Optimal power flow calculation in hybrid power system involving solar,wind,and hydropower plant using weighted mean of vectors algorithm

Abstract

One of the most complex and motivating issues in power system is optimal power flow (OPF), which is a constrained optimization problem characterized by non-linearity and non-convexity. From these specifications, researchers competed in the past decades to find optimal solutions to OPF problem while keeping system stability. This paper presents an efficient optimization approach to deal with OPF problem in the hybrid renewable energy systems involving wind turbines, solar photovoltaic and small hydropower plant using optimization method depends on weighted mean of vectors INFO. Total generation cost, active power losses, and combined cost and emission are the principle goal, taking into account both reserve and penalty cost appropriate to over and under estimation respectively in the generation cost model. To evaluate the performance of INFO in solving OPF problem, modified IEEE 30-bus and IEEE 57-bus test systems will be utilized. The obtained results are compared with several algorithms such as Gorilla troop optimizer GTO, artificial ecosystem-based optimization AEO, Barnacles Mating Optimizer BMO for the same test systems keeping the same conditions. Simulation results have indicated the superiority of INFO while respecting all constraints. INFO can minimize total generation cost to 788.9417 $/h for IEEE 30-bus and 5259.2040 $/h for IEEE 57-bus. The results demonstrate clearly that the INFO is a highly efficient algorithm that is an encouraging tool for solving OPF problem. The promising findings highlight the potential of the INFO algorithm to smoothest the integration of RES, and its role in promoting sustainable energy solutions. Furthermore, the one-way analysis of variance (ANOVA) test, a statistical approach, was employed to evaluate the superiority of the proposed algorithm and to highlight a certain level of confidence to our study.

Keywords

Optimal power flow hybrid renewable energy systems emission weighted mean of vectors wind power solar Photovoltaic (PV) power hydropower plant uncertainty analysis of variance (one-way ANOVA test)

Introduction

Optimal power flow (OPF) is one of the major subjects in power systems since its appearance. The primary objective of OPF is to minimize the generation cost by finding the best operating point with regard to control variables (Avvari and Kumar, 2023). Active power of generators and generator bus voltages represent control variables.

The increasing consumption of energy and the high price of its production through thermal generators led the world to the inevitability of energy transition through the inclusion of renewable energies in the electric grid, which in turn is considered clean, environmentally friendly and inexpensive (Hannan et al., 2020). The integration of renewable energies into the electrical network has increased the complexity of studying the OPF problem because of the intermittence nature of these sources (Riaz et al., 2021). The main objective behind integration of renewable generators (wind turbines, solar photovoltaic, small hydro-power) in the grids is to minimize the generation fuel cost, decrease the power losses and reduce environmental pollution (Rawa et al., 2021).

Many methods and various algorithms have been used to solve the OPF problem in the power system using traditional mathematical algorithms or metaheuristic approaches. Among the classical optimization techniques are Newton method (Bjelogrlic et al., 1990), non-linear programing (Habibollahzadeh et al., 1989) and interior point methods (Capitanescu and Wehenkel, 2013). It is undeniable that some of the aforementioned algorithms have strengths such as excellent convergence and have many uses in industrial applications. However, it shows weaknesses that cannot be overlooked because it leads to a reduction in its efficiency.

In the meantime, metaheuristic processes have become more and more used, specifically to fix the OPF problem. That is because of their reliance on easy principles that imitate from nature, which make them get away from the local optimality (Sulaiman and Mustaffa, 2020). Several metaheuristic optimization techniques have been used to solve classical OPF problem such as improved version of PSO (Singh et al., 2016), Moth Swarm Optimization (MSO) (Mohamed et al., 2017), backtracking-search algorithm (BSA) (Chaib et al., 2016), Glowworm Swarm Optimization (GSO) (Surender Reddy and Srinivasa Rathnam, 2016), artificial bee colony (ABC) (Armaghani et al., 2015; Rezaei Adaryani and Karami, 2013), and stud krill herd (SKH) (Pulluri et al., 2017, 2018). Despite this, the use of the mentioned algorithms were limited in the classical power systems that contain only thermal power generators.

In recent years, some studies have been carried out in order to solve OPF problem in hybrid power systems, which are formed from classical and renewable energy sources like wind, solar and hydropower (Khan et al., 2020). Researchers utilized probability density functions (PDF) such as Weibull and lognormal to predict wind speed and solar irradiance. Evolutionary particle swarm optimization (EPSO) was implanted to solve OPF problem taking into consideration wind generation (Chang et al., 2014). Among these recent studies proposed: Modified cuckoo search optimization (Mishra et al., 2015), Modified bacteria foraging algorithm (MBFA) (Panda et al., 2020), Barnacles mating optimization (Sulaiman and Mustaffa, 2021), hybrid of differential evolution (DE), and particle swarm optimization (PSO) (Duman et al., 2020).

In this paper, IEEE 30-bus and IEEE 57-bus test systems are modified in order to integrate solar, wind and small-hydro power generators with a limited number of thermal generators. The uncertainties of wind speed, solar irradiance and river flow are treated in detail and are modeled by probability density functions: Weibull, lognormal and Gumbel respectively (Zobaa and Aleem, 2021). In this work, two different scenarios are considered to manipulate the intermittence of sources: overestimation and underestimation by inserting both reserve cost and penalty cost in the generation cost.

Based on what has been mentioned from earlier studies, it is noted that the application of metaheuristic algorithms in finding optimal solutions to OPF problem with presence of renewable energy sources (RES) is very motivating to contribute to this field. Hence, this study presents an application of recent population-based optimization algorithm namely the INFO algorithm, which depends on the weighted mean of vectors (Ahmadianfar et al., 2022).

INFO uses three major operators in order to swap the location of population, modernizing rule step, vector combining, and finally a local search operator in the search space (Ahmadianfar et al., 2022). Wavelet function was used to compute the weight of vectors. The results obtained indicate clearly the high efficiency of the proposed algorithm regarding exploitation and exploration behavior, getting away local optimum and convergence speed. The contributions of this work are summarized as below

To the authors’ best knowledge, this is the first attempt to apply this optimizer to such a problem, that is, application of INFO algorithm in solving OPF problem taking into consideration the incorporation of wind, solar and small hydropower generators.

INFO was applied to resolve stochastic-OPF constrained problem applied on modified IEEE 30 and IEEE 57 test systems.

To authenticate the proficiency of the proposed INFO algorithm, achieved results were compared with other contemporary optimizers. It was observed that the proposed method for different objective functions provides better optimal cost, compared with other optimizers.

The optimal generation of multiples energy resources (MERs) and less consumption of fuel also play role in environmental features and regulatory institutions of emission as a result protecting environment.

Besides, a statistical study of different algorithms using standard approaches such ANOVA has been conducted for both test systems (IEEE 30-bus and IEEE 57-bus).

The structure of the paper is as follows: problem formulation is described in section 2. Section 3 explains the INFO algorithm. In section 4, we have presented a detail discussion concerning the treated cases and obtained results. Finally, section 5 expresses the conclusion of this paper.

Problem formulation

Optimal power flow model

The important goal of solving OPF problems is to determine the optimal values of control variables so that minimizing a certain objective function while respecting all the physical and security constraints. Mathematically, the OPF problem is expressed as follows:

M i n i m i z e F (x, u)

(1)

s u b j e c t t o : {\begin{matrix} g (x, u) = 0 \\ h (x, u) \leq 0 \end{matrix}

(2)

where: $F (x, u)$ presents the objective function, $x$ and $u$ are the vectors of control and state variables, respectively, $g (x, u)$ and $h (x, u)$ are the equality and inequality constraints, respectively.

Objective functions

In this study, three objective functions are proposed, minimization of total generation cost, total active power loss, and cost with emission effect.

