Energy Efficiency Optimization of Multi-Unit System

Abstract

Multi-unit systems refer to those systems that are composed of more than one device to complete the same task.

14.1 What is a Multi-Unit System?

Multi-unit systems refer to those systems that are composed of more than one device to complete the same task.

For example, a power plant composed of multiple generators, a pum** station composed of multiple water pumps, a high-speed train driven by multiple motors, a hydrogen production station composed of multiple hydrogen generators, an LNG station composed of multiple vaporizers, and multiple transformers composed of substations, and so on.

Only a system with multiple energy-consuming devices that work together to complete a task is called a multi-unit system. Multi-unit systems include numerous systems: power stations, transmission and distribution stations, energy-consuming systems, etc.

14.2 The Essence of Multi-Unit System Optimization

If you have on-site operating data and equipment data, can you complete the following work?

• (1) With the same input of water volume and water pressure, can you still increase the power generation of the largest hydropower station in the world? How much? By what method? What is the basis?

• (2) When transporting the same amount of water and the same water pressure, can you save more electricity for the world's largest water diversion pump station? How much can you save? By what method? What is the basis?

• (3) Under the same wind volume and wind speed, can the world's largest wind power hydrogen production station still increase the hydrogen production capacity? How much? By what method? What is the basis?

There are many such examples. It can be said that as long as there is more than one power generation or power consumption equipment in a system, there will be such a problem. This is the problem of energy efficiency optimization.

The so-called energy efficiency optimization is to complete the same task with the least energy. It can also be said: input the same primary energy and produce the most secondary energy.

Obviously, this is not a simple job.

Can you do this confidently?

Can you accurately predict optimal overall energy efficiency?

How much is it?

This chapter is intended to address these issues.

Energy efficiency optimization of multi-unit system mainly solves three problems:

• 1. How many devices are optimal?

• 2. What is the optimal output of each device?

• 3. What is the maximum operating energy efficiency of the system?

The first two are methods, the last one is the result.

14.3 Energy Efficiency Optimization of Multi-Unit System

A system C that converts energy A into energy B has m devices in total, and n devices are running. The total amount of energy A consumed by system C is P_t, and the total amount of energy B produced is W_t. The i-th device produces energy B is W_i, the energy A consumed by the i-th device is P_i, and the energy efficiency optimization of system C has two expressions.

For a fixed total amount of energy-A input P_t, the maximization of the total amount of output energy B W_t is expressed as:

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{n}{P}_{i}{\eta }_{Pi}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{n}{P}_{i}={P}_{t} \\ {P}_{im}\ge {P}_{i}\ge 0\end{array}\end{array}$$

(14.1)

For a fixed total energy B output W_t, the minimization of the input energy A total P_t can be expressed as:

$$\begin{array}{c}{P}_{t}=min\sum_{i=1}^{n}\frac{{W}_{i}}{{\eta }_{Wi}({W}_{i})} \\ \begin{array}{c}s.t. \sum_{i=1}^{n}{W}_{i}={W}_{t} \\ {W}_{im}\ge {W}_{i}\ge 0\end{array}\end{array}$$

(14.2)

Energy efficiency functions η_Pi (P_i) and η_Wi (W_i) are not the same function, their variables are different, and their function values are also different.

The Eq. (14.1) and Eq. (14.2) are two different manifestations of the same optimization problem, and both are running energy efficiency optimization problems. Only one of them is solved, the problem is solved. We choose to solve the optimal result of Eq. (14.1).

It can be seen from Eq. (14.1) that since the energy efficiency function is nonlinear and has a limit on the maximum input value P_im, this is a nonlinear optimization problem with constraints. Since the number n of optimized operating units is an integer, the optimal load distribution value P_i is a real number, which is another integer-real-number mixed optimization problem.

We call the method to solve this optimization problem the quantum optimization method.

14.4 Energy Efficiency Function

The energy efficiency function has a similar shape and some of the same characteristics. Let's take η_Pi(P) as an example, for the convenience of writing, we use η_i(P) instead of η_Pi(P). The energy efficiency curve of η_i(P) is shown in Fig. 14.1.

