Optimal operation of energy at hydrothermal power plants by simultaneous minimization of pollution and costs using improved ABC algorithm

Abstract

The aim of this paper is simultaneous minimization of hydrothermal units to reach the best solution by employing an improved artificial bee colony (ABC) algorithm in a multi-objective function consisting of economic dispatch (ED) considering the valve-point effect and pollution function in power systems in view of the hot water of the hydro system. In this type of optimization problem, all practical constraints of units were taken into account as much as possible in order to comply with the reality. These constraints include the maximum and minimum output power of units, the constraints caused by the balance between supply and demand, the impact of pollution, water balance, uneven production curve considering the valve-point effect and system losses. The proposed algorithm is applied on the studied system, and the obtained results indifferent operating conditions are analyzed. To investigate in various operating conditions, different load profiles in 12 h are taken into account. The obtained results are compared with those of the other methods including the genetic algorithm (GA), the Basu technique, and the improved genetic algorithm. Fast convergence is one of this improved algorithm features.

Keywords

practical constraints of units / pollution function / inlet steam valve / up-ramp rate of units / improved ABC algorithm

Introduction

Much attention has been paid to the economic dispatch (ED) problem considering the pollution function since 1990 with the enactment of the Clean Air Agency to control environmental pollutants such as carbon monoxide, carbon dioxide, and sulfide oxide. Much field research has been conducted and various techniques have been proposed. For example a Powell hybrid method has been applied to solve the HPS planning problem based on Newton-Raphson method [ 1]. The applied methods require complicated relationships and computing practices. A linear approach has been used that greatly reduces the accuracy of the method [ 2]. Various calculation and optimization techniques such as Lambda iteration, optimum point technique, Lagrangian method, participation factors, and gradient method have been used for solving the ED problems [ 1, 2]. However, these methods are not appropriate when the cost function is nonlinear. In addition, it is very difficult to achieve the optimum solution in some problems. In this paper, an attempt is made to propose an appropriate solution to cope with these problems and increase optimization efficiency. The use of this algorithm not only leads to an optimum solution, but also increases the problem dimensions with the linear raise of units which is a viable way with the current computers to solve practical problems in power systems. That is the reason why intelligent optimization methods, including particle swarm optimization [ 3– 5] and genetic algorithms [ 6], are widely used in ED problems. Moreover, the differential evolutionary algorithm (DE) has been implemented to ED problems [ 7– 9]. The experimental study of the colony algorithm using an improved artificial bee colony (ABC) algorithm in this paper has been used to solve the ED problem, considering the objective function which consists of fuel cost of units, the constraints of the valve-point effect, the transmission losses, the balance of supply and demand in the system, the production limits, the up-ramp and down-ramp rates, and the pollution issues. The resulting algorithm is implemented on the case study systems and the obtained results were compared with those of other algorithms. This algorithm has fast convergence and is less likely to be trapped in local minima compared to other algorithms.

Hydrothermal Systems

In hydrothermal systems, water is heated by contacting with hot rocks. These systems are divided mainly into systems operating based on water vapor and systems operating based on liquid.

The hot water and steam trapped in shallow to medium depth of the crust, in the faults, porous/fractured rocks are called hydrothermal sources, which are the only sources currently used commercially. These sources in terms of dominant fluid phase are categorized into two groups of vapor (most of the reservoir is occupied by the steam phase) and the liquid (most of the reservoir is occupied by hot water). Hydrothermal resources utilized for electricity generation should have temperatures ranging from 90°C to 350°C. Indeed, it is estimated that two-thirds of hydrothermal sources in nature have the average temperature between 150°C and 200°C [ 2]. The richest of the resources are ones with dry steam or with a small amount of liquid in the steam. Unfortunately, so far only two major sources of this type have been identified, one in Italy and other in the United States [ 3].

The increasing use of natural resources such as raw materials as well as energy production using fossil fuels and industrial development have increased air and water pollution, toxic substances and industrial waste, and in turn destruction of the environment. Degradation and environmental pollution, especially in the second half of the 21th century have convinced scientists that if economic growth and environmental protection are not compatible, it will be difficult for human beings to live on the earth in the future [ 4, 5].

The importance of environmental protection in recent years has led to consideration of environmental indecisions related to technologies, energy source selection, allocation of production factors, the pattern of economic growth, and the promotion of social welfare. In the system of prices and costs these factors are considered.

Problem formulation

One of the most important issues in the optimization problem is finding the production share of each generating units such that the required demand is met by the least costs, yet to increase the return on investment for companies to increase production.

One of the main issues in the ED problem is to determine the effect of hydrothermal energy. In addition, the valve-point effect on the cost function is another important point in solving the ED problem.