Generation cost of thermal generators considering the valve point effect is given as follows:

C_{T} (P_{T G}) = \sum_{i = 1}^{N_{T G}} a_{i} + b_{i} P_{T G i} + c_{i} P_{T G i}^{2} + | d_{i} \times s i n (e_{i} (P_{T G i}^{\min} - P_{T G})) |

(3)

Where $a_{i}$ , $b_{i}$ , $c_{i}$ , $d_{i}$ , and $e_{i}$ represent the cost coefficients related to i-th thermal generator $P_{T G i}$ , $P_{T G i}^{\min}$ is the minimum value of power corresponding to i-th thermal generator. $N_{T G}$ is the number of thermal generators.

Case 1: the first objective of OPF is the minimization of total generation cost considering both thermal generators and RESs can be formulated as:

\begin{matrix} F_{O b j}^{1} = C_{T} (P_{T G}) + [C_{W, j} (P_{W G s, j}) + C_{R W, j} (P_{W G, j} - P_{W A v, j}) + C_{P W, j} (P_{W A v, j} - P_{W G, j})] \\ + [C_{S, k} (P_{S G, k}) + C_{R S, k} (P_{S G, k} - P_{S A v, k}) + C_{P S, k} (P_{S A v, k} - P_{S G, k})] \\ + [C_{SHG} (P_{SHG}) + C_{R S H} (P_{SHG} - P_{SHAv}) + C_{P S H} (P_{SHAv} - P_{SHG})] \end{matrix}

(4)

Case 2: the second objective of OPF is the minimization of total active power loss in the electric grid, which can be formulated as:

F_{O b j}^{2} = \sum_{q = 1}^{nl} G_{q (i j)} [V_{i}^{2} + V_{j}^{2} - 2 V_{i} V_{j} c o s (δ_{i} - δ_{j})]

(5)

where $nl$ is the total number of transmission lines, $G_{q (i j)}$ is the conductance of the branch i-j, $V_{i}$ and $V_{j}$ are the voltages at bus i and j respectively, $δ_{i j} = δ_{i} - δ_{j}$ , is the difference in voltage angles between them.

Case 3: the final objective of OPF is to minimize both generation cost and emission by including the carbon tax as penalty, which can be formulated as:

F_{O b j}^{3} = F_{O b j}^{1} + C_{t a x} \times E

(6)

E = \sum_{i = 1}^{N_{T G}} [(α_{i} + β_{i} P_{T G i} + γ_{i} P_{T G i}^{2}) + δ_{i} \exp (ε_{i} P_{T G i})]

(7)

where $α_{i}$ , $β_{i}$ , $γ_{i}$ , $δ_{i}$ , and $ε_{i}$ are the emission coefficients of the i-th thermal generators whose values are provided in Table 2, $C_{t a x}$ denotes the tax of carbon, which is equal to 20($/h) (Sulaiman and Mustaffa, 2020).

System constraints

While solving OPF objectives, different equality and inequality constraints are to be respected. These constraints are expressed as follow:

Equality constraints

{\begin{matrix} P_{G i} - P_{d i} - V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i j} c o s (δ_{i j}) + B_{i j} s i n (δ_{i j})] = 0 \\ Q_{G i} - Q_{d i} - V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i j} s i n (δ_{i j}) - B_{i j} c o s (δ_{i j})] = 0 \end{matrix}

(8)

where $P_{G i}$ , $Q_{G i}$ are the active and reactive power of generation, $P_{d i}$ , $Q_{d i}$ are the active and reactive power of demand and $N B$ is the total number of bus of the power system.

Inequality constraints

Generator constraints

P_{T G i}^{\min} \leq P_{T G i} \leq P_{T G i}^{\max} i = 1, 2 . \dots . N_{T G}

(9)

P_{W G}^{\min} \leq P_{W G} \leq P_{W G}^{\max}

(10)

P_{S G}^{\min} \leq P_{S G} \leq P_{S G}^{\max}

(11)

P_{SHG}^{\min} \leq P_{SHG} \leq P_{SHG}^{\max}

(12)

Q_{T G i}^{\min} \leq Q_{T G i} \leq Q_{T G i}^{\max}

(13)

Q_{W G}^{\min} \leq Q_{W G} \leq Q_{W G}^{\max}

(14)

Q_{S G}^{\min} \leq Q_{S G} \leq Q_{S G}^{\max}

(15)

Q_{SHG}^{\min} \leq Q_{SHG} \leq Q_{SHG}^{\max}

(16)

Prohibited operating zones POZs

P_{T G 2}^{\min p o z, j} \leq {P O Z}_{T G 2}^{j} \leq P_{T G 2}^{\max p o z, j}

(17)

Security constraints

V_{G i}^{\min} \leq V_{G i} \leq V_{G i}^{\max} i = 1, 2 . \dots . N_{T G}

(18)

V_{L i}^{\min} \leq V_{L i} \leq V_{L i}^{\max} i \in N L

(19)

S_{i} \leq S_{i}^{\max} i \in N T L

(20)

Modeling the uncertainty of renewable energy generators

Due to stochastic nature of RES, it is required to present the model uncertainty adopted in this study in terms of operation and planning of power systems. The wind speed is a random variable, wind uncertainty is modeled using the Probability Density Function (PDF) is to obtain its distribution employing shape factor (k) and scale factor (c). Mathematically can be written as:

f_{υ} (S) = (\frac{k}{c}) {(\frac{S}{c})}^{(k - 1)} \times \exp - {(\frac{S}{c})}^{k} f o r 0 < S < \infty

(21)

The wind power model

The output wind-power according to wind speed, is expressed as:

P_{w} (υ) = {\begin{matrix} 0, for ν 〈 ν_{i n} and ν 〉 ν_{o u t} \\ P_{w r} (\frac{ν - ν_{i n}}{ν_{r} - ν_{i n}}) for ν_{i n} \leq ν \leq ν_{r} \\ P_{w r} for ν_{r} \leq ν \leq ν_{o u t} \end{matrix}

(22)

With $ν_{i n}$ defines the turbine cut-in, $ν_{r}$ is the rated wind speed and $ν_{o u t}$ pertains cut-out wind speed, and $P_{w r}$ rated output-power.

Wind power probability for different wind-speeds

The equation (22) states that, when $ν_{i n}$ is more than $ν$ and down than $ν_{o u t}$ , the output of power equal to zero. But, the wind turbine output the rated-power $P_{w r}$ for the condition of $ν_{r} \leq ν \leq ν_{o u t}$ . To model this fact, the probabilities of wind power is modeled as:

f_{w} (P_{w}) {P_{w} = 0} = 1 - \exp [- {(\frac{ν_{i n}}{α})}^{β}] + \exp [- {(\frac{ν_{o u t}}{α})}^{β}]

(23)

f_{w} (p_{w}) {p_{w} = p_{w r}} = 1 - \exp [- {(\frac{ν_{r}}{α})}^{β}] + \exp [- {(\frac{ν_{o u t}}{α})}^{β}]

(24)

Outside to the discrete zones, the output of wind power is remain continuous for the condition of $ν_{i n} \leq ν \leq ν_{r}$ , This probability is modeled by the following equation:

f_{w} (p_{w}) = \frac{β (ν_{r} - ν_{i n})}{α^{β} * P_{w r}} {[ν_{i n} + \frac{P_{w}}{P_{w r}} (ν_{r} - ν_{i n})]}^{β - 1} \exp [- {(\frac{ν_{i n} + \frac{P_{w}}{P_{w r}} (ν_{r} - ν_{i n})}{α})}^{β}]

(25)

As well, the solar-irradiance to energy conversion for the PV generator is modeled as follow:

P_{s} (G) = {\begin{matrix} P_{s r} (\frac{G^{2}}{G_{std} R_{c}}) for 0 \leq G \leq R_{c} \\ P_{s r} (\frac{G^{2}}{G_{std}}) for G \geq R_{c} \end{matrix}

(26)

The direct-cost function of wind and solar generators are given by (Sulaiman and Mustaffa, 2020):

C_{W, j} (P_{W G, j}) = g_{j} P_{W G, j}

(27)

C_{S, k} (P_{S G, k}) = h_{k} P_{S G, k}

(28)

where $g_{j}$ and $h_{k}$ present the coefficient of direct cost corresponding to jth wind generator and kth solar generator respectively. $P_{W G, j}$ and $P_{S G, k}$ are the scheduled power from the appropriate generator.

