Development of methodology for calculation of optimal distribution of electric power between power units of condensing power plant

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Introduction

Modern condensing power plants (CPPs) typically have several power units, each with its own experimentally determined actual fuel consumption dependence on generated power (discharge characteristic).

This work is aimed at developing a methodology for determining the redistribution of the total generating capacity of a power plant among individual power units to obtain the lowest total fuel consumption.

Research methods and tools

The problem under consideration relates to optimization. Such problems can only be solved by utilizing precise mathematical techniques. Usually, a solution exists but can only be found using numerical methods, which are currently implemented using a computer.

The simplex method [1], the gradient method, Newton’s method [2], etc., are among the numerical methods that have been previously used. However, these methods are only applicable for a specific type of optimization problem and have significant limitations in terms of the form and complexity of the equations, as well as the number of conditions imposed.

To avoid these limitations, heuristic methods have been used in recent years, in which the search for a solution follows a different pathway each time the program is run, but ultimately leads to the same result. These methods are usually based on analogies from living nature, such as studying the development of bee colonies or ant colonies, or, more generally, on the synthetic theory of evolution [3].

The differential evolution method [4], which is based on generating initial random values of factors and applying evolutionary principles, such as cross-breeding, mutation, and artificial selection of specimens with the best characteristics to them, is one of the most relevant and universal methods for finding an optimal solution. Constructing a population of factors refers to each stage of obtaining intermediate values (iteration). The differential evolution method is based on the simultaneous employment of regular and random processes, and as a result, it allows for the global rather than local extremum of the optimized function to be found.

When improving load redistribution between power units, it seems fair to employ the differential evolution method since it works reliably even with complex discharge characteristics. It is necessary to assess the degree of perfection of the method, and other advantages and disadvantages, as well as to develop recommendations for its further application to other optimization problems in the energy industry.

Results and discussion

The initial data for optimization are the discharge characteristics of the power units of the power plant, including the dependence of the specific (per unit of power) or physical consumption of fuel on the generated electric power.

Table 1 presents such dependences for a power plant with four power units based on condensing steam turbines K-100-90 for driving electric generators with the initial steam parameters (pressure 8.8 MPa; temperature 530 ºС, and final pressure 4 kPa), as an example. The maximum power at the electric generator terminals was 110 MW, and the minimum permissible power was 30 MW [5].

 

Table 1.

Consumption characteristics of power units of the investigated power plant

N, MW	b1, kg/(kW • h)	b2, kg/(kW • h)	b3, kg/(kW • h)	b4, kg/(kW • h)	B1, kg/s	B2, kg/s	B3, kg/s	B4, kg/s
30	0,488	0,471	0,527	0,443	4,07	3,93	4,39	3,69
40	0,434	0,427	0,445	0,408	4,82	4,74	4,94	4,53
60	0,369	0,358	0,336	0,344	6,15	5,97	5,60	5,73
80	0,338	0,341	0,321	0,329	7,51	7,58	7,13	7,31
100	0,32	0,333	0,325	0,312	8,89	9,25	9,03	8,67
110	0,313	0,331	0,339	0,302	9,56	10,11	10,36	9,23

 

Table 1 was compiled using operational power unit data, and the specific consumption of equivalent fuel b_i, kg/(kW • h), was measured for each generated power N, MW, where i is the power unit number. The physical consumption of the equivalent fuel B_i, kg/s was calculated and added to the table using these data and Microsoft Excel spreadsheets.

$B_i=\frac{N\cdot b_i}{3.6}$.

Subsequently, consumption characteristics graphs were plotted in Microsoft Excel for all power units, as presented in Figure 1.

 

Fig. 1. Consumption characteristics of the investigated power units

 

Using these characteristics, the optimization problem must be addressed, in which such powers of each power unit N_i should be determined for a known capacity of the power plant N, so that power plant B consumes the least amount of fuel (equal to the sum of fuel consumption for power units B_i).

