Optimal power flow solution based on gorilla troops optimization technique considering uncertainty of renewable energy sources: A case study of Adrar’s isolated power network

Abstract

This paper proposes an efficient Gorilla troops-inspired algorithm to cope optimal power flow (OPF) problem considering uncertainty of renewable energy sources (RES). The problem is formulated as large-scale constrained optimization problem with non-linear characteristics. Its degree of complexity increases with incorporation of intermittent energy sources, making it harder to be solved using conventional optimization techniques. However, could be efficiently resolved by nature-inspired optimization algorithms and solvers. The objective function is the overall cost of system, including reserve cost for over-estimation and penalty cost for under-estimation of two types of PV-solar and wind energy. To demonstrate the consistency and robustness of the developed algorithm a case study on the modified IEEE 30-bus system and and Adrar’s power network (isolated grid) is carried out. Simulation results show the capability of GTO to find high quality optimal feasible solutions and ranked first among the compared algorithms, and so, over different function landscapes.

Keywords

Optimal power flow emission renewable energy sources Adrar’s power system uncertainty gorilla troop optimizer orca predation algorithm

Introduction

Optimal power Flow (OPF) is one of primordial tools of electric power systems, offering electric power at minimum-cost and high quality. In short, it’s the backbone of electric grids due to the important role that plays in maintaining operational reliability and economics of power systems. OPF master objective is to find optimal adjustment of the decision variables so that a selected objective function is optimized while satisfying different physical and operational-constraints inflicted by electric power grid (equality and inequality constraints). The most commonly objective-function is minimization of overall generation cost. However, other functions are minimization of gas emission, real power loss, voltage stability-index (VSI), and bus voltage-deviation [1]. While used control-variables are: active power of generators outputs, generator voltages magnitudes, positions of the transformer taps, and contributions of the compensators in terms of reactive power.

In traditional electric networks, study of OPF was based on the conventional power generators which run on fossil-fuels. However, with electricity market liberalization, and integration of renewable energy sources (RESs), study of OPF is becoming more complicated leading in raise the complexity of its objectives, significantly. This is due to the diverse functions based on the variability and uncertain used in its problem formulation. The prime objective behind incorporation of renewable generators (WG + SG) in the grids is to reduce the active power losses in transmission lines in improving the reliability and quality of electric grids. Also they reduce environmental pollution (Biswas et al., 2017). In addition, with increasing of injected power from RES, specifying optimal contribution of each generator in the system is necessity. Thus (Also), energy management and optimal scheduling of different resources could facilitate diverse missions of electric power system operator, ultimately reducing total generation electricity cost.

In the few past decades, numerous conventional optimization techniques have been applied to solve different versions of OPF problem. The conventional solvers are the Newton method (Bjelogrlic et al., 1990; Santos and da Costa, 1995), non-linear programing (NLP) (Luo and Semlyen, 1989) and interior point methods (Momoh and Zhu, 1999). Despite the fact that some of abovementioned methods have excellent convergence characteristics and some of them are usually suitable for industry applications. However, they have some weaknesses, which are summarized as follows:

Sensitivity to the initial search point, that is, they might converge easily to local solutions as may converge to global ones.

Lack of flexibility with respect to practical systems, that is, each method is suit for a specific problem formulation in its proper objectives and/or constraints.

Besides the inflexibility aspect, they also encounter a huge difficult to set of uncertain and stochastic problems, such as OPF with application of renewable generation.

Therefore, developing new and effective optimization methods is necessity in effort to overcome the shortcomings of the traditional optimization techniques’ (Hinojosa and Araya, 2013). Thanks to the computational intelligence schemes and open access to optimization techniques have liberated considerable researches in the field of meta-heuristic algorithms to solve complex optimization problems during first decade. These optimizers have ability to provide near-global solutions and capability to escape local ones, avoiding in premature convergence. Many meta-heuristic optimization algorithms have been implemented to cope with classical OPF problem like improved version of PSO (Singh et al., 2016), moth swarm algorithm (MSA) (Mohamed et al., 2017), improved bacterial forging method (IBF) (Amjady et al., 2012), teaching-learning-based optimization (TLBO) technique (Bouchekara et al., 2014), backtracking-search algorithm (BSA) (Chaib et al., 2016), improved colliding-bodies optimizer (ICBO) (Bouchekara et al., 2016), and adaptive multiple teams perturbation-guiding Jaya (AMTPG-Jaya) algorithm (Warid, 2020). While aforementioned references are limited on the thermal power generators only. In the few past years, a system with mixed resources involving thermal, wind and solar generators have been studied in quest of provide electrical energy at minimum generation-cost with high-quality. As mentioned earlier, electricity market allows the incorporation of renewable energy sources into the electricity grids in order to minimize the environmental problems and enhancement of load relief on a transmission lines as well system voltage profile control by transmission line active power losses reduction. In that context, a few works have been published in literatures. For instance in Elattar and ElSayed (2019) modified Jaya algorithm is applied to solve OPF incorporating RES considering four different objective functions to improve recorded results against other optimizers while the RES is modeled as a negative load, but any forecasting technique was not employed to forecast wind and solar photovoltaic power output. The results show outperforms of MJAYA on the basic Jaya as well on other existing algorithms. Biswas et al. (2018) proposed an adaptive version of differential evolution-based technique (SHADE) to solve OPF problem in a system involving renewable power generators. To forecast wind power and solar-photovoltaic production, authors used weibull and lognormal probability-distribution-functions (PDF). In addition, the feasibility of results was discussed and checked that all control variables fell inside the allowed limits. Thus, findings clearly show the efficacy of the proposed model, but, unfortunately, it was applied only on medium-sized test system, IEEE 30-bus. In another publication Elattar (2019) proposed modified version of the moth swarm algorithm to solve OPF problem of combined heat and power system with presence stochastic wind farm. The model is well presented and results were discussion but only for IEEE 30-bus system in which feasibility of solution of large-scale test system IEEE 118-bus were not discussed. As well, application of suggested model on a practical power grid was not conducted. (Ullah et al. (2019) provide a new hybrid optimization algorithm PPSOGSA for OPF solution considering renewable energy generators. The model of stochastic behavior is based of PDF scheme. The results amply show the superiority of proposed hybrid method against basic PPSO and GSA. Again, however, the algorithm was not examined by applying it to real/large-sized power system. In Chang et al. (2014), evolutionary particle swarm optimization (EPSO) algorithm was used for solving OPF problem in a wind-thermal power system. The suggested wind model is based on the up-spinning and down-spinning reserves of the production units. But, the approach was also evaluated only on modified IEEE 30-bus system and large-scale power systems were not taken into consideration when validating the proposed model. A modified cuckoo search optimization technique employed for OPF solution incorporating wind power was proposed in Mishra et al. (2015). The proposed stochastic model of wind generation is based on the Weibull PDF. Again, however, the simulation was also conducted only on standard medium-sized test systems.

