Solution approach for optimal power flow considering wind turbine and environmental emissions

Abstract

The power grid is changing expeditiously with the increasing penetration of renewable energy sources (RES). Optimal utilization of RES reduces the burden on the primary grid and makes the grid more resilient. Traditional optimal power flow (OPF) is a complex problem in power management systems, and the complexity further increases with the integration of RES due to their intermittency. This paper presents the complete formulation of the OPF model incorporating wind turbines (WT) and environmental emissions for proper scheduling, planning, and efficient operation of thermal generating units (TGU) using the Ant Lion Optimization (ALO) algorithm. The formulation of the OPF problem comprises forecasted active power generation of WT, depending on the real-time measurement and probabilistic wind speed models. The results are analyzed from the perspective of operating cost, voltage profile, and transmission power losses in the system. The OPF approach and the solution methodology are tested on the IEEE 30 and IEEE 57-bus systems. The effectiveness of the proposed ALO algorithm is evaluated against well-established algorithms like Particle Swarm Optimization and Teaching-learning-based optimization. The comparison emphasizes the effectiveness of the ALO approach for solving various OPF problems with complex and non-smooth objective functions.

Keywords

Optimal power flow wind power forecasting ant lion optimization greenhouse gases carbon tax

Introduction

Humans have relied heavily on fossil fuels and other sources of energy. As a result, the greenhouse gases from this fuel combustion have reached ever-higher multi-decade peaks, leading to global warming and climate change. In debates regarding climate changes, RES perceives the best solution for reducing the detrimental impacts of climate change (Upadhyay and Sharma, 2015). Besides, the integration of RES makes the electric grid more robust, increases energy access for people in developing countries, and lowers energy costs. Several scientific and technical breakthroughs have helped drive renewable energy forward, and new milestones are being set (Sharma and Sood, 2020). India is currently undergoing a large-scale renewable energy capacity expansion program to grow 450 GW by 2030.

Wind power is regarded as a prominent energy resource, with the added benefit of being environmentally friendly. Wind power has grown its popularity worldwide as a result of urban development, industrial development, the need for sustainable growth, and energy efficiency (Upadhyay and Sharma, 2014). In contrast to conventional power plants, which involve the ongoing expense of fuel and frequent maintenance, wind power plants require no fuel and produce zero carbon emissions once they are in running condition (Kaur Saini et al., 2019). The problem associated with the wind turbine is that the power generation relies on unpredictable or erratic natural resources, which makes them difficult to control and poses challenges to grid operators. As power grids are becoming more wind-powered, grid operators need to know how much energy is being generated by WT and how to schedule the thermal generating units (TGU) optimally to ensure that supply and demand are appropriately balanced. Also, the size and location of the WT significantly impacted the power system performance, such as operating cost, power losses, and voltage profile (Kumar et al., 2021). So, formulating OPF problem incorporating WT becomes essential for the efficient and effective operation of the power system.

OPF is a critical problem in power management systems, and it is a complex, highly non-linear, non-convex, and constrained optimization problem. The complexity further increases with the integration of RES due to their intermittent nature. The OPF optimizes a particular objective function, subject to the network power flow equations and operating constraints. The desired solution of an OPF objective has been achieved by adequately tuning control parameters (Meyyappan, 2018). The basic approach to formulating and solving the OPF problem incorporating WT are as follows (i) the scope and relevance of the problem; (ii) define objective function; (iii) modeling of WT power output considering the uncertainty of wind speed; (iv) defining technical constraints; (v) solution methodology to solve OPF problem. Therefore, researchers used metaheuristic techniques to solve OPF problems.

Numerous methodologies for solving the OPF problem with and without RES are discussed in the literature. The conventional techniques are the Newton method, Gradient method, Quadratic Programming, Non-Linear Programming, etc., and many more. These conventional techniques are complicated and suffer from many disadvantages, such as the risk of being trapped at the local optima, some theoretically accepted assumptions that do not assure the global solution (Elattar and ElSayed, 2019).

In recent, the author (Daqaq et al., 2021) proposed a backtracking search algorithm for solving the OPF problem to minimize operating costs of thermal generators, voltage variance, and power losses. (Srilakshmi et al., 2020) suggested a more efficient most valuable player optimization algorithm to solve OPF problems considering similar objectives. (Islam et al., 2020) suggested a similar study considering environmental emission using the harris hawks optimization algorithm. (Mahdad and Srairi, 2016) proposed adaptive flower pollination algorithm to solve security OPF considering contingencies at critical generators. (Saha et al., 2017) proposed a water evaporation algorithm to solve the OPF problem considering numerous objectives to show the proposed algorithm’s efficacy. (Pulluri et al., 2017) proposed enhanced self-adaptive differential evolution to solve the multi-objective optimal power flow (MO-OPF) problem with competing goals such as production cost, environmental emissions, voltage profile, and power losses. (Surender Reddy and Rathnam, 2016) performs a similar study considering objectives such as fuel cost and emission minimization of thermal generators using glowworm swarm optimization. (El-Fergany and Hasanien, 2015) proposed gray wolf and differential evolution algorithms to solve single and MO-OPF problems. The above-mentioned articles consider only conventional generators.

The system comprising conventional generators incorporated with RES has recently been addressed in a few publications. (Syed et al., 2021) proffer an improved wind-driven algorithm for solving OPF problems incorporating wind and solar power with the power system. (Sulaiman and Mustaffa, 2021) proposed barnacles mating optimizer for solving OPF problem considering stochastic wind, solar, and small hydro generators. (Shaheen et al., 2021) proposed heap optimization algorithm to solve OPF problem considering wind and solar farms. (Khan et al., 2020) proposed a gray wolf optimization algorithm for solving OPF problems incorporating RES like wind and solar. Weibull and lognormal probability density function (PDF) is used to forecasting power from WT and solar farms. (Pandya and Jariwala, 2020) proposed moth flame optimization approach to solve MO-OPF incorporating wind and solar power plants. (Gonal and Sheshadri, 2019) solves OPF problem using hybrid bat–dragonfly algorithm incorporating wind-solar power. (Salkuti, 2019) proposed glowworm swarm optimization algorithm for solving MO-OPF problem integrating wind energy generator. (Kouadri et al., 2018) examines the effect of the power grid on integrating wind power generators using ant lion optimizer. (Teeparthi and Vinod Kumar, 2017) proposed hybrid particle swarm optimization (PSO) and artificial physics optimization for solving security-constrained optimization problems incorporating wind and thermal generators. (Khaled et al., 2017) implemented modified PSO algorithm to solve the hourly optimal load flow considering renewable distributed generation. (Mishra et al., 2015) investigates OPF solution incorporating wind power using a modified cuckoo search algorithm. The wind generator cost modeling incorporates underestimating and overestimating wind power costs, depending on the wind speed uncertainty.

