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Many power plants export electricity to the public grid, delivering it to factories and homes (Fig. 18.1). We should optimize the distribution and scheduling of the load rate of each line to minimize the power loss of the transmission system. The amount of energy wasted in the power grid transmission process is very considerable, and many scientists, scholars, students and engineers are engaged in research in this field.
Many optimization methods require the establishment of a power grid model. However, since accurate models of actual power grids are difficult to establish, these optimization methods will quickly become difficult to apply in actual power grids given the complexity of the models and algorithms, the curse of dimensionality, and the computational time.
An approach that avoids these complications would be highly advantageous, especially for business and industry.
18.1 Energy Consumption in a Power Grid
There are n transmission lines to supply power in an area, the power lines in parallel. The total apparent power required is St, VA, the line voltage U of each transmission line is identical. The total apparent current is I0, the resistance and current of i-th line are Ri and Ii, respectively, the total energy consumption Wt of n transmission lines is expressed as
where \(\theta_{i}\) represents the load rate of the i-th transmission line, expressed as
If the resistance of each line is the same as R0, Wt of the n transmission lines is expressed as
18.2 Optimal Dispatching of Power Grids Composed of Lines with the Identical Resistance
Theorem:
If the resistance of each line is i R0, for the optimization problem of Wt, the minimization of the total energy consumption of the power grid.
is given by
The optimal point is
That is
Proof:
We begin our inductive proof by considering the case where n = 2.
The constraint condition then becomes
where
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, that is
We then check the second derivative,
Therefore, the optimal point is the unique minimum.
The minimal value of the total energy consumption of the power grid is
We then assume that this holds for n = k. The above conclusion is extended to the case of n = k, and the optimal point is then
That is
The minimal value of Wt is
Our inductive case is then given by n = k + 1. For the total energy consumption of the power grid we have
and the minimum of the first item is
where
The expression for Wt becomes
The optimal condition is given for
The solution is
and
Therefore, the optimal point is then
That is
The minimal value of total energy consumption of the power grid is
Load distribution theorem: In a power grid which consists of n main lines that have the identical resistance, the optimal control method is to keep each line to have the identical current I0/n.
18.3 Optimal Scheduling of a Power Grid with the Different Power Line
Theorem:
If the resistances of all power lines are different, respectively R1, R2, …, Rn, for the optimization problem of Wt, the minimization of the total energy consumption of the power grid.
is given by
The optimal point is
Proof:
We begin our inductive proof by considering the case where n = 2.
The constraint condition then becomes
where
The objective function Wt is expressed as
The optimal condition is given for
The solution is
Checking the second derivative, we see
Thus, the optimal point is the unique minimum.
The minimal value of the total energy consumption of the power grid 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 minimal value of Wt is
Our inductive case is then given by n = k + 1. For the total energy consumption of the power grid we have
and the minimum of the first item is
The expression for Wt becomes
The optimal condition is given for
Based on above conclusion of n = 2, the solution is
Therefore, the optimal point is then
and the minimal value of total energy consumption of the power grid is
18.4 Optimal Scheduling of a Direct Current Power Grid
Theorem:
If not all resistances of direct current power lines are identical, that is, R1, R2, …, Rn, respectively, are different, for the optimization problem of Wt, the minimization of the total energy consumption of the power grid.
is given by
The optimal point is
The Proof is Omitted.
18.5 Conclusion
By assuming a fixed value of the total apparent current, we propose an optimal control method that does not rely on an accurate model of the grid.
The proofs of optimal control and switching methods given in this chapter are mainly based on the minimum energy consumption rule. Therefore, this optimal method has the following characteristics:
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(1) Including lined and unlined systems,
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(2) No systematic mathematical model is required,
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(3) High versatility.
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Yao, F., Yao, Y. (2024). Optimal Dispatch of Power Grid Load. In: Efficient Energy-Saving Control and Optimization for Multi-Unit Systems. Springer, Singapore. https://doi.org/10.1007/978-981-97-4492-3_18
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DOI: https://doi.org/10.1007/978-981-97-4492-3_18
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