In view of aforementioned works, published results are promising and encouraging. But bear in mind that in spite of all efforts carried out in this area since half-a-century ago, topic is remains open for research and also worthy of further attention. On the other hand, despite the success of many optimization methods in realizing satisfactory results, but still suffer from some limitations and shortcomings as far as their susceptibility of falling into local optima and the difficulty of tuning the main intrinsic parameters. More precisely, none of them can guarantee finding the optimal solution for all optimization problems. Moreover, application of these algorithms on larger scale or real-sized electric grids is uncommon. Consequently, these gaps give an opportunity to suggest or develop effective meta-heuristic techniques able deal different OPF formulations.

In this paper, GTO algorithm is proposed to deal with OPF problem in the presence RES and different objective functions. The proposed technique is examined on the modified- test system IEEE 30-bus and real power system of Adrar located at isolated site in Algerian desert. In addition, penalty function method (PF) method is used herein to handle different constraint of stochastic OPF problem. In June 2021 a novel metaheuristic algorithm titled Gorilla troops optimizer (GTO) has been proposed by Abdollahzadeh et al. (2021), which simulates the social organizations of gorillas in forest. GTO has excellent randomness properties, which makes it search for all optimal solutions in the search-space without trapping into local-optimum. In addition, it has a simple concept and is easy to implement.

Problem formulation and objective functions

The main objective of OPF is to find the optimal settings of control-variables so that the specified objective-function is minimized while satisfying all constraints imposed (equality and inequality). Mathematically is formulated as follows:

Minimize F_{obj} (x, u)

(1)

Subjet . to, g (x, u) {\begin{matrix} g (x, u) = 0 \\ h (x, u) \leq 0 \end{matrix}

(2)

where $F_{F_{Obj}} (x, u)$ is the objective function, $g (x, u)$ defines equality constraints, $h (x, u)$ inequality constraints. $x$ and $u$ are the vector of dependent variables and the vector of control variables, respectively. For obtaining the optimality and guarantee the feasibility of solutions, dependent variables also should be within the allowable limits, which play an essential role in the security of electric power system.

Objective functions

In this work, three objectives will be minimized, cost, power loss, and gas emissions of thermal units.

Thermal power only units

Fuel cost of thermal power units can be described as (Elattar, 2019)

C_{T 0} (P_{TG}) = \sum_{i = 1}^{N_{TG}} a_{i} + b_{i} P_{TGi} + c_{i} P_{TGi}^{2}

(3)

For more realistic pattern and precise modeling valve-point effect scheme is considered. The cost function C under valve-point loading rewritten as follows:

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

(4)

Where $a_{i}, b_{i}, c_{i}, d_{i}, and e_{i}$ are the cost coefficients of the i^-th thermal generators producing power output $P_{TGi}$ , $N_{TG}$ is the number of thermal generating and $P_{TG}^{min}$ is the minimum of power of conventional thermal generator. The cost and emission gas coefficients for the conventional units used here are provided in Biswas et al. (2018).

Since wind and solar generators does not require any fuel like conventional thermal generators, cost-function evaluation of the wind and solar obey of some norms. The first norm is direct-cost for wind generator $Cos t_{W, j}$ and solar generator $Cos t_{S, k}$ . Mathematically expressed as follows:

Cos t_{W, j} (P_{WG, j}) = g_{j} P_{WG, j}

(5)

Cos t_{S, k} (P_{SGs, k}) = h_{k} P_{SG, k}

(6)

where $g_{j}$ and $h_{k}$ respectively, are the coefficient of direct-cost attached with jth wind power plant and kth solar power unit. $P_{WG, j}$ and $P_{SG, k}$ are the scheduled power from the same power plants (wind, solar).

Under the uncertainties, there are two possible scenarios: (1) if actual power-delivered by the wind farm or solar generator is less than the estimated-values, this scenario called as overestimation of power, herein the system operator needs to the spinning reserve to ensure uninterrupted supply to the consumers. The cost of committing the reserve production units to meet overestimated quantity is named as reserve-cost (Biswas et al., 2017). The reserve cost for wind and solar power units is written with following equations:

\begin{matrix} Cos t_{RW, j} (P_{WG, j} - P_{Av, j}) = K_{RW, j} (P_{WG, j} - P_{WAv, j}) \\ = K_{RW, j} \int_{0}^{P_{WG, j}} (p_{WG, j} - P_{W, j}) f_{W} (P_{W, j}) d p_{W, j} \end{matrix}

(7)

\begin{matrix} Cos t_{RS, k} (P_{SG, k} - P_{SAv, k}) = K_{RS, k} (P_{SG, k} - P_{SAv, k}) \\ = K_{RS, k} * f_{S} (P_{SAv, k} < P_{SG, k}) * [P_{SG, k} - E (P_{SAv, k} < P_{SG, k})] \end{matrix}

(8)

where $K_{RW, j}$ denotes coefficient of reserve-cost pertaining to wind power plant, $P_{WA, j}$ is the actual available-power from the same plant. $f_{W} (P_{W, j})$ is the Wind power PDF for j^th power plant where and more detail is given in Biswas et al. (2017). $K_{RS, k}$ is the coefficient of reserve-cost for kth solar-generator. $P_{SAv, k}$ is the actual available power from the same power-plant.

$f_{S} (P_{SAv, k} < P_{SG, k})$ is the probability of solar power-shortage occurrence than the scheduled power ( $P_{SG, k}$ ), $f_{s} (P_{SAv, k} > P_{SG, k})$ is the expectation of solar-power above $P_{SG, k}$ .

