Abstract
A new hybrid decomposition-based multiobjective evolutionary algorithm is proposed for optimal power flow (OPF) including wind and solar generation uncertainty. This study recommends a novel constraint-handling method, which adaptively adds the penalty function and eliminates the parameter dependency on penalty function evaluation. The summation-based sorting and improved diversified selection methods are utilized to enhance the diversity of multiobjective optimization algorithms. The OPF problem is modeled as a multiobjective optimization problem with four objectives such as minimizing (i) total fuel cost (TC) including the cost of renewable energy source (RES), (ii) total emission (TE), (iii) active power loss (APL), and (iv) voltage magnitude deviation (VMD). The impact of RESs such as wind and solar energy sources on integration is considered in optimal power flow cost analysis. The costs of RESs are considered in the OPF problem to minimize the overall cost so that the impact of intermittence and uncertainty of renewable sources is studied in terms of cost and operation wise. The uncertainty of wind and solar energy sources is described using probability distribution functions (PDFs) such as Weibull and lognormal distributions. The efficiency of the algorithm is tested on IEEE 30-, IEEE 57-, and IEEE 118-bus systems for all possible conditions of renewable sources using Monte Carlo simulations.
1. Introduction
In recent times, RES penetration has drastically increased in the power system. The penetration of RESs has introduced many challenges to the power system. The intermittent nature of RESs makes the system more complex in terms of operation and control. The uncertain nature of RESs is required to be accurately modeled to examine the dynamic functioning of the power system. Due to its unpredictable nature, protection schemes need to be updated for operating the power system in a secure region. In a power system, the main aim is to operate it with optimal cost and simultaneously satisfy the operating and security constraints. The OPF determines the optimal control settings by the satisfying system and security constraints to economically operate.
A significant amount of research has been carried out in the domain of OPF with the incorporation of RESs in the power system using both deterministic and meta-heuristic optimization algorithms. The gradient method is proposed [1] to develop the dynamic OPF to include wind farms without considering the costs of wind power. For solving the OPF model in the presence of a wind plant, the authors [2] used the Newton method and interior-point methods. The uncertain nature of wind power has been estimated and is added to the overall cost function. However, deterministic methods are problem-specific, exhibit poor convergence characteristics, and are stuck at local optima points. Moreover, these methods are unable to solve real-world optimization issues. To overcome the drawbacks of deterministic methods, meta-heuristic methods have been introduced.
In [3], the authors used the SHADE algorithm with the SF method for arriving at the solution to OPF with RESs. Similarly, in [4–8] the authors proposed several meta-heuristic optimization methods for solving OPF with RESs. However, these are formulated as single-objective optimization problems. In the real world, the OPF problem is multiobjective and the trade-off between multiple objects gives better optimal conditions for operation.
In [9], the authors introduced a modified JAYA algorithm for solving the MOOPF problem incorporating RESs with four different objectives. In this study, the authors transformed multiple objectives into a single objective problem with price and weights. Similarly, in [10–12] the authors proposed a weighted sum-based MOOPF problem with various objectives. The weighted sum-based methods are simple in combining multiobjectives into a single objective with suitable weights. However, this approach heavily depends on the weights that are assigned to each objective value, and these, in turn, affect the optimal solution. Moreover, the weighted sum-based methods fail to obtain the best-compromised solution when needed.
In [13], the authors concentrated on the analysis of the MOOPF solution with RESs using the hybrid DE and SOS algorithms, which have been tested under different operating conditions. Similarly, in [14–16] the authors used the nondominated sorting (NDS) technique to pick the best solutions for parents in an elitist fashion. When the dominant solutions are removed from the population, the effective exploration capability will be lost. Besides this, the nondominated sorting selection is challenging and time-consuming. Moreover, the constraints are handled using the penalty factor method, which is inefficient.
In OPF, constraints play a key role to obtain feasible optimal solutions. The constraint-handling techniques used in optimization techniques are divided into two categories; (i) generic methods and (ii) specific methods. The generic methods are penalty function-based methods. These are simple and mostly used in optimization algorithms as they do not demand additional changes in the algorithm. When a constraint violation occurs, a penalty is added to its fitness. However, these methods may not provide satisfactory results for all types of constraints. On the other hand, specific constraint-handling methods can be applied to convex region problems and large variable problems. The cutting plane method and gradient method are the commonly used methods to handle specific constraints [17, 18]. However, the drawback of specific methods is that, as the number of variables increases, the computing time also increases. The performance of both methods depends on fine-tuning different parameters of constraint handling, which also affects the fitness value.