The direct cost function for the combination of solar photovoltaic and small hydro generation plant is given by (Sulaiman and Mustaffa, 2020):

C_{SHG} (P_{SHG}) = C (P_{S G} + P_{H G}) = g_{S G} P_{S G} + h_{H G} P_{H G}

(29)

where $P_{SHG}$ is the scheduled power from incorporated renewable resources, $P_{S G}$ and $P_{H G}$ present the electric power contribution from the solar and hydro generators, respectively. $g_{S G}$ and $h_{H G}$ are the coefficients of direct-cost corresponding to solar PV unit and small hydro unit, respectively.

Due to stochastic nature of renewable energy sources, there are two possibilities tendencies which is, overestimation and underestimation. First case, when the actual power generated by renewable generators (PV, or combined PV and hydropower) less than the estimated-quantity in term of power, called overestimation. Second case is underestimation realized when estimated-quantity power is less than actual power. The reserve cost caused with overestimation is presented as follows:

\begin{matrix} C_{R W, j} (P_{W G, j} - P_{Av, j}) = K_{R W, j} (P_{W G, j} - P_{W A v, j}) \\ = K_{R W, j} \int_{0}^{P_{W G, j}} (p_{W G, j} - P_{W, j}) f_{W} (P_{W, j}) d p_{W, j} \end{matrix}

(30)

\begin{matrix} C_{R S, k} (P_{S G, k} - P_{S A v, k}) = K_{R S, k} (P_{S G, k} - P_{S A v, k}) \\ = K_{R S, k} * f_{S} (P_{S A v, k} < P_{S G, k}) * [P_{S G, k} - E (P_{S A v, k} < P_{S G, k})] \end{matrix}

(31)

\begin{matrix} C_{R S H} (P_{SHG} - P_{SHAv}) = K_{R S H} (P_{SHG} - P_{SHAv}) \\ = K_{R S H} * f_{SH} (P_{SHAv} < P_{SHG}) * [P_{SHG} - E (P_{SHAv} < P_{SHG})] \end{matrix}

(32)

where $K_{R W, j}$ is the reserve cost coefficient corresponding to jth wind generation unit, $P_{W A v, j}$ is the actual power delivered by the same unit, $f_{W} (p_{W, j})$ is the PDF of wind power corresponding to jth wind power plant. $K_{R S, k}$ is the reserve cost coefficient corresponding to kth solar power unit, $P_{S A v, k}$ is the actual power delivered by the same unit, $f_{S} (P_{S A v, k} < P_{S G, k})$ is the probability of solar energy curtailment from the scheduled power ( $P_{S G, k}$ ) and $E (P_{S A v, k} > P_{S G, k})$ is the prediction of solar-power above $P_{S G, k}$ . $K_{R S H}$ is the coefficient of reserve cost corresponding to the combined system, $P_{SHAv}$ is the actual power delivered by the same plant, $f_{SH} (P_{SHAv} < P_{SHG})$ is the probability of combined system energy curtailment from the scheduled power $P_{SHG}$ and $E (P_{SHAv} < P_{SHG})$ is the prediction of delivered power below $P_{SHG}$ .

The underestimation of power is related to the penalty terms which defined by second case. Mathematically, can be expressed as follows:

C_{P W, j} (P_{W A v, j} - P_{W G, j}) = K_{P W, j} (P_{W A v, j} - P_{W G, j}) = K_{P W, j} \int_{P_{W G, j}}^{P_{Wr, j}} (p_{W, j} - p_{W G, j}) f_{W} (P_{W, j}) d p_{W, j}

(33)

C_{P S, k} (P_{S A, k} - P_{S G, k}) = K_{P S, k} (P_{S A v, k} - P_{S G, k}) = K_{P S, k} * f_{S} (P_{S A v, k} > P_{S G, k}) * [E (P_{S A v, k} > P_{S G, k}) - P_{S G, k}]

(34)

C_{P S H} (P_{SHAv} - P_{SHG}) = K_{P S H} (P_{SHAv} - P_{SHG}) = K_{P S H} * f_{SH} (P_{SHAv} > P_{SHG}) * [E (P_{SHAv} > P_{SHG}) - P_{SHG})]

(35)

where $K_{P W, j}$ is the coefficient of penalty cost relate to j-th wind power generator, $P_{Wr, j}$ is the rated output power from the plant, $K_{P S, k}$ is the penalty cost coefficient corresponding to kth solar power generator, $f_{S} (P_{S A v, k} > P_{S G, k})$ is the probability of solar energy surplus of the scheduled power $P_{S G, k}$ and $E (P_{S A v, k} > P_{S G, k})$ is the prediction of solar-power above $P_{S G, k}$ . $K_{P S H}$ is the coefficient of penalty cost corresponding to the combined system, $f_{SH} (P_{SHAv} > P_{SHG})$ is the probability of combined system energy surplus of the scheduled power $P_{SHG}$ and $E (P_{SHAv} > P_{SHG})$ is the prediction of the combined system power above $P_{SHG}$ .

INFO proposed algorithm

Weighted mean concept

The proposed algorithm presented in this paper depends on a weighted mean that illustrates a special position in a system or an object (Ahmadianfar et al., 2022). The detailed definition of this conception is introduced in this part.

Mathematical interpretation of weighted men

For a group of vectors, the mean is defined as the average of their locations $x_{i}$ , as weighted by the vector’s fitness $w$ . Figure 1 shows the Weighted Mean (WM) of the set of solutions, which can be presented by the following equation (36):

W M = \frac{\sum_{i = 1}^{N} x_{i} \times w_{i}}{\sum_{i = 1}^{N} w_{i}}

(36)

Where $N$ presents the number of vectors. $W M$ can be described as two vectors, as presented in equation (36a) (Ahmadianfar et al., 2022)

W M = \frac{x_{1} \times w_{1} + x_{2} \times w_{2}}{w_{1} + w_{2}}

(36a)

The weight of any vector was calculated by using wavelet function (WF) (Ahmadianfar et al., 2022) which is used to generate efficient fluctuations throughout the optimization procedure. Figure 2 shows the mother wavelet, which is presented by the following equation:

W = c o s (x) \times \exp (- \frac{x^{2}}{ω})

(37)

where: $ω$ presents the parameter of dilation, which is a constant number.

Figure 3(a) and (b) show three vectors and Figure 3(c) depicts the differences between them.

The WM of vectors is calculated by the following equation:

W M = \frac{w_{1} \times (x_{1} - x_{2}) + w_{2} \times (x_{1} - x_{3}) + w_{3} \times (x_{2} - x_{3})}{w_{1} + w_{2} + w_{3}}

(38)

in which:

w_{1} = c o s ((f (x_{1}) - f (x_{2})) + π) \times \exp (\frac{f (x_{1}) - f (x_{2})}{ω})

(38a)

w_{2} = c o s ((f (x_{1}) - f (x_{3})) + π) \times \exp (\frac{f (x_{1}) - f (x_{3})}{ω})

(38b)

w_{3} = c o s ((f (x_{2}) - f (x_{3})) + π) \times \exp (\frac{f (x_{2}) - f (x_{3})}{ω})

(38c)

where: $f (x)$ presents the fitness function.

INFO algorithm: Weighted mean of vectors

The Vectors Weighted Mean algorithm (INFO) is a population-based optimization method that computes the weighted average of an ensemble of vectors in the search-area. In the proposed algorithm, the population consists of an ensemble of vectors that reflect candidate solutions to a given problem. This optimizer finds the optimal solution after several consecutive generations.

In each generation, three operators are responsible for updating the position of the vector:

Updating rule step

Vector combination step

Local search step

The initialization step

The INFO algorithm is formed of a population of $N$ vector in $D$ dimensional search area. $(X_{l, j}^{g} = {x_{l, 1}^{g}, x_{l, 2}^{g}, \dots, x_{l, D}^{g}}, l = 1, 2, \dots, N)$ . INFO algorithm has two main parameters of control, weighted mean factor $δ$ and scaling factor $σ$ .