It is evident from the literature review that incorporating metaheuristic algorithms to solve OPF solutions, especially when RES are included, is highly encouraging. Thus, this paper presents a recent ALO algorithm for solving the OPF problem considering wind as a renewable source. ALO algorithm is inspired by nature and imitates antlion hunting behavior in nature with its favorite prey ants. It is considered a global optimizer in the sense that it avoids stagnation into local optima and maintains the optimal balance of exploration and exploitation. The proposed algorithm is used to solve several OPF objectives such as minimization of operating cost of TGU, voltage profile enhancement, minimizing power losses in the system. Thermal generating units are well-known for releasing greenhouse gases into the atmosphere and polluting the environment, which is one of the most critical issues nowadays (Biswas et al., 2017). Therefore, an objective to reduce thermal power plants emissions is necessary to incorporate with the operating cost to protect the environment. However, the problem becomes multi-objective, which adds complexity to the OPF problem significantly. This multi-objective problem transforms into a single objective problem by levying a carbon tax on per unit greenhouse gases emitted. The proposed algorithm results are tested against well-established PSO and Teaching-learning-based optimization (TLBO) algorithm by comparing the same system data, control variables (CV), and constraints. The contribution of the paper is as follows:

This article proposed a recent nature-inspired metaheuristic algorithm (ALO) for solving single and MO-OPF problems. Weight factors are used to transform the MO-OPF problem into a single objective problem.

Formulating OPF problem incorporated with the forecasted active power generation of WT and examining the power system’s performance in various indices like operating cost of the TGU, voltage profile, and power losses.

To investigate the effectiveness of wind-thermal cooperation in lowering greenhouse gas emissions, the emission cost is combined with the operating cost of the TGU.

By combining real-time measurements and probabilistic wind speed models, the expected power output of the most common types of utility-scaled WT is forecasted.

The proposed ALO algorithm’s efficacy is evaluated using the IEEE 57-bus system.

The remainder of the paper is set out as follows. The II section summarizes the mathematical formulation of the OPF problem, including the appropriate constraints. Section III presents the statistical analysis of WT and forecasted power output calculation. Discussion of the ALO algorithm is covered in the IV section. The results are presented in section V, and the paper is concluded in Section VI.

Mathematical models

Thermal generating unit cost model

The quadratic relationship roughly gives the relation between the cost of fuel ($/hour) and the power produced (MW):

C_{0} (P_{T}) = min \sum_{i = 1}^{NT} a_{i} + b_{i} {P_{T}}_{i} + c_{i} {({P_{T}}_{i})}^{2}

(1)

where a_i, b_i, and c_i denotes the cost coefficients of the ith thermal generator, generating active power output P_Ti and NT denotes the total number of TGU.

The multi-valve steam turbine of TGU shows a more significant variance in fuel-cost functions. For more practical and accurate cost-function modeling, the valve-point effect (VPE) must be considered. The operating cost considering VPE becomes:

C (P_{T}) = \sum_{i = 1}^{NT} a_{i} + b_{i} {P_{T}}_{i} + c_{i} {({P_{T}}_{i})}^{2} + | d_{i} \times \sin (e_{i} \times (P_{Ti}^{min} - P_{Ti})) |

(2)

where, e_i, d_i denotes the coefficient representing valve point loading and $P_{Ti}^{\min}$ is the minimum active power generation of ith thermal generator.

Emission and carbon tax

Thermal power plants have been accused of releasing toxic gasses into the air that increases with the increase in power generation (MW). Emissions in tons/hour (E_T) is measured by:

E_{T} = \sum_{i = 1}^{NT} | (α_{i} + β_{i} {P_{T}}_{i} + γ_{i} {({P_{T}}_{i})}^{2}) \times 0.01 + (φ_{i} . e^{ω_{i} \times {P_{T}}_{i}}) |

(3)

where $α_{i}, β_{i}, γ_{i}, ω_{i}$ and $φ_{i}$ are emission coefficient of ith TGU.

In recent years, growing awareness of global warming has caused a rush in countries to bring down carbon emissions. In this regard, a carbon tax is levied on the per-unit amount of GHG emitted to promote cleaner energy sources such as wind turbines. The carbon tax (C_tax) is assumed $30 in this work. The emission cost is represented as:

E_{cost} = C_{tax} \times E_{T}

(4)

Objective function

As discussed above, one of the most critical issues nowadays is to protect the environment from various toxic pollutants and gases. Therefore, an objective that focuses on only minimizing the operating cost of the thermal power plants would no longer be efficient. So, an objective function that incorporates the emission cost with the operating cost of TGU is efficient in today’s era. Considering this, the primary objective function becomes:

F_{1} (x, u) = C (P_{T}) + E_{cost} + penalty

(5)

where x and u denote the state vector and CV, respectively. The CV are self-controlling. The optimization algorithm modifies the CV within its range to achieve the desired goal of the OPF problem. Whereas the state variables represent the electrical state of the system. The control variables are the active power of a generation of TGU omit the slack bus, voltages at all the generator buses (including WT), tap setting of the transformers, and shunt var compensation (SVC). Whereas state variables considered are active power output of TGU at the slack bus, voltages at the load buses, the reactive power output of all the generators, and transmission limits of transmission lines.

The objective function is solved under various system equality and inequality constraints. These constraints are as follows:

Equality constraints

Equality constraints are typical load flow equations. These constraints maintain the balance between power generation and load demand.

P_{T i} - P_{D i} - V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i j} \cos (δ_{i j}) + B_{i j} [G_{i j} \sin (δ_{i j})] = 0 \forall i \in N B

(6)

Q_{T i} - Q_{D i} - V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i j} \sin (δ_{i j}) - B_{i j} [G_{i j} \cos (δ_{i j})] = 0 \forall i \in N B

(7)

where P_Di and Q_Di are the active and reactive power demands, respectively. G_ij and B_ij are transfer conductance and susceptance of ith and jth bus. $δ_{ij}$ is angle difference between ith bus and jth bus. V_i and V_j is the voltage of ith, and jth bus and NB denotes the total number of buses.

Inequality constraints

Inequality constraints are the operating limits imposed on several power system components such as the generator, transmission lines, and load buses. These constraints should be synchronized within their minimum and maximum operating limits.

Generator constraints

P_{T i}^{\min} \leq P_{T i} \leq P_{T i}^{\max} where i = 1 \dots \dots \dots \dots \dots N T

(8)

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

(9)

V_{Ti}^{min} \leq V_{Ti} \leq V_{Ti}^{max}

(10)

V_{W f}^{\min} \leq V_{W f} \leq V_{W f}^{\max} where f = 1 . \dots \dots \dots \dots . N W

(11)

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

(12)

Transformer constraint

T_{j}^{min} \leq T_{j} \leq T_{j}^{max} where j = 1 \dots \dots \dots \dots . . XR

(13)

Static var compensation constraint

Q_{k}^{min} \leq Q_{k} \leq Q_{k}^{max} where k = 1 . \dots . \dots \dots . NC

(14)

Security constraints

V_{Lp}^{min} \leq V_{Lp} \leq V_{Lp}^{max} where p = 1 \dots \dots \dots \dots NL

(15)

S_{q} \leq S_{q}^{max} where q = 1 \dots \dots \dots \dots . NTL

(16)

The equation (8) to (10) shows the active power, reactive power, and voltage limits of TGU. Equation (11) to (12) shows the voltage and reactive power limit of WT. Equation (13) shows the transformer tap setting limits. Equation (14) shows the shunt var compensation limits. Equations (15) and (16) defines the load bus voltage limits and line flow limits, respectively. NW, NC, NL, XR, and NTL are the total number of wind turbines; shunt var compensation; load buses; tap setting transformers and transmission lines.

