A Lagrange-Based Multi-Objective Framework for Wind–Thermal Economic Emission Dispatch

Abstract

Economic dispatch using wind power plants plays a role in reducing the price of electricity production by dispatching power among different generating units for thermal and wind power plants, and supplying load demand while meeting the power system equality and inequality constraints. Adding wind power plants to the economic dispatch model can significantly reduce electricity production costs and reduce carbon dioxide emissions. In this paper, fuel cost and emission minimization are considered as the objective function of the economic dispatch problem, taking into account transmission loss using the B matrix. The quadratic model of the fuel cost and emission criterion functions is modeled without considering a valve-point loading effect. The real power generation limits for both wind and conventional generating units are considered. In addition, a closed-form expression based on the incomplete gamma function is provided to define the impact of wind power, which includes the cost of wind energy, including overestimation and underestimation of available wind power using a Weibull-based probability density function. In this research work, Lagrange’s algorithm is proposed to solve the Wind–Thermal Economic Emission Dispatch (WTEED) problem. The developed Lagrange classical optimization algorithm for the WTEED problem is validated using the IEEE test systems with 6-, 10-, and 40-generation unit systems. The proposed Lagrange optimization method for WTEED problem solutions demonstrates a notable improvement in both economic and environmental performance compared to other heuristic optimization methods reported in the literature. Specifically, the fuel cost was reduced by an average of 4.27% in the IEEE 6-unit system, indicating more economical power dispatch. Additionally, the emission cost was lowered by an average 22% in the IEEE 40-unit system, reflecting better environmental compliance and sustainability. These results highlight the effectiveness of the proposed approach in achieving a balanced trade-off between cost minimization and emission reduction, outperforming several existing heuristic techniques such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE) under similar test conditions. The research findings report that the proposed Lagrange classical method is efficient and accurate for the convex wind–thermal economic emission dispatch problem.

1. Introduction

The production of electricity based on fossil fuels affects the atmosphere by releasing pollutants like sulfur dioxide (SO₂), nitrogen oxides (NOx), and carbon dioxide (CO₂). The goal of wind–thermal economic emission dispatch problems in electrical power system generation is to dispatch power accordingly to the committed generating units for both conventional and wind power plants by satisfying load demand at a particular time while meeting the system equality and inequality constraints of the power system. Wind power plants play an essential role in the production of electricity. Introducing these Distributed Energy Resources (DERs) can tremendously reduce the production cost and the amount of gases emitted to the atmosphere; however, the unpredictability and uncontrollability of wind speed result in uncertain results in wind power output. The integration of wind power with existing power systems requires further investigation due to its stochastic characteristics, which differ from those of other conventional energy sources.

The literature has investigated the economic dispatch problem formulation, including the quadratic cost function with valve-point loading effects. The wind power economic dispatch problem is modeled with the Weibull probability distribution function (PDF). The authors in [1] investigated the economic dispatch problem formulation, which includes the quadratic cost function with a valve-point loading effect. The penalty for not utilizing all available wind (underestimation) and the reserve cost due to unavailability (overestimation) of wind power are included in the cost function, along with the direct cost function, depending on the ownership of the wind farm. The emission cost function is represented in terms of the quadratic cost and an exponential function. Power balance, including transmission loss, is defined as equality constraints, while real power operating limits and prohibited operating zones of the generators are considered as inequality constraints. The Modified Teaching–Learning Algorithm (MLTA) is used to solve the economic emission dispatch problem.

This algorithm has the benefit of moving freely towards the optimal solution without any adjustment to the parameters. The best values of production cost and emission are calculated for each test system. For a 6-unit system, incorporating wind units, the probabilistic approach recorded a high cost function and emissions compared to the deterministic approach. Looking at the deterministic results shows that the injection of wind power can reduce the fuel cost and emissions.

In [2], the economic emission dispatch problem formulation includes the quadratic cost function with a valve-point loading effect. The wind underestimation and overestimation of wind power is included in the cost function, with the direct cost function that depends on the ownership of the wind farm. The emission cost function is represented in terms of the quadratic cost and an exponential function. Power balance and transmission losses are included as equality constraints, while real power operating limits for both wind and thermal generators are considered as inequality constraints. The power system used as a case study contains five thermal power generators integrated with large-scale farms, which consist of 160 wind turbines. Bi-Population Chaotic Differential Evolution (BPCDE) is considered as the optimization algorithm. The results of the DE are compared only to the values obtained from the heuristic algorithms in the literature.

Researchers in [3] formulated the economic emission dispatch problem, which includes a quadratic cost function with a valve-point loading effect. The PDF is used to model wind power uncertainties. The penalty for not utilizing all available wind (underestimation) and the reserve cost due to unavailability (overestimation) of wind power are included in the cost function, along with the direct cost function, depending on the ownership of the wind farm. The emission cost function is represented in terms of the quadratic cost and an exponential function. Power balance, including transmission loss, is represented as an equality constraint, while real power operating limits and prohibited operating zones of the generators are considered as inequality constraints. In this study, three test systems, each consisting of 6, 15, and 40 thermal units, are used in conjunction with two wind farms.

The Guided Artificial Bee Colony (GABC) algorithm is used to solve the considered optimization problem. The total fuel cost of this algorithm is compared with that of other heuristic algorithms for 6-unit, 15-unit, and 40-unit systems. For a 6-unit system, the results were obtained for different power demands of 1200 MW, 1400 MW, and 1600 MW. Case 4 is only analyzed here since it is a multi-objective optimization problem.

In [4], the study presents the economic emission dispatch problem formulation, which involves a linear quadratic cost function with a valve-point loading effect. The wind power output is carefully modeled with PDF. The wind underestimation and overestimation of wind power are included in the cost function, with the direct cost function that depends on the ownership of the wind farm. The emission cost function is represented in terms of the quadratic cost and an exponential function. Power balance is included as an equality constraint, while real power operating limits for both wind and thermal generators are considered as inequality constraints. The above multi-objective problem is converted to a single-objective optimization problem by taking care of the implied cost of emission through the concept of Price Penalty Factors (PPFs). The IEEE 30-bus test system is considered the test system. The IEEE 30-bus system consists of six thermal generators and two extra wind turbines.

The Gravitational Search Algorithm (GSA) is implemented in solving this dispatch problem. This algorithm also shows the importance of selecting GSA parameters, such as population size, Go, and φ, towards finding the total cost of the WTEED problem. These are selected as population size of 100, Go of 0.10 and φ of 100.The algorithm has better quality and convergence properties and better computation time to solve both single- and multi-objective WTEED problems than other algorithms.

In studies [5,6,7], the formulation of the wind–thermal economic emission dispatch (WTEED) problem included the quadratic cost function with a valve-point loading effect, and modeled the wind power with PDF. The underestimation and overestimation of wind power availability are incorporated into the fitness function and the direct cost function of wind power, which depend on the ownership of the wind farm. The emission cost function is represented in terms of the quadratic cost and an exponential function. Ref. [5] uses a modified 30-bus system having six thermal units. Wind generators replace the last two units. Ref. [6] uses two conventional test systems, including 6 and 40 thermal generators, with two wind farms with penetratiosn of 10 MW and 100 MW, respectively. Lastly, ref. [7] utilizes a test system configured as described in [1].

In [5], the Hybrid Flower Pollination Algorithm and Time-Varying Fuzzy Selection Mechanism (HFPA-TVFSM) is used to solve the WTEED problem. This hybrid algorithm is used for solving the wind–thermal dynamic multi-objective optimal dispatch problem for simultaneous minimization of cost, emission, and power loss. The selection of parameters such as population size, switch probability, Levy distribution factor, and control parameters is essential in this algorithm. Parameters are set as population sizes of 10 and 100 iterations, switch probability of 0.8, Levy factor of 1.5, crossover in DE of 0.75, and weighing factors W1 to W3 of 1/3. The solution of the dispatch problem is presented using four different solutions using HFPA.

In [6], Gravitational Acceleration Enhanced Particle Swarm Optimization (GAEPSO), is another hybrid optimization algorithm in which GSA is combined with PSO. Firstly, parameters need to be set, such as population size, both set to 30 for both systems, and iteration set to 100 for 6 units and 200 for 40 units. For best results, the acceleration factors c_1, c_2, and c_3 are 1.5, 2.5, and 2.0. This is because the production cost of fuel for thermal power is higher than that of wind power. Also, the difference in emissions is relatively small because wind power plants have no emissions. In [7], Online Learning Honey Bee Mating Optimization (OLHBMO) is used as a solution to the WTEED problem. This method combines the HBMO with the chaotic local search (CLS) to form IHBMO. Firstly, parameters need to be stated, such as population size. The authors [8,9] solved the Combined Emission Economic Dispatch (CEED) problem with quadratic fuel cost, excluding the cost function for overestimation and underestimation.

