Economic dispatch in a power system considering environmental pollution using a multi-objective particle swarm optimization algorithm based on the Pareto criterion and fuzzy logic
Abstract
In recent years, many studies have studied economic dispatch problem in power systems. However, most of them have not considered the environmental pollution caused by fossil fuels. In this study, the use of an evolutionary search algorithm called multi-objective particle swarm optimization algorithm is proposed to solve the economic dispatch problem in power systems while considering environmental pollution. The proposed method is validated in terms of its accuracy and convergence speed based on comparisons with the results obtained using the classic nonlinear programming method. The proposed strategy is applied to a realistic power system under various conditions. Overall, six generating units are investigated along the corresponding constraints. The results obtained reveal that costs of operation and pollution with/without power loss are reduced significantly by the proposed approach. Obtained results show a good compromise can be established between two contradicting functions of exploitation cost and pollution by optimizing them simultaneously. Values of these function without considering their loss is 46,112.09 $/h and 682.32 kg/h, respectively. And if losses are considered, these values would be 48,381.09 $/h and 726.52 kg/h, respectively.
Keywords
Economic dispatch Multi-objective optimization Multi-objective particle swarm optimization (MOPSO) algorithm Pareto criterion Power plant environmental pollutionList of symbols
- a_{i}, b_{i}, and c_{i}
ith generating unit coefficients
- F(P_{j})
ith generating unit cost function
- N
Number of generating units in operation
- P_{i}
ith generating unit output power
- α_{i}, β_{i}, and γ_{i}
Emissions coefficients for the ith generating unit
- P_{loss}
System power loss
- \( P_{i}^{ \hbox{min} } \) and \( P_{i}^{ \hbox{max} } \)
The minimum and maximum power levels generated by each generating unit
- \( {\text{RDR}}_{i} \) and \( {\text{RUR}}_{i} \)
Ramp-down and ramp-up rates for a generating unit
- X
Vector includes the output power of the generating units
- \( \mu_{i}^{k} \)
Membership function that shows the ith objective function’s optimality
- \( f_{i}^{\hbox{max} } \) and \( f_{i}^{ \hbox{min} } \)
Upper and lower boundaries of the ith objective function
- n
Number of non-dominated solutions
- m
Number of objective functions
Introduction
Installation of pollution removal devices in power plant sites;
Replacement of old devices with new ones;
Operation of power plants by considering environmental pollutants.
Various approaches have been proposed that consider emissions from power plants to address the ED problem. Finnegan and Fouad considered the emissions from power plants for the first time in 1974 [2], where they treated emissions as a constraint within a permissible range. Later, this strategy was used to control pollution in related studies [3]. The EMO algorithm was employed in [5] to speed up the convergence of an operational cost function. Many other solutions have been suggested to address this problem in previous studies. Analytical methods [6], a Lagrangian method [7], and the Newton–Raphson method [8] have all been employed as initial approaches. Evolutionary methods have also been employed for this purpose, such as a genetic algorithm [9], particle swarm optimization (PSO) [10], simulated annealing [11], artificial immune system [12], differential evolution [13], and the frog algorithm [14]. In [15], the PSO-SIL algorithm was used to obtain an economical power flow with the optimum costs.
These methods differ in terms of their speed and accuracy. The ED problem without consideration of environmental issues leads to increased costs. In addition, concerns over environmental pollutions are increasing constantly. However, the aforementioned techniques only consider the costs related to systems. In this study, we propose an analytical strategy for simultaneously minimizing costs and the emissions from power plants. This multi-objective problem is solved using a multi-objective PSO (MOPSO) algorithm. We applied the proposed strategy to a realistic six-bus test system. The results obtained were validated based on comparisons with those produced using the classic nonlinear programming (NLP) method.
The remainder of this paper is organized as follows. In Sect. 2, the ED problem is modeled for a power system by considering an emissions function. Additional constraints such as the ramp-rate limit and prohibited operating zone for generators, as well as network security are also considered. Furthermore, the problem-solving method is described in this section. In Sect. 3, MOPSO algorithm is proposed as a method for solving the problem. In Sect. 4, we present the simulations and the numerical results are discussed. Finally, we give our conclusions in Sect. 5.