The objective function incorporates a penalty function to account for any violations of the prescribed constraints. Mathematically, the penalty function is described as:

penalty = [δ_{P} {(P_{T 1} - P_{T 1}^{\lim})}^{2} + δ_{Q} \sum_{i}^{N T} {(Q_{T i} - Q_{T i}^{\lim})}^{2} + δ_{w} \sum_{f}^{N W} {(Q_{W f} - Q_{W f}^{\lim})}^{2} + δ_{V} \sum_{i}^{N L} {(V_{L p} - V_{L p}^{\lim})}^{2} + δ_{S} \sum_{i}^{N T L} {(S_{q} - S_{q}^{\lim})}^{2}]

(17)

where $δ_{P}, δ_{Q}, δ_{w}, δ_{V}$ , and $δ_{S}$ are weighted factors in the range of 10⁶–10⁹.

One of the prominent and essential features in the power system that indicates safety and service quality is the bus voltages. By reducing the voltage deviation (VD) from the 1.0 p.u., a flat voltage profile is achieved. However, an objective function that minimizes only the operational cost may give an undesirable voltage profile. Therefore, a dual objective function is considered that seeks a flat voltage profile and minimizes the operational cost. The mathematical representation of the dual objective function is as follows:

F_{2} (x, u) = F_{1} (x, u) + k \sum_{p = 1}^{NL} | V_{Lp} - 1 |

(18)

where V_L denotes the load bus voltage, k denotes the weighting factor, which must be chosen carefully to attain the desired relevance of the VD term with the operating cost.

Reducing the power loss in the power system is essential. So, two objective functions are also considered in this work.

Active power loss minimization

The objective function for minimizing the active power losses in the system is expressed as:

min P_{loss} = \sum_{N = 1}^{NTL} G_{ij} [V_{i}^{2} + V_{j}^{2} - 2 V_{i} V_{j} \cos δ_{ij}]

(19)

Reactive power loss minimization

The objective function for minimizing the reactive power losses in the system is expressed as:

min Q_{loss} = \sum_{N = 1}^{NTL} B_{ij} [V_{i}^{2} + V_{j}^{2} - 2 V_{i} V_{j} \cos δ_{ij}]

(20)

WIND turbine modeling

WT power generation

Due to the unpredictability of the WT’s power output characteristics, integrating the WT enhances the complexity of the OPF problem. Few assumptions have been considered in this research to compute the OPF while incorporating the uncertainties of WT power output, and are mentioned below:

WT active power generation is non-schedulable. Therefore, the system operator mandatorily uses the generation according to the forecasted values.

The OPF is conducted at repeated intervals t of 15 minutes. So, fifteen readings of the wind speed are considered having a sampling time of 1 minute. Based on wind speed data and probabilistic wind speed models, WT’s forecasted output power is calculated.

WT are capable of generating reactive power in the range of −0.4 to 0.5 p.u. times their active power. Therefore, the voltage magnitude of WT is treated as a control variable in the OPF problem.

Wind turbine power model

Wind dynamics can be analyzed using the probability density function (PDF). The study has found that Weibull PDF has been commonly used to fit actual wind data.

f (v) = \frac{k}{C} {(\frac{v}{C})}^{k - 1} . \exp {(- \frac{v}{C})}^{k}

(21)

where f(v) denotes the Weibull PDF, and k and C indicate its shape and scale factor. The integration of Weibull PDF gives the cumulative density function (CDF). The CDF indicates how much the wind velocity falls within a specific speed range (Kumar et al., 2019).

F (v) = 1 - . e^{- {(\frac{v}{C})}^{k}}

(22)

The inverse of CDF is considered to calculate the wind speed (v):

v = C . {(- \ln (r))}^{1 / k}

(23)

where r denotes the random number uniformly distributed between [0, 1]. The value of k and C can be determined by examining the mean and standard deviation (SD) of wind speed over time interval t:

k = {(\frac{σ_{v}^{t}}{μ_{v}^{t}})}^{- 1.086}

(24)

C = \frac{μ_{v}^{t}}{Γ (1 + 1 / k)}

(25)

where $σ_{v}^{t}$ and $μ_{v}^{t}$ denotes the SD and mean of the wind speed, respectively. Γ(x) represents the gamma function of parameter x. The wind turbine’s output power is a function of wind speed, as shown by equation (26). Figure 1 illustrates the power-speed characteristics of a WT.

P_{W} (v) = {\begin{matrix} 0 & v \leq v_{cutin} \\ \frac{v - v_{cutin}}{v_{rated} - v_{cutin}} & v_{cutin} \leq v \leq v_{rated} \\ P_{rated} & v_{rated} < v \leq v_{cutout} \\ 0 & v \geq v_{cutout} \end{matrix}

(26)

where v_cutin indicates the cut-in wind speed, v_rated indicates the rated wind speed, v_cutout denotes the cut-out wind speed. P_rated denotes the rated active power output of the WT. P_W(v) is the active power generation of WT.

Figure 1.

Characteristics between wind speed and power output of wind turbine.

The OPF is conducted at repeated intervals of 5–15 minutes (Surender Reddy and Bijwe, 2016). The time interval t is set to be 15 min in this work. So, fifteen readings of the wind speed are considered having a sampling time of one minute. The mean and SD of wind speed are calculated to determine the value of k and C using equations (24) and (25). The data of the wind speed for time interval t is the basis for predicting the wind speed and, consequently, the output power of WT in the next time interval of 15 min. In discrete form, the Weibull PDF is realized by dividing the time interval t into N states. The probability of each state $(s = 1 / N)$ and respective wind speed is determined by equations (21) and (23), respectively. The expected active output power of the WT for the time interval t is calculated as follows (Ullah et al., 2019):

P_{W} = \frac{\sum_{s = 1}^{N} P_{W_{s}} . f_{v} ({v_{s}}^{t})}{\sum_{s = 1}^{N} f_{v} (v_{s}^{t})}

(27)

where $v_{s}^{t}$ denotes the sth state of wind speed during time interval t. P_W is the forecasted active power generation of the wind turbine, $P_{W_{s}}$ is the power generated from WT using equation (25) for $v = v_{s}^{t}$ . $f_{v} (v_{s}^{t})$ is the probability of wind speed for state s during the specified time interval t.

Optimization algorithm

The ALO algorithm proposed by (Mirjalili, 2015) is inspired by nature and widely used to solve constrained engineering optimization problems. The proposed algorithm imitates antlion hunting behavior in nature with its favorite prey (ants). This behavior is simulated as ants are expected to walk randomly and explore the search space, and antlions hunt them by using traps to become fitter. Ants are representing a solution evaluated utilizing a fitness function. The random walk of ants in the search region is mathematically defined as follows:

X (t) = [0, cs (2 r (t_{1}) - 1), cs (2 r (t_{2}) - 1) . . . . . ., cs (2 r (t_{mi}) - 1)]

(28)

where cs denotes the cumulative sum; t is the step of the random walk; mi denotes the maximum iteration; r(t) is the stochastic function and is defined as:

r (t) = {\begin{matrix} 1 & if rnd > 0.5 \\ 0 & if rnd \leq 0.5 \end{matrix}

(29)

where rnd is the random number uniformly distributed between [0, 1].