These conditions can be presented as follows:

$N=N_1+N_2+N_3+N_4$, (1)

$B=B_1+B_2+B_3+B_4\to \min$. (2)

In the considered optimization problem, the condition under Eq. (1) is a constraint, and the condition under Eq. (2) is an objective function. To solve the problem for each power unit, the mathematical fuel consumption dependence on power B_i = f(N_i) should be determined. This can be done by determining the best approximation dependences for the graphs in Figure 1.

In Microsoft Excel, the approximation was performed by plotting trend lines for each graph. The polynomial approximation by third-order curves was the most suitable, providing the best approximation curves to the initial graph. The following dependencies were obtained for the discharge characteristics of the power units considered:

$\left\{\begin{array}{l}{B}_1=1\cdot {10}^{-6}{N}_1^3-0.0002N_1^2+0.0826N_1+1.768\\ B_2=1\cdot {10}^{-6}{N}_2^3-4\cdot {10}^{-5}{N}_2^2+0.0667N_2+1.9838\\ B_3=5\cdot {10}^{-6}{N}_3^3-0.0003N_3^2+0.0431N_3+3.337\\ B_4=-2\cdot {10}^{-6}{N}_4^3+0.0005N_4^2+0.0423N_4+2.1224\end{array}\right.$   (3)

To solve the aforementioned problem, it is convenient to use the Python programming language, for which there is a special library of mathematical methods called SciPy [6], which includes a differential evolution method implementation [7].

The Python language is easy to learn, and the differential evolution method used in this work, which is presented in the form of a special command, is also easy to use and does not require special knowledge of its characteristics. The command parameters can also be fine-tuned, and one of 12 strategies for solving and parallelizing resource-intensive tasks on several processors can be selected. The default settings of the method are suitable for achieving the aim of this work.

The algorithm of the developed Python program consists of the following parts:

1. Import of the required commands NonlinearConstraint (nonlinear limitation), Bounds (boundaries of the solution search), and Differential_evolution from the Scipy. optimize optimization library. The Numpy module [8] was also imported to perform operations with data arrays, as well as the Matplotlib.pylab [9] module for displaying intermediate and final results of calculations as graphs.
2. Assigning the value N = 200 MW to a variable representing the total capacity of a power plant as an example.
3. Input of the condition under Eq. (1) using the lambda function [10] and the NonlinearConstraint command.
4. Creation of the objective function under Eq. (2), which will be used as one of the parameters of the Differential_evolution command, along with the condition under Eq. (1). The values of the power N_i and the equivalent fuel consumption B, obtained at each iteration (approximation of the calculation process), were stored in arrays.
5. Using the Bounds command, define the smallest and largest allowable values for N_i variables, which are equal to 30 MW and 110 MW, respectively, according to the task.
6. Application of the differential evolution method (Differential_evolution) and displaying the optimization results for a given station power N on the screen.
7. Using the Matplotlib command module to display the calculated changes in the factors N_i as a function of the iteration number, as presented in Figure 2. The graph colors for each power unit correspond to those in Figure 1.

 

Fig. 2. Changes in factors (capacities of power units N1–N4) depending on the iteration number

 

The evolution of changes in the factors N₁−N₄ upon searching for their optimal values corresponding to the minimum value of the objective function B is presented according to these graphical dependencies. At each start of the program, the two pathways (Figure 2) were different, as will the number of iterations required to provide a solution.

Despite the different evolutionary pathways of the optimized system from the initial random values, the found optimal values of the factors are equivalent at each start. Since a high degree of result accuracy was selected, the values remained unchanged at the last iterations.

For the total power N = 200 MW, the optimal capacity of the power units and the consumption of equivalent fuel (accurate to the second decimal place) were calculated (N₁ = 30.00 MW, N₂ = 60.73 MW, N₃ = 68.86 MW, N₄ = 40.41 MW; B = 21.23 kg/s).

Figure 3 presents the evolution of the total fuel consumption B toward the minimum value, with changes in the factors N₁–N₄, as shown in Figure 2.