Nevertheless, these techniques mentioned above often work in a centralized way, gathering global data from each generator for improvement at a single node. This centralized method may not meet the dynamic requirements of contemporary innovative grid systems and can be computationally and communication-intensive, especially as power networks get bigger. The use of the Lagrange multiplier method for solving the WTEED problem can minimize the drawbacks that are inherent in these methods. Many researchers have utilized the Lagrange multiplier method in solving the economic emission dispatch problem. In [10], the authors evaluated the economic emission dispatch problem formulation, which involves a quadratic cost function. Power balance equation is included as an equality constraint, whereas real power operating limits for both wind and thermal generators are considered as inequality constraints. The stochastic nature of wind is modeled using the Weibull distribution function. The overestimation and underestimation penalty of wind power is also included in the objective function. The IEEE 39-bus system consists of nine thermal systems and one wind power system. A classical optimization method, named the alternating direction method of multipliers (ADMM), is an optimization algorithm that combines the advantages of dual decomposition and augmented Lagrangian methods and is utilized to solve the WTEED problem. This is done by converting the EDP into a two-layered structure, the inner and the outer layer. The outer layer is considered as a parallel ADMM where the simultaneous updates of lambda variables are performed, and by doing so, can better achieve the required computational time. The inner layer is considered as a finite-step consensus algorithm where the agents reach consensus in some communication rounds in a power system. This algorithm converged efficiently to the optimal economic dispatch solution. The optimal values were compared with the values for Consensus-Based Parallel ADMM (C-PADMM) and Distributed Gradient Descent ADMM (DGD-ADMM) reported in the literature. It is also recorded that as the network becomes bigger, the computational advantages of this method become more useful.

In [11], the researchers analyze the economic emission dispatch problem formulation, which involves the linear quadratic cost function of the thermal unit. The emission cost function is used in terms of the quadratic. Power balance and transmission loss are included as equality constraints, while real power operating limits for thermal generators are considered as inequality constraints. The bio-objective optimization problem is converted to a single-objective one by using Price Penalty Factors (PPFs) (h). The IEEE 30-bus system, which consists of six thermal generators, is used as a case study for this optimization problem. A Lagrange multiplier (classical optimization method) is used as an optimization algorithm to solve the economic emission dispatch problem. A significant optimization problem is converted into a two-stage dual problem with a master problem and subproblems using Lagrange multipliers between the two problems until the condition for optimality is achieved. The problem was simulated with different power demands of 410–700 MW. By monitoring the fuel and the emission as the change in power demand is carried out, it is concluded that the higher the power demand, the greater the increase in fuel cost and emissions. This was compared with existing algorithms found in the literature, and the method proved superior. It is recommended that the use of wind or solar can reduce the fuel cost and emissions for the economic emission dispatch problem.

In [12], the studies focus on the economic emission dispatch problem formulation, which includes the linear quadratic cost function for thermal units. The emission cost function is used in terms of the quadratic. Power balance and transmission loss are used as equality constraints, while real power operating limits, prohibited zones, and ramp rate are considered as inequality constraints. The inclusion of ramp rates and prohibited zones makes this optimization a nonconvex one. The multi-objective optimization problem is converted to a single-objective one by using Price Penalty Factors (PPFs) (h). IEEE 14 and 30 bus systems with prohibited operating zones are used as a case study for this optimization problem. A Lagrange multiplier (classical optimization method) is used as an optimization algorithm to solve the economic emission dispatch problem. A significant optimization problem is converted into a two-stage dual problem with a master problem and subproblems using Lagrange multipliers between the two issues until the condition for optimality is achieved. The problem was simulated with different power demands of 550 MW and 600 MW. Monitoring of the fuel and the emissions was conducted to determine the change in power demand, and it was found that the higher the power demand, the greater the increase in fuel cost and emissions. The trade-off curve for both load demands is presented, showing different weights for fuels versus emissions. The method efficiently finds optimal solutions independent of system size or load and can effectively handle practical constraints.

By using the ramp and prohibited zones, this algorithm can effectively find the optimal solution of the economic emission dispatch problem, independent of the size and load demand. Also, it can be used as a real-time application of the EED problem. Ref. [13] includes an economic dispatch problem formulation which involves only the linear quadratic cost function. Power balance is included as an equality constraint, while real power operating limits for thermal generators are considered as inequality constraints. The above multi-objective problem is converted to a single-objective optimization problem by taking care of the implied cost of emission through the concept of Price Penalty Factors (PPFs) (h). The IEEE 30-bus test system is considered the test system. The IEEE 30-bus system consists of 6, 15, and 40-unit thermal systems. A Lagrange multiplier (classical optimization method) is utilized in solve the economic dispatch problem. A large-scale linked problem could be converted into a two-stage dual problem with a master problem and subproblems using Lagrange multipliers to relax global restrictions.

In the Lagrange method, the multiplier initial guess plays a vital role in finding a global solution and effectiveness in computational time. Also, all convergence times were recorded accordingly, and were less than 1 sec even for the 40-unit generating system. The method was compared with other classical algorithms and found to be superior. It was also found that this power system can be integrated with other applications in solving the economic emission dispatch problem. Ref. [14] includes the economic emission dispatch problem formulation, which involves only the linear quadratic cost function. The emission cost function is represented in terms of the quadratic. Power balance is included as an equality constraint, while real power operating limits for thermal generators are considered as inequality constraints. The above multi-objective problem is converted to a single-objective optimization problem by taking care of the implied cost of emission through the concept of Price Penalty Factors (PPFs). The IEEE 30-bus test system is considered the test system. The IEEE 30-bus system consists of six thermal generators. A Lagrange multiplier (classical optimization method) and Particle Swarm Optimization (Heuristic optimization method) (PSO) are used as two optimization algorithms to solve the economic emission dispatch problem. A large-scale linked problem could be converted into a two-stage dual problem with a master problem and subproblems using Lagrange multipliers to relax global restrictions. PSO uses a population (swarm) and updates the velocity of the local particles until the global solution is met. It is evident from these results that the Lagrange multiplier achieved less fuel cost and emissions than PSO. In the Lagrange method, the multiplier initial guess plays a vital role in finding a global solution and efficiency in computational time. In PSO, the larger the population, the longer the computational time in finding the global solution. It is concluded that in this comparison, the Lagrange multiplier method is superior than PSO in finding the minimum fuel cost and the emissions for the combined economic emission dispatch.

In [10,11,12,13,14], the Lagrange multiplier method can break more complex mathematical problems into smaller, more manageable subproblems and handle “local” and “coupling” constraints. This is accomplished by solving dual and relaxed issues and forming the Lagrange function. The relative size of the duality gap between the primal and dual solutions measures the convergence of the dual optimization method.

Table 1. Comparison of the Lagrange multiplier method to the heuristic algorithms for the WTEED problem [14].

Criteria	Lagrange Multiplier Method	Heuristic Algorithms
Optimality	Yes (for convex/nonconvex and differentiable problems).	Cannot be mathematically guaranteed, since these algorithms mostly use random numbers in generating population values, which positively affects the calculation of active power.
Determinism	The initial value does not change the mathematical solution. Same solution every run, depending on the initial guess value of lambda variable.	These algorithms mostly use random numbers in generating population values, which positively affects the calculation of active power. This will yield different results on different runs.
Constraint Fulfillment	Can fulfill constraints for both nonlinear and linear optimization problems.	Approximate via penalty functions.
Computational Cost	Low for convex economic emission dispatch problems.	This may be high due to many iterations and evaluations of the fitness function in finding global solutions.
Interpretability	This is mostly achieved by breaking down the whole optimization problem into subproblems and using $\left(\lambda \right)$ in the subproblems to find a solution in the master.	Not easy to break down this algorithm to subproblems, while it is easy to find the solutions of nonlinear, nonconvex, and non-smooth economic emission dispatch problems.
Parameter Modification	This is not required in the classical method. The best solution and computational time are achieved by choosing an initial guess value $\left(\lambda \right)$.	Most parameters need proper tuning of the algorithm parameters. If this is not done, the optimal solution of the economic dispatch problem can prematurely converge.
Convergence Speed	This is fast if the economic emission dispatch is convex but depends on choosing an initial guess value $\left(\lambda \right)$.	Can be slower, and maybe stagnate near the optimum solution.

, below, compares the Lagrange multiplier method to heuristic algorithms based on the reviewed literature above. This table illustrates the benefits of using the Lagrange multiplier method as compared to the heuristic algorithms.