Updating rule step

The updating rule operator rises the diversity of population in the search process. In fact, this operator is specific to this algorithm and makes it distinct from the rest of the algorithms and it comprises two parts. Firstly, a mean-based rule is taken from the WM for a group of random vectors. Secondly, convergence acceleration increases convergence speed, in reinforcing the convergence speed which leading to avoid falling into the local optima.

Raising the diversity of population is considered the MeanRule, which depends on the worst, better and the best solutions. The MeanRule is presented by the following equation:

M e a n R u l e = r \times W M 1_{l}^{g} + (1 - r) \times W M 2_{l}^{g}, l = 1, 2, \dots, N p

(39)

W M 1_{l}^{g} = δ \times \frac{w_{1} (x_{a 1} - x_{a 2}) + w_{2} (x_{a 1} - x_{a 3}) + w_{3} (x_{a 2} - x_{a 3})}{w_{1} + w_{2} + w_{3} + ε} + ε \times r a n d, l = 1, 2, \dots, N p

(39a)

w_{1} = c o s ((f (x_{b s}) - f (x_{bt})) + π) \times \exp (- | \frac{f (x_{b s}) - f (x_{bt})}{ω} |)

(39b)

w_{2} = c o s ((f (x_{a 1}) - f (x_{a 3})) + π) \times \exp (- | \frac{f (x_{a 1}) - f (x_{a 3})}{ω} |)

(39c)

w_{3} = c o s ((f (x_{a 2}) - f (x_{a 3})) + π) \times \exp (- | \frac{f (x_{a 2}) - f (x_{a 3})}{ω} |)

(39d)

ω = \max (f (x_{a 1}), f (x_{a 2}), f (x_{a 3}))

(39e)

W M 2_{l}^{g} = δ \times \frac{w_{1} (x_{b s} - x_{bt}) + w_{2} (x_{b s} - x_{ws}) + w_{3} (x_{bt} - x_{ws})}{w_{1} + w_{2} + w_{3} + ε} + ε \times r a n d, l = 1, 2, \dots, N p

(39f)

w_{1} = c o s ((f (x_{1}) - f (x_{2})) + π) \times \exp (\frac{f (x_{1}) - f (x_{2})}{ω})

(39g)

w_{2} = c o s ((f (x_{b s}) - f (x_{ws})) + π) \times \exp (- | \frac{f (x_{b s}) - f (x_{ws})}{ω} |)

(39h)

w_{3} = c o s ((f (x_{bt}) - f (x_{ws})) + π) \times \exp (- | \frac{f (x_{bt}) - f (x_{ws})}{ω} |)

(39i)

ω = f (x_{ws})

(39j)

where $f (x)$ is the objective function value; $a 1 \neq a 2 \neq a 3 \neq l$ represent different integers that are chosen randomly from the range $[1, N p]$ . $ε$ represents a constant number with a very small value. $r a n d n$ denotes a random value normally distributed. $x_{ws}$ , $x_{bt}$ , and $x_{b s}$ are the worst, better and best solutions between every vectors in the population for the $g^{th}$ generation, respectively. $r$ denotes a randomly selected number in the range $[0, 0.5]$ . $w_{1}$ , $w_{2}$ , and $w_{3}$ represent three WFs.

The WFs are utilized to vary the MeanRule space based on the theory of wavelet, as follows:

δ = 2 β \times r a n d - β

(40)

β = 2 \exp (- 4 \times \frac{g}{M a x g})

(40a)

In (40), δ denotes the scale factor, while the parameter β is considered as an exponential function. Also, the maximum number of generations is denoted as $M a x g$ .

The convergence acceleration can be formulated by this equation:

C A = r a n d n \times \frac{(x_{b s} - x_{a 1})}{(f (x_{b s}) - f (x_{a 1}) + ε)}

(41)

where $r a n d n$ denotes a random number in the range $[0, 1]$ . The new vector is computed based on equation (42).

z_{l}^{g} = x_{l}^{g} + σ \times M e a n R u l e + C A

(42)

The proposed updating rule is presented by the following scheme:

\begin{matrix} i f r a n d < 0.5 \\ z 1_{l}^{g} = x_{a}^{g} + σ \times M e a n R u l e + r a n d n \times \frac{(x_{a 2}^{g} - x_{a 3}^{g})}{(f (x_{a 2}^{g}) - f (x_{a 3}^{g}) + 1)} \\ z 2_{l}^{g} = x_{b s} + σ \times M e a n R u l e + r a n d n \times \frac{(x_{a 1}^{g} - x_{b}^{g})}{(f (x_{a 1}^{g}) - f (x_{a 2}^{g}) + 1)} \\ e l s e \\ z 1_{l}^{g} = x_{a}^{g} + σ \times M e a n R u l e + r a n d n \times \frac{(x_{a 2}^{g} - x_{a 3}^{g})}{(f (x_{a 2}^{g}) - f (x_{a 3}^{g}) + 1)} \\ z 2_{l}^{g} = x_{bt} + σ \times M e a n R u l e + r a n d n \times \frac{(x_{a 1}^{g} - x_{a 2}^{g})}{(f (x_{a 1}^{g}) - f (x_{a 2}^{g}) + 1)} \\ e n d \end{matrix}

(43)

where $z 1_{l}^{g}$ , $z 2_{l}^{g}$ denote the new vectors corresponding to the $g^{th}$ generation, $σ$ represents the scaling rate of vector, as presented in equation (44). α is considered as exponential function presented in equation (44a):

σ = 2 α \times r a n d - α

(44)

α = c \times \exp (- d \times \frac{g}{M a x g})

(44a)

Note that the parameter c and d are assume as 2 and 4 respectively.

Vector combining step

To improve the diversity of population in INFO, $z 1_{l}^{g}$ and $z 2_{l}^{g}$ are combined with $x_{l}^{g}$ concerning the condition $r a n d < 0.5$ to create the new vector $u_{l}^{g}$ , the equation (45) is used.

\begin{matrix} i f r a n d < 0.5 \\ i f r a n d < 0.5 \\ u_{l}^{g} = z 1_{l}^{g} + μ . | z 1_{l}^{g} - z 2_{l}^{g} | \\ e l s e \\ u_{l}^{g} = z 2_{l}^{g} + μ . | z 1_{l}^{g} - z 2_{l}^{g} | \\ e n d \\ e l s e \\ u_{l}^{g} = x_{l}^{g} \\ e n d \end{matrix}

(45)

where $u_{l}^{g}$ represents the new vector utilizing the vector combining corresponding to the $g^{th}$ generation. $μ = 0.05 \times r a n d n$ .

Local search step

To avoid falling into the local optimal solutions, the proposed algorithm resorts to the effective local search capacity. The local operator employed the global position $x_{best}^{g}$ and the mean-based rule presented in equation (46), a new vector can be created around $x_{best}^{g}$ if $r < 0.5$ .

\begin{matrix} i f r a n d < 0.5 \\ i f r a n d < 0.5 \\ u_{l}^{g} = x_{b s} + r a n d n \times (M e a n R u l e + r a n d n \times (x_{b s}^{g} - x_{a 1}^{g})) \\ e l s e \\ u_{l}^{g} = x_{r n d} + r a n d n \times (M e a n R u l e + r a n d n \times (v_{1} \times x_{b s} - v_{2} \times x_{r n d})) \\ e n d \\ e n d \end{matrix}

(46)

in which

x_{r n d} = ϕ \times x_{a v g} + (1 - ϕ) \times (ϕ \times x_{bt} + (1 - ϕ) \times x_{b s})

(46a)

x_{a v g} = (\frac{x_{a} + x_{b} + x_{3}}{3})

(46b)

where $ϕ$ is a random number from the range $[0, 1]$ , $x_{r n d}$ indicates a new solution that incorporates $x_{b s}$ , $x_{a v g}$ , and $x_{bt}$ , randomly. $υ_{1}$ and $υ_{2}$ denote two random numbers presented as:

υ_{1} = {\begin{matrix} 2 \times r a n d & i f p > 0.5 \\ 1 & o t h e r w i s e \end{matrix}

(46c)

υ_{2} = {\begin{matrix} r a n d & i f p < 0.5 \\ 1 & o t h e r w i s e \end{matrix}

(46d)

where $p$ is a random number from the $[0, 1]$ range.

