Due to the involvement of chemical reactions, fission reactions and other factors, the energy output by some systems is not all converted from the energy consumed by the system. Such systems can only maximize the output as the optimal goal of the overall energy efficiency of the system. For example, when electrolyzing water to produce hydrogen, you cannot say that the hydrogen energy produced is completely converted from the electrical energy of the electrolyzer, because the hydrogen in the water itself has a certain amount of energy. See Chap. 2.

In order to achieve the dual-carbon goal of mankind, all countries are vigorously develo** green energy, and wind power is one of them (Fig. 22.1).

Fig. 22.1
A photo of the wind farm. Four wind mills are are arranged vertically.

Wind farm

Full size image

Due to the large randomness of wind energy, it is difficult to dispatch wind power to the grid. In order not to waste the electric energy generated by wind power, people transmit the wind power to the hydrogen production station. There are multiple hydrogen production machines in the hydrogen production station, and the hydrogen production machines convert the electric energy into hydrogen energy and store it. For the identical wind power, arrange the operating number of hydrogen generators and the load of each hydrogen generator to produce the most hydrogen. This is a multi-machine energy efficiency optimization system.

22.1 Energy Efficiency Curve of Hydrogen Generator

Due to the large amount of power generated by the wind farm, the power input of the hydrogen generator is measured in MW. The energy efficiency curve of a hydrogen generator is shown in Fig. 22.2.

Fig. 22.2
A graph of eta versus the W. The solid line starts at around 0, move vertically, and then follow a horizontal direction.

Energy efficiency curve of hydrogen generator

Full size image

In Fig. 22.2, W is the electric energy input of the hydrogen generator, η is the energy efficiency of the hydrogen generator. Wm is the maximum load input of the device, ηe is the highest energy efficiency of the device, We is the optimal energy input when the device has the highest operating energy efficiency.

Below we refer to the hydrogen generator as device for short, and the hydrogen production station as system.

22.2 Optimal Load Distribution Theorem for Hydrogen Production System

Assuming that there are m devices in the hydrogen production station, the total input power is Wt, the electric energy input by the i-th device is Wi, and the overall efficiency of the system is ηt, then the overall energy efficiency expression of the system is

$$\begin{array}{c}{\eta }_{t}=\sum_{i=1}^{m}\frac{{W}_{i}}{{W}_{t}}{\eta }_{i}({W}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{m}{W}_{i}={W}_{t} >0\\ {W}_{imax}\ge {W}_{i}>0\end{array}\end{array}$$
(22.1)

Assuming that n is optimal, Wt is constant, consider the following three situations:

  1. (1)

    n = 2

The system has two variables W1 and W2

$$\begin{array}{c}{W}_{1}+{W}_{2}={W}_{t}\\ \begin{array}{c}{W}_{1}>0\\ {W}_{2}>0\end{array}\end{array}$$
(22.2)

The objective function can be expressed as

$${\eta }_{t}=\frac{{W}_{1}}{{W}_{t}}{\eta }_{1}({W}_{1})+\frac{{W}_{2}}{{W}_{t}}{\eta }_{2}({W}_{2})$$
(22.3)

The optimization condition is

$${\eta }_{t}{\prime}\left({W}_{1}\right)=0$$
(22.4)

According to known conditions

$${W}_{2}={W}_{t}-{W}_{1}$$
(22.5)

We have

$${{\left( {\eta _{1} \left( {W_{1} } \right) + W_{1} \eta _{1}^{\prime } (W_{1} ) - \eta _{2} \left( {W_{2} } \right) - W_{2} \eta _{2}^{\prime } \left( {W_{2} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\eta _{1} \left( {W_{1} } \right) + W_{1} \eta _{1}^{\prime } (W_{1} ) - \eta _{2} \left( {W_{2} } \right) - W_{2} \eta _{2}^{\prime } \left( {W_{2} } \right)} \right)} {W_{t} }}} \right. } {W_{t} }} = 0$$
(22.6)

If two devices are devices with the identical energy efficiency, then

$${\eta }_{2}\left(W\right)={\eta }_{1}(W)$$
(22.7)

and it is easy to see that

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{2}\\ {W}_{2}={W}_{1}=\frac{{W}_{t}}{2}\end{array}$$
(22.8)

is an optimization point.