2. Formulation of the WTEED Problem

2.1. Formulation of the Thermal Power, Emissions, and Wind Power

This section provides the problem formulation of the wind–thermal economic dispatch problem using a single criterion function. The objective of the wind–thermal economic emission dispatch (WTEED) problem is to reduce the total operating cost and emission of the entire generation system, bounded by certain operational constraints. The total cost given by conventional and wind generators can be represented mathematically as [7]

$$F(P_g,P_w)={\sum }_{i=1}^{N_G}{F}_i\left(P_{gi}\right)+{\sum }_{j=1}^{N_F}{\sum }_{k=1}^{N_W}{W}_{j,k}\left(P_{w_{j,k}}\right)$$

(1)

where $F(P_g,P_w)$ is the total cost of all generating units, $N_G$ is the number of thermal generators, $N_F$ is the number of wind farms, $N_w$ is the number of wind generators, and $P_{gi}$ is the active power for thermal generators. The cost function $F_i\left(P_{gi}\right)$ is for thermal generator units. It is formulated using a quadratic function without a valve-point loading effect and is given in Equation (2) [15]. The thermal cost function can be presented with or without a valve-point loading effect, as shown in Figure 1. The arrows in Figure 1 represent the loading of the thermal generators.

$$F_i\left(P_{gi}\right)=a_i+b_i{P}_{gi}+c_i{P}_{gi}^2$$

(2)

where $a_i,b_i, \mathrm{and} c_i$ are the fuel cost co-efficient of the $i^{th}$ generator. The last term $W_{j,k}\left(P_{w_{j,k}}\right)$ from Equation (1) represents the output power of the wind generating units. The wind power output can be further represented by Equation (3) [7].

$$W_{j,k}\left(P_{w_{j,k}}\right)={\varphi }_{D_{j,k}}{P}_{E_{j,k}}+{\varphi }_{O{E}_{j,k}}\times (P_{A_{j,k}}-P_{E_{j,k}})+{\varphi }_{U{E}_{j,k}}\times \left(P_{E_{j,k}}{P}_{A_{j,k}}\right)$$

(3)

where $P_{E_{j,k}}$, and $P_{A_{j,k}}$ are the expected and actual/available wind power output of the turbine $j$ in wind farm $k$ in [MW], and ${\varphi }_{D_{j,k}}, {\varphi }_{O{E}_{j,k}},{and \varphi }_{U{E}_{j,k}}$ are the direct, overestimation, and underestimation cost co-efficients of the turbine $j$ in wind farm $k$ [USD/MWh].

2.1.1. Formulation of the Emission Cost Function

Atmospheric pollution has increased tremendously due to industrial advancement. The burning of coal from thermal plant utilities emits dangerous gaseous pollutants such as oxides of carbon ($\mathrm{C}{\mathrm{O}}_2$), oxides of sulfur ($\mathrm{S}{\mathrm{O}}_2$), and oxides of nitrogen ($\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$). The green power committee recommends that the economic emission dispatch problem be solved with low emissions. Therefore, there is a need for better optimum control methods that can guarantee minimum levels of pollution at a very reasonable fuel cost. In [16], it is reported that natural gas is a crucial transitional energy for Pakistan to attain the emission reduction goal for sustainable economic growth. Emission gases such as, $\mathrm{C}{\mathrm{O}}_2$, $\mathrm{S}{\mathrm{O}}_2$ and $\mathrm{N}{\mathrm{O}}_{\mathrm{x}}$ produced by thermal generators should be included in the optimization fitness function. Wind turbine generators are not subjected to pollution, and the output power of the wind plant depends on the wind speed, so the emission cost for a wind generator will be zero. The overall emission of these pollutants is the sum of the quadratic cost function and emission values, considering the exponential function of the valve-point loading effect. This can be represented mathematically as [17]

$$E_T={\sum }_{i=1}^{N_E}{E}_i\left(P_i\right)={\sum }_{i=1}^{N_E}({\alpha }_i+{\beta }_i{P}_i+{\gamma }_i{P}_i^2)$$

(4)

where $E_T$ is the total emissions in [kg/h], and ${\alpha }_i, {\beta }_i, {\gamma }_i,$ are emission coefficients of the generating units in kg/h.

Equality and inequality constraints

(a)

Real power balance constraints

The output power of the wind and thermal generation units will be equal to the total sum of the power demand and transmission line losses, and can be stated mathematically as given in Equation (5).

$${\sum }_{i=1}^{N_G}{P}_{gi}+{\sum }_{k=1}^{N_W}{P}_{w{h}_{j,k}}=P_D+P_{Loss}$$

(5)

where $P_D$ and $P_{Loss}$ are the total load demand and transmission line losses of the power system.

In electric power systems, losses are mainly caused by the transportation of electric power over a long distance, or in the case of relatively low load density, vast areas, and it is necessary that the transmission losses be included in the formulation of the problem for wind–thermal dynamic economic dispatch. In [18], the transmission loss formula is given by the first quadratic term of Equation (6), and it was later called George’s formula. The equation is also called the loss formula or B-coefficient method. To obtain a more accurate transmission loss equation, a linear term and a constant are added to the quadratic expression [18]; this is called Kron’s formula and is given in Equation (6) [19,20]. The B-coefficient method enables the coordination of power loss in scheduling the output of each plant for maximum economy for a given load.

$$P_L={\sum }_{m=1}^{N_G+N_W}{\sum }_{m^{\prime }=1}^{N_G+N_W}{P}_m{B}_{m{m}^{\prime }}{P}_{m^{\prime }}+{\sum }_{m=1}^{Ng+Nw}{B}_{om}{P}_m+B_{00}$$

(6)

where $m \mathrm{and} m^{\prime }$ are wind–thermal generator indexes and $B_{m{m}^{\prime }}{,}^{\prime }{B}_{0m} \mathrm{and} B_{00}$ are power loss coefficients.

The fitness function can be subjected to the following inequality constraints.

Inequality Constraints

(a)

Generation capacitor for conventional and wind power constraints

The operational output power of each generating unit is bounded by the minimum and maximum operating capacity and is given in Equations (7) and (8).

$$P_{gi}^{min}\le P_{gi}\le P_{gi}^{max},i=\mathrm{1,2},\dots N_{g }$$

(7)

$$0\le P_{j,h}\le P_{R,j,h} j=\mathrm{1,2},\dots N_w$$

(8)

where ${P_{gi}}^{max}$ and ${P_{gi}}^{min}$ are the maximum and minimum output power of the $i^{th}$ thermal, and $P_{R,j,h}$ is the rated power of the $j^{th}$ wind output generator.

2.1.2. Wind Power Probability Distribution Function

The representation of short-term wind speeds is very crucial for monitoring of wind energy potential. PDFs are mostly used to describe wind speed interpretations. The appropriateness of numerous PDFs has been studied for a number of areas in the world. The selection of the PDF is crucial in wind energy investigations because wind power is formulated as a clear function of wind speed distribution parameters.

A PDF that fits the wind speed data more closely will reduce doubts about wind power output approximations. The Weibull two-parameter PDF and the Rayleigh PDF are the most frequently used probabilities in wind speed data analysis. Most literature has modeled wind speed in terms of the Weibull PDF to formulate the wind–thermal economic dispatch problem. Weibull PDFs offer several advantages over other probability distributions. To mention significant benefits, it only requires two parameters to estimate, and it is a positively skewed probability density distribution that favors moderate wind speeds.

The PDF can be represented using the WPDF [21] and is given in Equation (9).

$$f(V)={\displaystyle \frac{k}{c}}{\left({\displaystyle \frac{V}{c}}\right)}^{k-1}{e}^{-{\left(V/c\right)}^k}\hspace{1em}0\le V<\infty$$

(9)

where $c$ is the scale parameter at a given location in m/s and $k$ is the shape parameter at a given location (dimensionless). Based on Equation (9) above, the Cumulative Distribution function (CDF) can be found as given in Equation (10).

$$F(V)=P(v\le V)={\int }_{V=0}^{V=x}\left({\displaystyle \frac{k}{c}}{\left({\displaystyle \frac{V}{c}}\right)}^{k-1}{e}^{-{\left(V/c\right)}^k}\right)dv=1-\mathit{exp}\left(-{\left({\displaystyle \frac{V}{c}}\right)}^k\right)$$

(10)

According to the PDF, the $n^{th}$ moment for any probability density function $f\left(V\right)$ is defined by Equation (11) [22].

$$\overline{V^n}={\int }_{V_{min}}^{V^{max}}f\left(V\right)dV=C^n\Gamma \left({\displaystyle \frac{n}{k}}+1\right), n=\mathrm{1,2}\dots$$

(11)

where $\Gamma$ is the Gamma function and is given in Equation (12).

$$\Gamma (\beta )={\int }_0^{\infty }{x}^{\beta -1}\mathit{exp}(-x)dx$$

(12)

The probability of wind power distribution can be derived by using Equations (9)–(12). Now, for each power unit, a simplified linear piecewise function is adopted to express the relationship between the output power of the wind generation and wind speed, and is given in Equation (13) [23]:

$$\left\{\begin{array}{l}{P}_w=0, v<v_C or v> v_F\\ P_w=P_R{\displaystyle \frac{v-v_C}{v_r-v_C}}, v_C\le v\le v_R\\ P_w=P_R, v_R\le v\le v_F\end{array}\right.$$

(13)

where $P_w$ and $P_R$ are power delivered and rated power by a wind turbine in MW, and $v_C, v_R$ and $v_F$ are cut in, rated, and cut-out/furling wind speed in m/s. The wind power curve is defined by either a continuous or discrete probability function [24,25]. In a continuous probability function, the wind speed is between the cut-in and rated wind speeds, as shown in Equation (13) above. Using the Weibull distribution function in Equation (9), the PDF in the continuous range is given by Equation (14).

$$f_{P_w}(p_w)={\displaystyle \frac{k{h}_w{v}_C}{P_{R}c}}{\left[{\displaystyle \frac{\left(1+\frac{hw}{P_R}\right)v_C}{c}}\right]}^{k-1}\mathit{exp}\left\{-\left[{\left({\displaystyle \frac{\left(1+\frac{hw}{P_R}\right)v_C}{c}}\right)}^k\right]\right., v_C\le v\le v_R$$

(14)

where $h_w$ is the intermediary parameter given by: $h_w={\displaystyle \frac{v_R-v_C}{v_C}}$.