Contrary of overestimation, the second scenario called the under estimation of wind/solar power plant. In this scenario the actual power produced is higher than the estimated one, yielding the surplus power. This situation requests introduce the penalty cost against each surplus amount of power, where expressed by the following equations.

\begin{matrix} C_{PW, j} (P_{WAv, j} - P_{WG, j}) = K_{PW, j} (P_{WAv, j} - P_{WG, j}) \\ = K_{PW, j} \int_{P_{WG, j}}^{P_{Wr, j}} (P_{W, j} - p_{WG, j}) f_{W} (P_{W, j}) d p_{W, j} \end{matrix}

(9)

\begin{matrix} C_{PS, k} (P_{SA, k} - P_{SG, k}) = K_{PS, k} (P_{SAv, k} - P_{SG, k}) \\ = K_{PS, k} * f_{S} (P_{SAv, k} > P_{SG, k}) * [E (P_{SAv, k} > P_{SG, k}) - P_{SG, k}] \end{matrix}

(10)

where $K_{PW, j}$ is the penalty cost coefficient for the j-th wind power plant, $P_{Wr, j}$ is the rated output-power from the same wind-farm. $K_{PS, k}$ is the coefficient of penalty-cost for k^th solar PV plant. $f_{S} (P_{SAv, k} > P_{SG, k})$ is the probability of solar power surplus that is, actual power above the scheduled power ( $P_{SG, k}$ ), $f_{S} (P_{SAv, k} > P_{SG, k})$ is the expectation of solar power above $P_{ss, k}$ . $E (P_{SAv, k} > P_{SG, k})$ .

It is important to indicate that the cost evaluation for wind generator and solar PV unit are depends on the wind Weibull probability-distribution-functions (PDF) and solar radiation by lognormal PDF respectively (Sulaiman and Mustaffa, 2020).

Objective 1: In the first objective function, OPF is formulated with presence of RES, whereby all the cost-functions aforementioned are included. This objective minimizes the generation cost without including emission cost. Mathematically can be expressed as:

\begin{matrix} F_{Obj}^{1} = C_{T} (P_{TG}) = \sum_{j = 1}^{N_{WG}} [C_{W, j} (P_{WGs, j}) + C_{RW, j} (P_{WG, j} - P_{WAv, j}) + C_{PW, j} (P_{WAv, j} - P_{WG, j})] \\ + \sum_{k = 1}^{N_{SG}} [C_{S, k} (P_{SG, k}) + C_{RS, k} (P_{SG, k} - P_{SAv, k}) + C_{PS, k} (P_{SAv, k} - P_{SG, k})] \end{matrix}

(11)

Where $N_{WG}$ and $N_{SG}$ denote the number of wind generators and PV solar in the grid.

Objective 2: Emission gases and Carbon tax

Conventional thermal power generators emits harmful gases into the environment such as $S O_{X}$ , $N O_{X}$ , and $C O_{2}$ , which pollute the atmosphere. To reduce emission of greenhouse gases, the carbon tax was imposed as penalty. This end can be achieved through minimization of generation and emission cost which can be expressed as follows:

Second objective function Minimize-

F_{Obj}^{2} = F_{Obj}^{1} + C_{tax} \times E

(12)

E = \sum_{i = 1}^{N_{TG}} [(α_{i} + β_{i} P_{TGi} + γ_{i} P_{TGi}^{2}) \times 0.01 + ω_{i} \exp (μ_{i} P_{TGi})]

(13)

where $α_{i}, β_{i}, γ_{i}$ ω_i, µ_i are the emission-coefficients corresponding to the i-th generator and $C_{tax}$ is the carbon tax, which is equal to 20 ($/h).

Objective 3: minimization of voltage stability index

Power network monitoring and control has acquired great significance in the design, planning, and operation of modern electric power systems throughout voltage stability enhancement in an effort to get more information’s on the voltage drops especially with penetration of renewable energies. The operating interval of index L was set in [0, 1]. So, third objective function minimizes the voltage-stability-index in the transmission grid can be formulated as follows:

F_{Obj}^{3} = \min (VSI) = min (max (L_{j}))

(14)

where $L_{j}$ of the $j^{th}$ bus is given by the following expression:

L_{j} = | 1 - \sum_{i = 1}^{N_{PV}} F_{ji} \times \frac{V_{i}}{V_{j}} ∠ {θ_{ji} + (δ_{i} - δ_{j})} | j = 1, 2, \dots, N_{PQ}

(15)

with F_{ji} = | F_{ji} | ∠ θ_{ji}, V_{i} = | V_{i} | ∠ δ_{i}, V_{j} = | V_{j} | ∠ δ_{j}, F_{ji} = - {[Y_{1}]}^{- 1} \times [Y_{2}]

(16)

Y₁, Y₂, Y₃, and Y₄: are the sub-matrices of the system $Y_{bus}$ obtained after rearranging the PQ and PV-bus bar parameters as shown in equation (17).

[\begin{matrix} I_{PQ} \\ I_{PV} \end{matrix}] = [\begin{matrix} Y_{1} Y_{2} \\ Y_{3} Y_{4} \end{matrix}] \times [\begin{matrix} V_{PQ} \\ V_{PV} \end{matrix}]

(17)

Where $V_{i}, V_{j}$ Voltage magnitude of bus i and j, $G_{k}$ is the conductance of kth branch connected between bus i and j. $δ_{ij}$ is the voltage angle difference between $i$ and bus $j$ . $NTL$ is the number of transmission lines.