The pseudo-code of the INFO algorithm
• Set parameters

N p

and

M a x g

• Generate an initial population

PO P^{0} = {X_{j}^{0}, \dots, X_{N p}^{0}}

• The objective function value for each vector

f (X_{j}^{0}) j = 1, \dots, N p

• Determine

x_{b s}

f o r g = 1 t o M a x g

f o r i = 1 t o N p

• Choose

a \neq b \neq c \neq i

randomly in the range

[1, N p]

• Compute

z 1_{g}^{l}

and

z 2_{g}^{l}

by equation (43)
• Compute

u_{i}^{g}

by eq 45
• Determine the local search operator by equation (46)
• Compute

f (u_{i, j}^{g})

i f f (u_{i, j}^{g}) < f (x_{i, j}^{g})

x_{i, j}^{g + 1} = u_{i, j}^{g}

e l s e x_{i, j}^{g + 1} = x_{i, j}^{g}

e n d f o r

• Update

x_{b s}

e n d f o r

• Return the final solution

x_{best, j}^{g}

Numerical results and analysis

In the first part, three scenarios were considered on the modified IEEE 30-bus test system by MATLAB. The modification is to insert wind generator, solar generator and combined solar PV and small hydropower at bus 5, 11, and 13, respectively. Table 1 indicates the characteristics of the modified IEEE 30-bus and IEEE 57-bus test systems.

➢ In scenario I, total generation cost is minimized.

➢ In second scenario II, total active power loss minimization is considered.

➢ In the last one, both of generation cost and pollution were minimized while respecting all imposed constraints.

Table 1.

Characteristics of the modified test systems.

Elements	IEEE 30-bus	IEEE 57-bus test system
No. of buses	30	57
No. of transmission lines	41	88
No. of generators	06	07
No. of thermal generators	03	04
Number of renewable generators	03	03
No. of load buses	24	41
No. of control variables	12	14
Initial real and reactive power demand	283.4 MW; 126.2 Mvar	1250.80 MW; 336.40 Mvar

The proposed INFO is executed to solve the three cases, and some other algorithms are chosen to compare with INFO to further indicate its performance. Tables 2 and 3 present the coefficients of thermal generators and limitations of both soft and hard-variables.

Table 2.

Thermal generators cost and emission coefficients for test system I.

Generator	Bus	b	c	d	e	α	β	γ	δ	ε
TG1	1	2	0.0038	18	0.037	0.04091	−0.05554	0.06490	0.0002	6.667
TG2	2	1.75	0.0175	16	0.038	0.02543	−0.06047	0.05638	0.0005	3.333
TG3	8	3.25	0.0083	12	0.045	0.05326	−0.03550	0.03380	0.0020	2.000

a ($/h); b ($/MWh); c ($/MW²h); d ($/h); e (MW⁻¹); α (t/h); β (t/p.u.MWh); γ (t/p.u. MW²h); δ (t/h); ε (p.u.MW⁻¹).

Table 3.

Upper and lower bounds of control and state variables for test system I.

Variables	Control variables							State variables
Variables	P_TG1	P_TG2	P_TG3	P_WG	P_SG	P_SHG	V_gi	Q_TG1	Q_TG2	Q_TG3	Q_WG	Q_SG	Q_SHG
Min	50	20	10	0	0	0	0.95	−20	−20	−15	−30	−25	−20
Max	140	80	35	75	60	50	1.1	150	60	40	35	30	25

POZ of P_TG2 (MW): [30–40]; [55–65].

All algorithms have been coded and simulated in Matlab R2013b platform, and run on an Intel (R) Pentium (R) CPU B950 @2.10 GHz 2.00 GB RAM personal computer. After an empirical study on size of population, that is, (30, 40, 60, and 100), we found the chosen of 30 individuals gives better results. Consequently, the population size for all case studies have been fixed of 30 individuals’ and the maxima number of iterations approved for showing the convergence curve, is 300. Moreover, to demonstrate the consistency and robustness of INFO, 30 independent runs were conducted for each case study and the best value of the objective function obtain with corresponding decision-variables are registered.

Case 1: Total generation cost minimization

In the first case, the primary goal is to minimize the total generation cost considering wind, solar and combined solar and small hydro generators. Obtained results are based on the Weibull, lognormal and Gumbel PDF parameters. Table 4 indicates PDF parameters of renewable energy sources that have been presented in Sulaiman and Mustaffa (2020). Weibull fitting and wind speed frequency distributions are presented in Figure 4 reached from the simulation of 8000 Monte Carlo scenarios. Figure 5 presents the lognormal fitting and solar irradiance frequency distributions obtained from the simulation of 8000 sample size of Monte Carlo. Figure 6 presents active power distribution of solar PV generator at bus 11.

Table 4.

PDF parameters of renewable energy sources.

Wind-power generating unit
No of turbines	Rated power, $P_{w r}$ (MW)	Weibull PDF parameters
25	75	l = 9; p = 2
Photovoltaic power plant
Rated power, $P_{s r}$ (MW)	Lognormal PDF parameters
50	μ = 5.2 σ = 0.6
Combined solar and small hydro-power
Photovoltaic rated power $P_{s r}$ (MW)	Lognormal PDF parameters
45	μ = 5.0 σ = 0.6
Small hydro rated power $P_{hr}$ (MW)	Gumbel PDF parameters
5	λ = 15 γ = 1.2

In this study, combined solar PV with small hydro generator is used in place of thermal generator at bus 13. Figure 7 indicates the lognormal fitting and solar irradiance accessible of solar PV generator at the same bus. while Figure 8 indicates Gumbel fitting and river flow rate frequency distribution from small hydro generator. Like last time, diagrams are generated after the simulation of 8000 Monte Carlo scenarios. The capacity of solar PV generator is 45 MW while for small hydropower is 5 MW. Figures 9 and 10 present the histograms of both available solar power and hydropower for the site and from the solar PV generator and small hydro generator respectively at bus 13. Table 5 indicates direct, penalty and reserve cost coefficients of renewable energy sources.

Table 5.

The different cost coefficients of renewable energy sources.

Direct cost coefficient			Penalty cost coefficient			Reserve cost coefficient
Wind	Solar	Small hydro	Wind	Solar	Small hydro	Wind	Solar	Small hydro
$g_{j} = 1.7$	$h_{k} = 1.6$	$h_{H G} = 1.5$	$K_{P W, j} = 1.4$	$K_{P S, k} = 1.4$	$K_{P S H} = 1.4$	$K_{R W, j} = 3$	$K_{R S, k} = 3$	$K_{R S H} = 3$

Table 6 compares the statistical results based on the minimum, average, maximum and standard deviation of the total generation cost obtained by INFO with the chosen algorithms: Black Widow Optimization Algorithm (BWOA), PSO, Gravitational Search Algorithm (GSA), Moth Flame Optimization (MFO), Harmony Search (HS), and BMO given in ref (Sulaiman and Mustaffa, 2020). as well as with implemented GTO and AEO. Through this comparison, it is clear that INFO outperformed the rest of the algorithms in all statistical results within 30 independent runs of simulation. The minimum value of total generation cost obtained by INFO is 788.9417 $/h. The proposed algorithm exceeds all optimization techniques, BWOA (791.4748), PSO (789.4849 $/h), GSA (790.3496 $/h), MFO (789.5271 $/h), HS (800.5362 $/h), BMO (789.1248 $/h), GTO (789.2231 $/h) and AEO (789.3185 $/h). The difference between the value obtained by INFO and the worst value obtained by HS is 11.5945 $/h which is very important around 278.268 $ cost saving per day and 101,567.82 $ cost saving per year.