Fig. 14.1

Energy efficiency curve η_i(P)

In Fig. 14.1, P_im is the maximum A energy input of the i-th device, η_ie is the highest operating energy efficiency of the i-th device, P_ie is the A energy input when the i-th device has the highest operating energy efficiency, and η_i(P) has the following characteristics:

$$\begin{array}{c}0\le \text{P}\le {P}_{im} \\ {\eta }_{ie}={\eta }_{i}\left({P}_{ie}\right)\\ \begin{array}{c}{0\le \eta }_{i}\left(P\right)\le {\eta }_{ie}\\ \begin{array}{c}{\eta }_{i}\left(0\right)=0\\ {\eta }_{i}^{{\prime}{\prime}}(P)<0\end{array}\end{array}\end{array}$$

(14.3)

14.5 Similar Energy Efficiency Device

Assume the energy efficiency curves η₁(P) and η_i(P) of the first and i-th device, as shown in Fig. 14.2.

Fig. 14.2

Energy efficiency curves η₁(P) and η_i(P)

If the following equation holds:

$${\eta }_{i}\left({P}_{i}\right)={\eta }_{1}\left(\frac{{P}_{1}}{{\beta }_{i}}\right)$$

(14.4)

where β_i is a constant, we say that the i-th device and the first device are devices with similar energy efficiency, referred to as “similar energy efficiency device”.

β_i < 1, the i-th device is a device with a smaller output capability than the first device;

β_i = 1, then the i-th device and the first device are the same device, recorded as β₁ = 1;

β_i > 1, the i-th device has a larger output capability than the first device.

Device with similar energy efficiency has the following characteristics:

$$\begin{array}{c}{\eta'_i}\left({P}_{i}\right)=\frac{{\eta }_{1}{\prime}({P}_{1})}{{\beta }_{i}}\\ \begin{array}{c}{P}_{ie}={\beta }_{i}{P}_{1e}\\ {\beta }_{i}=\frac{{P}_{ie}}{{P}_{1e}}\end{array}\end{array}$$

(14.5)

14.6 Optimal Load Distribution Theorem of Multi-Unit System, Yao Theorem 1

Assume that Pi is greater than 0, that is, every operating device does work. The Eq. (14.1) is simplified as

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{n}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{n}{P}_{i}={P}_{t} >0\\ {P}_{imax}\ge {P}_{i}>0\end{array}\end{array}$$

(14.6)

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

• (1) n = 2

System C has two variables P₁ and P₂

$$\begin{array}{c}{P}_{1}+{P}_{2}={P}_{t}\\ \begin{array}{c}{P}_{1}>0\\ {P}_{2}>0\end{array}\end{array}$$

(14.7)

The objective function can be expressed as

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

(14.8)

The optimization condition is

$${W'_t}\left({P}_{1}\right)=0$$

(14.9)

According to known conditions

$${P}_{2}={P}_{t}-{P}_{1}$$

(14.10)

We have

$${\eta }_{1}\left({P}_{1}\right)+{P}_{1}{\eta'_1}({P}_{1})-{\eta }_{2}\left({P}_{2}\right)-{P}_{2}{\eta'_2}\left({P}_{2}\right)=0$$

(14.11)

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

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

(14.12)

easy to see

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{2}\\ {P}_{2}={P}_{1}=\frac{{P}_{t}}{2}\end{array}$$

(14.13)

is an optimization point.

The optimal control method is to keep

$${P}_{1}={P}_{2}=\frac{{P}_{t}}{n}$$

(14.14)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{1}(\frac{{P}_{t}}{n})$$

(14.15)

The maximum value of the overall operating energy efficiency η_t2(P_t) of system C is

$$\text{max}{\eta }_{t2}\left({P}_{t}\right)={\eta }_{1}(\frac{{P}_{t}}{2})$$

(14.16)

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

$${W}_{t}^{{\prime}{\prime}}\left({P}_{1}\right)<0$$

(14.17)

W_t has the only maximum value, and the overall energy efficiency has the only maximum value also.