 

Fig. 3. Change in the value of the objective function (fuel consumption at power plant B) depending on the iteration number

 

To obtain the optimal redistribution of power unit capacities in the entire range of the possible total capacity of the power plant (N = 120−440 MW), the calculation program comprises a cycle for calculating the above parameters depending on N. The results are also displayed graphically.

Figure 4 presents the main dependence of the power unit capacities (factors N₁−N₄) on the power of station N, which is the study aim.

 

Fig. 4. Redistribution of the capacities of power units N1–N4 depending on the capacity of the station N

 

Figure 4 shows how to determine the power redistribution of individual power units N₁−N₄ at which the total fuel consumption at station B, reaches a minimum value for any station power N. The resulting diagram should be used to develop a control program for CPP power units.

In Figure 5, the actual consumption of the equivalent fuel В = f(N), is graphically represented for all possible plant capacities.

 

Fig. 5. Consumption of equivalent fuel at power plants with optimal power distribution between power units

 

In the following research stage, the relationship between relative fuel savings and optimal redistribution of power unit capacities in comparison to the case of their uniform loading was assessed. With uniform loading, each power unit had a capacity N_unif = N/4, and fuel consumption B_unif is determined by Eq. (3), where all N_i are equal to N_unif. The relative fuel saving was determined by the following equation:

$\delta =\frac{B_{равн}-B}{B_{равн}}100\%$. (4)

Figure 6 shows the results of calculations using Eq. (4) for all possible power plant capacities.

 

Fig. 6. Fuel savings with optimal power redistribution between power units versus an even power distribution

 

Thus, a decrease in the relative fuel consumption at power plants when using the obtained diagram of the optimal power distribution between power units (Fig. 4) compared with their uniform loading can reach 2.4%, resulting in a significant increase in the economic efficiency of the considered powerful condensing power plant with steam turbines.

Conclusion

To achieve the aim of this work, the following tasks were solved:

• a brief comparison of the capabilities of widespread optimization methods and modern evolutionary methods, in particular, the differential evolution method used in this work;
• software tools for the implementation of the task of optimizing power redistribution between the power units of the power plant to ensure the minimum fuel consumption were selected and described;
• a methodology for the implementation of the specified optimization and visual display of the results using the Python programming language was fully developed;
• an example was calculated, demonstrating the relative simplicity and efficiency of applying the differential evolution method to solve the problem;
• a diagram of the optimal power redistribution between the power units of the condensing power plant was drawn up.

The code for the Python program developed during the research and discussed in this article is available for free study, application, and modification on the authors’ website [11].

The method considered can be used to optimize power redistribution between the power units of any power plant (not only steam turbine CPPs but also thermal power plants, including gas turbine and steam gas turbines), as well as to solve other problems in the energy industry, such as:

• redistribution of liquid and gaseous fuel flows delivered from the point of production to the consumer [12];
• redistribution of electrical energy or energy carriers (steam or hot water flows) in the networks;
• optimization of investments in various energy facilities to maximize income [13].

A wide range of optimization problems can be solved using the methods provided and based on the algorithm of the developed program. The studies performed by the authors prove that the Python language is useful for creating not only programs for optimizing the characteristics of power systems, but also for solving other problems [14]. The differential evolution method has many advantages when compared to other numerical methods and differs from them in universality.

In addition to the fact that the automation of calculations in solving the problem considered is necessary, its implementation should result in a significant increase in the efficiency of power plant operation. The program developed can be used by university students and power plant operators to study the basics of power plant operation, and as a visual illustration of how to use the most relevant functions of the Python language for performing calculations in the energy industry.

About the authors

V. Y. Ilichev

Kaluga Branch of Bauman Moscow State Technical University

Author for correspondence.
Email: patrol8@yandex.ru

PhD in Engineering

Russian Federation, Kaluga

E. A. Yurik

Kaluga Branch of Bauman Moscow State Technical University

Email: patrol8@yandex.ru

PhD in Engineering

Russian Federation, Kaluga

References

Supplementary files