System constraints

Equality-constraints: Are the power flow equations which are given below

{\begin{matrix} P_{Gi} - P_{di} - \sum_{j = 1}^{NB} | V_{i} | \times | V_{j} | \times | Y_{ij} | \times \cos \times (θ_{ij} - δ_{i} + δ_{j}) = 0 \\ Q_{Gi} - Q_{di} - \sum_{j = 1}^{NB} | V_{i} | \times | V_{j} | \times | Y_{ij} | \times \sin \times (θ_{ij} - δ_{i} + δ_{j}) = 0 \end{matrix}

(18)

Inequality constraints: Represent the limits applied on the following variables

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

(19)

P_{wsj}^{min} \leq P_{ws j} \leq P_{wsj}^{max}, j = 1, 2 \dots . . N_{WG}

(20)

P_{ss, k}^{min} \leq P_{ss, k} \leq P_{ss, k}^{max} k = 1, 2 \dots . . N_{SG}

(21)

Q_{TGi}^{min} \leq Q_{TGi} \leq Q_{TGi}^{max},

(22)

Q_{wsj}^{min} \leq Q_{wsj} \leq Q_{wsj}^{max}, i = 1, 2 \dots . . N

(23)

Q_{ss, k}^{min} \leq Q_{ss, k} \leq Q_{ss, k}^{max} k \in N_{SG}

(24)

Q_{Ci}^{min} \leq Q_{Ci} \leq Q_{Ci}^{max} i \in N_{C}

(25)

V_{Gi}^{min} \leq V_{Gi} \leq V_{Gi}^{max},

(26)

V_{Li}^{min} \leq V_{Li} \leq V_{Li}^{max} i \in NL

(27)

Security constraints

T_{k}^{min} \leq T_{k} \leq T_{k}^{max} k \in NT

(28)

S_{i} \leq S_{i}^{max} i \in NTL

(29)

Equations (19)–(21) are the active power limits of conventional power-plants, wind and solar power generators, respectively. equations (22)–(25) are the reactive power capabilities of conventional power-plants, wind-/solar generators and shunt reactive power sources.

Equation (26) shows the constraints applied at the generators voltage, whereas equation (27) represents the voltage limits constraining load-buses, NL being the number of load-buses. Security-Constraints of: tap changing transformer and line capacity are given by equations (28) and (29), respectively. NTL is the number of lines in the electric grid.

In handling constraints, one of the first widely adopted approaches employed is static-penalty function method. If the constraints are violated, a large penalty term is added to the objective-function to punish these violation, thus guaranteeing the feasibility of solutions.

GTO-based proposed algorithm

In June 2021 a novel metaheuristic algorithm titled Gorilla troops optimizer (GTO) has been proposed by Abdollahzadeh et al. (2021), which simulates the social organizations of gorillas in forest. Its life style is based on five strategies, which ordered as follows: migration to other unknown regions, moving toward other group of gorillas, migration into direction of determined position, adhering silver-back, and competition for adult-females. These strategies are employed to explain the optimization process which based on the exploration and exploitation phases. Exploration phase uses the three first aforementioned mechanisms, while latter two mechanisms which are: adhering silver-back, and competition for adult-females are dedicated to explain exploitation phase (Abdollahzadeh et al., 2021). One of the most interesting characteristics of gorillas is that they very intelligent in which last study finds today’s gorillas may be smarter than human. They can use simple tools and learn sign language. Such characteristic guarantees a smart exploration that allows to discover different regions in search-space while avoiding to falling into local optima. Each individual in the gorillas group has a specific task’s to accomplish, for instance Silverback is the leader element in group, which plays an important role in the group by making all the decisions, takes responsibility for the group’s safety well-being, mediates the fights, determines the movements of group, and guides the gorillas to food-sources. (Abdollahzadeh et al., 2021)

In the GTO algorithm, each gorilla represents a possible solution to the optimization problem, and the best- candidate solution is regarded as a silverback-gorilla. For better understanding the optimization process of GTO, different mechanism in exploration and exploitation abilities are described below.

Exploration phase

All gorillas in GTO are considered as candidate solutions of problem, and at every optimization process stage, the Silver-back gorilla is considered as the best candidate solution. Exploration stage is based on three different behaviors, which are: migration to a non-identified position for raising exploration ability of GTO. Gorillas movement’s with others gorillas for balancing exploration and exploitation, and migration in an identified location’s direction to enhance the GTO capability to search for various optimization-spaces.

When (p) superior of random value (rand), only migration to a non-identified position is chosen. When rand≥0.5, gorilla’s movement’s toward others ones is ensured. However, when rand <0.5, a migration in an identified location’s direction is selected. These three behaviors to the exploration framework can be mathematically formulated by the following equation:

GX (t + 1) = {\begin{matrix} (ub - lb) \times r_{1} + lb, rand < p \\ (r_{2} - C) \times X_{r} + L \times H, rand \geq 0.5 \\ X (i) - L \times (L \times (X (t) - G X_{r} (t)) + r_{3} \times (X (t) - G X_{r} (t))), rand < 0.5 \end{matrix}

(30)

Where $GX (t + 1)$ is the gorilla-candidate location vector in the forthcoming-iteration. $X (t)$ is the present vector of the gorilla location, while, the indices $r_{1}$ , $r_{2}$ , $r_{3}$ , and rand are the random numbers in the range [0, 1]. $p$ parameter represents the probability of choosing between two behaviors, that takes value in a range 0–1. $ub, lb$ , respectively represent the upper and lower limits of control variables. $X_{r}$ and $G X_{r}$ is one of the gorillas’ individuals in the group randomly chosen from the population. The variables C, H, and L can be mathematically represented by the following formulas:

C = F \times (1 - iter / Maxiter)

(31)

F = \cos (2 \times r_{4}) + 1

(32)

L = C \times l

(33)

H = Z \times X (t)

(34)

Z = [- C . C]

(35)

where the current iteration and the maximum number of iteration are indicated by $iter$ and $Maxiter$ , respectively during optimization process, while $r_{4}$ is a randomly generated value with a range of [0, 1], and the symbol “ $\cos$ “ is the cosine function. Furthermore, symbols $Z$ and $l$ are the random numbers with a range of [−C, C] and [−1, 1], respectively.

In ending stage of exploration phase, the cost of all solutions is evaluated, and if the cost of $GX (t)$ has inferior than $X (t)$ , it is replaced with the $X (t)$ solution and becoming the best solution (Silverback) so far.

Exploitation phase

At the exploitation stage, two strategies are executed which are track the silver-back and competition on adult-females. Based on the comparison selection between the value of $C$ in equation (31) and the setting parameter (W), one of two aforementioned strategies can be selected, as indicated in flowchart of GTO below.

The leader of the group is the silverback, which is the responsible on other individuals and guide them toward different food sources. This behavior is chosen when the value of parameter C is greater or equal than setting parameter W. It can be represented mathematically by equation (36). Figure 1, shows how the location of search agent vectors shifts during the exploration stage.

GX (t + 1) = L \times M \times (X (t) - X_{Silverback}) + X (t)

(36)

Where $X (t)$ is the position-vector of the gorillas, and $X_{Silverback}$ is the position vector of the best solution, whereas symbols L and M can be calculated by using equations (33) and (37), respectively.

