MULTIDISCIPLINARY TECHNOVATION Wind Driven Optimization Approach based Multi-objective Optimal Power Flow and Emission Index Optimization

This paper proposes one of the optimization methods based on atmospheric motion. It is a global optimization nature-inspired method such as Wind Driven Optimization (WDO) approach to solve the Optimal Power Flow (OPF) and Emission Index (EI) in electric power systems. Our main aim is to minimize an objective function necessary for a best balance between the energy production and its consumption, which is presented as a nonlinear function, taking into account of the equality and inequality constraints. The WDO approach is nature-inspired, population based iterative heuristic optimization algorithm for multi-dimensional and multi-modal problems. WDO method have been examined and tested on the standard IEEE 30-bus system and IEEE 57-bus system with different objectives that reflect total active power generation cost, the active power losses and the emission index. The results of used method have been compared and validated with known references published recently. The results are promising and show the effectiveness and robustness of proposed approach.


Emission coefficients f(x, u)
Objective function. f1 Cost function w/o valve-point. f2 Cost function with valve-point f3 Power losses function. f4 Emission index function f5 Cost and active loss function f6 Cost and emission function f7 Cost, power loss and emission g (

Introduction
Electric power systems engineering has the longest history of development compared to the various fields of engineering. In electrical supply systems, there are a wide range of problems involved in system optimization [1]. Among these problems, power system scheduling is one of the most important in system operation, control, and management.
Electric power plants that operate on Esotericfuel are among the most prominent sources of air pollution and contribute to causing great harm to the environment due to the burning of raw fuels such as coal, gas, and oil [2]. The burning of coal contributes a large proportion of polluting gases to the Earth's atmosphere, as it produces large amounts of Carbon oxides CO2, and some toxic and dangerous gases such as emissions of Sulfur oxides SOx, and Nitrogen oxides, NOx. The quantity and nature of the pollutant depends on the type and quality of the used fuel [2,3].
After implementation of the 1990 amendment to the United States Clean Air Act and increasing public awareness of environmental protection and public utilities, electricity production companies were obligated to adapt their designs and making strategy to reduced pollution rate and emissions of electric power plants [3,4].
The WDO is a natural-inspired algorithm based on heuristics techniques [105]. This promising algorithm is implemented firstly to solve the electromagnetic problems in communication engineering studies [106].

Problem Formulation
The OPF and EI are nonlinear optimization problems, represented by a predefined objective function f, subject to a set of equality and inequality constraints [18,64]. Generally, these problems can be expressed as follows.
,..., , ,... , ,... , Where, G P , G Q , L V and Sk are the generating active power at slack bus, reactive power generated by all generators, magnitude voltage of all load buses and apparent power flow in all branches, respectively. g n , L n and br n are, respectively, the total number of generators, the total number of load buses and the total number of branches.
The set control parameters are represented in terms of the decision vector u as follows:

Single-Objective Function
In general, the single-objective function is a nonlinear programming problem. In this paper, four single objectives commonly found in OPF and EI have been considered who are the generation cost without and with valve-point effect, 1 f and 2 f , respectively, the active power losses 3 f , and the emission index optimization 4 f .

Cost Without Valve-Point Optimization
The objective function of cost optimization 1 f of quadratic cost equation for all generators as given below

Active Power Loss Optimization
The active power loss function 3 f in (MW) to be minimized can be expressed as follows Where, Vk and Vj are the voltage magnitude at buses k and j, respectively, Gkj is the conductance of line kj, kj  is the voltage angle between buses k and j and b n is total number of buses.

Emission optimization
The emission function is the sum of exponential and quadratic functions of real power generating. Using Where, 4 f is the emission function in (ton/h), and k  are the emission coefficients of the generator k.

Cost and active power loss optimization
When the optimization is the cost and the active power losses together, the bi-objective function as given below Where 1  and 2  are the weighting factors.

Cost and Emission Optimization
Emission is needs to minimize the generation cost and emission. The objective function is f is the total cost-emission in ($/h), and D is the price penalty factor in ($/ton).

Multi-Objective Optimization
All objective functions discussed before are used to solve the multi-objective OPF and EI problems. Therefore, the multi-objective problems can be stated as follows The function used in the case of weighted aggregation is given by equation (12).

Equality Constraints
These equality constraints are the sets of nonlinear load flow equations that govern the power system, i.e.: ✓ Voltage magnitudes and angles limits:

Wind Driven Optimization Technique
The WDO algorithm was first introduced in 2010 [108]. The WDO is one of the optimization methods based on atmospheric motion, and it is global optimization nature-inspired method. This technique works on population based global heuristic algorithms for multi-dimensional and multi-dimensional models in the research field to apply constraints [104,107].