The optimal control method is to keep

$${W}_{1}={W}_{2}=\frac{{W}_{t}}{n}$$
(22.9)

The maximum overall energy efficiency of system is

$${\text{max}}\eta _{{t2}} \left( {W_{t} } \right) = \eta _{1} \left( {\frac{{W_{t} }}{n}} \right)$$
(22.10)

If two devices are similar energy efficiency devices, then

$$\eta _{2} \left( {W_{2} } \right) = \eta _{1} \left( {\frac{{W_{1} }}{{\beta _{2} }}} \right)$$
(22.11)

and it is easy to see that

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{1+{\beta }_{2}}\\ {W}_{2}={\beta }_{2}{W}_{1}\end{array}$$
(22.12)

is an optimization point.

The optimal control method is to keep

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{1+{\beta }_{2}}\\ {W}_{2}={\beta }_{2}{W}_{1}\end{array}$$
(22.13)

The maximum value of the overall operating energy efficiency ηt2(Wt) of the system is

$$\text{max}{\eta }_{t2}\left({W}_{t}\right)={\eta }_{1}\left(\frac{{W}_{t}}{1+{\beta }_{2}}\right)$$
(22.14)

Since the shape of the overall efficiency curve of the system is the same as that of a single device, the second derivative is also less than zero. The overall energy efficiency is a maximum value.

  1. (2)

    n = 3

The system has three variables W1, W2 and W3

$$\begin{array}{c}{W}_{1}+{W}_{2}+{W}_{3}={W}_{t}\\ \begin{array}{c}\begin{array}{c}{W}_{1}>0\\ {W}_{2}>0\end{array}\\ {W}_{3}>0\end{array}\end{array}$$
(22.15)

ηt expression is

$${\eta }_{t}=\frac{{W}_{1}}{{W}_{t}}{\eta }_{1}\left({W}_{1}\right)+\frac{{W}_{2}}{{W}_{t}}{\eta }_{2}\left({W}_{2}\right)+\frac{{W}_{3}}{{W}_{t}}{\eta }_{3}({W}_{3})$$
(22.16)

If the three devices have the identical energy efficiency, assuming that W3 is the optimal point and fixed, W1 and W2 are variables, leading to

$${W}_{1}+{W}_{2}={W}_{t}-{W}_{3}$$
(22.17)

Based on the conclusion of n = 2, we have

$${W}_{2}={W}_{1}$$
(22.18)

Similarly, assuming that W2 is the optimal point and has been fixed, we have

$${W}_{3}={W}_{1}$$
(22.19)

Assuming that W1 is the optimal point and it has been fixed, we have

$${W}_{3}={W}_{2}$$
(22.20)

where

$${W}_{1}={W}_{2}={W}_{3}=\frac{{W}_{t}}{3}$$
(22.21)

is an optimization point.

The optimal control method is to keep

$${W}_{1}={W}_{2}={W}_{3}=\frac{{W}_{t}}{n}$$
(22.22)

The overall maximum operating energy efficiency ηt3(Wt) of the system is

$$\text{max}{\eta }_{t3}\left({W}_{t}\right)={\eta }_{1}(\frac{{W}_{t}}{3})$$
(22.23)

If the three devices have similar energy efficiency, assuming that W3 is the optimal point and fixed, based on the conclusion of n = 2, we have

$${W}_{2}={\beta }_{2}{W}_{1}$$
(22.24)

Similarly, assuming that W2 is the optimal point and has been fixed, we have

$${W}_{3}={\beta }_{3}{W}_{1}$$
(22.25)

where

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{1+{\beta }_{2}+{\beta }_{3}}\\ \begin{array}{c}{W}_{2}={\beta }_{2}{W}_{1}\\ {W}_{3}={\beta }_{3}{W}_{1}\end{array}\end{array}$$
(22.26)

is an optimization point.