M = {({| (\frac{1}{N}) \sum_{i = 1}^{N} GX (t) |}^{g})}^{\frac{1}{g}}

(37)

g = 2^{L}

(38)

$GX (t)$ represents the position of vector of each candidate gorilla’s in iteration t. $N$ gives total number of gorillas. The second strategy is chosen when $C < W$ . After a period, when young-gorillas reach a puberty age, they start to fight with other males enlarging their group to choose adult females, and that competition-behavior is usually violent. This behavior is expressed mathematically by equation (39):

G X_{i} (t) = X_{Silverback} - (X_{Silverback} \times Q - X (t) \times Q) \times A

(39)

Q = 2 \times r_{5} - 1

(40)

A = β \times E

(41)

E = {\begin{matrix} N_{1} rand \geq 0.5 \\ N_{2} rand < 0.5 \end{matrix}

(42)

In equation (39), $X_{Silverback}$ is the position vector of the best silverback while X(t) is the current gorilla-position vector. Symbol Q, reflects the impact force. It is calculated using equation (40), whereas, r5 takes random values within range [0, 1]. Coefficient vector A is used to measure the degree of violence in conflicts. Parameter $β$ is a determined-value before the optimization maneuver. And E is employed to reflect the violence effect on the solutions’ dimensions. In ending stage of exploitation phase, the cost of all solutions is evaluated, and if the cost of $GX (t)$ has inferior than $X (t)$ , it is replaced with the $X (t)$ solution and becoming the best solution (Silverback) so far. The flowchart of proposed algorithm is presented in Figure 2.

Simulation result and discussions

Test system I: Modified IEEE 30-bus test system

In this section, the proposed GTO is applied to some cases of OPF problem, and some other algorithms are selected for comparison with GTO to further validate its performance. To test the capacity and feasibility of the proposed GTO algorithm in solving stochastic OPF problem, it examined on the modified IEEE 30-bus test system under different scenarios. The modification is to insert two wind generators at buses #5 and #11along with one solar generator at bus #13. All data ca be retrieved in Sulaiman and Mustaffa (2020) and Zimmerman et al. (2011). All implemented algorithms have been coded and solved under Matlab R2014a platform, and run on an Intel^® Core™ i5-4300U 2.50 GHz 4.00 GB RAM personal computer. For all selected cases the population-size is set as 30 individuals; the maximum number of iterations is fixed 300 for IEEE 30-bus and 200 for Adrar power network. Moreover, to demonstrate the consistency and robustness of GTO, 30 independent runs were conducted for each case and the best value of the objective-function obtain with corresponding control-variables are registered. Table 1 reports description of all test-systems characteristics used in this article. For the purpose of a fair comparison, all control variables of test systems were considered as continuous. Table 2 gives description of renewable energy source characteristics used herein.

Table 1.

Electric power-networks characteristics description.

Parameter	IEEE 30-bus	Adrar Power Network
Buses, NB	30	26
Generators, NG	6	7 + 2 RES (WG + PV)
Branches, NE	41	171
Control variables	12	18
Base case for PLoss, p.u.	5.880	4.373
Base case for VD, p.u.	1.4942	0.3727
Base case for L-index	0.1798	0.2564
Algorithms parameter setting
Population size Np	30	30
Max Iteration	300	200

Table 2.

PDF parameters of Wind-power and solar power system, IEEE 30-Bus.

Wind-power generating units
Position of Windfarm	No of turbines	Rated power, $P_{wr}$ (MW)	Weibull PDF parameters	Weibull mean, $M_{wbl}$
Bus # 5	25	75	c = 9 k = 2	v = 7.976 m/s
Bus #11	20	60	c = 10 k = 2	v = 8.862 m/s
Photovoltaic power plant
Position of Solar system	Rated power, $P_{rv}$ (MW)		Lognormal PDF parameters	Lognormal mean
(JB 13)	50		l = 6 r = 0.6	G = 483 W = m²

Simulation results of test system I

Results on case 1

In this case, the objective-function is minimization of the total cost of generation. Obtained findings by using proposed algorithm GTO are based on the Weibull PDF parameters. Figures 3 to 5 represent Weibull fitting and wind distribution obtained from the simulation of 8000 Monte Carlo scenarios, while the stochastic power-output of solar photovoltaic unit is illustrated by Figure 6. Table 3 gives all PDF parameters used of renewable energy sources.

Table 3.

Optimal results comparison for different algorithms for IEEE 30-bus Case 1.

Algorithm	Min	Max	MFO (Sulaiman and Mustaffa, 2020)	BMO (Sulaiman and Mustaffa, 2020)	SHADE-SF (Biswas et al., 2017)	JS (Farhat et al., 2021)	OPA	AEO	HGS	GTO
^*P_TG1	50	140	134.90	134.907	134.908	134. 905	134.91	134.91	134.907	134.907
P_TG2	20	80	28.471	26.6024	28.564	29.0226	27.0785	27.4771	28.8317	28.1779
P_WG1	0	75	44.644	43.8165	43.774	43.9696	43.0040	42.3477	42.8527	43.2909
P_TG3	10	35	10.000	10.0001	10	10.0006	10.0003	10.0000	10.0000	10.0000
P_WG2	0	60	36.588	36.0467	36.949	37.0193	36.4374	37.3504	35.3547	36.5917
P_SG1	0	50	34.460	37.8118	34.976	34.2532	37.7603	37.1256	37.2989	36.1438
V₁	0.95	1.1	1.0798	1.0814	1.072	1.07725	1.0724	1.0727	1.0757	1.0725
V₂	0.95	1.1	1.0647	0.95	1.057	1.05698	1.0568	1.0585	1.0606	1.0578
V₅	0.95	1.1	1.0428	1.0453	1.035	1.03507	1.0346	1.0352	1.0385	1.0374
V₈	0.95	1.1	1.1	1.0492	1.04	1.03705	1.0392	1.0390	1.0422	1.0395
V₁₁	0.95	1.1	1.1	1.1	1.1	1.0983	1.0981	1.0987	1.1000	1.1000
V₁₃	0.95	1.1	1.0583	1.0680	1.055	1.04571	1.0566	1.0594	1.0582	1.0548
Q_TG1	−20	150	−1.4418	18.050	−1.903	−0.6835	−0.8914	−3.67768	1.97011	−2.6423
Q_TG2	−20	60	12.139	−20	13.261	11.0011	12.3311	16.87073	19.89941	12.5121
Q_WG1	−30	35	22.405	30.344	23.181	22.6673	23.1802	22.88825	26.51019	4.65643
Q_TG3	−15	40	40.00	40.0	35.101	40.0	34.3065	32.37013	39.23979	32.047
Q_WG2	−25	30	28.294	27.8687	30	30	29.9846	29.99762	30.000	29.7031
Q_SG	−20	25	14.436	19.9978	17.346	14.0246	18.1398	18.99885	17.94308	16.09073
F_Cost ($/h)			781.69	781.651	782.503	781.638	782.076	781.5219	781.86	781.2626
VD (p.u.)`			NR	NR	NR	0.4421	0.46629	0.47949	0.48809	0.4838
PL (MW)			NR	NR	NR	5.771	5.7882	5.8091	5.8460	5.7117
Emission (t/h)			NR	NR	NR	1.76199	1.76241	1.76237	1.7622	1.76209

