for} G>0$$

(42)

The solar system’s overall goal is to alter solar energy into electrical energy. Equation (43) describes the solar system’s output power, P_s(G), as a function using the solar irradiance calculation in Equation (42) [25].

$$P_s\left(G\right)=\left\{\begin{array}{l}{P}_{sr}\frac{G^2}{G_{std}{R}_c}0<G<R_c\\ P_{sr}\frac{G}{G_{std}{}_c}G\ge R_c\end{array}\right.$$

(43)

The overall cost of solar power generation is computed using three components, similar to wind power units (i.e., direct, overestimation, and penalty fees) to reduce the influence of doubt in solar power fee estimation. Equations (44)–(46) compute these three components in (USD/h), correspondingly [25].

$$C_{s,k}\left(P_{ws,j}\right)=h_k{P}_{ss,k}$$

(44)

$$\begin{array}{l}{C}_{Rs,k}\left(P_{ss,k}-P_{sav,k}\right)=K_{Rs,k}\left(P_{ss,k}-P_{sav,k}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=K_{Rs,k}\ast f_s\left(P_{sav,k}<P_{ss,k}\right)\ast \left[P_{ss,k}-E\left(P_{sav,k}<P_{ss,k}\right)\right]\end{array}$$

(45)

$$\begin{array}{l}{C}_{Ps,k}\left(P_{sav,k}-P_{ss,k}\right)=K_{Ps,k}\left(P_{sav,k}-P_{ss,k}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=K_{Ps,k}\ast f_s\left(P_{sav,k}>P_{ss,k}\right)\ast \left[E\left(P_{sav,k}>P_{ss,k}\right)-P_{ss,k}\right]\end{array}$$

(46)

$$C_S^T={\displaystyle \sum _{k=1}^{N_S}\left[C_{s,j}\left(P_{ss,k}\right)+C_{Rs,k}\left(P_{ss,k}-P_{sav,k}\right)+C_{Ps,k}\left(P_{sav,k}-P_{ss,k}\right)\right]}$$

(47)

where $h_k$ is the direct cost coefficient associated with k-th solar PV plant and $P_{ss,k}$ is the scheduled power from the same plant. $K_{Rs,k}$ is the reserve cost coefficient pertaining to k-th solar PV plant and $P_{sav,k}$ is the actual available power from the same plant. $f_s(P_{sav,k}<P_{SS,k})$ represents the probability of solar power shortage occurrence than the scheduled power. By contrast, $f_s(P_{sav,k}>P_{SS,k})$ shows the expectation of solar PV power below $P_{SS,k}$.

4.2.4. Case 7

Case 7 optimized the generating program for thermal and renewable energy producers in order to reduce the overall generation cost as determined by (50). The cost coefficients, in this case, were the same as in Case 1, and the probability density function (PDF) parameters are shown in Table 9. Moreover, Table 10 summarizes the optimal settings for all decision variables, reactive generator power (Q), total generating cost, and other important computed metrics. In the Table, V_i indicates the voltage at the ith bus. P_loss and V.D. were computed using (30) and (31). It is worth noting that P_ws,1 denotes the scheduled power from wind generator W_G1, and so on. The lowest generation cost obtained using the generation schedules indicated in the table was 782.2795 USD/h.

$$\begin{array}{l}{J}_7={\displaystyle \sum _{i=1}^{NG}{F}_i(P_{Gi})=}{\displaystyle \sum _{i=1}^{N_T}({\alpha }_i+b_i{P}_{Gi}+c_i{P}_{Gi}^2)}\\ +{\displaystyle \sum _{j=1}^{N_W}\left[C_{w,j}\left(P_{ws,j}\right)+C_{Rw,j}\left(P_{ws,j}-P_{wav,j}\right)+C_{Pw,j}\left(P_{wav,j}-P_{ws,j}\right)\right]}\\ +{\displaystyle \sum _{k=1}^{N_S}\left[C_{s,j}\left(P_{ss,k}\right)+C_{Rs,k}\left(P_{ss,k}-P_{sav,k}\right)+C_{Ps,k}\left(P_{sav,k}-P_{ss,k}\right)\right]}\end{array}$$

(48)

Table 9. PDF parameters of wind power and solar PV plants.

Wind Power Generating Plants					Solar PV Plant
Wind Farm	No. of Turbines	Rated Power, Pwr (MW)	Weibull PDF parameters	Weibull Mean, Mwbl	Rated Power, Psr (MW)	Lognormal PDF parameters	Lognormal Mean, Mlgn
1 (bus 5)	25	75	c = 9, k = 2	v = 7.976 m/s	50 (bus 13)	µ = 6, σ = 0.6	G = 483 W/m2
2 (bus 11)	20	60	c = 10, k = 2	v = 8.862 m/s

Table 10. The optimal values by HFAJAYA for Case 7.

Variables	Optimal Values
		PG1 (MW)	134.90791
PG2 (MW)	28.7687
Pws1 (MW)	43.8862
PG3 (MW)	10
Pws2 (MW)	37.049
Pss (MW)	34.5606
VG1 (p.u.)	1.0717
VG2 (p.u.)	1.0568
VG5 (p.u.)	1.0348
VG8 (p.u.)	1.0547
VG11 (p.u.)	1.0982
VG13 (p.u.)	1.049
QG1 (MVAR)	−2.33741
QG2 (MVAR)	11.7952
Qws1 (MVAR)	22.4203
QG3(MVAR)	40
Qws2 (MVAR)	30
Qss (MVAR)	15.1297
Fuelvlvcost (USD/h)	441.4626
Wind gen cost (USD/h)	247.0741
Solar gen cost (USD/h)	93.7428
Total Cost (USD/h)	782.2795
Emission (t/h)	1.76202
Power losses (MW)	5.7724
V.D. (p.u.)	0.45423

4.2.5. Case 8

This case focuses on the minimization of the generation cost incorporating stochastic wind and solar power with the carbon tax.