Cost function

Most researchers have used quadric function to estimate the cost of the production of each generating unit, leading to less adjustment of the optimization problem with the reality. According to Eq. (1), the cost function for each generating unit with regard to the valve-point effect is expressed as

(1) F_i\left(p_i\right)=a_i+b_i{p}_i+c_i{p}_i^{\mathrm{2}}+\left|e_i\sin \left(f_i\left(p_i^{\min }-p_i\right)\right)\right|,

where F_i, e_i, c_i, b_i, and a_i are the coefficients of cost function related to the ith generating unit. As mentioned earlier, the valve-point effect on cost function is in the form of sinusoidal function integrated to quadric function. The valve-point effect on cost function is illustrated in Fig. 1.

Pollution function

Consumption of fossil fuels as the primary sources for power generation units leads to the emission of nitrous oxide as an important pollutant in the environment for which, according to the Environmental Protection Agency, companies are obliged to reduce this type of fuel consumption for electricity generation. Here, the amount of nitrous oxide has been studied as an environmental pollutant. Emissions of each generating unit are described by nonlinear Eq. (2).

(2) F_{\mathrm{2}}={\displaystyle \sum _{m=\mathrm{1}}^M{\displaystyle \sum _{i=\mathrm{1}}^{N_i}{t}_m\left[{\alpha }_i+{\beta }_i{P}_{mi}+{\gamma }_i{P}_{mi}^{\mathrm{2}}+{\zeta }_i{e}^{\left({\zeta }_i{P}_{mi}\right)}\right]}},

where α_i, γ_i, β_i and ξ_i are the objective function coefficients for environmental pollution and t_m is the time period of production for each unit.

Since it is critical to consider the constraints of the problem for any optimization problem, in EED problem, all operating constraints of plants are introduced and considered.

Balance of system supply and demand

The total power generated by all units in the circuit must be equal to the total demand of the system.

(3) {\displaystyle \sum _{i=\mathrm{1}}^{N_i}{P}_{mi}+{\displaystyle \sum _{h=\mathrm{1}}^{N_h}{P}_{mh}-P_{md}-P_{lm}=\mathrm{0}}},

where P_mh is the produced power of the hth hydrothermal unit in the mth laterals and P_md is the total amount of load demand for the mth lateral. P_lm is the total active power loss in transmission lines in the mth lateral which can be calculated based on Eq. (4) [ 10].

(4) P_{lm}={\displaystyle \sum _{i=\mathrm{1}}^{N_i+N_h}{\displaystyle \sum _{j=\mathrm{1}}^{N_i+N_h}{P}_{mi}{B}_{ij}{P}_{mj}}}.

Water availability

Water limitation in the HPS model is given by Eq. (5).

(5) {\displaystyle \sum _{m=\mathrm{1}}^M{t}_m\left(a_{\mathrm{0}h}+a_{\mathrm{1}h}{p}_{mh}+a_{\mathrm{2}h}{p}_{mh}^{\mathrm{2}}\right)-w_h=\mathrm{0}},

where a_0h, a_1h and a_2h are the coefficients of the limitations employed on the required water in the EED optimization problem for the HPS system. w_h is the availability of water for the hth unit.

Generation constraint

For each generating unit, the maximum and minimum limits of power, reactive power, and voltage are defined as inequality (6).

(6) p_i^{\min }\le p_i\le p_i^{\max }.

Improved ABC algorithm

Standard ABC algorithm

ABC algorithm is an optimization technique used to solve problems based on the behavior of bees in nature. In this method, bees have a direct cooperation and share information, trying to get the best solution according to the laws of probability. Honey bees utilize a complex system to find information about the location and quality of food sources outside their hive. The ABC algorithm is based on the amount of rotation of a bee and the movement of onlooker bees toward high quality food source [ 10– 12].

The coding procedure of this algorithm is as follows:

1) Initialization of population for first solutions X_ij.

2) Calculation of initial solutions in objective function.

3) Initial iteration cycle= 1. Providing new solutions based on finding new food resource, V_ij in neighborhood of X_ij.

4) Equation (7) is used to find new solutions.

(7) V_{ij}=X_{ij}+{\mathrm{\Phi }}_{ij}\left(X_{ij}-X_{kj}\right).

5) Selection of the best source or solution between V_ij and X_ij.

6) Calculation of probability amount for X_ij solutions based on Eq. (8).

(8) p_i={\displaystyle \frac{f_i{t}_i}{{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{SN}}{f}_i{t}_i}}}.

Equation (9) is used to find the fitness function of solutions.