Table 6.

Comparison of statistical results of case 1 for different algorithms.

Algorithms	Minimum	Maximum	Average	Std
$B W O A$ (Sulaiman and Mustaffa, 2020)	791.47	796.53	793.70	$1.20 E + 00$
$P S O$ (Sulaiman and Mustaffa, 2020)	789.48	799.67	793.78	$3.57 E + 00$
$G S A$ (Sulaiman and Mustaffa, 2020)	790.35	802.44	794.17	$3.19 E + 00$
$M F O$ (Sulaiman and Mustaffa, 2020)	789.53	793.94	790.89	$1.32 E + 00$
$H S$ (Sulaiman and Mustaffa, 2020)	800.54	802.25	801.83	$1.49 E - 01$
$B M O$ (Sulaiman and Mustaffa, 2020)	789.12	793.09	790.44	$1.04 E + 00$
GTO	789.22	791.48	790.13	$5.90 E - 01$
AEO	789.32	791.27	790.11	$5.76 E - 01$
INFO	788.94	790.61	789.73	$4.46 E - 01$

Table 7 presents optimal results of objective function as well as the optimal results of control and state variables related to case 1 obtained by different algorithms. It is clear that these last optimal values obtained by INFO, AEO and GTO are all within the permissible range as mentioned in Table 3. Figure 11 shows voltage profile of PQ buses for different algorithms related to case 1. It clearly appears that the voltages are within the allowed limits. The convergence curve of total generation cost utilizing the proposed INFO and the two other optimization techniques is depicted in Figure 12.

Table 7.

Optimal results obtained of case 1 for different algorithms.

Variable	BWOA (Sulaiman and Mustaffa, 2020)	PSO (Sulaiman and Mustaffa, 2020)	GSA (Sulaiman and Mustaffa, 2020)	MFO (Sulaiman and Mustaffa, 2020)	HS (Sulaiman and Mustaffa, 2020)	BMO (Sulaiman and Mustaffa, 2020)	GTO	AEO	INFO
P_TG1	134.9088	134.9079	134.7760	134.9079	134.9079	134.9128	134.91	134.91	134.93
P_TG2	41.6666	40.0000	45.2135	40.0048	65.0000	40.0000	42.58	40.45	41.99
P_WG	45.4876	46.4294	47.1036	46.2024	36.6795	46.9372	48.18	47.55	48.20
P_TG3	11.8272	10.0000	10.8455	10.0000	10.0000	10.0000	10.00	10.01	10.01
P_SG	41.8719	44.7704	38.4926	44.7271	31.8026	43.7995	39.66	41.54	40.09
P_SHG	13.8799	13.0326	12.8524	13.3049	11.6731	13.4839	13.97	14.79	14.06
V₁	1.0520	1.0835	1.0816	1.0831	1.0843	1.0835	1.077	1.076	1.075
V₂	1.0429	1.0689	1.0678	1.0686	1.0704	1.0691	1.064	1.060	1.061
V₅	1.0351	1.0467	1.0450	1.0463	1.0447	1.0476	1.039	1.037	1.041
V₈	1.0367	1.0489	1.0471	1.0488	1.0472	1.0493	1.040	1.041	1.041
V₁₁	1.0380	1.1000	1.1000	1.1000	1.1000	1.1000	1.098	1.100	1.099
V₁₃	1.0380	1.0527	1.0535	1.0530	1.0567	1.0524	1.052	1.067	1.061
Q_TG1	−15.4003	−0.2965	−1.4807	−0.4564	−0.2154	−0.7352	1.35	4.88	0.35
Q_TG2	10.9783	14.6635	15.5110	14.4673	14.8635	14.5828	25.81	15.88	19.74
Q_WG	35.6096	23.2576	22.7903	23.2508	24.4993	23.7295	24.11	24.79	27.66
Q_TG3	56.3782	36.8604	36.8058	37.0509	36.4881	37.0644	38.38	38.83	38.84
Q_SG	13.7126	28.5785	28.4866	28.6084	27.7191	28.4257	30.00	29.96	29.80
Q_SHG	18.9927	14.7221	15.4696	14.8853	16.3498	14.4774	17.04	22.46	20.30
F_Cost ($/h)	791.4748	789.4849	790.3496	789.5271	800.5362	789.1248	789.2231	789.3185	788.9417

P_TGi (MW), V_i (p.u.), and Q_gi (MVAr).

Case 2: Total Active Power Loss Minimization (TAPLM)

The minimization of total active power loss is the secondary aim in the OPF problem. Again, the comparison of statistical results of INFO with other different algorithms are illustrated in Table 8. Based on the results presented in this table, the minimum power loss of 2.0938 MW is obtained via INFO. The proposed algorithm exceeds all optimization techniques, AEO (2.1244 MW), BMO (2.1669 MW), MFO (2.1669 MW), PSO (2.1868 MW), BWOA (2.1876 MW), GTO (2.2212 MW), HS (2.5114 MW), and GSA (2.8962 MW).

Table 8.

Comparison of statistical results of case 2 for different algorithms.

Algorithms	Minimum	Maximum	Average	Standard deviation
$B W O A$ (Sulaiman and Mustaffa, 2020)	2.1876	$2.6771$	2.4063	$9.66 E - 02$
$P S O$ (Sulaiman and Mustaffa, 2020)	$2.1868$	3.8005	$2.4511$	$3.05 E - 01$
$G S A$ (Sulaiman and Mustaffa, 2020)	$2.8962$	4.3537	$3.3340$	$3.59 E - 01$
$M F O$ (Sulaiman and Mustaffa, 2020)	2.1669	$2.5215$	2.2151	$7.59 E - 02$
$H S$ (Sulaiman and Mustaffa, 2020)	$2.5114$	2.6544	$2.5789$	$3.90 E - 02$
$B M O$ (Sulaiman and Mustaffa, 2020)	2.1669	$2.5209$	2.1906	$6.50 E - 02$
GTO	2.2212	2.8313	2.4273	$3.50 E - 01$
AEO	2.1244	2.1813	2.1452	$3.14 E - 02$
INFO	2.0938	2.1790	2.1047	$1.87 E - 02$

Table 9 presents optimal results of objective function as well as the optimal results of control and state variables related to case 2 obtained by different algorithms while satisfying all constraints. The convergence curve of total active power loss is depicted in Figure 13.

Table 9.

Optimal results obtained of case 2 for different algorithms.

Variable	BWOA (Sulaiman and Mustaffa, 2020)	PSO (Sulaiman and Mustaffa, 2020)	GSA (Sulaiman and Mustaffa, 2020)	MFO (Sulaiman and Mustaffa, 2020)	HS (Sulaiman and Mustaffa, 2020)	BMO (Sulaiman and Mustaffa, 2020)	GTO	AEO	INFO
P_TG1	50.0000	50.0000	53.9396	50.0000	50.00134	50.0000	50.00	50.02	50.01
P_TG2	27.4484	40.0000	67.8207	30.0000	65.0000	30.0000	25.62	29.80	25.16
P_WG	73.2263	75.0000	61.1047	75.0000	75.0000	75.0000	75.00	74.95	74.99
P_TG3	45.0533	35.0000	23.3276	35.0000	33.9369	35.0000	35.00	34.91	34.99
P_SG	50.3881	50.0000	46.9904	50.0000	45.1813	50.0000	50.00	59.86	59.99
P_SHG	39.4714	35.5868	33.1132	45.5669	16.7919	45.5669	50.00	35.98	40.35
V₁	1.0521	1.0610	1.0638	1.0597	1.0639	1.0597	1.061	1.058	1.060
V₂	1.0479	1.0560	1.0594	1.0543	1.0600	1.0543	1.055	1.053	1.055
V₅	1.0402	1.0455	1.0435	1.0444	1.0479	1.0444	1.049	1.040	1.045
V₈	1.0476	1.0497	1.0474	1.0496	1.0498	1.0496	1.046	1.045	1.049
V₁₁	1.0581	1.1000	1.1000	1.1000	1.1000	1.1000	1.085	1.092	1.096
V₁₃	1.0568	1.0571	1.0564	1.0606	1.0536	1.0606	1.066	1.056	1.063
Q_TG1	−7.7466	−4.6581	−4.9102	−4.8064	−4.3041	−4.8052	−1.04	0.38	−0.89
Q_TG2	8.7748	7.3152	8.7765	6.9443	7.8094	6.9432	15.78	17.16	14.43
Q_WG	23.3476	20.7149	21.7427	20.5941	20.9463	20.5960	24.43	21.70	24.34
Q_TG3	44.3276	36.6244	36.4820	36.2915	37.4288	36.2921	40.00	39.23	39.45
Q_SG	17.3799	29.8316	29.7941	29.8880	29.1509	29.8884	25.00	29.96	29.98
Q_SHG	21.1666	17.1485	16.9013	18.6070	16.1932	18.6040	22.01	19.35	20.41
F_Loss (MW)	2.1876	2.1868	2.8962	2.1669	2.5114	2.1669	2.2212	2.1244	2.0938