In the discrete probability function, the wind speed is between the rated and furling wind speed and also where the output power has constant zero value, i.e., below the cut-in and cut-out wind speeds, as given in Equation (13) above. The mathematical representation of the two events $P_w=0$ and $P_w=P_R$ are given in Equations (15) and (16) separately [26].

$$\begin{array}{l}\mathit{Pr}(P_w=0)=\mathit{Pr}(V<v_C)+\mathit{Pr}(V\ge v_F)\\ =1-\mathit{exp}\left[-{\left({\displaystyle \frac{v_C}{c}}\right)}^k\right]+1-\mathit{Pr}(V<v_F=1-\mathit{exp}\left[-{\left(v_C/c\right)}^k\right]+1-F(V)\\ =1-\mathit{exp}\left[-{\left(v_C/c\right)}^k\right]+1-\left[1-\mathit{exp}\left[-{\left(v_F/c\right)}^k\right]\right]\\ =1-\mathit{exp}\left[-{\left(v_C/c\right)}^k\right]+\mathit{exp}\left[-{\left(v_F/c\right)}^k\right]\end{array}$$

(15)

$$\begin{array}{l}\mathit{Pr}(P_w=P_R)=\mathit{Pr}(v_R\le V\le v_F)\\ =\mathit{Pr}(V\le v_F)-\mathit{Pr}(V\ge v_R)\\ =1-\mathit{exp}\left[-{\left(v_F/c\right)}^k\right]-\left[1-\mathit{exp}\left[-{\left(v_R/c\right)}^k\right]\right]\\ =\mathit{exp}\left[-{\left(v_R/c\right)}^k\right]-\mathit{exp}\left[-{\left(v_F/c\right)}^k\right]\end{array}$$

(16)

The Cumulative Distribution Function (CDF) of wind power is represented by both continuous and discrete probabilities. The following Equation (17) represents the CDF of the output power random variable $P_w$ [24,25].

$$\left\{\begin{array}{l}0 P<0\\ 1 P\ge P_R\\ 1-\mathit{exp}\left\{-\left[{\left(1+{\displaystyle \frac{\left(1+\frac{hw}{P_R}\right)v_C}{c}}\right)}^k\right]+\mathit{exp}\left\{-{\left[{\displaystyle \frac{v_F}{c}}\right]}^k\right\}; 0\le P<P_R\right.\end{array}\right.$$

(17)

Since the immediate wind speed is a random variable at any time, the system operator may overestimate or underestimate the exact availability of wind power output from wind generating systems. Due to this nonlinear behavior of wind power output, it is necessary to include the underestimation and overestimation of wind power in the cost function. These two concepts were mentioned in Equation (3) above and are presented mathematically using the relationship in Equation (17). The overestimation and underestimation of wind power is given by Equations (18)–(21), respectively [26]. The overestimation is given by Equation (18).

$$C_{r,w,i}(P_{w_i}-P_{W_{i,av}})=k_{r,i}(P_{w_i}-P_{W_{i,av}})=k_{r,i}{\int }_0^{w_i}(P_{w_i}-P_W)f_{P_W}(w)dw$$

(18)

where $k_{r,i}$ is the reserve cost coefficient for the $i^{th}$ wind power generator, $P_{w_i}$ is the predicted wind power, $P_{W_{i,av}}$ is the available/generated random wind power and $f_{P_W}(w)dw$ is the probability distribution function for the random variable $P_W$. Now, using the relationship found in Equation (17), the overestimation Equation (18) is further simplified as shown in Equation (19).

$$\begin{gathered} E(Y_{oe,j,t})=w_{j,t}\left[1-{\mathit{exp}\left(-{\displaystyle \frac{{v_C}_{j,t}}{c_{j,t}}}\right)}^{k_{j,t}}+{\mathit{exp}\left(-{\displaystyle \frac{{v_F}_{j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right]+ \\ +\left({\displaystyle \frac{P_{R,j,t}{v}_{C,j,t}}{v_{R,j,t}-v_{C,j,t}}}+w_{j,t}\right)\left[{\mathit{exp}\left(-{\displaystyle \frac{{v_C}_{j,t}}{c_{j,t}}}\right)}^{k_{j,t}}-{\mathit{exp}\left(-{\displaystyle \frac{{v_1}_{j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right]+ \\ +\left({\displaystyle \frac{P_{R,j,t}{c}_{j,t}}{v_{R,j,t}-v_{C,j,t}}}\right)\left[\Gamma \left(1+{\displaystyle \frac{1}{k_{j,t}}},{\left({\displaystyle \frac{v_{1,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right)-\Gamma \left(1+{\displaystyle \frac{1}{k_{j,t}}},{\left({\displaystyle \frac{{v_R}_{,j,t}}{c^{k,j,t}}}\right)}^{k_{j,t}}\right)\right] \end{gathered}$$

(19)

The analyses of underestimation are done exactly the same way as that of the overestimation. The underestimation of wind power is given by Equation (20).

$$C_{p,w,i}(P_{W_{i,av}}-P_{w_i})=k_{p,i}(P_{W_{i,av}}-P_{w_i})=k_{p,i}{\int }_0^{w_i}(P_W-P_{w_i})f_{P_W}(w)dw$$

(20)

where $k_{p,i}$ is the reserve cost coefficient for the $i^{th}$ wind power generator. The underestimation Equation (20) is further simplified as shown in Equation (21).

$$\begin{gathered} E(Y_{ue,j,t})=(P_{R,j,t}-w_{j,t})\left[{\mathit{exp}\left(-{\displaystyle \frac{v_{R,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}-{\mathit{exp}\left(-{\displaystyle \frac{v_{F,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right]+ \\ +\left({\displaystyle \frac{P_{R,j,t}{v}_{C,j,t}}{v_{R,j,t}-v_{C,j,t}}}+w_{j,t}\right)\left[{\mathit{exp}\left(-{\displaystyle \frac{v_{R,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}-{\mathit{exp}\left(-{\displaystyle \frac{v_{1,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right]+ \\ +\left({\displaystyle \frac{P_{R,j,t}{c}_{j,t}}{v_{R,j,t}-v_{C,j,t}}}\right)\left[\Gamma \left(1+{\displaystyle \frac{1}{k_{j,t}}},{\left({\displaystyle \frac{v_{1,j,t}}{c_{j,t}}}\right)}^{k_{j,t}}\right)-\Gamma \left(1+{\displaystyle \frac{1}{k_{j,t}}},{\left({\displaystyle \frac{{v_R}_{,j,t}}{c^{k,j,t}}}\right)}^{k_{j,t}}\right)\right] \end{gathered}$$

(21)

2.1.3. Multi-Criteria Optimization Problem

The WTEED problem is a multi-criteria optimization problem with two distinct objectives: to minimize the fuel cost of the wind and thermal economic dispatch problem and to minimize emissions, subject to the system equality and inequality constraints given in Equations (4)–(8). The multi-objective optimization is converted into a single fitness function by using the max–max price penalty factor given in Equation (22) [17,19,27,28]. The fuel cost function of the wind–thermal dispatch problem is given in Equation (2), and the emission function given in Equation (4).

$$h_i={\displaystyle \frac{a_i+b_i{P}_{gimax}+c_i{P}_{gimax}^2}{{\alpha }_i+{\beta }_i{P}_{gimax}+{\gamma }_i{P}_{gimax}^2}}$$

(22)

Equation (22) above represents the price penalty factor that is used to convert the bio-objective wind–thermal economic dispatch problem. For this case, the penalty factor is used as a ratio of the maximum fuel cost function to the maximum emission function. This can also be used as either max/min, min/min, or min/max, depending on the application of the optimization method. The optimization problem is solved by using a function of Lagrange based on a Lagrange multiplier $\lambda$ in the following way, as given in Equation (23).

$$L=F_T+\lambda \left(P_D+P_{Loss}-{\sum }_{i=1}^{N_G}{P}_{gi}-{\sum }_{k=1}^{N_W}{P}_w\right)$$

(23)

where

$$F_T=a_i{P}_{gi}^2+b_i{P}_{gi}+c_i+h_i(d_i{P}_{gi}^2+e_i{P}_{gi}+f_i)$$

(24)

2.2. Proposed Algorithm for Solving the WTEED Problem

The Lagrange algorithm is one of the best methods for solving optimization problems involving coupled structures. A large-scale linked problem could be converted into a two-stage dual problem with a master problem and subproblems using Lagrange multipliers to relax global restrictions. A suboptimal viable solution close to the dual optimal point is anticipated to be recognized as a valid solution for the primal problem based on the sharp bound given by the Lagrange dual optimum. In a parallel framework, the subproblems can be resolved quickly and easily because they are significantly less than the main problem. The dual fitness function is optimized in the master problem stage by updating the Lagrange multipliers. The performance of LR processes is strongly influenced by the Lagrange multiplier initialization and updating methodologies.