Ants updates their positions at each stage of optimization in the search region. The ants are normalized using the following equation to ensure that they remain within the search region (Mouassa et al., 2017).

X_{i}^{t} = \frac{(X_{i}^{t} - A_{i}) . (D_{i}^{t} - C_{i}^{t})}{(B_{i} - A_{i})} + C_{i}^{t}

(30)

where A_i is the minimum and B_i is the maximum of random walk of ith variable. $C_{i}^{t}$ denotes the minimum and $D_{i}^{t}$ denotes the maximum value of ith variable at tth iteration. The random walk within the search space is ensured by equation (30) and should be implemented in each iteration.

Trapping in antlion’s pits

It is believed that Antlions are lurking in the search space and build their traps as per their fitness value. These traps influence the random walk of ants. This supposition is mathematically formulated as:

C_{i}^{t} = {Antlion}_{j}^{t} + C^{t}

(31)

D_{i}^{t} = {Antlion}_{j}^{t} + D^{t}

(32)

where, ${Antlion}_{j}^{t}$ denotes the jth antlion position at tth iteration. C^t represents the smallest value of all variables at tth iteration, and D^t represents a vector consisting of maximum values of all variables at tth iteration. The equations (31) and (32) illustrate how ants randomly walk around an ant lion in a hypersphere defined by two vectors C and D.

Building traps

The ALO algorithm selects antlions based on their fitness value during the optimization process; to accomplish this, the ALO algorithm employs a roulette wheel operator. This process improves the possibility of fitter antlions hunting ants.

Sliding ants toward antlion

Antlions build traps according to their fitness. When antlions perceive that an ant falls in the trap and trying to escape, antlions fire sand outward from the bottom of the pit to slide down the ant in their trap. Such behavior is modeled as the radius of random walks of the ants in a hypersphere is decreased adaptively. The following equations are being suggested to describe these events:

C^{t} = C^{t} / R

(33)

D^{t} = D^{t} / R

(34)

where R is the ratio defined as

R = 10^{x} \times \frac{t}{mi}

(35)

where x is a fixed value that depends on the current iteration’s value.

The equations (34) and (35) reduce the radius of ant’s positions being updated, which resembles ants’ sliding mechanism inside the pits. This enables the rapid exploitation of the search region.

Catching preys and rebuilding the traps

Antlion catches the ant when it reaches the bottom of a pit. After that, the ant is dragged into the sand and eaten by the antlion. This natural process is reflected by assuming that catching prey is only considered when ants outperform the ant lion in terms of fitness. To maximize prey capture, the antlion’s location is constantly updated to the hunt’s current location. It increases the chance that it can capture a new prey. In this regard, the following equation is proposed:

{Antlion}_{j}^{t} = {Ant}_{i}^{t} if f ({Ant}_{k}^{t}) > f ({Antlion}_{j}^{t})

(36)

where, ${Ant}_{k}^{t}$ shows the position of the kth ant at tth iteration.

Elitism

Each iteration saves the fittest antlion obtained so far and is treated as an elite. Due to the elite’s ability to control all ants’ movement during each iteration, it is assumed that the ants roundabout their random walks around the selected antlion determined by Roulette wheel and the fittest ant simultaneously as follows.

{Ant}_{i}^{t} = \frac{R_{A}^{t} + R_{E}^{t}}{2}

(37)

where $R_{A}^{t}$ denotes the random walk around the antlion at tth iteration picked by roulette wheel and $R_{E}^{t}$ denotes the random walk of around the elite at tth iteration. The complete flowchart of the ALO algorithm applied to the OPF problem is presented in Figure 2.

Figure 2.

Flow chart of ALO algorithm.

Results and discussion

The proposed ALO is applied on IEEE 30-bus and IEEE 57-bus systems to perform the OPF simulations. Table 1 presents the basic parameters of both the bus systems that are considered in this work. The bus and branch data, cost, and emission coefficients of TGU of both test systems are taken from reference (Chaib et al., 2016). The simulation program is executed on MATLAB platform on PC having configuration of 8 GB RAM and Intel core i5 processor.

Table 1.

Summary of IEEE 30-bus and IEEE 57-bus system.

Items	IEEE 30-bus system	IEEE 57-bus system
Thermal generating units	6	7
Buses	30	57
Transmission lines	41	80
Shunt var compensation	9	3
Connected active and reactive load	283.4 MW and 126.2 MVAr	1250.8 MW and 336.4 MVAr
Transformers with off-nominal tap ratio	4	17

Deterministic OPF (IEEE 30-bus system)

To prove the effectiveness and efficacy of the proposed ALO algorithm, the simple IEEE 30-bus case system without wind farm is considered. Each objective function is run on 30 trials. The results generated by using the ALO algorithm in deterministic OPF are compared to those produced using the PSO and the TLBO algorithm in Table 2. The parameters such as population size (40), maximum iteration (200), the total number of executions are the same for each algorithm.

Table 2.

Simulation result obtained for case studies: IEEE 30-bus system.