Many nations have put significant pressure on the whole energy industry to decrease carbon emissions in recent years because of global warming [22]. The carbon tax (C_tax) is levied on every unit quantity of released greenhouse gases to stimulate investment in greener types of power such as wind and solar. The cost of emission (in dollars per hour) is expressed as:

$$C_E=C_{tax}E$$

(49)

$$J_8=J_7+C_{tax}E$$

(50)

This case study sough to reduce total generating costs by imposing a carbon tax on emissions from traditional thermal power producers. As calculated by (50), the total cost must be kept to a minimum. The C_tax, or carbon tax rate, is estimated to be USD20 per tonne [22]. Because wind and solar are spotless sources of energy, the carbon tax component is intended to boost their adoption. Table 11 shows the optimal generating schedule, reactive generator power, entire generation cost (counting C_tax), and other computed factors. When a carbon price was imposed in Case 8, the penetration of both solar and wind energy was higher than in Case 7 where no carbon tax was considered. The volume of emissions and the pace at which a carbon price is applied determine the amount of growth in the ideal renewable producing schedule.

Table 11. The optimal values by HFAJAYA for Case 8.

Variables	Optimal Values
		PG1 (MW)	122.91239
PG2 (MW)	31.4629
Pws1 (MW)	45.1894
PG3 (MW)	10
Pws2 (MW)	38.0721
Pss (MW)	41.0406
VG1 (p.u.)	1.0701
VG2 (p.u.)	1.0566
VG5 (p.u.)	1.0354
VG8 (p.u.)	1.0403
VG11 (p.u.)	1.0999
VG13 (p.u.)	1.0573
QG1 (MVAR)	−2.82122
QG2 (MVAR)	12.1528
Qws1 (MVAR)	22.9933
QG3(MVAR)	35.0868
Qws2 (MVAR)	30.0
Qss (MVAR)	18.2947
Fuelvlvcost (USD/h)	422.6759
Wind gen cost (USD/h)	255.1889
Solar gen cost (USD/h)	115.4701
Total Cost (USD/h)	810.5517
Emission (t/h)	0.86084
Power losses (MW)	5.2775
V.D. (p.u.)	0.47175
Carbon tax (USD/h)	17.2168

Constraints on load bus voltage are critical in the OPF problem since load bus operating voltages are frequently near their limitations. In our investigation, the load bus voltage had to be kept between 0.95 and 1.05 p.u.

4.3. Discussion on Results

In this section, the statistical results of the proposed method including max, mean, min, Std., and computational time, compared to the FA algorithm for all cases are given. The results for all single-objective cases are given in Table 12. The proposed HFAJAYA in all cases had a good advantage over the FA algorithm and on the other hand its optimization process was simple so that no additional time was required to perform the optimization process. HFAJAYA was also the most robust algorithm between the two algorithms. The important point is that as the dimension increased, the issue of HFAJAYA superiority over the FA algorithm became more prominent. Moreover, the ability of this algorithm to solve Optimal Power Flow Problem Considering Wind and Solar Power Generations can be seen.

Table 12. Statistical results to show the performance of the proposed HFAJAYA and FA algorithms.

Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 1
HFAJAYA	800.4800	800.5095	800.5378	0.0095	20.6
FA	800.7502	8001.0924	801.6420	1.82	20.6
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 2
HFAJAYA	646.5020	646.6267	646.7001	0.017	20.7
FA	647.2612	647.5939	647.2614	0.823	20.7
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 3
HFAJAYA	832.1798	832.3673	832.6049	0.0097	20.7
FA	832.5596	832.8512	833.1725	0.746	20.7
Optimizer	Min	Mean	Max	Std.	Time (s)
	Case 7
HFAJAYA	782.2795	782.5472	782.7363	0.0648	21.4
FA	783.8409	784.6110	785.4923	1.84	21.5

In our proposed HFAJAYA, the population initialization, and the firefly position update by the combinatorial movement were included. Let O(F) be the computational time complexity of the objective function F(X). In the HFAJAYA, the computational time complexity of population initialization was O(F*N), and the computational time complexity of position update was O(F*N²). Hence, by analyzing other references (Yang, 2008), we know the total computational complexity of the HFAJAYA was the same as the basic FA that was O(F*N²).

5. Conclusions

In this study, a new efficient improved FA algorithm combined with the JAYA algorithm, namely HFAJAYA, was proposed to solve the OPF problem considering solar and wind energy systems with different formulations in 30 bus tests systems. This problem includes optimally selecting control variables such as transformer pulses, generator outputs and reactive power sources, and generator voltages. Various cases with different combinations of the important objective functions and constraints were simulated, and the proposed algorithm results were compared with the state-of-the-art methods in the literature. The results demonstrated that the proposed HFAJAYA method had effective and reliable results compared with all recent studies. In addition, the computational analysis showed that the proposed HFAJAYA algorithm without imposing extra complexity to the main algorithm, only by parallelizing the motion vector, was able to strengthen the global search of the main algorithm and increase its robustness. Hence, it can be utilized for other engineering optimization problems with a challenging search space. The future work of this study will involve an extension of the proposed model for the optimal management of a hub energy system with various kinds of energy sources and storage systems.