Context Theory and Destination Of WDO
In the atmosphere, wind blows in an effort to make equal air pressure [106]. More exclusively, the air is used to move from high pressure to low pressure at a velocity, which is proportional to the pressure gradient [104].
Furthermore, some assumptions and simplifications are formulated in derivation of the WDO algorithm. The starting point in the development of WDO is with Newton's second law of motion, which is known to provide very accurate results when applied to the analysis of atmospheric motion [106].
Where, →  is the acceleration vector, ρ is the air density for an infinitesimal air parcel, and i F → are the all forces acting on the air parcel [108]. The equation that relates air pressure to its density and temperature is given by the ideal gas law, formulated as follows In Eq. (24), P, R and T are, respectively, the pressure, the universal gas constant, and the temperature. In Eq. (23), there are four main forces that either cause the wind to move in a specific direction or deflect it from its path [102]. The most observable force causing the air to move is the pressure gradient force By assuming air has a finite volume (δV), the physical force equation because of pressure gradient can be expressed as [102].
The frictional force oppose the air parcel motion started by FPG, and can be expressed as The gravitational force pull the air parcel to the center of the earth expressed as The Coriolis is caused by the rotation of earth, and deflects the path of wind from one dimension to another. This force will work in such a way that velocity in one direction is influenced by velocity of another direction [108]. It can be expressed as Taking for simplicity, the acceleration equal to (Δu/Δt), the time step Δt =1 and δV = 1. Therefore, the in the total force described in Eq. (25) can be rewritten as The change in velocity in Eq. (29) can be extracted from modifying the Eq. (30) based on Eq. (24) and division by ( ) ) (k P RT [109]. The vector g can be written as g = |g|(0x(k)) [103,109]. The pressure gradient is the force that attempts to move an air parcel from its current position into optimal pressure. It can be expressed as . All coefficients in the last term of Eq. (30) are collected to be a single term as c = −2|Ω|RT [111]. Eq. (30) can be modified as in Eq. (31).
On the basis of ideal gas law equation from Eq. (24), and for simplicity, assuming that a single time step (Δt=1), the air density, ρ can be written as the pressure [104]. Based on Newton's second law of motion, the velocity vector, v is The updated velocity of the next iteration v(k+1) shown in Eq. (32) depends on the velocity of current iteration (v(k)), the air parcel of current position in search space (x(k)), the distance from the highest pressure point that has been found (x(opt)), the maximum pressure (P(opt)), the pressure at the current location (P(k)), the temperature (T), the gravitational acceleration (g), the universal gas constant R, the frictional coefficient α, and the Coriolis constant, c [102][103][104][105][106][107]. Air parcel position is updated, after the velocity of parcel given by Eq. (32) is updated. This can be expressed as represent that the air parcel vector would continue to move in its previous path with some opposition that is created due to friction.
is an attractive force that pulls against the center of coordinate system. The time step Δt supposing that is the global best position.
x is a vector represent the deflecting force [107 -109]. The WDO permits the air parcels to move only in the interval [-1, 1] for each dimension [110]. To check that the velocity amplitude is within the maximum and minimum limits in any dimension, the following equation is used [108].

Implementation of WDO In OPF Problem
In order to implement the WDO method to solve the OPF and EI problems, the decision variables must be specified. The first step to execute the WDO method is the initialization, i.e. (the algorithm starts by randomly initializing the position and the velocity vectors). In the second step, after the execution of the optimization practice based on the WDO algorithm, the populations of air parcels are distributed randomly over the search space and at random velocities. In the third step, the values of the position and the velocity of each air parcel chosen in the previous step must be evaluated (objective function). The velocity would be updated and check the limits using Eq's. (32) and (34), respectively. In the fifth step, the position of each air parcel must be updated and outgoing air parcels are verified to avoid violating limits. The updating iterations are tested according Eq. (33). Then, the above procedure would be repeated until reaching the maximum iterations.

Simulation & Results
The proposed WDO-based algorithm for solving OPF and EI problems has been applied to the IEEE 30bus and IEEE 57-bus test systems. The numerical and graphical results are represented in these sections.

IEEE 30-bus test system
The five generators system, IEEE 30-bus system is used throughout this work to test the proposed Int. Res. J. Multidiscip. Technovation, 4(2) (2022) 21-41| 26 algorithm. This system consists, 30 buses, 6 generators units and 41 branches, 37 of them are the transmissions lines and 4 are the tap changing transformers. One of these buses is chosen like as a reference bus (slack bus), the buses containing generators are taken the PV buses, the remaining buses are the PQ buses or loads buses. It is assumed that 9 capacitors compensation is available at buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. The network data, the cost and emission coefficients of the five generators are referred in [122]. The one-line diagram of IEEE 30-bus system is shown in Figure 1.
The total loads of active and reactive powers are 283.4 (MW) and 126.2 (MVAr), respectively, with 24 control variables. The basis apparent power used in this paper is 100 (MVA). The simulation results of load flow problem of test system are summarized in Table 1.

Case 1: Cost optimization
The objective functions of cost 1 f given in Eq.
(7) is optimized. Therefore, in this case, the cost has resulted in 801.1347 ($/h), which is considered 8.3608 % lower than the initial case (load flow). Figure 2 shows the convergence characteristic of cost using WDO algorithm. Table 1 summarizes the optimal control variables of this case.