The conventional generators are subjected to different costs as they run on fuel. RESs such as wind and solar do not require any fuel. Therefore, fuel costs are not considered for wind and solar power generation. In the case of wind and solar generations owned by anyone other than ISO, direct cost needs to be added to the total cost, which is in the form of maintenance costs and renewal charges [19]. The direct prices are agreed by ISO to pay for the scheduled wind and solar energy. Direct prices have not been addressed in most of the literature.
The above literature review reveals the following:(i)Most of the authors designed the OPF problem as single-objective optimization. In real time, multiple objectives play a key role in the economic viability of the power system.(ii)The weighted sum-based methods depend on weights assigned to each objective, and it affects the optimal solution.(iii)In most of the literature, the Pareto dominance method is used, and in the Pareto dominance method, nondominated sorting (NDS) technique to select the best solutions is used, which improves the diversity and convergence. When all the dominant solutions have been removed, the diversity of the population is lost. NDS selection is complex and time-consuming.(iv)The constraints are handled using the penalty factor method, a specific method that is inefficient, due to parameter dependency.(v)In calculating the uncertainty cost of RESs, only overestimation and underestimation costs are considered, while the direct cost is neglected.
In this study, a new hybrid MOEA based on decomposition and summation of normalized objectives with an improved diversified selection method is used for the MOOPF problem. An SF strategy is employed to tackle various constraints (i.e., equality and inequality) of the MOOPF problem.
The major contributions of the research work include the following:(1)Proposing a novel MOEA based on decomposition and summation of normalized objectives with improved diversified selection for the MOOPF problem.(2)Integrating RESs like wind and solar power plants with conventional OPF to consider the impact of the uncertain nature of these sources.(3)Modeling the uncertain nature of wind and solar power plants using PDF and calculating the uncertain cost using Monte Carlo simulations.(4)Multiobjective OPF (MOOPF) with TC, TE, APL, and VMD as four objectives.(5)Utilizing an efficient constraint-handling technique (CHT) called the superiority of feasible solution (SF) to tackle complex constraints in MOOPF problems.
The study is structured as follows: Section 2 presents a wind and solar uncertainty modeling. Section 3 describes the problem formulation of MOOPF with RES. Section 4 presents the framework of the proposed algorithm. In Section 5, simulation case studies are discussed and conclusions are made in Section 6.
2. Wind and Solar Power Uncertainty Modeling
The wind speed at a given geographical area is most likely distributed according to Weibull distributions. Mathematically, the Weibull PDF is written as follows:
The PDFs for two different shape and scale factors are given in [20]. The relationship between wind speed and power generation is as follows:
The probability of obtaining a rated and zero power output is given by the following:
The probability for the linear part of the wind speed is given by the following:where Weibull PDF parameters = 2 and c = 10. The wind speeds = 3 m/sec, = 25 m/sec, and = 16 m/sec.
Similarly, the power output of a solar energy system is a factor of solar irradiance () and it likely follows the lognormal distribution [21]. The PDF for the lognormal distribution is as follows:
The PV unit’s solar irradiance to energy generation is [22]as follows:where lognormal PDF parameters = 6 and = 0.6. The standard solar irradiance () = 800 W/m2, and particular irradiation point () = 120 W/m2.
3. Problem Formulation with Renewable Energy Sources
In this study, a wind generator and solar generator are located at two different buses in the test system. Since wind and solar powers are intermittent, the Monte Carlo simulations are used to account for uncertainty and to calculate the uncertainty cost. The estimated price for the intermittency of wind and solar power is reflected in three ways: direct price, reserve price, and penalty price. Whenever power is underestimated, extra unusable power is wasted; however, in practical power system applications, such power can be saved in an energy storage system and thus counted as the reserve price. The price of overestimating power that is lower than the scheduled power is considered a penalty price in the case of overestimation.
3.1. Direct Price Calculation of Wind and Solar Power Plants
In contrast to conventional generators, wind and solar power generators do not require any fuel. When an ISO owns wind/PV facilities, the direct fuel cost may not occur except if the ISO intends to allocate any compensation for setting up or charging it as a renewal cost and repair work [22]. When private agencies own wind/PV plants, however, ISO proportionally pays for the agreed-upon scheduled power.