The ideal value is found using the LR approach. By temporarily loosening coupling constraints and considering each unit independently, it can solve the dimension issue that arises in dynamic programming [29]. Therefore, the LR approach can effectively handle the dual problems of minimizing fuel expense and pollution. The following is the Lagrange technique solution procedure.

The demand and the emission constraints are added to the original cost function using Lagrange multiplier $\lambda$, resulting in Equation (25).

$$L=F_T+\lambda P_D+\lambda P_{Loss}-{\sum }_{i=1}^{N_G}\lambda P_{gi}-{\sum }_{k=1}^{N_W}{\lambda P}_w$$

(25)

where

$$F_T=a_i{P}_{gi}^2+b_i{P}_{gi}+c_i+h_i{d}_i{P}_{gi}^2+{h_{i}e}_i{P}_{gi}+h_i{f}_i$$

(26)

Now given the loss Equation (6) and substituting into Equation (25) will result in Equation (27)

$$\begin{gathered} L=a_i{P}_{gi}^2+b_i{P}_{gi}+c_i+h_i(d_i{P}_{gi}^2+e_i{P}_{gi}+f_i)+ \\ +\lambda \left(\begin{array}{c}{P}_D+{\sum }_{m=1}^{N_G+N_W}{\sum }_{m^{\prime }=1}^{N_G+N_W}{P}_m{B}_{m{m}^{\prime }}{P}_{m^{\prime }}+{\sum }_{m=1}^{Ng+Nw}{B}_{om}{P}_m\\ +B_{00}-{\sum }_{i=1}^{N_G}{P}_{gi}-{\sum }_{k=1}^{N_W}{P}_w\end{array}\right) \end{gathered}$$

(27)

The necessary conditions for optimality for the solution of the problem (27) are:

$$\mathrm{According} \mathrm{to} P_{gi,} {\displaystyle \frac{\partial L}{\partial P_{gi}}}=0, i=\overline{1,N_g}$$

(28)

$$\mathrm{According} \mathrm{to} P_{w,} {\displaystyle \frac{\partial L}{\partial P_w}}=0, k=\overline{1,N_w}$$

(29)

$$\mathrm{According} \mathrm{to} \lambda , {\displaystyle \frac{\partial L}{\partial \lambda }}=0$$

(30)

The necessary conditions for optimality for thermal generators presented in Equation (26), are derived as follows:

$$\begin{gathered} {\displaystyle \frac{\partial L}{\partial P_{gi}}}=2a_i{P}_{gi}+b_i+h_i\left(2d_i{P}_{gi}+e_i\right)+ \\ +\lambda \left(2{\sum }_{j=1}^{N_g}{B}_{ij}{P}_j+B_{oi}-1\right)=0=i=\overline{1,n} \end{gathered}$$

(31)

This can be simplified to the following:

$$\begin{gathered} {\displaystyle \frac{\partial L}{\partial P_{gi}}}=\left(2a_i+h_{i}2d_i\right)P_{gi}+2\lambda B_{ii}{P}_{gi}+h_i{e}_i+b_i+ \\ +\lambda \left(2{\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_g}{B}_{ij}{P}_j+B_{oi}-1\right)=0=i=\overline{1,n} \end{gathered}$$

(32)

Simplifying Equation (32) gives the following Equation (33).

$${\displaystyle \frac{\partial L}{\partial P_{gi}}}=\left({\displaystyle \frac{a_i+h_i{d}_i}{\lambda }}+B_{ii}\right)P_{gi}+{\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{B}_{ij}{P}_j+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{h_i{e}_i+b_i}{\lambda }}+B_{oi}-1\right)=0,i=\overline{1,n}$$

(33)

Equation (33) can be written in matrix form in the following way:

$$\left[\begin{array}{ccc}{\displaystyle \frac{a_1+h_1{d}_1}{\lambda }}+B_{11}& B_{12}& B_{1n}\\ B_{21}& {\displaystyle \frac{a_2+h_2{d}_2}{\lambda }}+B_{22}& B_{2n}\\ B_{n1}& B_{n2}& {\displaystyle \frac{a_{1n}+h_n{d}_n}{\lambda }}+B_{nn}\end{array}\right]\left[\begin{array}{c}{P}_1\\ P_2\\ P_3\\ ⋮\\ P_n\end{array}\right]={\displaystyle \frac{1}{2}}\left[\begin{array}{c}\left[1-\left({\displaystyle \frac{b_1+h_1{e}_1}{\lambda }}\right)-B_{01}\right]\\ \left[1-\left({\displaystyle \frac{b_1+h_1{e}_1}{\lambda }}\right)-B_{02}\right]\\ \left[1-\left({\displaystyle \frac{b_1+h_1{e}_1}{\lambda }}\right)-B_{0n}\right]\end{array}\right]$$

(34)

Equation (34) can be written in the following simple matrix form.

$$BP=D$$

(35)

The matrix vector P can be found with the known value of the Lagrange multiplier, in a MATLAB environment using Equation (35).

$$P=B/D$$

(36)

The necessary conditions for optimality for wind generators presented in Equation (27) are derived as follows: The total wind power is expressed as in Equation (37).

$${\displaystyle \frac{\partial L}{\partial P_w}}=\lambda \left(2{\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N_w}{B}_{ij}{P}_j+B_{oi}-{\displaystyle \frac{\partial }{\partial P_w}}\left(OE+UE+d_i{P}_w\right)\right)=0, i=\overline{1,n}$$

(37)

Also, Equation (37) can be written in matrix form as follows:

$$\left[\begin{array}{ccc}{B}_{11}& B_{12}& B_{13}\\ ⋮& ⋮& ⋮\\ B_{n1}& B_{n2}& B_{nn}\end{array}\right]\left[\begin{array}{c}{P}_1\\ ⋮\\ P_{nw}\end{array}\right]={\displaystyle \frac{1}{2}}\left[\begin{array}{c}{\displaystyle \frac{\partial }{\partial P_w}}(C_p.UE+C_r.O.E+d_i{P}_w)-B_{o1}\\ ⋮\\ {\displaystyle \frac{\partial }{\partial P_w}}(C_p.UE+C_r.O.E+d_i{P}_w)-B_{onw}\end{array}\right]$$

(38)

The necessary conditions for optimality for Lagrange multiplier $\lambda$ in Equation (27) are found as follows:

$${\displaystyle \frac{\partial L}{\partial \lambda }}=\left(P_D+P_{Loss}-{\sum }_{i=1}^{N_G}{P}_{gi}-{\sum }_{k=1}^{N_W}{P}_w\right)=0=\Delta \lambda$$

(39)

Necessary conditions for optimality according to Lagrange multiplier are not the function of $\lambda$ but represent the gradient of the functional $L$ according to $\lambda$. The optimality conditions for the gradients have to be zero. Equation (39) cannot be solved using analytical methods; hence, the gradient procedure has to be deployed in the following way.

$${\lambda }^{(k+1)}={\lambda }^k+\alpha \Delta {\lambda }^k,\lambda \ne 0$$

(40)

where $\alpha$ is the steps of the gradient procedure and $\Delta {\lambda }^k$ is found using Equation (40). The gradient method will always start with an initial value of a Lagrange multiplier. Once the value of ${\displaystyle \frac{\partial L}{\partial \lambda }}=0,$ the optimal value for the Lagrange variable is found. It will determine the solution for the energy that has to be produced by the generators as a solution of Equations (34) and (38). The obtained solution of every step of the gradient procedure must fit to the constraint’s domain in the following way:

$$P_i^k=\left(\begin{array}{ccc}{P}_{i,min}^{k}if& P_i^k<& P_{i,min}\\ P_i^k if& P_{i,min}^k\le & P_i^k<P_{i,max}\\ P_{i,min}if& P_i^k>& P_{i,max}\end{array}\right)$$

(41)

The condition for the end of gradient and the iterations is given by the following Equation (42) below.

$$\Delta {\lambda }^k\le \epsilon , \mathrm{and} k=iter Max$$

(42)

where $\epsilon >0$ is a smaller number and $iter Max$ is the maximum number of iterations. The algorithm of the method is presented as follows:

(1)

The initial value of the Lagrange multiplier ${\lambda }^0$, and the value of the condition for optimality $\epsilon$ is given.

(2)

In Equation (34), matrices $B^0 \mathrm{and} D^0$ are formed.

(3)

Equation (36) is solved, and $P^0=B^0/D^0$ is determined.

(4)

Equations (18)–(21) are used for wind power calculation. Wind power calculation is done using the trapezoidal method within the MATLAB environment.

(5)

The optimum power calculations from Equations (18)–(21) and (36) are added to form the total power generated by both wind and thermal generators.

(6)

The obtained vector of $P^0$ is fitted to constraint Equation (41).