(9) f_i{t}_i=\{\begin{array}{ll}{\displaystyle \frac{\mathrm{1}}{\mathrm{1}+f_i}}\hfill & f_i\ge \mathrm{0},\hfill \\ \mathrm{1}+abc\left(f_i\right)\hfill & f_i\le \mathrm{0.}\hfill \end{array}

The obtained solutions are in the (−1, 1).

7) Production of new solutions (new sources) V_i based on onlooker bees from solutions X_i and determining their probability P_i.

8) Selection of the best solution (the most gluttonous bee) between solutions X_ij and V_ij.

9) Determination of the rotten sources and replacing it by random sources created using X_i by pathfinder bees.

(10) X_{ij}={\min }_j+\mathrm{rand}(\mathrm{0},\mathrm{1})\times \left({\max }_j-{\min }_j\right).

10) Storing the best solution (high quality supply source) obtained so far.

\mathrm{Cycle=Cycle+1}.

11) Iterating all steps to meet the termination criterion.

Improved ABC (IABC)

The IABC algorithm is established based on the gravitational forces between objects. By the following steps, this algorithm is pursued.

The percentage is selected as initial solutions of the search space.

Forming initialization:

Movement of foraging bees: The study of the probability for the selected resources is based on Eq. (12). However, the selection of a food source to use the roulette wheel for each foraging bee and determination of the amount of nectar for each of the mare are based on the model developed by the mutual gravitational force among foraging bees obtained by Eqs. (11) to (15) [ 12].

(11) P_i=\mathrm{f}\mathrm{i}{\mathrm{t}}_i/{\displaystyle \sum _{\mathit{n}=1}^{N_{\mathrm{e}}}\mathrm{f}\mathrm{i}{\mathrm{t}}_{\mathit{i}}}.

The mutual force between two objects expressed in Eqs. (12) and (13) is illustrated in Fig. 2.

(12) F_{\mathrm{12}}G{\displaystyle \frac{m_{\mathrm{1}}{m}_{\mathrm{2}}}{r_{\mathrm{21}}^{\mathrm{2}}}}{r}_{\mathrm{21}},

(13) r_{\mathrm{21}}={\displaystyle \frac{r_{\mathrm{2}}-r_{\mathrm{1}}}{|r_{\mathrm{2}}-r_{\mathrm{1}}|}}.

where F(θ_j) and F(θ_i) are the fitness function for scout bees, respectively. Considering the mutual effect of all bees for the selected bee, Eq. (14) is extended as Eq. (15) [13]:

(14) F_{ikj}=G{\displaystyle \frac{F\left({\theta }_i\right)\times F\left({\theta }_k\right)}{{({\theta }_{kj}-{\theta }_{ij})}^2}}·{\displaystyle \frac{{\theta }_{kj}-{\theta }_{ij}}{|{\theta }_{kj}-{\theta }_{ij}|}} ,

(15) x_{ij}\left(t+\mathrm{1}\right)={\theta }_{ij}\left(t\right)+{\displaystyle \sum _{k=\mathrm{1}}^n{F}_{ikj}\left[{\theta }_{ij}\left(t\right)-{\theta }_{kj}\left(t\right)\right]}.

Movement of path finder bees: If the fitness function will not be corrected in the next iterations of the algorithm, it is called “Limit” and named as abandoned resources, replacing/retrieving, with the help of the movement of path finder bees, the abandoned resources by new resources. The movement of bees is

(16) {\theta }_{ij}={\theta }_{ij\min }+r\left({\theta }_{ij\max }-{\theta }_{ij\min }\right).

Placement: If a food source found in the next steps is better than earlier ones, this will be stored in memory.

Program termination: The program will continue until the end of iterations. In the case of finding a satisfactory amount, the program will be terminated; otherwise it will go to the second step. Figure 3 shows the flowchart of the proposed algorithm.

Objective function of studied system: The objective function of this case study is simultaneously minimizing of the generation cost of each generating unit and environmental pollution based on Pareto criterion in simultaneous optimization of multi-objective function. The desired objective function of this study is given as

(17) \min \left\{F_{\mathrm{1}}\left(p_{mi}\right),F_{\mathrm{2}}\left(p_{mi}\right)\right\},

(18) F_{\mathrm{1}}\left(p_{mi}\right)={\displaystyle \sum _{m=\mathrm{1}}^M{\displaystyle \sum _{i=\mathrm{1}}^{N_i}{t}_m\left(a_i+b_i{p}_i+c_i{p}_i^{\mathrm{2}}+e_i\sin (F_i\left(p_i^{\min }-p_i\right)\right)}},

(19) F_{\mathrm{2}}\left(p_{mi}\right)={\displaystyle \sum _{m=\mathrm{1}}^M{\displaystyle \sum _{i=\mathrm{1}}^{N_i}{t}_m\left\{{\alpha }_i+{\beta }_i{p}_{mi}+{\gamma }_i{p}_{mi}^{\mathrm{2}}+{\xi }_i{e}^{\left({\zeta }_i{p}_{mi}\right)}\right\}}}.