The direct price associated with wind plants is as follows:
Similarly, the direct price of PV plant is as follows:
3.2. Uncertainty Price Calculation of the Wind and Solar Power Plants
If the actual output power of the wind farm is lower than the predicted value, to ensure a constant supply of electricity to the consumers, the operator requires some spinning reserve. It is called the overestimation of power from unreliable sources. The cost incurred to maintain the spinning reserve is known as the reserve cost [23].
The reserve price of the wind plant is as follows:
In contrast to the overestimation scenario, when the actual power output of wind exceeds the predicted output, the surplus power generated by WT cannot be used and is wasted. This is called the underestimation of power from uncertain sources. In this case, ISO must pay a penalty for excess power.
The penalty price of the wind plant is as follows:
In the same way as the wind plant, the PV plant also has intermittency in power output. The reserve and penalty price equations for PV plants are described as follows [24].
Reserve price for PV plant is as follows:
The penalty price for a PV plant is as follows:where the direct, penalty, and reserve price coefficients of wind and PV plants are 1.6, 1.5, and 3, respectively.
3.3. Objective Functions
The MOOPF problem assumed the minimization of four objectives: (i) TC, (ii) TE, (iii) APL, and (iv) VMD. The objectives can be described as follows:where p.u., i.e., reference voltage.
3.4. Constraints
3.4.1. Equality Constraints
The overall demand and losses throughout the system are equal to the total real and reactive power delivered.
3.4.2. Inequality Constraints
Generator constraints
Transformer constraints
Shunt VAR compensator constraints
Security constraints
Two equality constraints (equations (18) and (19)) are automatically satisfied when the power flow converges to an optimal solution. The generator buses’ real power (excluding slack bus), transformer tap ratios, voltage limits, and shunt compensator ranges are considered as control variables that are self-limiting. The remaining inequality constraints require constraint-handling techniques.
In OPF, generator reactive power capacities are significant. In the case of thermal generators, the ranges are considered as in [25, 26]. In recent years, WTs with complete reactive power capability have become commercially viable [27]. Enercon FACTS-WT can deliver reactive power in the range of -0.4p.u.to 0.5p.u. The negative sign signifies the generator’s ability to absorb. Rooftop solar PV is designed as load buses with zero reactive power. However, because utility-based solar PVs have converters built-in, full generator modeling is required due to the converters’ dynamic behavior [28]. In this study, the reactive power capabilities of solar PV are assessed between −0.4p.u and 0.5p.u.
3.5. Superiority of Feasible Solution (SF) Method
The most commonly used constraint-handling technique is the penalty function method. When a constraint violation occurs, its solution is penalized. Owing to its simplicity and ease of operation, the outcome of this method is strongly contingent on the penalty factor, which is to be chosen using trial and error, going to cause the fitness value to deteriorate. This study deployed a new CHT called the SF technique [29], which does not require any penalty coefficient.
Since MOOPF is a constrained optimization problem, it requires a better-constrained handling method. In this study, the SF technique [29] was employed to solve the MOOPF problem with RESs. The steps followed when comparing two solutions are as follows:(1)While comparing two nonfeasible solutions, the solution having the smallest constraint violation is selected.(2)When two feasible solutions are compared, the one with a better fitness solution is selected.(3)When a feasible solution is compared to a nonfeasible solution, the feasible solution is selected.
By incorporating these three rules into the proposed algorithm to solve the MOOPF problem, two situations arise, the first of which is when the population size is lower than the number of feasible solutions, and the second method is to ignore nonfeasible solutions. The use of the summation-based method is to select feasible solutions if the number of feasible solutions is greater than the population size.
4. Proposed Algorithm
The MOEAs are normally modeled to handle different conflicting goals, such as maximizing the spread of solutions along the Pareto front (i.e., diversity) and minimizing the distance between the solutions along the Pareto front (i.e., convergence) [30]. The trade-off between convergence and diversity is important to choose the best solution among the obtained solutions. Therefore, to attain a balance between exploration and exploitation in this study, a new method is proposed.