(7)

$\Delta {\lambda }^0$ is calculated using Equation (39) where $P^0$ is substituted.

(8)

Condition (41) is checked. If it is fulfilled, the operation stops. But if it is not, the improved value of $\lambda \to {\lambda }^1$ is calculated using Equation (40).

(9)

Calculation of the improved $P_i\to {P_i}^1$ is done following step 5, and the process will repeat itself until the conditions of Equation (42) are met.

The optimal solution is used to calculate the total cost of generation and emission pollutants using Equations (2) and (4). The flow chart of the algorithm is presented in Figure 2 below.

3. Description of the Test Systems

3.1. Test System 1

The system consists of six thermal generators and one cluster of wind turbines. Figure 3 represents a line diagram of a simple interconnected power system. It can then be assumed that the power generated by the wind turbines is 125 MW, and the power generated by the six thermal units is 270 MW. However, power can be transferred from one area to another through interconnected transmission lines. The research work focuses on the wind–thermal economic dispatch problem, as illustrated by the simplified power system model in Figure 1. The power demand varies from 125 to 300 MW during the day. The solution for the wind–thermal economic dispatch problem is given in

Table 2. The WTEED problem using LMM with different power demand for a 6-unit system.

PD [MW]	125	150	175	200	225	250	283.4
P1 [MW]	5	5	6.65	9.54	12.44	15.35	19.24
P2 [MW]	28.29	30.37	32.32	34.17	36.03	37.89	40.37
P3 [MW]	14.5	20.87	26.85	32.54	38.24	43.96	51.63
P4 [MW]	50.68	57.51	63.91	69.98	76.06	82.15	90.31
P5 [MW]	13.72	19.98	25.85	31.42	37.01	42.61	50.11
P6 [MW]	13.15	16.9	20.4	23.71	27.02	30.32	34.72
P7 [MW]	118.28	118.28	118.28	118.28	118.28	118.28	118.28
PL [MW]	1.32	1.63	1.97	2.36	2.79	3.26	3.96
FC [USD/h]	25,391	29,234	33,164	37,160	41,167	45,183	50,562.51
ET [kg/h]	22.26	21.62	21.05	20.55	20.17	19.89	19.69
CEED [kg/h]	28,361	33,066	37,966	43,039	48,265	53,644	61,070

.

A cluster consists of turbines situated in a geographic area, represented by a group of identical turbines in a large wind farm. The fuel cost coefficients are based on the ones reported in [30,31], and, for wind power, [24,25,32]. The WTEED problem is solved considering a load demand of 2.834 MW and using a wind farm that consists of 50 identical wind turbines, each of 2.5 MW power capacity, therefore making a total wind farm installed capacity of 125 MW [28]. Different power demand values are presented to test the system’s behavior in response to changes in power demand. The solution of the WTEED problem is shown in Table 2. All programs are developed and simulated on Windows 11 Enterprise, using an Intel(R) Core (TM) i3-7020U CPU @ 2.30 GHz, with 12.0 GB (11.9 GB usable) of installed RAM, running MATLAB 2024a.

Table 2 shows the real power of the thermal (P1–P7) units, wind (P7) generators, transmission losses (${\mathit{P}}_{\mathit{L}}$) in [MW], fuel cost (${\mathit{F}}_{\mathit{C}}$) for thermal generators) in MW, total emissions (${\mathit{E}}_{\mathit{T}}$) in kg/h, and CEED in kg/h.

Results and Discussion of the WTEED Problem for the Six-Unit System

This system utilized 50 available wind turbines that supply the power dispatch problem with 118.28 MW. The following parameters for the Lagrange multiplier method are used. The initial value of lambda is 200, the total number of iterations m is 100, the tolerance value epsilon is 0.01, and the tolerance value of delta-lambda epsilon is 0.1. The proposed algorithm for a 6-unit system is used and compared with other optimization algorithms in the literature, namely, the Genetic Algorithm (GA) [10], the Particle Swarm Optimization (PSO), the Gravitational Search Algorithm (GSA), and the hybrid optimization algorithm Gravitational acceleration Enhanced Particle Swarm Optimization Algorithm (GAEPSO) [6]. The comparison is of the simulation results shown in

Table 3. The WTEED problem using LMM, compared with other optimization algorithms for PD = 283.4 [MW].

Algorithm	PSO MW	GSA MW	GAEPSO MW	GA MW	GPSOA Without Wind	Proposed LMM MW
P1 [MW]	2	14.45	20.75	36.62	11.29	19.24
P2 [MW]	30.06	53.46	30.09	42.58	29.95	40.37
P3 [MW]	28.24	23.87	34.63	30.31	52.66	51.63
P4 [MW]	31.29	6	35.06	31.36	101.69	90.31
P5 [MW]	40.32	25.48	34.53	53.44	52.68	50.11
P6 [MW]	41.61	22.67	32.93	43.52	35.95	34.72
Pw [MW]	110.2	137.71	95.47	45.57	--	118.28
PL [MW]	---	---	---	---	---	3.96
FC [USD/h]	62,497	58,846	57,219	66,582	60,029	50,562.51
% Deviation FC [USD/h] solutions from the literature with reference to the proposed LMM	5.27	3.78	3.08	6.83	4.10	4.27
ET [kg/h]	20.43	21.33	20.49	19.72	22.21	19.69
% Deviation ET [kg/h] solutions from the literature with reference to the proposed LMM	0.922	1.99	0.99	0.038	3.007	3.007

.

The fuel cost results for PSO, GSA, GAEPSO, GA, and GPSOA algorithms are 5.27, 3.78, 3.08, 6.83, and 4.27 compared to the LMM algorithm. When using the LLM algorithm, there is a 3–6% benefit in fuel consumption when wind power is added to the economic emission dispatch as compared to other optimization algorithms. Additionally, the resultant emission pollutants of NOx are 0.922, 1.99, 0.99, 0.038, and 3.01 compared to the proposed algorithm. The effectiveness of the Lagrange algorithm in emission reduction, when wind power is added to the conventional generation, decreased by 1–3%. The optimization algorithm only took 84 iterations to find the optimal solution of the WTEED problem. The calculated value for lambda is 203.6459, compared to the initial guess of 200. The elapsed time is 11.042308 s, which can be regarded as long, and this is due to the properties of the personal computer (PC) on which MATLAB is installed. The fuel cost and emissions are shown in Figure 4 and Figure 5 below.

3.2. Test System 2

This study of the WTEED power system problem has ten thermal generators and one cluster of wind turbines. The cluster consists of turbines situated in a specific geographic area, represented by a group of identical turbines in a large wind farm. The fuel cost, emission, and loss coefficients are based on those reported in [21]. The wind parameters are adopted from [24,25,33], as in Test System I above. The study focuses on 2000 MW power demand as presented in [34]. Various power demand values are presented to test the system’s behavior in response to changes in power demand. The solution of the WTEED problem is shown in

Table 4. The WTEED problem using LMM with different power demands for 10-unit generators.

PD [MW]	650	800	1000	1200	1500	2000
P1 [MW]	16.9077	43.2946	55.0000	55.0000	55.0000	55.0000
P2 [MW]	20.4349	45.0263	59.3561	69.8398	80.0000	80.0000
P3 [MW]	47.0000	47.0000	60.9549	71.9979	87.5874	111.9414
P4 [MW]	20.0000	44.1126	59.4937	70.7465	86.6321	111.4487
P5 [MW]	50.0000	50.0000	50.0000	50.0000	67.6960	111.5112
P6 [MW]	70.0000	70.0000	70.0000	70.0000	78.8096	134.2497
P7 [MW]	60.0000	62.8499	106.5857	138.5827	183.7531	254.3184
P8 [MW]	70.0000	70.0000	114.6430	152.1590	205.1205	287.8573
P9 [MW]	135.0000	135.0000	175.7180	229.6450	305.7741	424.7033
P10 [MW]	150.0000	150.0000	173.2323	228.4474	306.3949	428.1648
P11 [MW]	24.028	100.116	100.116	100.116	100.116	100.116
PL [MW]	13.3715	17.4007	25.1010	36.5356	56.8851	99.3099
FC [USD/h]	37,639.687359	41,146.509	50,742.1419	60,746.3	76,945.943152	107,422.761421
ET [kg/h]	1399.058217	1333.454716	1501.85983	1803.17	2440.652768	3910.593584
CEED [kg/h]	63,739.681832	66,384.319239	80,126.5769	97,793.9	131,033.780343	203,261.325452

.

It shows the real power of the thermal (P1–P11) units, wind (P11) generators, and transmission losses (${\mathit{P}}_{\mathit{L}}$) in MW, fuel cost ${\mathit{F}}_{\mathit{C}}$ for thermal generators, total emissions (${\mathit{E}}_{\mathit{T}}$) in kg/h, and CEED in kg/h.