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
P_T1	50	200	170.990	169.961	170.970	172.248	180.264	171.343	51.630	51.917	51.682	51.633	52.659	51.757
P_T2	20	80	49.834	50.470	49.702	49.955	50.220	49.791	80.000	79.955	80.000	80.000	79.928	80.000
P_T5	15	50	21.908	21.852	21.895	22.072	20.428	21.866	50.000	49.980	50.000	50.000	50.000	50.000
P_T8	10	35	24.293	23.234	24.133	20.890	12.088	24.252	35.000	34.999	35.000	35.000	34.999	35.000
P_T11	10	30	13.194	13.638	13.298	14.689	14.671	13.238	30.000	29.990	30.000	29.998	29.347	30.000
P_T13	12	40	12.323	13.401	12.391	13.056	15.867	12.003	40.000	40.000	40.000	40.000	39.999	40.000
V_T1	0.95	1.1	1.050	1.050	1.050	1.034	1.039	1.050	1.050	1.050	1.050	1.050	1.050	1.050
V_T2	0.95	1.1	1.039	1.031	1.033	1.018	1.028	1.032	1.049	1.050	1.050	1.050	1.050	1.050
V_T5	0.95	1.1	1.013	0.993	1.001	0.998	1.000	1.003	1.032	1.027	1.037	1.044	1.046	1.043
V_T8	0.95	1.1	1.022	1.000	1.005	1.002	1.000	1.003	1.039	1.038	1.043	1.049	1.046	1.050
V_T11	0.95	1.1	1.014	1.045	1.023	1.045	1.043	1.037	1.044	1.037	1.050	1.050	1.005	0.992
V_T13	0.95	1.1	1.022	1.042	1.050	1.036	1.032	1.026	1.050	0.978	1.037	1.050	1.036	1.019
T₁₁	0.9	1.1	1.042	1.082	0.995	1.029	0.983	1.056	0.982	1.099	1.071	1.023	1.100	1.079
T₁₂	0.9	1.1	0.984	0.939	0.903	0.925	0.952	0.900	1.077	0.965	1.053	0.946	0.962	1.019
T₁₅	0.9	1.1	1.034	1.003	0.991	0.989	1.007	1.014	1.062	1.041	1.028	0.991	1.006	1.026
T₃₆	0.9	1.1	1.008	0.996	0.953	0.961	0.953	0.969	1.006	1.025	1.014	1.003	1.052	1.047
Q₁₀	0	5	4.160	1.941	3.219	2.952	1.484	0.242	4.983	0.141	4.917	4.426	0.202	5.000
Q₁₂	0	5	3.784	4.732	4.936	0.191	4.997	0.001	4.921	4.985	4.442	4.970	0.106	5.000
Q₁₅	0	5	3.860	4.533	4.981	1.454	2.090	4.986	4.978	0.401	4.526	4.987	0.001	4.999
Q₁₇	0	5	4.245	1.988	4.953	3.293	2.482	2.993	4.945	4.995	4.897	4.975	3.621	5.000
Q₂₀	0	5	5.000	1.369	4.934	1.799	3.285	4.992	4.661	0.448	4.932	4.979	0.002	5.000
Q₂₁	0	5	2.705	4.013	5.000	4.976	2.916	5.000	4.976	0.000	3.896	4.941	0.146	5.000
Q₂₃	0	5	1.103	3.373	3.827	0.996	0.006	4.803	4.996	0.006	4.199	5.000	4.925	4.999
Q₂₄	0	5	4.336	4.408	5.000	4.899	4.976	4.997	4.981	4.950	4.992	4.211	4.971	5.000
Q₂₉	0	5	4.648	0.726	3.082	2.113	1.244	2.214	4.859	4.846	3.981	4.259	4.994	3.405
Cost			813.030	814.205	813.418	814.051	816.683	813.692	974.134	974.894	974.315	974.117	973.811	974.911
VD			0.3143	0.357	0.495	0.1292	0.169	0.130	0.503	0.817	0.354	0.801	0.611	0.534
P_losses			9.0861	9.257	9.114	9.4026	10.239	9.200	3.207	3.441	3.283	3.225	3.532	3.357
Q_losses			2.5769	8.007	11.234	4.5784	13.982	8.312	−19.295	−16.466	−17.136	−20.057	−18.284	−19.138

Qty.: quantity.

Case (a) focuses on minimizing the operating cost, which includes the emission cost. The primary objective function mentioned in equation (5) is determined. After the runs mentioned above of the simulation program, the best value of the minimum operating cost obtained is $813.030. Figure 3 shows the shift in the active power generation schedule of TGU in the presence and absence of emission cost with the operating cost. It is noticed that the active power generation of all generators is within tolerable bounds both in the presence and absence of emission costs. Case (b) focuses on the minimization of the VD in the system. The objective function specified in equation (18) is to be minimized. It is observed from Table 2. the voltage profile is substantially improved over Case (a). The minimum value of the VD obtained is 0.1292 p.u., but this causes maximum power losses in the system. Power loss is also an essential parameter of the power system. The leftover cases are concerned with minimizing the system’s power losses. In Case (c), the system’s active power losses are minimized. The objective function specified in equation (19) is to be minimized, and the best value obtained is 3.207 MW. Case (d) focuses on minimizing the reactive power losses in the system. The objective function given in equation (20) is to be minimized. The minimum reactive power loss obtained is −20.057 MVAr, which causes the maximum value of VD in the system. Concentrating exclusively on the objectives of power loss minimization leads to high operating costs of the system. It is perceived that in all the cases discussed, the proposed ALO algorithm’s results align with the examined objective functions and satisfy all the used constraints.

Figure 3.

Active power generation of thermal generating units in the presence and absence of emission cost.

The curves of convergence for each case are shown in Figures 4 to 7. Figure 8 displays all load buses (PQ buses) voltage profiles for all cases, and it can be concluded that voltages for all load buses are within their defined constraint (0.95–1.05) p.u., that is, 5%. Table 3 summarizes the reactive power generated by each generator, and it is observed that reactive power is confined within its prescribed boundaries. Table 4 summarizes the results for all cases obtained by ALO, PSO, and TLBO algorithm.

Figure 4.

Operating cost convergence curve of ALO, PSO, and TLBO.

Figure 5.

Convergence curve of voltage deviation along with operating cost of ALO, PSO, and TLBO.

Figure 6.

Convergence curve of active power loss of ALO, PSO, and TLBO.

Figure 7.

Convergence curve of reactive power loss of ALO, PSO, and TLBO.

Figure 8.

Voltage profile of load buses (PQ buses) for deterministic OPF cases obtained from ALO algorithm.

Table 3.

Reactive power generation of each generator: IEEE 30 bus system.

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
Q_T1	−20	150	−19.095	1.905	−2.766	−7.571	−17.905	−0.993	−16.182	−13.954	−17.766	−17.211	−16.643	−17.472
Q_T2	−20	60	25.461	21.082	24.347	11.514	35.491	20.191	6.190	19.700	3.262	0.532	3.939	0.471
Q_T5	−30	35	28.367	20.740	27.354	34.999	31.460	29.239	22.179	18.731	25.005	32.354	34.991	29.817
Q_T8	−15	40	36.088	16.760	30.560	37.344	34.836	23.963	26.219	36.503	22.493	39.999	39.725	40.000
Q_T11	−20	30	9.825	27.757	0.000	18.786	11.982	18.412	6.029	28.515	22.685	7.0295	13.066	4.695
Q_T13	−20	25	14.923	18.407	10.119	15.546	13.863	11.963	19.968	1.652	11.700	1.548	16.190	6.451

Table 4.

Summary of results obtained by ALO, PSO, and TLBO algorithm.

Algorithm		Case (a)	Case (b)	Case (c)	Case (d)
ALO		–	0.1292 p.u.	3.207 MW	−20.057 MVAr
	Operating Cost ($)	813.030	814.051	974.134	974.117
PSO		–	0.169 p.u.	3.441 MW	−18.284 MVAr
	Operating Cost ($)	814.205	816.683	974.894	973.811
TLBO		–	0.130 p.u.	3.283 MW	−19.138 MVAr
	Operating Cost ($)	813.418	813.692	974.315	974.911

It is observed from Table 4 and the convergence curves that the suggested ALO algorithm provides more robust solutions and converges with fewer iterations. These facts demonstrate the proposed algorithm’s ability to solve more complex OPF problems, such as incorporating stochastic WT power generation.

Probabilistic OPF (IEEE 30-bus system)

The IEEE 30-bus is modified by incorporating two wind farms. The WF connected at bus 21 has a capacity of 50 MW and consists of 25 WT of 2 MW, whereas the WF connected at bus 30 has a capacity of 30 MW consisting of 15 WT of the same capacity. All the WT have nominal, cut-in, and cut-out wind speeds of 10, 2.7, and 25 m/s, respectively.