Economic dispatch considering emissions
In power distribution systems, electrical engineers attempt to improve the power system efficiency by increasing the number of generating units and making profits to obtain the maximum benefits with the least cost. In addition, they should satisfy the total load requirements of the network and observe all the operating constraints on power plants, as well as transmission lines. The ED problem is an optimization problem regarding environmental emissions and the operating costs for generating units. The aim when solving this problem is to meet the load demand with the least cost and emissions, while also satisfying the constraints on the problem.
Operating cost function
Emission function
Constraints
The power system considered operates subject to technical constraints.
Power balance constraint
Generating units operating constraint
Ramp-rate limit
MOPSO algorithm
The following steps are required to apply the algorithm to the problem considered in this study.
Step 1) Enter the input data
First, the input data required by the program are entered in detail: the power system configuration, operating characteristics of the generating units, and pollution coefficients for each generating unit.
Step 2) Define the initial population.
Step 4) Determine the non-dominated solutions.
The non-dominated solutions are determined by Eq. (8).
Step 5) Separate the non-dominated solutions and store them in an archive.
To access non-dominated solutions, they should be stored.
Step 6) Select the best particle from the archive of non-dominated solutions as a leader.
The leader selection process is as follows. First, the search space is divided into equal parts. Next, a probability distribution is assigned to each search space division. Finally, the best particle is selected as the leader by a roulette wheel method.
Step 7) Obtain the new velocity and position for each particle using Eqs. (10) and (11).
Step 9) Integrate the current non-dominated solutions into the archive.
Step 10) Eliminate dominated solutions from the archive.
Dominated-solutions are eliminated from solution circle of the archive.
Step 11) If the number of solutions in the archive exceeds a predefined value, eliminate extra solutions.
Step 12) Check the program termination criterion.
Step 13) Select the best interactive solution.
Case study and simulation results
Fuel costs for generating units
Power plant | Fuel cost coefficients | \( P_{{G{ \hbox{min} }}} \) | \( P_{{G{ \hbox{max} }}} \) | |||
---|---|---|---|---|---|---|
c_{i} | b_{i} | a_{i} | ||||
\( P_{{G{ \hbox{max} }}} \) | G_{1} | 0.15247 | 38.5397 | 756.79886 | 10 | 125 |
G_{2} | 0.10587 | 46.1591 | 451.32513 | 10 | 150 | |
G_{3} | 0.02803 | 40.3965 | 1049.3251 | 40 | 250 | |
2 | G_{4} | 0.03546 | 38.3055 | 1243.5311 | 35 | 210 |
G_{5} | 0.02111 | 36.3278 | 1658.5696 | 130 | 325 | |
3 | G_{6} | 0.01799 | 38.2704 | 1356.6592 | 125 | 315 |
Pollution coefficients for generating units
Power plant | Pollution cost coefficients | \( P_{{G{ \hbox{min} }}} \) | \( P_{{G{ \hbox{max} }}} \) | |||
---|---|---|---|---|---|---|
d_{i} | e_{i} | f_{i} | ||||
1 | G_{1} | 0.00419 | 0.32767 | 13.85932 | 10 | 125 |
G_{2} | 0.00419 | 0.32767 | 13.85932 | 10 | 150 | |
G_{3} | 0.00683 | −0.54551 | 40.2669 | 40 | 250 | |
2 | G_{4} | 0.00683 | −0.54551 | 40.2669 | 35 | 210 |
G_{5} | 0.00461 | −0.51116 | 42.89553 | 130 | 325 | |
3 | G_{6} | 0.00461 | −0.51116 | 42.89553 | 125 | 315 |
Transmission power loss coefficients
0.000091 | 0.000031 | 0.000029 |
0.000031 | 0.000062 | 0.000028 |
0.000029 | 0.000028 | 0.000072 |
Operating cost function
Economic dispatch without transmission system power loss
Unit | PSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 32.45 | 32.49 |
2 | 10.72 | 10.81 | |
3 | 143.69 | 143.64 | |
4 | 143.15 | 143.03 | |
5 | 287.16 | 287.10 | |
6 | 282.80 | 282.90 | |
Total fuel costs ($/h) | 45463.49 | 45463.49 | |
Total emissions (kg/h) | 795.11 | 795.07 |
Economic dispatch with transmission system power loss
Unit | PSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 33.77 | 33.99 |
2 | 12.54 | 12.97 | |
3 | 153.82 | 151.79 | |
4 | 148.53 | 147.27 | |
5 | 296.21 | 294.23 | |
6 | 293.30 | 298.05 | |
Total fuel costs ($/h) | 47170.92 | 47328.74 | |
Total emissions (kg/h) | 856.99 | 863.29 | |
Transmission loss (MW) | 38.23 | 38.32 |
The results in Tables 4 and 5 indicate that generating units 5 and 6, which had the lowest start-up cost among all the generating units, contributed enormously to meeting the load demand. Generating unit-5 produces 2,871,629 MW and generating unit-6 generates 2,828,004 MW without considering power losses. Obviously, these values increase when power losses are considered (see Table 5).