P_G (MW), V (p.u.), Q_G (MVAr), NR: means not reported, Bold results indicate the best one.

The maximum, minimum values of control variables and detail optimal results are shown in Table 3. That reflect the better performance of GTO in solving the stochastic OPF problem. Figure 7 illustrates comparison between the convergence curves of different OPF optimization algorithms.

Numerical results of case 1 show the effectiveness of GTO algorithm, rapid convergence, and high-solution quality when comparing it with other optimization methods. The minimum value of total generation cost achieved by GTO is 781.2626 MW. Consequently, GTO exceeds all published algorithms and all other applied optimization techniques. It is worth to note that P_WG1 and P_WG2 indicate the scheduled powers from wind generators #W_G1 and #W_G2, respectively. The emission rate is calculated by using the optimal scheduled power of thermal generators, where reserve is assumed an alternate source that does not add to the emission. From Table 3, it can be seen that the proposed GTO reached the smallest cost with 781.2626 $/hour, and outperformed all optimization techniques, PSO (784.3400 $/hour), TLBO (782.6767 $/hour), SHADE-SF (782.50 $/hour), JADE, NBA (784.038), ALO (782.81 $/hour), jellyfish (781.6387 $/hour), artificial ecosystem optimizer (781.5219 $/hour), and hunger games search (781.86 $/hour) as well as vs. a recently optimization technique which introduced October 1, 2021 entitled, orca predation algorithm (782.0760 $/hour).

Based on the results obtained in the literature regarding solution of classical OPF problem and Table 3, we can state that with insertion of renewable energy sources, the total generation-cost decreased from 800.00 $/hour as a reference cost to 781.2626 $/hour, that is, around 18.7 $/hour. More precisely, if every hour can save the cost of 18.7 $, and the operating time per-year is supposed as 7500 hours, then operating time from the propose optimizer GTO can save 140,250 Dollars in total every year. Consequently, the insertion of wind generators and solar power plant significantly contributes on the reduction on total fuel cost compared with the original system configuration (i.e. without RES).

Case 2: Optimized-cost against reserve-cost

In second case, all parameters are retained the same as in case 1 except reserve cost-coefficients. These coefficients for both wind generators and solar photovoltaic unit are varied by discrete-step of 1 starting from 4 to 6, that is, =4, (case2-a), = 5, (case2-b) = 6, (case2-c). The penalty cost-coefficients for all intermittent sources are similar to the first case. The optimal power scheduled of generators is presented by bar graph in Figure 8 and compared with those found for the base case (case 1). For clarification purpose, Case 2-a describes case when reserve coefficient $K_{r} = 3$ , case 2-b represents for $K_{r} = 4$ , and case 2-c represents $K_{r} = 5$ . In this case study, when the coefficient of reserve cost increases, the contribution of both wind and solar generators reduces resulting in a shortage of scheduled power. So, an amount of spinning reserve is urgently needed in order to fill this shortage.

This shortage in power automatically compensated by thermal generators which result in increasing the cost of thermal power generators due to the increase of the output power illustrated in Figure 8. In summary, total generation cost raises with the increase in the reserve-cost coefficient.

Figure 9 presents voltage profile of load buses for all sub-case 2 and case 1. it can be seen that all the voltages are within the limits predefined by the system-operator.

Case 3: Optimized-cost against penalty-cost

Unlike to the second case, in this case study, all parameters of reserve cost are keeping as in first case excluding penalty cost-coefficients. Coefficients of penalty-cost for all wind generators and photovoltaic power plant are raised from 1.5 to 5 by the following order, that is, = 3 (case 3-a), = 4 (case 3-b), = 5 (case 3-a). The optimal power scheduled of six-generators is represented by bar-graph in Figure 10 and compared with those found for the case 1 in the same figure.

When penalty cost coefficient increases, the optimum power scheduled from wind and solar PV generators increases too, leading to decrease the output of thermal generating units with a not uniform manner. This different can be justified by the economic dispatch between three thermal generators, in which considerable part of power is dispatched on the generator which having the lower production cost. On the other hand, the scheduled output for all renewable energy sources also seems not to uniform, which can be interpreted by the highly nonlinear relation between PDF and reserve / penalty cost of both solar and wind generators. It is also seen that the thermal generators cost (ThG_x) is constant and a steady rise in total cost is observed.

Figure 11 illustrates the schedule reactive power of generators, for sub-cases of third case. By comparing obtained the reactive power displayed in Figure 11 with their imposed upper and lower limits, generators Th_G3 and W_G2 operate at maximum limits of reactive capability for many cases. So, following the reactive power constraints is a must. Consequently, achieved results confirm that the proposed solver is able to maintain the solution feasibility.