Case 2: Cost with valve-point effect optimization
The cost function 2 f given in Eq. (8) is optimized. Therefore, in this case, the cost has resulted in 826.37 ($/h), which is considered 5.4742 % lower than the initial case. The convergence characteristic of cost optimization for this case is introduced in Figure 2. Table  1 summarizes the optimal control variables of this case.

Case 3: Active Power Loss Optimization
The optimal control variables of this case are introduced in Table 1.       Figure 3 shows the trend for convergence characteristics of active power losses using WDO algorithm. The active power loss minimization has dramatically decreased to 3.2327 (MW) and 3.2771 (MW) without and with valve-point effect, respectively, which is considered 81.5905 % and 81.3376 % lower than the basic case, that is, the case without optimization.

Case 4: Emission optimization
In this case, the emission reduction yielded 0.1763 (ton/h), which is considered 97.7962 % lower than initial case. The optimal settings of control variables for individual objective functions are detailed in Table 1. The convergence characteristics of emission using WDO method is shown in Figure 4.

Case 5: Cost and active loss optimization
The control variables of this case are tabulated in detail in Table 2

Case 6: Cost and emission optimization
The bi-objective optimization considering the cost and the emission are tabulated in Table 2. The control variables of this case are tabulated in detail in Table 2. The cost has resulted in 801.41 ($/h) and 826.29 ($/h) w/o and with valve-point effect, respectively. Figure 5 shows the convergence characteristics obtained in cases 5 and 6.

Case 7: Cost, Active Power Loss and Emission
The IEEE 30-bus control variables of multiobjective considering cost, active power loss and emission are presented in detail in Table 2.     When the valve-point is not in consideration, de generation cost is the 822.58 ($/h) and 863.03 ($/h) with valve-point effect is in consideration. The active power losses and emission w/o and with valve-point effect for this case are, respectively, 6.039 (MW), 6.4499 (MW), 0.2499 (ton/h) and 0.1783 (ton/h). Figure 6 shows the convergence characteristics of multi-objective optimization obtained in case 7 without and with valve-point effect with respect the number of generation under cost optimization, losses optimization and emission optimizations using proposed method.
For the IEEE-30 bus system, 24 control variables (5 generators outputs excluding slack bus, 6 generators magnitude voltages, 4 transformers tap and 9 reactive powers compensators) were optimized. Under the same conditions i.e. control variables limits, constraints and system data, the optimal solutions of IEEE 30-bus test system using the WDO algorithm reported in this paper are compared to some other techniques reported in the literature.
The parameters of WDO method used in this paper are the friction coefficient, α=0.4, the gravitational constant g=0.2, the wind velocity vector, v=3, the coefficient RT=3 and the Coriolis constant, c=0.4.
The developed WDO has been implemented and used to solve the OPF and EI problems of IEEE 30bus system under varying operating conditions. Figure 7 shows the convergence characteristics of WDO method for case 6 with various population sizes applied to IEEE 30-bus system.
It is clearly shown that the WDO could effectively find the optimum solution before the maximum iteration was reached.
The proposed method to solve the OPF and the EI problems is considered to have given the best results because the results obtained using the WDO method are better compared to those published recently in several researches papers.
From Figures 5 and 6, all cases study of biobjective and multi-objective results obtained the minimum values after 120 iterations.

Case 1: Cost optimization
The optimal settings of control variables for individual objective functions are detailed in Table 3. The convergence characteristic of this case is shown in Figure 8.

Case 2: Cost and power losses optimization
The optimal settings of control variables for biobjective functions are detailed in Table 3. The convergence characteristic of this case is shown in Figure 8.

Case 3: Cost and emission optimization
The optimal settings of control variables for biobjective functions are detailed in Table 3. The convergence characteristic of this case is shown in Figure 8.

Case 4: Cost, losses and emission optimization
The optimal settings of control variables for multi-objective functions are detailed in Table 3. The convergence characteristic of this case is shown in Figure 8. Tables 4 and 5 shows a comparison between the obtained single and multi-objective results of costs, power losses and emission with the results obtained in literature.

Conclusion
The WDO approach is successfully implemented in this paper to find the optimum control variables of OPF and EI problems for several cases studies using two power systems which are IEEE 30-bus and IEEE 57-bus test systems.
The versatility of the OPF and the EI are illustrated by different cases by changing of the parameters of the WDO approach such as the friction coefficient, α, the gravitational constant g, the velocity vector of the wind, v, the RT coefficient and the Coriolis constant, c.
The WDO approach is considered to have the capacity to get global solutions with stable convergence, and this is clear from the results obtained from all cases of simulations mentioned previously. Therefore, it can be recommended to future researchers as a promising this algorithm for solving some more complex engineering optimization problems. However, we have to mention that it becomes slow if the numbers of system variables are increased. It is found that the CPU time increases rapidly as system size increases (number of variables augmented) and the convergence slows down.
Finally, the result obtained by WDO approach is quite comparable with other methodology used for the OPF and EI problems.  Table 5 Comparison of obtained results for the cases 6 and 7 of IEEE 30-bus system.