The optimal control method is to keep

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{{\sum }_{i=1}^{3}{\beta }_{i}}\\ \begin{array}{c}{W}_{2}={\beta }_{2}{W}_{1}\\ {W}_{3}={\beta }_{3}{W}_{1}\end{array}\end{array}$$
(22.27)

The maximum value of the overall operating energy efficiency ηt3(Wt) of the system is

$$\text{max}{\eta }_{t3}\left({W}_{t}\right)={\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{3}{\beta }_{i}}\right)$$
(22.28)
  1. (3)

    n = k

If k devices are devices with the identical energy efficiency, the above conclusion is extended to the case of n = k, and the optimal point is

$${W}_{1}={W}_{2}=\dots ={W}_{k}=\frac{{W}_{t}}{k}$$
(22.29)

The optimal control method is to keep

$${W}_{1}={W}_{2}=\dots ={W}_{k}=\frac{{W}_{t}}{k}$$
(22.30)

The maximum value of the overall operating energy efficiency ηtk(Wt) of the system is

$$\text{max}{\eta }_{tk}\left({W}_{t}\right)={\eta }_{1}(\frac{{W}_{t}}{k})$$
(22.31)

If k devices are devices with similar energy efficiency, the above conclusion is extended to the case of n = k, and the optimal point is

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\\ \begin{array}{c}\begin{array}{c}{W}_{2}={\beta }_{2}{W}_{1}\\ \dots \end{array}\\ {W}_{k}={\beta }_{k}{W}_{1}\end{array}\end{array}$$
(22.32)

The optimal control method is to keep

$$\begin{array}{c}{W}_{1}=\frac{{W}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\\ \begin{array}{c}\begin{array}{c}{W}_{2}={\beta }_{2}{W}_{1}\\ \dots \end{array}\\ {W}_{k}={\beta }_{k}{W}_{1}\end{array}\end{array}$$
(22.33)

The maximum value of the overall operating energy efficiency ηtk(Wt) of the system is

$$\text{max}{\eta }_{tk}\left({W}_{t}\right)={\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right)$$
(22.34)

Whether it is device with the identical energy efficiency or device with similar energy efficiency, their optimal load distribution methods have one thing in common, that is, the energy efficiency of all operating device is identical, that is,

$${\eta }_{1}\left({W}_{1}\right)={\eta }_{2}\left({W}_{2}\right)=\dots ={\eta }_{n}\left({W}_{n}\right)$$
(22.35)

Optimal load distribution theorem (Yao Theorem 1): The optimal load distribution method is to keep the operating energy efficiency of each operating device equal.

$${\eta }_{1}\left({W}_{1}\right)={\eta }_{2}\left({W}_{2}\right)=\dots ={\eta }_{n}\left({W}_{n}\right)$$
(22.36)

22.3 Optimal Switching Theorem for Hydrogen Production System

The optimal methods of load distribution obtained above are all obtained under the assumption that n is already optimal, but is the n optimal? We analyze two cases.

The highest overall energy efficiency of system is

$$\text{max}{\eta }_{tn}\left({W}_{t}\right)={\eta }_{1}(\frac{{W}_{t}}{n})$$
(22.37)

For the same Wt, there are (n − 1) running devices with the identical energy efficiency, and the highest overall energy efficiency of system is

$$\text{max}{\eta }_{t(n-1)}\left({W}_{t}\right)={\eta }_{1}(\frac{{W}_{t}}{n-1})$$
(22.38)

For the same Wt, there are (n + 1) running devices with the identical energy efficiency, and the highest overall energy efficiency of the system is

$$ {\text{max}}\eta_{t(n + 1)} \left( {W_{t} } \right) = \eta_{1} \left( {\frac{{W_{t} }}{n + 1}} \right) $$
(22.39)

where

$$ \frac{{W_{t} }}{n - 1} > \frac{{W_{t} }}{n} > \frac{{W_{t} }}{n + 1} $$
(22.40)

On the η1( ) energy efficiency curve, the Wt/(n − 1) point is on the right side of the Wt/n point, and the Wt/(n + 1) point is on the left side of the Wt/n point.

If the following formula holds

$$ \eta_{1} \left( {\frac{{W_{t} }}{n}} \right) = max\left\{ {\eta_{1} \left( {\frac{{W_{t} }}{n - 1}} \right),\eta_{1} \left( {\frac{{W_{t} }}{n}} \right),\eta_{1} \left( {\frac{{W_{t} }}{n + 1}} \right)} \right\} $$
(22.41)

we say that n is truly optimal.

