Abstract
The electricity generated by the power plant is sent to the distribution station, and then transmitted to factories, mines and households through the public power grid. An enterprise power distribution station may also have many transformers, and the number of transformers used and power distribution are adjusted to transmit power to each workshop. The electricity consumed by power transmission and distribution is very considerable, and a large number of scientists, scholars, students and engineers are engaged in research in this field. This chapter will discuss this issue.
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The electricity generated by the power plant is sent to the distribution station, and then transmitted to factories, mines and households through the public power grid. An enterprise power distribution station may also have many transformers, and the number of transformers used and power distribution are adjusted to transmit power to each workshop (Fig. 17.1). The electricity consumed by power transmission and distribution is very considerable, and a large number of scientists, scholars, students and engineers are engaged in research in this field. This chapter will discuss this issue.
17.1 Energy Relations for Multiple Transformers in a Distribution Station
Assume that there are n transformers in a distribution station to supply power to all devices in a factory area, the total power output of the station is P0, assuming that P0 is a fixed value, the power output by the i-th transformer is Pi, and all operating transformers consume the total electric energy is Pt, Pt is the total input of the station, and the expression of Pt is
The unit of measurement for electric energy is watts.
Assume the energy efficiency curve of the i-th transformer is shown in Fig. 17.2.
In Fig. 17.2, Pim is the maximum power output of the i-th transformer, ηie is the maximum energy efficiency of the i-th transformer, Pie is the power output when the i-th transformer has the maximum energy efficiency, and ηi(P) has the following characteristics:
Assume the energy efficiency functions η1(P) and ηi(P) of the first and the i-th transformer as shown in Fig. 17.3.
If the following formula holds:
where βi is a constant, we say that the i-th transformer and the first transformer are the transformers with similar energy efficiency, referred to as “similar energy efficiency transformer”, and β1 = 1.
The transformer with similar energy efficiency has the following characteristics:
17.2 Optimal Scheduling of Multiple Transformers in a Grid
-
1.
If the n transformers are identical, the energy expression of the power supply becomes
$$ P_{t} = P_{0} \sum\limits_{i = 1}^{n} {\frac{{\theta_{i} }}{{\eta (\theta_{i} P_{0} )}}} $$(17.5)
where
Consider the minimization problem of total power consumption
This problem can also be written as
We consider three cases:
-
(1)
n = 2
The system has two variables and has
The objective function Pt can be expressed as
The optimization condition is
and it is easy to see that
is an optimization point. Then we have
That is, the optimal control method is to keep
The total power dissipation is
Since the shape of the overall efficiency curve of the distribution station is the same as that of a single transformer, so the second derivative of the Pt is also greater than zero
Pt is the only minimum value.
The overall energy efficiency ηt of the distribution station is the only maximum value.
-
(2)
n = 3
The system has three variables, based on known conditions, we have
The Pt expression becomes
Assuming that θ3 is fixed and an optimization point, only θ1 and θ2 are variables, we have
Based on the conclusion of n = 2 above, we have
to be the optimal point.
Assuming that θ2 is fixed and is an optimization point, only θ1 and θ3 are variables, there are
According to the conclusion of n = 2 above, we have
to be the optimal point.
Similarly, assuming thatθ1 is fixed and is an optimization point, only θ2 and θ3 are variables, we have
According to the conclusion of n = 2 above, we have
to be the optimal point.
So, we have the optimal point
That is, the optimal control method is to keep
The minimum total power dissipation is
The maximum overall energy efficiency ηt is
-
(3)
n = k
The system has n variables, and the above conclusion can be extended to the case of n = k, the optimal point is
That is, the optimal control method is to keep
The minimum total power dissipation is
The maximum overall energy efficiency ηt is
Conclusion: When there are n sets of identical transformers supplying power to a factory area on the same power grid, and the load allocated to each transformer is the same, it is the optimal dispatching scheme.
-
2.
If the n transformers are the energy efficiency similarity transformers, the energy expression of the power supply becomes
where
Consider the minimization problem of total power consumption
This problem can also be written as
We consider three cases:
-
(1)
n = 2
The system has two variables and has
The objective function Pt can be expressed as
The optimization condition is
and it is easy to see that
for the optimization point. Then we have
That is, the optimal control method is to keep
The total power dissipation is
Since the shape of the overall efficiency curve of the distribution station is the same as that of a single transformer, so the second derivative of the Pt is also greater than zero
Pt is the only minimum value.
The overall energy efficiency ηt of the distribution station is the only maximum value.