Case 3: Generation cost with emission minimization

Minimizing the generation cost taking into account the emission effect is the focus of attention. As previously mentioned, reducing greenhouse gas emissions resulting from classical energy sources is a challenge. For this reason, the imposition of a carbon tax was resorted to as a penalty. Table 10 compares statistical results related to this case for different algorithms. Based on these results, the minimum value obtained by INFO is 820.5593 $/h which is close to the minimum value obtained by BMO 820.4852 $/h as the difference is negligible 0.0741 $/h. However, the proposed algorithm outperformed BMO in average, maximum and standard deviation.

Table 10.

Comparison of statistical results of case 3 for different algorithms.

Algorithms	Minimum	Maximum	Average	Std
$B W O A$	822.58	826.82	824.44	$1.11 E + 00$
$P S O$	820.81	835.15	824.55	$3.34 E + 00$
$GSA$	822.13	829.41	825.11	$1.90 E + 00$
$M F O$	820.81	825.89	821.34	$9.90 E - 01$
$H S$	827.02	827.78	827.67	$2.07 E - 02$
$B M O$	820.49	824.95	821.19	$8.30 E - 01$
GTO	821.75	822.66	822.25	$3.37 E - 01$
AEO	820.58	821.85	821.46	$5.27 E - 01$
INFO	820.56	822.08	821.19	$4.13 E - 01$

Renewable energy sources are clean, so it makes that electric power generation from these sources will increase due to the inclusion of a carbon tax. In the first case, the active power produced from the solar generator was 40.09 MW. While in the third case, when the carbon tax was imposed, the active power produced from the solar generator amounted to 45.52 MW with a significant increase of about 5.43 MW.

Table 11 presents optimal results related to case 3 obtained by different algorithms. The convergence curve of generation cost and emission is depicted in Figure 14.

Table 11.

Optimal results obtained of case 3 for different algorithms.

Variable	BWOA	PSO	GSA	MFO	HS	BMO	GTO	AEO	INFO
P_TG1	126.0739	126.5950	127.0432	126.5938	123.4563	126.69	126.42	127.21	126.63
P_TG2	43.1541	44.2211	45.6540	44.2121	65.0000	44.231	44.43	46.88	44.35
P_WG	46.6268	48.4997	45.5324	48.5097	43.4951	48.491	48.53	49.76	48.66
P_TG3	12.7203	10.0000	14.3060	10.0000	10.0000	10.000	10.00	10.00	10.01
P_SG	44.8928	45.0261	40.7565	45.0261	34.9329	45.026	45.33	41.38	45.52
P_SHG	15.5858	14.4528	15.5839	14.4528	12.4211	14.452	14.17	13.75	13.71
V₁	1.0587	1.0816	1.0818	1.0816	1.0824	1.0816	1.076	1.078	1.076
V₂	1.0437	1.0682	1.0683	1.0681	1.0703	1.0682	1.062	1.063	1.061
V₅	1.0353	1.0466	1.0460	1.0466	1.0469	1.0466	1.039	1.039	1.041
V₈	1.0359	1.0488	1.0493	1.0488	1.0483	1.0489	1.041	1.040	1.041
V₁	1.0587	1.1000	1.1000	1.1000	1.1000	1.1000	1.097	1.097	1.098
V₁₃	1.0437	1.0536	1.0539	1.0536	1.0559	1.0535	1.053	1.051	1.058
Q_TG1	−0.6057	−0.9694	−0.9707	−0.9946	−1.7444	−1.1055	4.93	5.08	5.10
Q_TG2	−2.5917	13.7453	13.9367	13.7523	14.5427	13.8389	19.25	22.26	15.50
Q_WG	34.0173	22.9881	23.2131	23.0018	23.9643	23.0134	25.20	24.29	27.24
Q_TG3	48.5122	37.1349	36.8448	37.1307	36.8996	36.9844	39.24	37.53	39.20
Q_SG	19.9778	28.6308	28.1882	28.6359	27.7266	28.5148	30.00	29.98	29.84
Q_SHG	19.2986	15.0950	14.9932	15.0995	15.8818	15.2158	17.51	16.76	19.32
F_CE ($/h)	822.5772	820.8064	822.1316	820.8071	827.0182	820.4852	821.749	820.5813	820.5593

The rest of study tested the performance of INFO again, but with the modified IEEE 57-bus test system. The results obtained by the proposed algorithm were compared again with AEO and GTO, while BMO, MFO, and PSO were selected from the rest of algorithms due to their efficiency. Combined solar PV and small hydro power generator are located at bus 6, while solar generator and wind generator are located at bus 9 and 12, respectively. The number of control variables to be minimized is 14 as mentioned in Table 1. Upper and lower bounds of control and state variables for modified IEEE 57-bus test system can be found in MATPOWER package.

Case 4: Total generation cost minimization

Table 12 presents the optimal results obtained by INFO and different algorithms. The minimum value of total generation cost obtained by INFO is 5259.2040 $/h. it is noticeable that the proposed algorithm outperforms all other algorithms GTO (5260.0009 $/h), AEO (5260.2497 $/h), BMO (5300.457 $/h), MFO (5316.14 $/h), and PSO (5417.538 $/h). The difference between the value obtained by INFO and the worst value obtained by PSO is 158.334 $/h which is very important around 3800.016 $ cost saving per day and 1,387,005.84 $ cost saving per year. It should be noted that GTO and AEO achieve results close to INFO, however, the difference between INFO and GTO is 0.7969 $/h that is, 6980.844 $/year. Figure 15 shows voltage profile of PQ buses for different algorithms related to case 4. Again, it clearly appears that the voltages are within the allowed limits. Figure 16 presents the convergence curve related to this case.

Table 12.

Optimal results obtained of case 4 for different algorithms.