Test system II: Adrar’s power network (APN) DZ 26-bus

To evaluate applicability of the GTO on the practical power system, Adrar’s power network (APN) 28-bus (Makhloufi et al., 2016) has been considered as test system. Adrar’s power network (APN) is situated in the southwest of Algeria, is isolated grid from the Algerian power grid. It designed to meet the local energy demand in the isolated-site, with a total peak-load up to 291.2 MW, operated in summer 2015. The latter system consists of seven gas turbines based power plants (ThG) with a total capacity of 425.6 MW and two kinds of renewable energy sources, wind farm with capacity of 10.2 MW along with a PV solar generator with 20 MW, respectively. The slack-bus is Bus no 19. All data of test system are available in (Naidji and Boudour, 2020). Therefore, there are a total of 18 variables to be optimized, including nine active power of generators, and nine voltage magnitudes of generators. The Gamesa G52–850 kW machines has been set as convenient turbines for this site, where wind farm feed the 30/220 kV sub-stations of Adrar, Kaberten and Timimoun. Its characteristics are: Rated power, $P_{wr}$ =850 kW, 12 wind turbines ( $0.85 \times 12 = 10.2 MW$ ), the cut-in, rated and cut-out wind speeds values are $ν_{in}$ =4 m/s, $ν_{r}$ =13 m/s, and $ν_{out}$ =24 m/s. Single line diagram of this power network is presented in Figure 12 above. The Weibull PDF scale & shape parameters of Adrar area for July arek are given by to c = 7.2 and k = 2.1, respectively (Naidji and Boudour, 2020). Further details are found in Chellali et al. (2012); Diaf et al. (2013); Makhloufi et al. (2016).

The wind characteristics at Adrar region as follow: mean wind speed of 7 m/s for 10 m height, making this zone best in terms of wind energy potential in Algeria during all the year (Diaf and Notton, 2013; Naidji and Boudour, 2020). Minimum and maximum of active and reactive power of the generators for this test system is given in Table A1 at Annex. The present study aims to solve real world problem, which is finding the solution of optimal power flow problem in Adrar power network using meta-heuristic optimization techniques.

To this end, four algorithms utilized to optimize different objective-functions under different system constraints. Two cases are considered herein, with these objective functions, case 1—minimization of total generation cost given by equation (12); Case 2—is dedicated to voltage stability index minimization equation (14).

Results on case 1 of test system II

At this case, the objective-function (12) is minimized. The optimal results obtained by the proposed GTO algorithm and other methods, shown in Table 4, which are the best results achieved over 30 independent test runs. The findings clearly show that the proposed GTO provides high quality and stable solutions in comparison with other metaheuristic algorithms. The optimal settings of control variables significantly reduced total generation cost. Also, it is worth noted that all specified constraints are met, which is reflect the feasibility of solutions.

Table 4.

Optimization Results of the stochastic OPF in the Adrar’s power network.

	Min	Max	GTO-Base Case	AHA	OPA	AEO	HGS	GTO
P_TG18	10	24	23.9993	20.7105	21.391	23.3350	24.00	24.00
^*P_TG19	30	220	51.7842	36.00	38.6234	39.600	35.824	35.8471
P_TG20	4	13.6	13.5999	12.7810	10.7606	9.0743	13.60	13.60
P_TG21	10	44	34.9167	27.8416	44.00	32.1041	30.930	30.9632
P_TG22	10	50	50	49.4101	42.8772	49.6168	50	49.9999
P_TG23	20	100	82.2716	81.1966	84.2356	79.3583	79.638	79.5636
P_TG24	10	44	34.84488	33.2979	19.3779	28.1442	27.232	27.2519
P_SG25	0	20	-	19.9976	20.00	20.00	20.00	20
P_WG26	0	10.2	-	10.1979	10.20	10.1999	10.20	10.20
V₁₈	0.95	1.1	1.1000	1.0839	1.0847	1.0985	1.1000	1.0964
V₁₉	0.95	1.1	1.0900	1.0885	1.1000	1.0990	1.0996	1.0843
V₂₀	0.95	1.1	1.0740	1.0523	1.0714	1.0416	1.0551	1.0712
V₂₁	0.95	1.1	1.0737	1.0707	1.1000	1.0602	1.0687	1.0746
V₂₂	0.95	1.1	1.0885	1.0688	1.0847	1.0842	1.0840	1.1000
V₂₃	0.95	1.1	1.09826	1.0921	1.0711	1.0819	1.0911	1.0937
V₂₄	0.95	1.1	1.081380	1.0457	1.0711	1.0750	1.0717	1.1000
V_SG25	0.95	1.1	-	1.014	1.028	1.022	1.026	1.034
V_SG26	0.95	1.1	-	1.040	1.058	1.049	1.053	1.065
Q_TG18	−18	42	34.1397	34.84602	22.91479	46.31534	41.2303	29.43962
Q_TG19	−54	126	11.416	16.75753	16.10999	18.16565	16.7809	8.22447
Q_TG20	−16	8	4.386	0.05509	7.82950	−15.9324	−8.3092	1.32126
Q_TG21	−18	42	7.14782	12.43248	17.50813	5.56596	7.1868	5.02494
Q_TG22	−18	42	18.8208	17.81550	18.99003	20.74775	19.0492	21.63883
Q_TG23	−36	84	22.3062	24.62807	20.63987	20.91106	21.4045	19.83981
Q_TG24	−16	42	11.3564	2.13090	5.25867	12.46691	8.7168	18.02804
Q_SG25	−8	10	-	0.00	0.00	0.00	0.00	0.00
Q_WG26	−5.1	5.1	-	0.00	0.00	0.00	0.00	0.00
Fuel Cost (US$/hr)			9085.3510	8285.335	8286.713	8284.855	8284.445	8284.3024
$\sum P_{G}$ (MW)			291.41658	291.4332	291.4657	291.4326	291.424	291.4257
VD (p.u.)`			0.95169	0.6757	0.8714	0.7716	0.8595	1.0078
PL (MW)			0.2168	0.2339	0.2657	0.2332	0.2261	0.2258
NOx (kg/hr) ×1 US$/kg			256.79	237.01	239.14	236.89	236.08	236.05

P_G (MW), V (p.u.), Q_G (MVAr). *P_TG: Slack-bus.

Figure 13 represents Weibull fitting and wind distribution obtained from the simulation of 8000 Monte Carlo scenarios, while lognormal PDF of solar irradiance and the stochastic power-output of solar photovoltaic unit are illustrated by Figures 14 and 15, respectively. The spotted line illustrates the scheduled power supposed to be delivered to the grid by the solar-PV generator. The limits on all control variables are listed in the table of results.