3.3. Results and Discussion of the WTEED Problem for the Ten-Unit System

The Lagrange method is implemented for solving the economic emission dispatch problem with wind power, utilizing 10 units of conventional generators and 50 available wind turbines on a wind farm, which consists of 50 identical wind turbines. The following parameters are used for the Lagrange multiplier method. The initial value of lambda is 200, the total number of iterations m is 3000, the tolerance value (epsilon) is 0.01, and the tolerance value of delta-lambda epsilon is 0.1. The Lagrange multiplier method is compared with the Salp Swarm Algorithm (SSA), Grey Wolf Optimization (GWO), Moth–Flame Optimization (MFO), Whale Optimization Algorithm (WOA), Search and Rescue Algorithm (SAR), and Genetic Algorithm (GA) [33]. The optimization algorithm took 1013 iterations to find the optimal solution of the WTEED problem. The calculated value of lambda after the optimization algorithm is recorded as 151.2992, compared to the initial guess of 200. The elapsed time is 11.657372 s, which can be considered long, and this is due to the properties of the personal computer (PC) on which MATLAB is installed. The proposed algorithm has a lower fuel cost and is observed to be superior to all other optimization algorithms, as shown in

Table 5. The WTEED problem with Lagrange compared with other optimization algorithms PD = 2000 [MW].

Algorithm	MFO	WOA	SAR	GA	Proposed LMM
P1 [MW]	37.16	46.96	53.49	34.17	55.0000
P2 [MW]	25.84	68.61	77.10	63.49	80.0000
P3 [MW]	120	69.24	64.41	95.75	111.9414
P4 [MW]	93.90	129.21	83.05	121.67	111.4487
P5 [MW]	160	50	158.73	145.83	111.5112
P6 [MW]	240	226	239.50	229.86	134.2497
P7 [MW]	244.61	298.21	275.88	281.21	254.3184
P8 [MW]	286.25	337.93	332.28	307.35	287.8573
P9 [MW]	470	390.40	409.12	391.63	424.7033
P10 [MW]	404.92	468	388.56	410.56	428.1648
P11 [MW]	30	30	30	30	100.116
Fc [USD]	116,690.89	114,678.84	114,106.19	115,802.24	107,422.761421
%Deviation of Fc for LMM compared to the literature	2.07	1.63	1.508477	1.876913	1.876
FCT [USD]	239,158.66	244,987.20	226,506.78	231,903.60	110,572.517314-
PL [MW]	83.70	84.44	82.81	83.9	99.310
ET [T]	4080	4220	4103.3	4390	3910.593584
%Deviation of Fc for LMM compared to the literature	1.060036	1.90273	1.202327	2.897008	2.887784
CEED [USD]	239,158.66	244,987.20	226,506.78	231,903.60	203,261.325452

.

The fuel cost obtained for the MFO, WOA, SAR, and GA algorithms are 2.07%, 1.63%, 1.5%, and 1.89% compared to the LMM algorithm. The proposed algorithm is 1–2% superior to other algorithms used to solve the wind–thermal economic emission dispatch problem. The resultant emission pollutants of NOx are 1.06%, 1.9%, 1.2%, and 2.89%, compared to the proposed LM algorithm. The LM method has effectively decreased the pollutants by 1–2.89% compared to other optimization algorithms. Fuel cost and emissions are shown graphically in Figure 6 and Figure 7 below.

The fuel cost and emission values are further compared to the statistical values of different algorithms, as presented in

Table 6. Comparison of LMM with different algorithms for the ten-unit system [34].

Algorithm	Best Cost (USD/h)	Best Cost (%) Deviation	Best Emission (kg/h)	Best Emission (%) Deviation
DE	111,500.000	0.931205	3923.40	0.081736
QPSO	119,005.3030	2.557665	4032.38	0.76663
GQPSO	112,424.7444	1.137603	4011.9244	0.639511
SGO	111,497.6302	0.930674	3932.243252	0.138022
MSGO	111,497.6301	0.930674	3932.243252	0.138022
LMM	107,422.761421	1.297564	3910.593584	0.352784

.

In Table 6, fuel cost and emission values for different algorithms used to solve the thermal economic problem are compared with the proposed LMM. It is found that the LMM is 1.29% and 0.353% better in fuel and emission minimization.

3.4. Test System 3

The test system consists of a 40-unit thermal power generator, with a power demand of 10,500 MW, considering power losses without the valve-point loading effect. The simulation data for power limits, fuel cost coefficients, emission coefficients, and B-loss coefficients used in this test system are identical to those used in [34]. The cluster consists of turbines situated in a geographic area, represented by a group of identical turbines in a large wind farm, as presented in Test System One. Various power demand values are given to test the system’s behavior in response to changes in power demand. The solution of the WTEED problem is shown in

Table 7. The WTEED problem using LMM with different power demands for a 40-unit generator.

PD [MW]	10,500	11,000	11,200	11,300
P1 [MW]	114.0000	114.0000	114.0000	114.0000
P2 [MW]	114.0000	114.0000	114.0000	114.0000
P3 [MW]	120.0000	120.0000	120.0000	120.0000
P4 [MW]	190.0000	190.0000	190.0000	190.0000
P5 [MW]	97.0000	97.0000	97.0000	97.0000
P6 [MW]	140.0000	140.0000	140.0000	140.0000
P7 [MW]	300.0000	300.0000	300.0000	300.0000
P8 [MW]	300.0000	300.0000	300.0000	300.0000
P9 [MW]	300.0000	300.0000	300.0000	300.0000
P10 [MW]	236.1173	300.0000	300.0000	300.0000
P11 [MW]	279.1346	369.8932	375.0000	375.0000
P12 [MW]	277.8458	369.5691	375.0000	375.0000
P13 [MW]	383.4729	500.0000	500.0000	500.0000
P14 [MW]	422.9295	500.0000	500.0000	500.0000
P15 [MW]	421.4242	500.0000	500.0000	500.0000
P16 [MW]	421.4242	500.0000	500.0000	500.0000
P17 [MW]	500.0000	500.0000	500.0000	500.0000
P18 [MW]	500.0000	500.0000	500.0000	500.0000
P19 [MW]	550.0000	550.0000	550.0000	550.0000
P20 [MW]	550.0000	550.0000	550.0000	550.0000
P21 [MW]	550.0000	550.0000	550.0000	550.0000
P22 [MW]	550.0000	550.0000	550.0000	550.0000
P23 [MW]	550.0000	550.0000	550.0000	550.0000
P24 [MW]	550.0000	550.0000	550.0000	550.0000
P25 [MW]	550.0000	550.0000	550.0000	550.0000
P26 [MW]	550.0000	550.0000	550.0000	550.0000
P27 [MW]	17.9820	23.6373	101.6061	142.1987
P28 [MW]	17.9820	23.6373	101.6061	142.1987
P29 [MW]	17.9820	23.6373	101.6061	142.1987
P30 [MW]	97.0000	97.0000	97.0000	97.0000
P31 [MW]	190.0000	190.0000	190.0000	190.0000
P32 [MW]	190.0000	190.0000	190.0000	190.0000
P33 [MW]	190.0000	190.0000	190.0000	190.0000
P34 [MW]	90.0000	90.0000	90.0000	90.0000
P35 [MW]	90.0000	90.0000	90.0000	90.0000
P36 [MW]	90.0000	90.0000	90.0000	90.0000
P37 [MW]	110.0000	110.0000	110.0000	110.0000
P38 [MW]	110.0000	110.0000	110.0000	110.0000
P39 [MW]	110.0000	110.0000	110.0000	110.0000
P40 [MW]	550.0000	550.0000	550.0000	550.000
P41 [MW]	150.386	150.386	150.386	150.386
FC [USD/h]	133,540.341	143,551.62517	159,778.38	175,659.5
PL [MW]	1038.682	1152.8	1197.2	1218.984
ET [kg/h]	105,263.403	136,510.06316	190,426.05	244,979.8
CEED [kg/h]	254,190.3053	283,453.10221	317,220.20	350,503.7

below.

It shows the real power of the thermal (P1–P40) units, wind (P41) generators, and transmission losses (${\mathit{P}}_{\mathit{L}}$) in MW, fuel cost (${\mathit{F}}_{\mathit{C}}$) for thermal generators, total emissions (${\mathit{E}}_{\mathit{T}}$) in kg/h, and CEED in kg/h.

Results and Discussions of the WTEED Problem for the Forty-Unit System

This is a fourth-generation unit system used to study the effectiveness of the Lagrange multiplier method compared to other optimization methods in minimizing fuel costs and atmospheric emission levels. The following parameters are used for the Lagrange multiplier method. The initial value of lambda is 8.9914, the total number of iterations m is 35,000, the tolerance value (epsilon) is 0.01, and the tolerance value of delta-lambda epsilon is 0.1. The Lagrange multiplier method is compared with different heuristic algorithms, i.e., PSO, GSA, GAEPSO [6] and MSGO [34]. The comparison is made for the best cost and the lowest emissions. The summary of the comparison between the Lagrange method and the mentioned heuristic algorithms is given in

Table 8. The WTEED problem with Lagrange compared with other optimization algorithms for an IEEE 40-unit system for PD = 10,500 MW.