-
(2)
n = 3
The system has three variables, based on known conditions, we have
The Pt expression becomes
Assuming that θ3 is fixed and is an optimization point, only θ1 and θ2 are variables, there are
Based on the conclusion of n = 2 above, we have
to be is the optimal point.
Assuming that θ2 is fixed and is an optimization point, only θ1 and θ3 are variables, there are
According to the conclusion of n = 2 above, there are
is the optimal point.
Similarly, assuming that θ1 is fixed and is an optimization point, only θ2 andθ3 are variables, we have
According to the conclusion of n = 2 above, we have
to be the optimal point.
So, we have the optimal point, and the optimal control method is to keep
The minimum total power consumption is
The maximum overall energy efficiency ηt is
-
(3)
n = k
The system has n variables, the above conclusion can be extended to the case of n = k, the optimal point and the optimal control method is to keep
The minimum total power dissipation is
The maximum overall energy efficiency ηt is
17.3 Optimum Number of Transformers Operating in a Distribution Station
-
1.
A distribution station has m identical transformers, if the following equations are satisfied
Then n is the number of transformers with optimal operation.
-
2.
A distribution station has m energy efficiency similarity transformers, if the following equations are satisfied
n1 is any combination other than the optimal combination of n units this time, and also include other combinations of n units. The number n is the number of transformers with optimal operation.
17.4 Optimal Switching Rules for Multiple Transformers in Distribution Stations
-
1.
A distribution station has m identical transformers, and the number n is the number of transformers currently in optimal operation. If P0 increases to P01, the following relation holds:
Then P01 is the optimal switching point between the operation of n transformers and the operation of n + 1 transformer. When the total required power P0 is greater than P01, the optimal number of transformers in operation is switched from n to n + 1. If P0 increases until P0/n = P1m, there isn’t the point P01, then P0/n = P1m is the switching point from n to n + 1.
If P0 is reduced to P02, the following relation is established
Then P02 is the optimal switching point between the operation of n transformers and the operation of n−1 transformers. When the total required power P0 is less than P02, the optimal number of transformers in operation is switched from n to n−1. If P0 reduces until P0/(n−1) = P1m, there isn’t the point P02, then P0/(n−1) = P1m is the switching point from n to n−1.
The analysis process is shown in Fig. 17.4.
-
2.
A distribution station has m energy efficiency similarity transformers, and the number n is the number of transformers currently in optimal operation. If P0 increases to P01, the following relation holds:
Then P01 is the optimal switching point between the operation of n transformers and the operation of k1 transformers. When the total required power P0 is greater than P01, the optimal number of transformers in operation is switched from n to k1. If P0 increases until P0/\({\sum }_{l=1}^{n}{\beta }_{l}\)=P1m, there isn’t the point P01, then P0/\({\sum }_{l=1}^{n}{\beta }_{l}\)=P1m is the switching point from n to k1.
If P0 is reduced to P02, the following relation is established
Then P02 is the optimal switching point between the operation of n transformers and the operation of k2 transformers. When the total required power P0 is less than P02, the optimal number of transformers in operation is switched from n to k2. If P0 reduces until P0/\({\sum }_{l=1}^{k2}{\beta }_{l}\) =P1m, there isn’t the point P02, then P0/\({\sum }_{l=1}^{k2}{\beta }_{l}\) =P1m is the switching point from n to k2.
The analysis process is shown in Fig. 17.5.
k1 and k2 are any combination other than the optimal combination of n units this time, and also include other combinations of n units. Point \({P}_{0}/{\sum }_{l=1}^{k1}{\beta }_{l}\) is the point closest to \({P}_{0}/{\sum }_{l=1}^{n}{\beta }_{l}\) to the left of point \({P}_{0}/{\sum }_{l=1}^{n}{\beta }_{l}\). Point \({P}_{0}/{\sum }_{l=1}^{k2}{\beta }_{l}\) is the point closest to \({P}_{0}/{\sum }_{l=1}^{n}{\beta }_{l}\) to the right of point \({P}_{0}/{\sum }_{l=1}^{n}{\beta }_{l}\). If all transformers are identical, we have k1 = n + 1 and k2 = n−1.
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Yao, F., Yao, Y. (2024). Optimal Control and Scheduling of Distribution Stations. In: Efficient Energy-Saving Control and Optimization for Multi-Unit Systems. Springer, Singapore. https://doi.org/10.1007/978-981-97-4492-3_17
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DOI: https://doi.org/10.1007/978-981-97-4492-3_17
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