In this study, a summation of normalized objective values (SNOVs) with improved diversified selection (IDS) is proposed and integrated with the multiobjective evolution algorithm based on the decomposition (MOEA/D) [31] method to solve the MOOPF problem with RES. The MOEA/D method decomposes the multiobjective optimization problem into several single scalar optimization problems and optimizes them all at the same time using weight vectors. The weight vectors’ distance is used to create neighborhoods. In every population evolution, information from the neighborhood is used to find a solution. The nondominated sorting used in MOEA/D is complex and time-consuming. Some useful information may be lost if the dominant solutions are completely discarded. In addition, diversity may be lost during the search process and lead to local optima. To overcome these problems, the summation of normalized objective values with IDS [32] is employed in this study instead of nondominated sorting selection to get uniformly distributed Pareto front and improved convergence characteristics.
A new constraint-handling strategy called the superiority of feasible solution (SF) method is employed to handle the various constraints (i.e., equality and inequality) of the MOOPF problem. The proposed algorithm utilizes the fuzzy method to get the best-compromised values. The outcomes of the proposed method are compared with popular methods like MOEA/D [33], NSGA-II [34], and MOPSO [35] for different cases.
The pseudocode of the proposed method is as follows:
Step 1. Initialization: Generate the initial population () of size N. Using SSA [36], generate uniformly distributed weights, and the number of weight vectors is defined as follows:
Step 2. Run the load flow and evaluate the fitness values of the selected objective functions and total constraint violations.
Step 3. Using angle criteria [37], locate neighbors with the smallest angles for each weight vector. The following is an example of the angle criteria:where , , and , = angle between and .
Step 4. Evaluate the smaller objective values to form the present ideal point.
Step 5. Evaluate the larger objective values to form the present nadir point.
Step 6. Reproduction: Angle criteria are used to choose pairs of mating parents. A set of mating parents is picked with a probability of each weight.
Step 7. To generate the new population (), use two-point crossover and mutation.
Step 8. The new population is formed by combining the original population () with the newly generated population ().
Step 9. For each objective and solution, calculate the normalized objective values.
Step 10. By adding all of the normalized objective values for each solution, obtain the sum of the normalized objective values [32].
For m = 1 to M,
Calculate the max and min objectives of the mth objective and find its range.
Normalize the mth objective values using the expression:End.
For = 1 to N.
Add up all normalized objectives to get a unique value.
End.
Step 11. Calculate the Euclidean space between all of the solutions and the reference point.
Step 12. Set a stopping point for the individual with the shortest path to the original point.
Step 13. Divide the objective range into 100 bins, and scan all bins till you reach the stopping point. The solution having the least summation value will be picked to enter into the preferential set for each scanned bin.
Step 14. The solutions are dominated by stopping points, and also the individuals who were not selected will be sent to the backup set.
Step 15. Apply the fuzzy min-max method [38] to get the best-compromised values.
5. Simulation Results
In this study, to tackle the MOOPF problem including wind and photovoltaic uncertainties, the proposed method, MOEA/D [33], NSGA-II [34], and MOPSO [35] are demonstrated on IEEE 30-, IEEE 57-, and IEEE 118-bus power systems. It is implemented in MATLAB R2016a and runs on an i3 processor with 4 GB RAM.
In general, more than two objectives are treated as a multiobjective optimization (MOO) problem. While formulating the MOO problem, the objectives are chosen such that the objectives conflict with each other. The conflict between objectives depends on the correlation among the objectives. Different objectives will have different degrees of correlation among the combination of objectives. To formulate the combination of objectives, four different objective functions are considered, which are as follows: (i) TC, (ii) TE, (iii) APL, and (iv) VMD. A total of ten different case studies are considered on three standard test systems to test the efficiency of the proposed method for the MOOPF problem. The various case studies considered in this study are given in Table 1.
Numerous trials with various control parameters were conducted, and the best findings obtained are summarized in this study. The parameters chosen for each method are listed in Table 2.
5.1. Modified IEEE 30-Bus System
The IEEE 30-bus power system has 6 thermal generators placed at buses 1, 2, 5, 8, 11, and 13 (# 1 generator as a slack generator), 41 lines. In this study, 4 off-nominal transformers are considered between lines 6–10, 6–9, 4–12, and 27–28, and 9 shunt VAR compensators are placed at the buses. The whole real and reactive power demand on the system is 238.40 MW and 126.20MVAR, respectively. In addition to the above thermal generators, one wind generator and one solar generator are added to buses 22 and 25, respectively. Detailed information about the test system is provided in [39, 40].