As shown in Fig. 22.3, η1(Wt/n) has the highest operating energy efficiency, and the Wt/n point is closer to the W1e point than Wt/(n − 1) and Wt/(n + 1),

Fig. 22.3
A line graph of eta versus the W. The line starts at around 0 and then follow a concave slope. The dashed lines follow a vertical decline.

Energy efficiency comparison curve

Full size image

In the above discussion, we have always regarded Wt as an invariable constant. In practical applications, Wt changes with the size of the air volume.

When Wt increases, η1(Wt/(n + 1)) also increases, then the following conditions are met

$${\eta }_{1}\left(\frac{{W}_{t}}{n+1}\right)={\eta }_{1}\left(\frac{{W}_{t}}{n}\right)$$
(22.42)

The switching point is reached, if Wt continues to increase, then

$${\eta }_{1}\left(\frac{{W}_{t}}{n+1}\right)>{\eta }_{1}\left(\frac{{W}_{t}}{n}\right)$$
(22.43)

It should be switched the number of running devices from n to n + 1.

Similarly, when Wt decreases, η1(Wt/(n-1)) increases, when the following conditions are met

$${\eta }_{1}\left(\frac{{W}_{t}}{n-1}\right)={\eta }_{1}\left(\frac{{W}_{t}}{n}\right)$$
(22.44)

The switching point is reached, if Wt continues to decrease, then

$${\eta }_{1}\left(\frac{{W}_{t}}{n-1}\right)>{\eta }_{1}\left(\frac{{W}_{t}}{n}\right)$$
(22.45)

should switch the number of running devices from n to n-1.

Due to the limitation of W1m, when Wt increases, Wt/n also increases until Wt/n = W1m, which is still not satisfied

$${\eta }_{1}\left(\frac{{W}_{t}}{n+1}\right)={\eta }_{1}\left(\frac{{W}_{t}}{n}\right)$$
(22.46)

If Wt/n continues to increase, the device will be overloaded, and it is necessary to force switching to (n + 1) devices at the point Wt/n = W1m.

The total value of the input energy is Wt, and there are n devices with similar energy efficiency running, and the maximum value of the overall operating energy efficiency ηtn(Wt) of system is

$$\text{max}{\eta }_{tn}\left({W}_{t}\right)={\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)$$
(22.47)

For the same Wt, there are k devices with similar energy efficiency operating, k is any feasible combination except n, and the highest overall operating energy efficiency of system is

$$\text{max}{\eta }_{tk}\left({W}_{t}\right)={\eta }_{1}(\frac{W}{{\sum }_{i=1}^{k}{\beta }_{i}})$$
(22.48)

If the number of running units n is optimal, it must satisfy

$${\eta }_{1}\left(\frac{{W}_{t}}{n{\sum }_{i=1}^{n}{\beta }_{i}}\right)=max\left\{{\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right),{\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right),\dots \right\}$$
(22.49)

When Wt increases, the condition is satisfied

$${\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)={\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{{k}_{1}}{\beta }_{i}}\right)$$
(22.50)

The switching point has been reached. If Wt continues to increase, it should switch the number of operating devices from n to k1.

When Wt decreases, the condition is satisfied

$${\eta }_{1}\left(\frac{W}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)={\eta }_{1}\left(\frac{{W}_{t}}{{\sum }_{i=1}^{{k}_{2}}{\beta }_{i}}\right)$$
(22.51)

The switching point has been reached. If Wt continues to decrease, it should switch the number of operating devices from n to k2.

k1 and k2 are any combination other than the optimal combination of n units this time, and also include other combinations of n units.

If there is no point of equal energy efficiency, the optimal switching point is at the maximum load point of the equipment.

Optimal switching theorem (Yao Theorem 2): The optimal switching point for the number of operating units is at the point of equal efficiency or at the maximum output point of the devices.

$${\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)={\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{{k}_{2}}{\beta }_{i}}\right) or {\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)={\eta }_{1m}$$
(22.52)

22.4 Engineering Optimal Versus Theoretical Optimal

In actual engineering, because the electrolytic cell needs several hours to start up, there is energy loss in preheating. Thus it can only achieve engineering optimization, not theoretical optimization.