As discussed above, the OPF is conducted at a repeated interval of 15 minutes. WT forecasted output power is calculated by considering fifteen real-time wind speed readings, having a sampling time of 1 minute. The real-time wind speed data is taken from NREL (Andreas, 2021). The WT forecasted output power is computed based on the wind speed data and probabilistic wind speed models using equation (27). The structural outline to calculate the WT forecasted power output is shown in Figure 9. Similar cases as discussed above are evaluated, and the estimated results are shown in Table 5. In Case (a), the best value of the minimum operating cost obtained is $624.336. In Case (b), the minimum value of the VD obtained is 0.108 p.u., resulting in the system’s maximum power loss. In Case (c), the minimum active power loss obtained is 1.997 MW, resulting in the system’s highest operating costs. In Case (d), the minimum reactive power loss obtained is −27.537 MVAr, which causes maximum VD in the system. The negative reactive power generation shows the power absorption potential of the generators. The proposed ALO algorithm successfully optimizes the considered objective function, and all the required constraints are satisfied.

Figure 9.

Structural outline to calculate the forecasted power of WT.

Table 5.

Simulation result obtained for case studies: IEEE 30-bus system.

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
P_T1	50	200	142.533	138.950	143.291	135.267	133.020	142.549	50.000	50.018	50.002	50.000	51.398	50.011
P_T2	20	80	42.576	40.971	42.287	40.088	41.914	42.271	33.876	58.147	31.026	55.157	50.515	54.090
P_T5	15	50	19.663	19.708	18.716	18.604	19.643	19.414	50.000	50.000	49.939	50.000	46.247	49.999
P_T8	10	35	10.000	12.235	10.004	20.156	19.111	10.002	35.000	23.981	34.660	35.000	23.312	34.998
P_T11	10	30	10.000	12.149	10.000	10.000	10.095	10.001	23.854	18.645	28.642	16.218	21.749	13.132
P_T13	12	40	12.000	12.066	12.061	12.000	12.097	12.000	38.952	31.158	37.328	25.488	38.755	29.578
P_W21	0	50	33.631	33.663	33.654	33.605	33.610	33.608	33.702	33.656	33.698	33.569	33.667	33.664
P_W30	0	30	20.174	20.162	20.157	20.170	20.223	20.196	20.172	20.154	20.189	20.179	20.189	20.185
V_T1	0.95	1.1	1.036	1.044	1.050	1.024	1.034	1.047	1.022	1.034	1.014	1.050	1.050	1.050
V_T2	0.95	1.1	1.018	1.021	1.035	1.009	1.022	1.030	1.015	1.028	1.008	1.046	1.046	1.050
V_T5	0.95	1.1	0.983	0.991	1.008	0.984	0.998	1.002	0.998	1.007	0.991	1.027	1.034	1.041
V_T8	0.95	1.1	0.990	1.001	1.010	1.000	0.999	1.005	1.011	1.017	1.004	1.038	1.045	1.050
V_T11	0.95	1.1	1.026	0.951	0.953	1.008	0.951	1.024	1.006	0.998	1.050	1.028	1.004	0.984
V_T13	0.95	1.1	1.010	1.037	1.013	1.028	1.018	1.005	1.021	1.016	1.044	1.027	1.011	1.006
V_W21	0.95	1.1	0.995	1.020	0.996	1.009	1.012	1.012	1.000	0.992	1.034	1.032	1.006	0.993
V_W30	0.95	1.1	0.993	1.048	1.008	1.021	1.017	1.014	1.016	1.017	1.050	1.038	1.035	1.006
T₁₁	0.9	1.1	0.931	0.968	0.987	0.957	0.936	1.052	0.986	1.051	0.927	1.030	1.082	1.078
T₁₂	0.9	1.1	1.029	0.926	0.981	1.048	1.098	0.923	1.011	1.002	1.059	0.939	1.028	1.086
T₁₅	0.9	1.1	0.954	1.071	1.056	0.988	0.964	0.979	1.028	1.045	0.974	0.994	1.022	1.027
T₃₆	0.9	1.1	0.994	0.950	1.018	0.999	1.009	0.989	1.003	1.036	0.969	1.003	1.073	1.056
Q₁₀	0	5	0.238	4.927	0.872	2.889	0.010	0.000	3.210	2.307	2.549	4.342	3.149	5.000
Q₁₂	0	5	0.287	4.998	4.237	1.033	0.013	1.374	0.240	2.499	2.526	4.456	4.904	4.995
Q₁₅	0	5	3.122	4.524	4.172	3.800	4.892	5.000	1.556	1.933	5.000	3.137	2.457	5.000
Q₁₇	0	5	0.780	0.541	3.176	3.273	4.702	0.008	2.936	4.541	4.122	1.407	2.292	4.974
Q₂₀	0	5	0.360	4.394	2.781	4.003	3.022	5.000	2.542	3.550	4.917	3.872	1.436	4.930
Q₂₁	0	5	3.602	1.646	2.738	1.110	3.373	0.004	4.668	3.270	1.269	1.451	0.782	0.011
Q₂₃	0	5	2.911	4.474	3.852	3.996	3.402	3.942	2.067	3.599	2.904	0.384	3.302	3.472
Q₂₄	0	5	2.781	0.971	1.532	4.622	4.975	5.000	0.020	3.353	4.993	3.046	2.275	5.000
Q₂₉	0	5	0.844	4.926	3.320	0.384	4.233	0.001	4.772	0.633	4.963	2.493	1.599	1.186
Cost			624.336	625.060	624.718	625.130	626.622	624.480	760.278	747.860	768.684	747.893	742.940	746.372
VD			0.308	0.224	0.350	0.108	0.130	0.127	0.542	0.391	0.503	0.795	0.390	0.467
P_losses			6.782	6.802	6.929	7.027	6.488	6.863	1.997	2.360	2.043	2.179	2.432	2.264
Q_losses			−9.731	−0.813	−6.550	−8.061	−3.627	−5.443	−25.582	−23.118	−11.054	−27.537	−24.780	−27.259

Note that, in OPF formulation, the WT active power generation is not considered a control variable. Therefore, in all the cases, the WT active power generation is different because it is considered a stochastic variable. Figure 10 shows the shift in thermal generators’ generation schedule with WT when emission cost is added with the fuel cost. Active power generation of TGU is within tolerable limits in the presence as well as in the absence of emission cost. It is be concluded from Figure 11 that voltages for all load buses are within their set constraint (0.95–1.05 p.u.), tolerance limit. The results in Table 6 reveal that the generated reactive power of each generator is within the specified bounds.

Figure 10.

Active power generation of thermal generating units in the presence and absence of emission cost with wind farms.

Figure 11.

Voltage profile of load buses (PQ buses) for deterministic OPF cases obtained from ALO algorithm.

Table 6.

Reactive power generation of each generator: IEEE 30-bus system.