5.1.1. Case-1: Simultaneously Minimize TC and TE
In this case, TC and TE are the objectives considered to simultaneously minimize. The optimal decision variables obtained by the suggested method are included in Table 3. The best-compromised values that could be found using the proposed algorithm have a TC of 794.0907$/h and a TE of 0.2166ton/h, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 4. The Pareto optimal fronts of all the methods are depicted in Figure 1.
5.1.2. Case-2: Simultaneously Minimize TC and APL
In this case, TC and APL are the objectives considered to simultaneously minimize. The optimal decision variables obtained by the suggested method are included in Table 3. The best-compromised values that could be found using the proposed algorithm have a TC of 798.6845$/h and an APL of 3.9899 MW, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 4. The Pareto optimal fronts of all the methods are depicted in Figure 2.
5.1.3. Case-3: Simultaneously Minimize TC, TE, and APL
In this case, TC, TE, and APL are the objectives considered to simultaneously minimize. The optimal decision variables obtained by the suggested method are included in Table 3. The best-compromised values that could be found using the proposed algorithm have a TC of 838.0936 $/h, a TE of 0.2049ton/h, and an APL of 3.2506 MW, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 4. The Pareto optimal fronts of all the methods are depicted in Figure 3.
5.1.4. Case-4: Simultaneously Minimize TC, TE, and VMD
In this case, TC, TE, and VMD are the objectives considered to simultaneously minimize. The optimal decision variables obtained by the suggested method are included in Table 3. The best-compromised values that could be found using the proposed algorithm have a TC of 799.7880$/h, a TE of 0.2172ton/h, and a VMD of 0.0902p.u., which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 4. The Pareto optimal fronts of all the methods are depicted in Figure 4.
5.1.5. Case-5: Simultaneously Minimize TC, TE, APL, and VMD
In this case, TC, TE, APL, and VMD are the objectives considered to be simultaneously minimized. The optimal decision variables obtained by the suggested method are included in Table 3. The best-compromised values that could be found using the proposed algorithm have a TC of 851.9069$/h, a TE of 0.2057ton/h, APL of 3.1972 MW, and VMD of 0.1038p.u., which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 4.
5.2. Modified IEEE 57-Bus System
To show the scalability of the proposed algorithm, the IEEE 57-bus system is used for solving the MOOPF problem. It contains 7 thermal generators placed at buses 1, 2, 3, 6, 8, 9, and 12 (# 1 generator as a slack generator), 80 lines. In this study, 15 off-nominal transformers are considered along with 3 shunt VAR compensators. The entire real and reactive power demand on the system is 1250.80 MW and 336.40MVAR, respectively. In addition to the above thermal generators, one wind generator and one solar unit are added at buses 45 and 46, respectively. Detailed information about the test system is provided in [39, 40].
5.2.1. Case-6: Simultaneously Minimize TC and TE
In this case, TC and TE are the objectives that need to be simultaneously minimized. The optimal decision variables obtained by the suggested method are included in Table 5. The best compromise solution that could be found using the proposed algorithm has a TC of 36195.21$/h and a TE of 1.0182ton/h, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 6. The Pareto optimal fronts of all the methods are depicted in Figure 5.
5.2.2. Case-7: Simultaneously Minimize TC, TE, and APL
In this case, TC, TE, and APL are the objectives that need simultaneous minimizing. The optimal decision variables obtained by the suggested method are included in Table 5. The best-compromised values that could be found using the proposed algorithm have a TC of 36096.69$/h, a TE of 1.0238ton/h, and an APL of 10.3303 MW, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 6. The Pareto optimal fronts of all the methods are depicted in Figure 6.
5.2.3. Case-8: Simultaneously Minimize TC, TE, APL, and VMD
In this case, TC, TE, APL, and VMD are the objectives that need to be simultaneously minimized. The optimal decision variables obtained by the suggested method are included in Table 5. The best-compromised values that could be found using the proposed algorithm have a TC of 36207.21$/h, a TE of 1.0916ton/h, APL of 9.9732 MW, and VMD of 0.6848p.u., which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 6.
5.3. Modified IEEE 118-Bus System
To show the scalability of the proposed algorithm for a large-scale test system in solving the MOOPF problem, the IEEE 118-bus system is considered. It contains 54 thermal generators (# 69 generator as a slack generator) and 186 lines. In this study, 9 off-nominal transformers and 12 shunt VAR compensators are considered. The sum of real and reactive power demand on the system is 4242.00 MW and 1439.00MVAR, respectively. In addition to the above thermal generators, one wind generator and one solar unit are added to buses 63 and 64, respectively. Detailed information about the test system is provided in [39, 40].