Variables	BMO	MFO	PSO	GTO	AEO	INFO
P_TG1	555.5707	554.2932	500.4827	556.35	556.88	556.81
P_TG2	100	100	100	100.00	100.00	100.00
P_TG3	76.62421	76.62421	140	76.62	76.62	76.61
P_SHG	100	100	100	100.00	100.00	100.00
P_TG4	51.27178	54.47699	38.48657	51.68	51.23	51.23
P_SG	200	200	200	200.00	200.00	200.00
P_WG	210	210	210	210.00	210.00	210.00
V₁	1.1	1.1	1.1	1.059	1.051	1.067
V₂	1.1	0.95	1.1	1.048	1.043	1.055
V₃	0.95	1.1	0.95	1.014	1.020	1.019
V₆	1.024522	0.95	1.1	1.012	1.014	1.015
V₈	1.021023	1.052336	1.015266	1.016	1.013	1.014
V₉	0.95	1.1	1.1	0.987	0.985	0.984
V₁₂	0.98036	0.958536	1.1	0.976	0.975	0.968
Q_TG1	200	200	92.10943	125.02	98.26	143.76
Q_TG2	50	−17	50	49.98	50.00	49.99
Q_TG3	−10	60	−10	21.83	48.70	27.89
Q_SHG	25	−8	25	21.95	24.22	24.96
Q_TG4	106.2492	179.9646	18.25645	117.75	109.32	113.89
Q_SG	−3	9	9	−1.63	−1.48	−1.69
Q_WG	−1.47679	−46.2699	155	41.06	47.28	16.80
F_Cost ($/h)	5300.457	5316.14	5417.538	5260.0009	5260.2497	5259.2040

Case 5: Total active power loss minimization

The minimization of total active power loss is the secondary aim in the OPF problem using modified IEEE 57-bus test system. Again, the comparison of optimal results of INFO with other different algorithms are illustrated in Table 13. Based on the results presented in this table, the minimum power loss of 19.7040 MW is obtained via INFO. The proposed algorithm exceeds all optimization techniques, AEO (19.7633 MW), GTO (19.7703 MW), BMO (20.785 MW), MFO (21.3031 MW), and PSO (21.3621 MW). The convergence curve of total active power loss is depicted in Figure 17.

Table 13.

Optimal results obtained of case 5 for different algorithms.

Variables	BMO	MFO	PSO	GTO	AEO	INFO
P_TG1	262.6827	259.8313	273.4183	293.00	298.32	302.96
P_TG2	39.12372	39.25049	34.07384	7.57	0.99	3.32
P_TG3	140	140	140	140.00	139.98	139.93
P_SHG	100	100	100	100.00	100.00	100.00
P_TG4	319.7785	323.0214	314.67	320.00	321.26	314.31
P_SG	200	200	200	200.00	200.00	200.00
P_WG	210	210	210	210.00	210.00	209.99
V₁	1.045745	1.050806	1.0514	1.021	1.021	1.023
V₂	1.1	1.1	1.1	1.013	1.014	1.015
V₃	1.1	1.1	1.1	1.010	1.016	1.017
V₆	1.1	1.1	1.1	1.021	1.025	1.022
V₈	1.03229	1.030651	1.032615	1.038	1.038	1.029
V₉	0.95	1.099932	0.95	1.005	1.003	0.999
V₁₂	0.95768	0.950289	0.95	0.981	0.977	0.979
Q_TG1	105.8136	123.7601	123.7423	63.65	61.36	62.36
Q_TG2	50	50	50	50.00	49.91	48.10
Q_TG3	60	60	60	17.62	31.10	35.75
Q_SHG	25	25	25	−4.14	0.12	2.81
Q_TG4	66.64419	60.39987	71.64968	91.72	87.95	72.39
Q_SG	−3	9	−3	9.00	9.00	8.92
Q_WG	−21.5476	−43.207	−42.0513	53.93	42.13	51.22
F_Loss (MW)	20.785	21.3031	21.3621	19.7703	19.7633	19.7040

Case 6: Generation cost with emission minimization

In the last case, minimizing the generation cost considering the emission effect is the target. Table 14 presents optimal results related to this case for different algorithms. Based on these results, the minimum value obtained by INFO is 5295.8597 $/h. The proposed algorithm outperforms all optimization algorithms, AEO (5298.1921 /h), GTO (5299.6942 $/h), BMO (5320.851 $/h), PSO (5332.054 $/h), and MFO (5332.379 $/h). The difference between the value obtained by INFO and the worst value obtained by MFO is 36.5193 $/h which is very important around 876.4632 $ cost saving per day and 319,909.068 $ cost saving per year. It should be noted that AEO and GTO achieve results close to INFO, however, the difference between INFO and AEO is 2.3324 $/h that is, 20,431.824 $ /year. Figure 18 presents the convergence curve related to this case.

Table 14.

Optimal results obtained of case 6 for different algorithms.

Variables	BMO	MFO	PSO	GTO	AEO	INFO
P_TG1	551.8909	553.3165	554.1651	554.41	556.41	553.97
P_TG2	100	100	100	100.00	99.17	100.00
P_TG3	76.6242	76.62421	76.62421	76.63	76.63	76.55
P_SHG	100	100	100	100.00	100.00	100.00
P_TG4	52.19387	52.55224	51.77639	53.44	52.43	53.67
P_SG	200	200	200	200.00	200.00	200.00
P_WG	210	210	210	210.00	210.00	210.00
V₁	1.090978	1.078181	1.1	1.046	1.067	1.065
V₂	1.1	1.1	0.95	1.038	1.056	1.053
V₃	0.961542	1.036156	1.1	1.016	1.020	1.019
V₆	1.1	1.1	1.1	1.012	1.008	1.014
V₈	1.044066	1.011965	1.001535	1.012	1.018	1.013
V₉	0.95	1.019763	1.1	0.989	0.988	0.985
V₁₂	1.1	0.985229	0.985591	0.979	0.964	0.968
Q_TG1	65.39634	130.9053	200	87.78	146.51	138.29
Q_TG2	50	50	−17	48.33	49.94	49.62
Q_TG3	−10	42.54005	60	45.12	34.95	30.31
Q_SHG	25	25	25	24.93	5.78	24.98
Q_TG4	63.4618	78.66775	61.37461	101.07	126.76	109.10
Q_SG	−3	9	9	9.00	8.80	3.02
Q_WG	155	27.21733	26.59974	59.21	3.09	18.53
F_CE ($/h)	5320.851	5332.379	5332.054	5299.6942	5298.1921	5295.8597

Statistical study analysis

Since the nature of stochastic algorithms, the algorithm must be executing numerous times on the same treated problem to get result values, because the results may vary from run to another. To make a fair comparison, a statistical tool based on a one-way analysis of variance (ANOVA) test was used to evaluate the superiority of the proposed algorithm as well as to highlight a certain level of confidence to our study. In short, to assess the mean of different implemented algorithms in which there is a significant difference. This test is necessary to assess the mean of different implemented algorithms in which there is a significant difference. In this study, the significance level is set to $0.05$ ; if the $P_{Value}$ is lower than 5%, a significant difference between the proposed and other optimizers happens. Tables 15 and 16 list the findings of one-way ANOVA on all optimizers for both electric grids.

Table 15.

Statistical results of ANOVA test, for IEEE 30 bus.

Test System I: Modified IEEE 30-bus
variance	SS	df	MS	F	p-Value
Between groups	3.0463	2	1.5233	5.49	0.0074
Within groups	25.5162	87	0.29329
Between groups	28.5625	89

Table 16.

Statistical results of ANOVA test, for IEEE 57 bus.

Test System II: Adrar power Network
variance	SS	df	MS	F	p-Value
Between groups	1237.5	2	618.768	3	0.0549
Within groups	17,934.2	87	206.14
Total	19,171.7	89

Conclusion

This paper presents a recent population-based algorithm called INFO for solving stochastic OPF problem in the hybrid power system. To verify the consistency of the suggested algorithm to find near-optimal solution of OPF problem, three single objective-functions were examined. The main target is to minimize the total generation cost, total active power loss and generation cost considering emission effect on two modified test systems while satisfying equality and inequality constraints. The obtained results demonstrate the superior capability of INFO in all cases compared to other algorithms, such as AEO, BMO, GTO, MFO, and PSO. The total generation cost obtained via INFO is 788.9417 $/h for IEEE 30-bus test system and 5259.2040 $/h for IEEE 57-bus test system. This means a cost reduction of 1.47% and 3.01% per hour as compared to the worst results obtained by HS and PSO respectively. The results proved that the proposed algorithm outperformed all the compared algorithms while ensuring the feasibility of solutions.

Footnotes

Acknowledgements

I would like to thank very much the Laboratory of Engineering Materials, Energy Systems, Renew-able Energies and Energy-Management (LMSEERGE) in Laghouat, and Special thanks to Bouira University of represented by the Department of Electrical Engineering where the work was fully achieved under supervision of Prof. S, MOUASSA.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Souhil Mouassa

Francisco Jurado

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