Abstract
There are many power stations in the world with different generators. These include thermal power stations, hydropower stations, etc. They convert the energy of raw materials into electrical energy. In these power stations, with the same input of raw materials, there is maximum power generation. To achieve the maximum, we should decide how many generators to use and how many tasks should be assigned to each generator.
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There are many power stations in the world with different generators. These include thermal power stations, hydropower stations, etc (Fig. 16.1). They convert the energy of raw materials into electrical energy. In these power stations, with the same input of raw materials, there is maximum power generation. To achieve the maximum, we should decide how many generators to use and how many tasks should be assigned to each generator.
16.1 Power Generation in ESPS
A generalized generator must reach the rated speed n0 before it can be connected to the grid. Taking the input amount of raw materials as a variable, such as the amount of coal input in a thermal power station or the water flow in a hydropower station, the efficiency of a generalized generator is shown in Fig. 16.2.
In Fig. 16.2, L is the adjustable input of the generator, η is the efficiency function of the generator, L0 is L at zero load of the generator, ηe is the maximum efficiency corresponding to Le, Le is the optimal input, and Lm is the maximum L.
We define the load factor γ as
We define ηN(γ) to be the normalized efficiency function. The normalized efficiency function ηN(γ) has a shape shown in Fig. 16.3.
In Fig. 16.3, γ is the load rate and a variable, ηN(γ) is the normalized efficiency. ηN(γ) and η(L) have the following relationship
If the normalized efficiency functions of two generators are identical, then
We define them as efficiency similarity generators. The power station which includes the efficiency similarity generators is called the efficiency similarity power station (ESPS).
The electricity energy generated by an ESPS is expressed as
where n is the total number of generators, W0 is the ideal energy input from raw materials, Wt is the total power generation of the power station, Wi is the power generation of the i-th generator, Lt is the total input of the power station, η0 represents the total power generation efficiency, ηi(Li) represents the operating efficiency of the i-th generator, ηN(\(\gamma_{i}\)) represents the normalized efficiency of the i-th generator, Li represents the adjustable input of the i-th generator, and Lie represents the maximum input of the i-th generator. Optimal input, Li0 represents L when the i-th generator is zero load, γi represents the load rate of the i-th generator, there is
16.2 Optimal Control in an ESPS
Load distribution theorem: For the optimization problem of Wt at the fixed Lt, the maximization of the total electricity energy output Wt of the ESPS
is given by
That is the maximization of the overall efficiency η0 of the power station
is given by
Proof
We begin our inductive proof by considering the case where n = 2.
The constraint condition then becomes
where
That is
The objective function Wt is expressed as
The optimal condition is given for
We have
It is then easily verified that
is an optimal point.
We then check the second derivative,
So, the optimal point is only maximum.
The maximal value of the total electricity energy output Wt of the power station is
The maximal value of the overall efficiency η0 of the power station is
We then assume that this holds for n = k. The above conclusion is readily extended to the case of n = k, and the optimal point is then
The maximal value of Wt is
The maximal value of the overall efficiency η0 of the power station is
Our inductive case is then given by n = k + 1. For the total electricity energy output Wt we have
and the maximum of the first item is
where
The expression for Wt becomes
The optimal condition is given for
We have
It is then easily verified that
is an optimal point.
That is
and
Therefore, the optimal point is then
and the maximal value of the total electricity energy output Wt of the power station is
The maximal value of the overall efficiency η0 of the power station is
Load distribution theorem: In an ESPS which consists of n generators that do not have all the identical model, the optimal control method is to keep each generator to have the same load rate.
16.3 Optimization Discriminant of the ESPS
We define \(\gamma \left( {L_{t} ,n} \right)\) as
In an ESPS, suppose that it has M-unit generators in total, if n is the optimal, as shown in Fig. 16.4, there must be
Namely
In Fig. 16.4, n1 and n2 are arbitrary number which corresponds to different device combinations. n1 is less than or equal to M, and n2 is less than or equal to M also.
Note: Even though the same n1, there are many different generator combinations.