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
Q_T1	−20	150	7.369	12.637	−3.430	−5.278	−5.882	2.562	−4.485	−4.471	−3.980	−7.182	−10.362	−18.234
Q_T2	−20	60	28.743	0.399	19.083	9.427	25.658	17.739	5.967	3.955	5.966	13.915	−1.287	5.591
Q_T5	−30	35	25.896	26.513	31.512	27.492	34.968	29.092	21.470	18.850	21.988	20.706	24.930	26.908
Q_T8	−15	40	34.954	33.259	30.341	36.096	31.783	24.682	29.474	19.904	26.210	32.096	25.478	28.953
Q_T11	−20	30	2.633	−23.711	−17.218	−4.014	−24.980	11.711	0.464	5.792	1.285	2.463	6.506	0.619
Q_T13	−20	25	4.041	24.999	16.953	12.420	4.628	−1.503	18.718	17.133	8.840	1.264	4.298	3.348
Q_W21	−20	25	2.208	16.429	13.397	13.201	25.000	13.913	9.408	9.844	11.579	12.412	21.782	15.533
Q_W30	−25	15	−0.159	−1.367	0.775	3.051	0.547	0.107	0.139	5.434	−0.040	0.124	8.067	1.792

Table 7 presents the ALO algorithm’s results for deterministic and probabilistic OPF in all cases. It is observed that when the WT is considered, the operating cost is highly reduced, and overall system performance is improved in terms of VD and power loss.

Table 7.

Summary of results obtained for probabilistic and deterministic OPF by ALO algorithm.

Objective function	Probabilistic OPF	Deterministic OPF
Operating cost minimization ($)	624.336	813.030
Operating cost and VD (p.u.) minimization	0.108	0.1292
Active power loss minimization (MW)	1.997	3.207
Reactive power loss minimization (MVAr)	−27.537	−20.057

Probabilistic OPF (IEEE 57 bus system)

The probabilistic OPF is performed on a more extensive bus system to prove the proposed ALO algorithm’s scalability. The IEEE 57 bus is modified by incorporating two WF connected at bus 16 has a capacity of 100 MW consisting of 50 WT and at bus 17 has a capacity of 60 MW consisting of 30 WT of similar ratings mentioned above. Similar cases are evaluated under this test system, and the results are tabulated in Table 8. Each algorithm’s selected parameters are considered population size (100) and iteration number (1000). The results satisfy all the desired constraints taken in the study. In Case (a), the minimum fuel cost incorporated with the emission cost obtained is $4971.34. In Case (b), the minimum VD obtained is 0.787 p.u., responsible for maximum power losses in the system. In Case (c), the minimum active power loss is 14.860 MW. In Case (d), the best value of reactive power loss achieved is -51.038 MVAr. The objective of reducing power loss results in a high operating cost for the system. The Figure 12 shows the change in the active power generation schedule of TGU integrated with wind turbine in the presence and absence of the emission cost incorporated with the operating cost of TGU. The active power generation of TGU is within its limits both in the presence and absence of emission cost. As seen in Figure 13, the voltages on all load buses are within their specified limits (0.95–1.05 p.u.). The results in Table 9 indicate that each generator generates reactive power within the specified limits.

Table 8.

Simulation result obtained for case studies: IEEE 57-bus system.

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
P_T1	0	578.88	344.460	345.324	343.465	341.549	352.085	343.014	193.366	355.230	342.883	212.464	325.470	342.640
P_T2	0	100	100.000	99.996	100.000	100.000	99.690	100.000	72.080	42.507	100.000	56.008	76.498	99.997
P_T3	0	140	113.015	104.491	108.659	118.212	102.542	109.320	119.277	118.311	109.346	115.176	108.452	108.981
P_T6	0	100	100.000	100.000	100.000	100.000	99.835	100.000	100.000	59.627	99.999	100.000	67.660	100.000
P_T8	0	550	66.414	68.365	66.660	64.730	73.930	66.585	133.736	109.276	66.694	133.776	90.444	66.774
P_T9	0	100	100.000	99.994	99.997	100.000	46.272	100.000	100.000	68.185	99.999	100.000	100.000	100.000
P_T12	0	410	297.156	302.835	298.941	299.341	351.095	299.772	384.759	366.502	298.739	387.198	348.980	299.467
P_W16	0	100	67.284	67.283	67.402	67.281	67.089	67.152	67.320	67.202	67.397	67.116	67.324	67.327
P_W17	0	60	40.378	40.452	40.379	40.396	40.393	40.415	40.226	40.460	40.361	40.373	40.417	40.326
V_T1	0.95	1.05	1.043	1.050	1.050	1.002	1.024	1.050	1.043	1.049	1.050	1.042	1.022	1.050
V_T2	0.95	1.05	1.033	1.038	1.047	0.987	1.017	1.042	1.040	1.033	1.050	1.036	1.019	1.045
V_T3	0.95	1.05	1.023	1.012	1.034	0.988	1.000	1.017	1.034	1.003	1.048	1.024	1.015	1.029
V_T6	0.95	1.05	1.010	1.009	1.005	0.997	1.014	0.978	1.024	1.001	1.036	1.027	1.020	0.995
V_T8	0.95	1.05	1.039	1.025	0.995	1.036	1.046	0.961	1.023	1.044	1.031	1.031	1.040	0.986
V_T9	0.95	1.05	1.008	0.995	0.985	0.998	1.003	0.956	1.006	0.998	1.013	1.005	1.010	0.979
V_T12	0.95	1.05	0.999	0.986	1.005	0.991	0.999	0.987	1.018	0.981	1.025	1.010	1.004	1.004
V_W16	0.95	1.05	0.996	0.980	1.030	1.015	1.023	1.015	1.030	0.998	1.050	1.014	1.011	1.034
V_W17	0.95	1.05	1.041	1.016	1.050	1.043	1.014	1.034	1.045	1.022	1.036	1.032	1.031	1.041
T₁₉	0.9	1.1	1.029	1.050	0.902	0.940	1.076	0.900	0.996	1.034	0.966	1.024	1.023	0.908
T₂₀	0.9	1.1	0.944	0.976	1.008	1.000	1.022	1.022	0.999	0.930	0.983	0.939	1.010	0.969
T₃₁	0.9	1.1	1.076	0.990	1.004	0.999	0.950	0.992	0.994	0.950	0.998	0.936	1.003	1.002
T₃₅	0.9	1.1	0.946	0.964	0.956	0.919	0.960	0.975	0.919	0.956	0.926	0.950	0.944	0.950
T₃₆	0.9	1.1	1.049	0.900	0.914	1.018	0.902	0.900	0.936	1.007	0.922	0.988	0.962	0.900
T₃₇	0.9	1.1	1.089	0.967	0.976	1.065	1.044	0.982	1.025	1.042	0.978	1.046	1.080	0.962
T₄₁	0.9	1.1	0.949	0.943	0.931	0.961	0.984	0.920	0.992	0.952	0.965	0.949	0.953	0.923
T₄₆	0.9	1.1	0.988	0.975	0.925	0.971	0.900	0.933	0.986	0.905	0.912	0.923	0.915	0.913
T₅₄	0.9	1.1	0.997	0.955	0.904	0.981	0.900	0.900	0.968	0.920	0.901	0.948	0.961	0.900
T₅₈	0.9	1.1	0.964	0.945	0.970	0.933	0.952	0.949	0.968	0.943	0.977	0.956	0.955	0.962
T₅₉	0.9	1.1	0.974	0.983	0.960	0.934	1.012	0.959	0.963	0.985	0.992	0.936	0.984	0.962
T₆₅	0.9	1.1	0.955	0.949	0.941	0.940	0.926	0.943	0.954	1.030	0.980	0.933	0.973	0.949
T₆₆	0.9	1.1	0.921	0.900	0.900	0.922	0.908	0.911	0.934	0.905	0.948	0.956	0.934	0.919
T₇₁	0.9	1.1	1.012	0.925	0.929	0.969	0.964	0.902	0.941	0.924	0.957	0.941	0.949	0.926
T₇₃	0.9	1.1	0.959	0.977	0.998	0.997	1.034	1.045	0.965	0.972	0.975	1.019	0.979	0.990
T₇₆	0.9	1.1	0.982	1.033	0.960	0.997	0.975	0.961	1.023	1.024	0.942	1.029	0.996	0.954
T₈₀	0.9	1.1	0.985	1.043	0.932	0.975	1.019	0.919	0.955	0.989	0.959	0.952	1.024	0.928
Q₁₈	0	5	1.729	0.735	4.927	3.464	4.057	4.883	4.270	1.203	4.865	1.844	0.814	0.026
Q₂₅	0	5	4.884	5.000	5.000	2.192	3.869	4.981	2.435	4.762	4.987	1.757	4.676	5.000
Q₅₃	0	5	3.377	4.336	5.000	1.742	4.406	4.988	4.429	3.461	5.000	1.214	3.641	5.000
Cost			4971.340	4996.559	4977.018	4977.789	5311.99	4985.719	6000.000	5824.312	4976.695	6000.0	5375.684	4978.851
VD			1.080	1.108	1.069	0.787	1.013	0.861	0.982	0.861	1.116	0.940	0.948	1.041
P_losses			30.388	32.940	29.705	31.101	37.132	30.459	14.860	31.500	29.618	15.422	29.444	29.712
Q_Losses			29.897	47.123	16.473	43.137	54.893	23.965	−31.039	33.747	28.536	−51.038	21.698	25.078