Pareto efficiency: Pareto optimization is an economic concept which has applications in engineering and social sciences. This term is named after Vilfredo Pareto, an Italian economist who utilized this concept in his studies of economic efficiency and income distribution. It is an initial allocation of goods among a set of unique individuals. A change to a different allocation of the minimum conditions such that a person’s condition becomes better without deterioration in others called a Pareto improvement. An assignment is defined as Pareto efficiency or Pareto optimal when Pareto improvements cannot excel further.

Simulation results

The studied system consists of four thermal generators and two hydrothermal units whose characteristics are available in Ref. [ 2]. The HPS system investigation is designed for three different loads of 900–1100 and 1000 MW. It is compared with two different operation scenarios. The proposed algorithm has a population of 40, a limit of 200, and the number of iterations of 100. The obtained results are compared with those of other methods, indicating the high capability of the proposed algorithm.

First scenario

The first scenario is based on the minimization of the best obtained cost. The proposed method shows better performance than the other algorithms in solving the problem. The results obtained are listed in Table 1. Figure 4 demonstrates the variation of the objective function in the first scenario.

Second scenario

In the second scenario, simulations are conducted based on the minimization of pollution function. The obtained results for 12-h load profile with various loads are compared with the other techniques. Table 2 and Figs. 5, 6, 7, 8 and 9 present the obtained results.

Conclusions

In this paper, a novel improved algorithm based on particle warm intelligence and the mutual gravitational forces among them was utilized to find the solution to the ED problem by taking into account the practical constraints. For this problem, a multi-objective function consisting of the cost and pollution functions was used. A case study was conducted on a 6-unit system consisting of four thermal units and two hydrothermal units. Tests were performed under different operating conditions with 12-h load profile. In the proposed objective function, simultaneous minimization of generation cost and pollution was considered. The obtained results proved the high capability and appropriate speed of this algorithm in optimization problem. Moreover, the presented values indicated that the standard deviation of the algorithm was low.

References

[1] Wu L H, Wang Y N, Yuan X F, Zhou S W. Environmental/economic power dispatch problem using multi-objective differential evolution algorithm. Electric Power Systems Research, 2010, 80(9): 1171–1181

[2] Lamont J W, Obessis E V. Emission dispatch models and algorithms for the 1990’s. IEEE Transactions on Power Systems, 1995, 10(2): 941–947

[3] Basu M. A simulated annealing-based goal attainment method for economic emission load is patch of fixed head hydrothermal power systems. International Journal of Electrical Power & Energy Systems, 2005, 27(2): 147–153

[4] Venkatesh P, Gnanadass R, Padhy N P. Comparison and application of evolutionary programming techniques to combined economic emission dispatch with line flow constraints. IEEE Transactions on Power Systems, 2003, 18(2): 688–697

[5] Huang C M, Yang H T, Huang C L. Bi-Objective power dispatch using fuzzy satisfaction-maximizing decision approach. IEEE Transactions on Power Systems, 1997, 12(4): 1715–1721

[6] Coello Coello C A. A comprehensive survey of evolutionary-based multi objective optimization techniques. Knowledge and Information Systems, 1999, 1(3): 269–308

[7] Muslu M. Economic dispatch with environmental considerations: tradeoff curves and emission reduction rates. International Journal of Electrical Power & Energy Systems, 2004, 71: 153–158

[8] El-hawary M E, Landrigan J K. Optimum operation of fixed-head hydro-thermal electric power systems: Powell’s hybrid method versus Newton-Raphson method. IEEE Transactions on Power Apparatus and Systems, 1982, PAS-101(3): 547–554

[9] Zaghlool M F, Trutt F C. Efficient methods for optimal scheduling of fixed head hydrothermal power systems. IEEE Transactions on Power Systems, 1988, 3(1): 24–30

[10] Abido M A. Environmental/economic power dispatch using multiobjective evolutionary algorithms. IEEE Transactions on Power Systems, 2003, 18(4): 1529–1537

[11] Chiang C L. Optimal economic emission dispatch of hydrothermal power systems. International Journal of Electrical Power & Energy Systems, 2007, 29(6): 462–469

[12] Karaboga D, Basturk B. A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization, 2007, 39(3): 459–471

[13] Tarafdar hagh M, Ebrahimian H, Ghadimi N. Hybrid intelligent water drop bundled wavelet neural network to solve the islanding detection by inverter based DG. Frontiers in Energy, 2015, 9(1): 75–90

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