16.4 Optimal Switch in the ESPS
Now we consider the optimal switch point for an ESPS. Then we take n to be less than or equal to M and the optimum, the total electricity energy output Wt of the ESPS is the maximum.
The optimal overall efficiency is then
If the Lt varies, the optimal n may change as well. The optimal switching point depends on whether Lt is increasing or decreasing.
Theorem
The optimal switch point is at ηN(Lt, n) = ηN(Lt, n2), if Lt is increasing.
Proof
From Figs. 16.5 and 16.6, the ηN(Lt, n1) is less than ηN(Lt, n) and greater than other efficiency values on the right side of γ = 1,the ηN(Lt, n2) is less than ηN(Lt, n) and greater than other efficiency values on the left side of γ = 1, we see that
As seen in Fig. 16.5, if γ (Lt, n) < 1 and Lt increases, then ηN(Lt, n) will continue to increase until γ(Lt, n) reaches 1, immediately after which point it will begin to decrease. At the same time, ηN(Lt, n2) will increase until it will eventually become greater than ηN(Lt, n) and will be the new maximum. This change of the maximum efficiency defines the optimal switch point at ηN(Lt, n) = ηN(Lt, n2).
If γ (Lt, n) > 1, then ηN(Lt, n) will decrease with the increase of Lt while ηN(Lt, n2) will increase as can be inferred from Fig. 16.6. Ultimately, ηN(Lt, n2) will increase to such a point that it will become the new maximum and ηN(Lt, n) = ηN(Lt, n2) is the optimal switch point.
If γ (Lt, n) > 1, then ηN(Lt, n) will decrease as Lt increases, while ηN(Lt, n2) will increase, as shown in Fig. 16.6. Eventually, ηN(Lt, n2) will increase to the point where it becomes the new maximum, and ηN(Lt, n) = ηN(Lt, n2) is the optimal switching point.
Theorem
The optimal switch point for an M-unit system is at ηN(Lt, n) = ηN(Lt, n1) if Lt is decreasing.
Proof
Similarly, if Lt decreases and γ(Lt, n) > 1, Fig. 16.6 shows that ηN(Lt, n) will increase until γ(Lt, n) reaches the value 1, at which point it will immediately start decreasing. At the same time, ηN(Lt, n1) will increase. As before, there is a point where ηN(Lt, n1) exceeds ηN(Lt, n) as a new maximum. This is our optimal switching point, given by: ηN(Lt, n) = ηN(Lt, n1).
As shown in Fig. 16.5, for a given γ(Lt, n) < 1, ηN(Lt, n) will decrease as Lt decreases. We also find that ηN(Lt, n1) will increase to the point of maximum efficiency. Then the optimal switching point is ηN(Lt, n) = ηN(Lt, n1).
16.5 ESPS Operates at Maximum Efficiency
For the total Lt, if the optimal run-unit of the ESPS is n as described above, the maximal value of the overall efficiency η0 of the power station is
If the following equation holds
Then
Thus, the ESPS can work at the maximum efficiency ηe.
16.6 Conclusion
By assuming a fixed value of the total load, we propose an optimal control method that does not rely on an accurate model of the ESPS. By varying the total load, we also propose an optimal switching method. This method depends only on the efficiency function of the generator.
The proofs of optimal control and switching methods given in this chapter are mainly based on the characteristics of the efficiency function, which can be approximately considered as a concave and non-negative function. Therefore, this optimal method has the following characteristics:
-
(1)
High versatility,
-
(2)
Includes linear and nonlinear systems,
-
(3)
No mathematical model of the system is required.
Acknowledgements
In order to solve the optimization problem of general equipment, the author has conducted long-term research. We would like to thank Zhang Yanfang from Hebei Automation Company and Yao Bosheng from Beijing IAO Technology Development Company for their valuable help and suggestions during the development and experiment of this theory.
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Yao, F., Yao, Y. (2024). Efficiency Optimization of Power Stations with Different Generating Units. In: Efficient Energy-Saving Control and Optimization for Multi-Unit Systems. Springer, Singapore. https://doi.org/10.1007/978-981-97-4492-3_16
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DOI: https://doi.org/10.1007/978-981-97-4492-3_16
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