Figure 16 provides comparison of convergence characteristics for different solvers. Comparing the results given Table 4 (GTO-Base Case) and GTO with RES, it is evident that the total generation cost decreases significantly whith integration of RES into grid. Moreover, rate of NOx emissions decreased too from 256.7 kg/hour to 236 kg/hour, with up to 20 kg/hour. More precisely, if each hour can clean 20 kg of atmospheric pollution from NOx emissions, and the operating time per-year is supposed as 5500 hours, then pollution rate decreased up to 150 tons every year. The explanation is that the hybrid generation system highly contributes to reduce atmospheric pollution with compared with the classical generation system (i.e. without RES). In addition, the total generation-cost decreased from 9085.35 US$/h as a reference cost to 8284.3 (US$/h), that is, around 801 US$/hour, that is, up to 4.40 million US$ per year. Base case reports the simulation without considering renewable energy sources. In this case study, minimization of total generation cost is performed and the obtained results were listed in the first column of Table 4.

Results on case 2 of test system II

In this case, the voltage stability index given by equation (14) is minimized. The optimal control variables obtained from different methods as well as the P_Loss, VD, and NOX emissions, shown in Table 5. The findings clearly show that the proposed GTO outperformed all other applied optimizers in terms of VSI enhancement and solution convergence. From numerical results can see that both of hard and soft variables are within admissible limits, reflecting the feasibility of solutions. Figure 17 describes voltage profile for case 2 of Adrar’s test system. Voltage deviation and VSI were reduced by 72.48% and 17.56% respectively compared with the base case. As a result, even with the realistic power network, the GTO algorithm gives the best adjustment of control variables and converges well to the optimal solutions. In short, it has been also proven that the proposed GTO is more suitable to solve stochastic OPF than the other optimization technique, in which it can not only benefit from best solutions, but more importantly guarantee the feasibility of solutions achieved in two test power systems.

Table 5.

Optimization results of the stochastic OPF in the Adrar’s power network; case 2.

	Min	Max	AHA	OPA	AEO	HGS	GTO
P_TG18	10	24	17.7942	18.4462	21.5470	20.9327	21.0308
^*P_TG19	30	220	64.7148	86.3359	67.7936	59.8538	54.8319
P_TG20	4	13.6	9.9768	4.8901	10.6136	13.60	13.60
P_TG21	10	44	34.3989	31.9191	29.0884	35.8004	44.00
P_TG22	10	50	49.7963	36.8829	48.5560	50.00	50.00
P_TG23	20	100	57.0157	53.8387	61.4375	59.1584	58.8493
P_TG24	10	44	29.9201	38.3322	25.4113	22.1768	19.2309
P_SG25	0	20	19.9418	13.2302	19.9997	20	20
P_WG26	0	10.2	8.2354	8.1857	7.2339	10.20	10.20
V₁₈	0.95	1.1	1.0988	1.100	1.100	1.100	1.100
V₁₉	0.95	1.1	1.0937	1.100	1.100	1.100	1.100
V₂₀	0.95	1.1	1.0850	1.0844	1.0876	1.0878	1.0879
V₂₁	0.95	1.1	1.0968	1.0839	1.0999	1.100	1.100
V₂₂	0.95	1.1	1.0994	1.100	1.100	1.100	1.100
V₂₃	0.95	1.1	1.0985	1.100	1.0999	1.100	1.100
V₂₄	0.95	1.1	1.0971	1.0956	1.100	1.100	1.100
V_SG25	0.95	1.1	1.043	1.043	1.046	1.046	1.046
V_SG26	0.95	1.1	1.073	1.070	1.076	1.076	1.076
Q_TG18	−18	42	24.57464	27.99437	23.60179	22.8938	22.59585
Q_TG19	−54	126	8.27416	10.92259	9.79545	9.71178	9.74462
Q_TG20	−16	8	7.61067	7.57468	7.96711	8.00	8.00
Q_TG21	−18	42	10.21049	6.38929	10.00186	10.36648	11.0402
Q_TG22	−18	42	18.96742	18.54758	18.10950	18.23043	18.2590
Q_TG23	−36	84	13.03104	13.02627	13.69070	13.18074	13.1306
Q_TG24	−16	42	12.04952	13.12018	11.87327	11.68040	11.6580
Q_SG25	−8	10	0.00	0.00	0.00	0.00	0.00
Q_WG26	−5.1	5.1	0.00	0.00	0.00	0.00	0.00
VSI			0.212579	0.213934	0.21175	0.21135	0.21137
VD (p.u.)`			1.1566	1.1223	1.2029	1.2050	1.2033
PL (MW)			0.5939	0.8611	0.4809	0.5267	0.5429
NOx (kg/hr) ×1 US$/kg			236.14	247.50	238.03	233.63	233.32

Conclusion

This article presents an efficient and robust gorilla troops-inspired algorithm to solve the stochastic optimal power flow (OPF) problem in the modern power system. Uncertainty nature of both solar PV power plant and wind energy sources have been modeled based on the Weibull and lognormal PDFs distribution, respectively. To investigate the performance of GTO algorithm, three optimization techniques: HGS, AEO, and orca predation algorithm- (OPA) are applied on the modified IEEE 30-bus test system and a real isolated power network DZ-26 bus in the Adrar region. Numerical results of GTO algorithm are compared with the findings found by three optimization algorithms and by other competitive algorithms in the literature. The results revealed that the GTO significantly gives a superior solution, while insuring the feasibility of solutions, where outperformed AEO, JS, HGS, OPA and AHA methods in the base case and other sub-cases whatever the constraints of test system. The findings suggest that the proposed GTO algorithm can be successfully applied to solve highly nonlinear problems and real world problems.

Footnotes

Appendix

Acknowledgements

The authors deeply appreciate the support of “Electric Operator System” of Algeria for providing access to the Adrar power system data. To Memory of my Supervisor in PhD dissertation, Prof., Tarek Bouktir my, I ask Allah Almighty to forgive and have mercy upon him, pardon him and make honorable and grant him the highest degrees in Jannah.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by Ministry of Higher Education of Algeria and scientific research progects PRFU-2021 (Grant no. A01L07UN100120210002). The research of Dr. Souhil Mouassa has received a support from the higher polytechnic school of Linares, University of Jaen-(Spain).

ORCID iDs

Souhil Mouassa

Francisco Jurado

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