Emission cost function
Economic dispatch of emissions without transmission system power loss
Unit | PSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 116.99 | 116.99 |
2 | 116.98 | 116.99 | |
3 | 135.69 | 135.69 | |
4 | 135.69 | 135.66 | |
5 | 197.31 | 197.31 | |
6 | 197.31 | 197.31 | |
Total fuel costs ($/h) | 48051.22 | 48051.51 | |
Total emissions (kg/h) | 646.12 | 646.18 |
Economic dispatch of emissions with transmission system power loss
Unit | PSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 124.60 | 122.75 |
2 | 127.51 | 122.75 | |
3 | 140.27 | 139.22 | |
4 | 140.30 | 141.96 | |
5 | 204.11 | 206.64 | |
6 | 204.19 | 207.78 | |
Total fuel costs ($/h) | 50223.97 | 50262.99 | |
Total emissions (kg/h) | 697.03 | 701.12 | |
Transmission loss (MW) | 40.99 | 41.12 |
Based on the results in Tables 6 and 7, we can state that generating units 1 and 2, which had the lowest emissions, operated near their maximum value and these units made major contributions to meeting the load demand. Generating unit-1 produces total power of 116.99 MW and generating unit-2 generates 116.98 MW without power loss. This will definitely increase when power loss is considered (See Table 7).
Simultaneous minimization of the operating cost and emission cost functions
Simultaneous minimization of the operating cost and emission cost functions without transmission system power loss
Unit | MOPSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 68.86 | 36.02 |
2 | 66.77 | 16.66 | |
3 | 143.77 | 147.79 | |
4 | 156.01 | 146.54 | |
5 | 244.55 | 278.73 | |
6 | 220.01 | 274.24 | |
Total fuel costs ($/h) | 46112.09 | 46248.23 | |
Total emissions (kg/h) | 682.32 | 775.48 |
Simultaneous minimization of the operating cost and emission cost functions with transmission system power loss
Unit | MOPSO | NLP | |
---|---|---|---|
Generating unit (MW) | 1 | 81.71 | 37.53 |
2 | 79.57 | 18.81 | |
3 | 152.51 | 154.58 | |
4 | 162.82 | 151.32 | |
5 | 232.58 | 286.6 | |
6 | 228.84 | 289.4 | |
Total fuel costs ($/h) | 48381.09 | 48181.49 | |
Total emissions (kg/h) | 726.52 | 843.19 | |
Transmission loss (MW) | 38.1 | 38.34 |
According to the results in Tables 8 and 9, when we simultaneously optimized the operating and emission cost functions, there was a trade-off between the two conflicting functions to operate the system at the optimum point.
Conclusions
In this study, the MOPSO algorithm was used to solve the ED problem considering emissions in a power system with various constraints. To obtain satisfactory results, the problem was solved by taking into account the operating and emission costs in a separate mode. Next, the problem was solved by trading off between two contrasting objective functions, i.e., operating and emission cost functions, for simultaneous minimization. This process was analyzed in two modes, without and with transmission system power loss. As results indicate when two objective functions are optimized simultaneously the related costs are highly reduced. Comparing the results obtained with those produced by the NLP method demonstrated that the MOPSO algorithm outperformed the NLP method in terms of its accuracy and convergence speed as results show, if two pollution and exploitation cost functions are optimized simultaneously, without considering losses, their values would be 46,112.09 $/h and 682.32 $/h and if losses are considered, these values would be 48,381.09 $/h and 726.52 $/h.
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