Figure 12.

Active power generation of thermal generating units in the presence and absence of emission cost with wind farms.

Figure 13.

Voltage profile of load buses (PQ buses) for probabilistic OPF cases obtained from ALO algorithm.

Table 9.

Reactive power generation of each generator: IEEE 57 bus system

Qty.	Min	Max	Case (a)			Case (b)			Case (c)			Case (d)
			ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO	ALO	PSO	TLBO
Q_T1	−140	200	70.508	117.283	36.784	39.281	25.099	84.149	29.254	115.057	−8.384	58.472	4.341	44.295
Q_T2	−17	50	23.469	34.475	49.909	−4.118	44.278	49.298	50.000	42.538	49.961	49.823	45.914	49.786
Q_T3	−10	60	39.871	11.658	59.699	20.100	−5.740	49.950	34.327	−9.793	59.970	15.766	29.384	58.961
Q_T6	−8	25	−5.795	14.117	16.113	−0.163	10.308	15.621	10.009	−7.866	24.474	21.886	24.995	12.258
Q_T8	−140	200	158.533	139.369	59.511	184.474	185.755	39.264	69.105	198.310	78.852	100.338	143.678	53.126
Q_T9	−3	9	9.000	8.935	8.717	8.982	7.292	5.764	8.986	−3.000	8.971	9.000	9.000	8.951
Q_T12	−150	40	38.227	39.755	39.835	39.999	27.637	37.798	31.804	−24.149	39.966	36.491	32.928	39.082
Q_W16	−40	50	−25.138	−39.937	21.932	37.426	29.938	17.661	4.384	−5.990	32.306	−13.651	2.873	31.036
Q_W17	−24	30	30.000	14.872	29.714	30.000	30.000	29.861	26.829	29.976	29.985	−0.978	30.000	29.211

The findings of probabilistic OPF for each objective function using the ALO, PSO, and TLBO lgorithms are summarized in Table 10 for the IEEE 30-bus and IEEE 57-bus systems, respectively. As illustrated in Table 10, the proposed technique achieves better outcomes when compared to previous techniques. To obtain the best results, the proposed optimization technique is run 30 times for each case; the best (min(objective)), mean, and worst outcomes, as well as the standard deviation and time of execution, are listed in Table 11.

Table 10.

Summary of results obtained for probabilistic OPF by ALO algorithm.

	IEEE 30-bus system			IEEE 57-bus system
	ALO	PSO	TLBO	ALO	PSO	TLBO
Operating cost ($)	624.336	625.06	624.718	4971.34	4996.559	4977.018
Voltage deviation (p.u.)	0.108	0.130	0.127	0.793	1.013	0.861
Active power losses (MW)	1.997	2.360	2.043	14.86	33.747	29.618
Reactive power loses (MVAr)	−27.537	−24.78	−27.259	−51.038	21.698	25.078

Table 11.

Statistical analysis of results of all the cases obtained from ALO.

Statisticalresults	IEEE 30-bus system (deterministic OPF)				IEEE 30-bus system (probabilistic OPF)				IEEE 57 bus system (probabilistic OPF)
	Case (a)	Case (b)	Case (c)	Case (d)	Case (a)	Case (b)	Case (c)	Case (d)	Case (a)	Case (b)	Case (c)	Case (d)
Best (Min(objective))	813.032	0.1292	3.207	−20.057	624.336	0.108	1.997	−27.537	4971.340	0.787	14.860	−51.038
Mean	814.013	0.161	3.269	−19.822	625.962	0.129	2.235	−26.263	4986.015	0.8571	15.809	−46.301
Worst	819.485	0.195	3.458	−19.268	631.136	0.157	2.807	−22.622	5011.350	1.0666	17.530	−40.871
Standard deviation	1.329	0.016	0.046	0.168	1.272	0.013	0.187	1.048	9.761	0.0683	0.740	3.119
Time(seconds)	24.151	24.434	25.381	25.768	22.848	22.484	22.109	24.643	572.667	548.5787	624.039	625.897

Conclusion

In this paper, a novel nature-inspired Ant Lion algorithm is successfully implemented to solve optimal power flow problems incorporated with wind turbine generators, and its results are compared with Particle Swarm Optimization and Teaching-Learning-Based Optimization algorithm. The proposed algorithm is validated on IEEE 30-bus and IEEE 57-bus systems. The simulation results demonstrated that the proposed algorithm provides better solutions and ensures good convergence characteristics in single and multi-objective optimal power flow problems. The results show that incorporating wind turbines enhances the overall power system’s performance, such as operating cost of thermal generating units, power losses, voltage profile, and greenhouse gas emissions. Uncertainties of wind turbines are modeled using the Weibull probability density function. The carbon tax is considered, which affects the scheduling and the operating cost of thermal generating units, further minimizing greenhouse gas emissions. The active power forecasting model of wind turbines is advantageous in the power system’s real-time optimal power flow problems. This modeling can also be applied to large-scale test systems in future research work.

Footnotes

Appendix

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 iD

Ankur Maheshwari

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