5.3.1. Case-9: Simultaneously Minimize TC and APL
In this case, TC and APL are the objectives that need to be simultaneously minimized. The optimal decision variables obtained by the suggested method are included in Table 7. The best compromise solution that could be found using the proposed algorithm has a TC of 132958.66$/h and an APL of 31.2916 MW, which is the lowest value compared with MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 8. The Pareto optimal fronts of all the methods are depicted in Figure 7.
5.3.2. Case-10: Simultaneously Minimize TC, APL, and VMD
In this case, TC, APL, and VMD are the objectives that need simultaneous minimizing. The optimal decision variables obtained by the suggested method are included in Table 7. The best compromise solution that could be obtained using the proposed algorithm has a total cost of 135774.93$/h, APL of 39.6333 MW, and VMD of 0.4299p.u., which is the lowest value compared to MOEA/D [33], NSGA-II [34], and MOPSO [35] as shown in Table 8. The Pareto optimal fronts of all the methods are depicted in Figure 8.
5.4. Computational Time
In this study, the MOOPF problem was executed on a 2.00 GHz, i3 processor, with a 4 GB RAM computer. The computational (CPU) times of the proposed method, MOEA/D, NSGA-II, and MOPSO for all the cases are given in Table 9. The computational times of the proposed method are significantly faster than those of other studied methods for all cases. Hence, the proposed method outperformed the other methods in terms of solution quality and computing time.
6. Conclusions
This study proposes a solution to the MOOPF problem with a combination of thermal, wind, and PV systems using MOEA based on decomposition and summation of normalized objectives with an improved diversified selection method. The method also deals with tackling various constraints in the MOOPF problem using the superiority of the feasible solution (SF) technique. The fuel costs of thermal generators and uncertainty prices associated with wind and PV energy systems are minimized along with carbon emission, active power losses, and voltage magnitude deviation. Monte Carlo simulations were used to assess the uncertainty of wind and solar power. Apart from the conventional cost minimization, this study selects factors to account for the uncertain price of available wind and solar power. It depicts the OPF formulation along with factors affecting wind and PV power’s intermittency. To show the efficacy of the suggested method, simulations were performed on the same test systems as with MOEA/D, NSGA-II, and MOPSO algorithms. The results show the superiority of the proposed method compared to other methods. Hence, the proposed method can be effectively used in operation and control when wind and solar power generation are included in the power system.
Abbreviations
| : | Shape and scale factors, respectively |
| : | Wind speed (m/sec) |
| , , : | Cut-in, cut-out, and rated wind speeds, respectively |
| : | Solar irradiance |
| : | Mean |
| : | Standard deviation |
| : | Rated power of wind and solar plants, respectively |
| : | Scheduled power of wind and solar plants, respectively |
| : | Direct price coefficients of a windmill and solar plant, respectively |
| , : | The actual power output of windmill and PV plant, respectively |
| , : | Reserve and penalty price coefficients of windmill, respectively |
| , : | Reserve and penalty price coefficients of PV plant, respectively |
| : | Generator cost coefficients |
| , , and : | Number of thermal, wind, and solar power plants, respectively |
| : | generator emission coefficients |
| , : | Conductance and susceptance between buses and |
| , , , , and : | Number of buses, thermal generators, shunt VAR compensators, PQ buses, and transformers, respectively |
| , : | Min-max limits on generator real power |
| , : | Min-max limits on generator reactive power |
| : | Apparent power flow and its maximum limit, respectively |
| , : | Min-max limits of transformer tap positions |
| , : | Min-max limits of bus voltages |
| : | Voltage angle between buses and |
| , : | Real and reactive power injection at bus |
| , : | Real and reactive power demand at bus |
| : | Weight vector |
| : | Number of lines |
| : | Number of objectives |
| : | Normalized objective |
| MOEA: | Multiobjective evolutionary algorithm |
| MOOPF: | Multiobjective optimal power flow |
| PV: | Photovoltaic |
| WT: | Wind turbine |
| ISO: | Independent system operator |
| NSGA-II: | Nondominated sorting genetic algorithm-II |
| MOPSO: | Multiobjective particle swarm optimization. |
Data Availability
The data supporting these findings are from previously reported studies and datasets, which have been cited.
Conflicts of Interest
The authors declare that they have no conflicts of interest.