Algorithm	MGSO Without Wind Power	MSGO	PSO	GSA	GAEPSO	Proposed LMM 150 MW Wind Power	Proposed LMM 1100 MW Wind Power
P1 [MW]	114.000000	549.9997828	114.0000	105.5679	106.3768	114.0000	114.0000
P2 [MW]	113.999999	549.9997818	104.0000	88.2574	113.2637	114.0000	114.0000
P3 [MW]	120.000000	119.9988116	120.0000	105.9739	108.5784	120.0000	108.3202
P4 [MW]	189.999995	179.7374211	169.3671	150.3464	188.6623	190.0000	165.3147
P5 [MW]	96.999999	96.99883835	97.0000	82.0595	77.0086	97.0000	97.0000
P6 [MW]	140.000000	105.4022758	124.2630	119.5704	132.6636	140.0000	130.6214
P7 [MW]	300.000000	299.9999742	299.6931	248.5154	288.3156	300.0000	285.1828
P8 [MW]	300.000000	285.7135868	297.9093	276.3936	235.0264	300.0000	300.0000
P9 [MW]	299.999998	287.9519754	297.2578	244.2866	285.7760	300.0000	300.0000
P10 [MW]	279.599683	204.8086989	130.0007	262.1424	264.6783	236.1273	166.9658
P11 [MW]	168.799860	243.6012566	298.4210	293.2579	312.0426	279.1439	214.3810
P12 [MW]	94.0000003	243.6003501	298.0264	264.5149	112.8675	277.8553	212.4040
P13 [MW]	484.039161	394.2851366	433.5590	432.2395	486.8320	383.4865	289.4015
P14 [MW]	484.039166	394.2799802	421.7360	391.8179	341.2265	422.9446	318.7482
P15 [MW]	484.039164	394.2841752	422.7884	422.8119	428.9123	421.4392	317.4792
P16 [MW]	484.039178	484.0292251	422.7841	414.8810	436.8761	421.4392	317.4792
P17 [MW]	489.279372	489.2692227	439.4078	428.5659	426.2216	500.0000	462.7621
P18 [MW]	489.279372	399.5209877	409.4132	422.1613	442.5531	500.0000	464.5057
P19 [MW]	511.279600	506.0067276	439.4111	440.9423	454.6218	550.0000	506.5800
P20 [MW]	511.279490	421.5364187	429.4155	435.9092	491.0606	550.0000	506.5812
P21 [MW]	526.732209	433.5530376	439.4421	451.8724	450.2265	550.0000	550.0000
P22 [MW]	550.000000	514.2353651	439.4587	436.1765	406.2236	550.0000	550.0000
P23 [MW]	523.279384	433.5329212	429.7822	437.2970	464.5563	550.0000	550.0000
P24 [MW]	523.279383	433.5268561	439.7697	442.0350	444.5589	550.0000	550.0000
P25 [MW]	524.239856	433.5342317	430.1191	445.9564	455.3451	550.0000	550.0000
P26 [MW]	523.815577	433.7489257	440.1219	429.3785	468.2267	550.0000	550.0000
P27 [MW]	10.000020	10.05540017	28.9738	124.0932	103.8549	17.9826	13.9471
P28 [MW]	10.000113	10.24418652	29.0007	117.8668	146.2286	17.9826	13.9471
P29 [MW]	10.000007	10.0022597	28.9828	94.5359	148.8421	17.9826	13.9471
P30 [MW]	87.800155	89.80426078	97.0000	89.6485	96.7263	97.0000	97.0000
P31 [MW]	190.000000	189.998884	172.3348	153.1318	189.2261	190.0000	175.9692
P32 [MW]	190.000000	189.9955403	172.3327	159.0102	86.6516	190.0000	175.9692
P33 [MW]	190.000000	189.9978565	172.3262	148.7814	89.2264	190.0000	175.9692
P34 [MW]	200.000000	199.9994215	200.0000	176.4061	93.9663	90.0000	90.0000
P35 [MW]	199.999999	199.9994151	200.0000	170.0710	102.2234	90.0000	90.0000
P36 [MW]	164.799870	199.998299	200.0000	181.6662	168.2261	90.0000	90.0000
P37 [MW]	110.000000	110.0000000	100.8441	96.8108	78.2663	110.0000	110.0000
P38 [MW]	109.999999	109.9997758	100.8346	94.3094	107.8912	110.0000	110.0000
P39 [MW]	110.000000	109.9962344	100.8362	82.4816	98.0368	110.0000	110.0000
P40 [MW]	550.000000	509.5675741	439.3868	456.2560	442.4562	550.0000	506.5800
P41 [MW]	-	1100	66.2736	83.099	151.9009	150.294	1100
FC [USD/h]	136,454.3369	126,645.13420	133,995.0611	142,675.36	158,269.6617	133,541.733580	121,045.419972
%Deviation of Fc for LMM compared with the literature	2.992	1.13	2.54	4.17	6.66	2.454	3.313174 (Average)
PL [MW]	958.6206217	962.815076	-	-	-	1038.679	1063.127
ET [kg/h]	347,578.4905	239,155.73184	352.766.822	366,190.13	156,826.1213	105,267.750200	75,880.539978
%Deviation of Fc for LMM compared with the literature	32.08	25.91371319	32.29	32.83	17.39	8.11	24.77 (Average)

.

The fuel costs obtained for the MGSO (without wind), MGSO, PSO, GSA, GAEPSO, and LMM (1500 MW) algorithms are 2.99%, 1.33%, 2.54%, 4.17%, 6.66%, and 2.454% compared to the LMM (1100 MW) algorithm. The proposed algorithm is 1–6% superior to other algorithms used to solve the WTEED problem. The resultant emission pollutants of NOx are 32.08%, 25.91%, 32.29%, 32.83%, 17.39%, and 8.11%, compared to the proposed LM algorithm. The LM method has effectively decreased the pollutants by 1–32% compared to other optimization algorithms.

Table 9. Comparison of LMM with different algorithms for the forty-unit system [34].

Algorithm	Best Cost (USD/h)	Best Cost (%) Deviation	Best Emission (kg/h)	Best Emission (%) Deviation
PSO	123,607.9479	0.523706	193,313.7047	21.8119754
DE	123,804.0394	0.56333	193,953.9668	21.87886
SCA	125,895.2706	0.981987	210,484.9674	23.50221
SGO	123,289.7874	0.45928	193,311.54075	21.81175
MGSO	123,161.8867	0.433334	193,311.54075	21.81175
LMM	121,045.419972	0.592327	75,880.5399	22.16331

compares the fuel cost and emission values for different algorithms used to solve the thermal economic problem with the proposed LMM. It is observed that the LMM is 0.59% and 22.163% better in fuel and emission minimization.

The optimization algorithm took 72 iterations to find the optimal solution of the WTEED problem. The calculated value of lambda after the optimization algorithm is recorded as 33.9059, compared to the initial guess of 8.9914. The elapsed time is 16.145170 s, which can be considered long, and this is due to the properties of the personal computer (PC) on which MATLAB is installed. The LM method performed better than other optimization algorithms in solving the WTEED problem. It can also be seen from Table 9 that increasing power generation from renewable energy resources can significantly decrease emissions and fuel costs. The relationship between the fuel cost [USD/h], Power demand [MW], and emissions is shown graphically in Figure 8 and Figure 9 for different optimization algorithms.

4. Conclusions

In this paper, the Lagrange multiplier method is successfully applied to solve a wind–thermal economic emission dispatch problem. Economic emission dispatch optimization is presented as a bi-objective problem aimed at minimizing both fuel and emission costs. The inequality constraints, emission, and transmission line loss are presented in this paper. The algorithm is tested on three different test systems, i.e., 6-unit, 10-unit, and 40-unit systems. The fuel cost and emissions for 6-unit systems are 3–6% and 1–3% lower than those of other algorithms. Also, the fuel cost and emissions for the 10-unit system are 1–2% and 1–2.89% compared to other optimization algorithms. Lastly, for the bigger system, there is a very significant decrease in fuel and emissions by 1–6% and 1–32%.

It is evident from all the test systems that by choosing the best initial guess for lambda values, the Lagrange optimization method will converge to an optimal solution efficiently. The following computational times were recorded for the 6-unit, 10-unit, and 40-unit models: 11.042308 s for 84 iterations, 11.657372 s for 1013 iterations, and 16.145170 s for 72 iterations. In comparison, the optimal values for the 6-unit, 10-unit, and 40-unit systems are found to be within the expected values noted in the literature for fuel cost, emissions, and total generation of the WTEED problem. It can also be seen that, irrespective of load demand or the types of generators used, the Lagrange multiplier will always find an optimal solution precisely and quickly if the initial guess of lambda is chosen carefully. In addition, the results of the LM algorithm have been compared to those of other heuristic algorithms published in recent literature. The classical LM algorithm can easily achieve optimal results for fuel and emission reduction as compared to heuristic algorithms reported in the literature. Lagrange’s classical methods are efficient and accurate for simple, convex problems. Future research should consider implementing Particle Swarm Optimization (PSO) to solve the Wind–Thermal Economic Emission Dispatch (WTEED) problem and benchmark the solutions between classical and heuristic optimization methods for the wind–thermal economic dispatch problem.