If two devices are similar energy efficiency devices, then we have

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

(14.18)

and it is easy to see that

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{1+{\beta }_{2}}\\ {P}_{2}={\beta }_{2}{P}_{1}\end{array}$$

(14.19)

is an optimization point.

The optimal control method is to keep

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{1+{\beta }_{2}}\\ {P}_{2}={\beta }_{2}{P}_{1}\end{array}$$

(14.20)

The maximum total output energy W_t of system C is

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

(14.21)

The maximum value of the overall operating energy efficiency η_t2(P_t) of system C is

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

(14.22)

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. W_t has a maximum value, and the overall energy efficiency is a maximum value also.

• (2) n = 3

The system has three variables P₁, P₂ and P₃

$$\begin{array}{c}{P}_{1}+{P}_{2}+{P}_{3}={P}_{t}\\ \begin{array}{c}\begin{array}{c}{P}_{1}>0\\ {P}_{2}>0\end{array}\\ {P}_{3}>0\end{array}\end{array}$$

(14.23)

W_t expression is

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

(14.24)

If the three devices have the identical energy efficiency, using the commutative law of addition, assuming that P₃ is the optimal point and fixed, P₁ and P₂ are variables, there is

$${P}_{1}+{P}_{2}={P}_{t}-{P}_{3}$$

(14.25)

Based on the conclusion of n = 2, there are

$${P}_{2}={P}_{1}$$

(14.26)

Similarly, assuming that P₂ is the optimal point and has been fixed, there is

$${P}_{3}={P}_{1}$$

(14.27)

Assuming that P₁ is the optimal point and it has been fixed, we have

$${P}_{3}={P}_{2}$$

(14.28)

where

$${P}_{1}={P}_{2}={P}_{3}=\frac{{P}_{t}}{3}$$

(14.29)

is an optimization point.

The optimal control method is to keep

$${P}_{1}={P}_{2}={P}_{3}=\frac{{P}_{t}}{n}$$

(14.30)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{1}(\frac{{P}_{t}}{n})$$

(14.31)

The overall maximum operating energy efficiency η_t3(P_t) of system C is

$$\text{max}{\eta }_{t3}\left({P}_{t}\right)={\eta }_{1}(\frac{{P}_{t}}{n})$$

(14.32)

If the three devices have similar energy efficiency, using the commutative law of addition, assuming that P₃ is the optimal point and fixed, based on the conclusion of n = 2, we have

$${P}_{2}={\beta }_{2}{P}_{1}$$

(14.33)

Similarly, assuming that P₂ is the optimal point and has been fixed, there is

$${P}_{3}={\beta }_{3}{P}_{1}$$

(14.34)

where

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{1+{\beta }_{2}+{\beta }_{3}}\\ \begin{array}{c}{P}_{2}={\beta }_{2}{P}_{1}\\ {P}_{3}={\beta }_{3}{P}_{1}\end{array}\end{array}$$

(14.35)

is an optimization point.

The optimal control method is to keep

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{{\sum }_{i=1}^{3}{\beta }_{i}}\\ \begin{array}{c}{P}_{2}={\beta }_{2}{P}_{1}\\ {P}_{3}={\beta }_{3}{P}_{1}\end{array}\end{array}$$

(14.36)

The maximum total output energy W_t of the system is

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

(14.37)

The maximum value of the overall operating energy efficiency η_t3(P_t) of system C is

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

(14.38)

• (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

$${P}_{1}={P}_{2}=\dots ={P}_{k}=\frac{{P}_{t}}{k}$$

(14.39)

The optimal control method is to keep

$${P}_{1}={P}_{2}=\dots ={P}_{k}=\frac{{P}_{t}}{k}$$

(14.40)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{1}(\frac{{P}_{t}}{k})$$

(14.41)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

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

(14.42)

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

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

(14.43)

is an optimization point.

The optimal control method is to keep

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

(14.44)

The maximum total output energy W_t of system C is

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

(14.45)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

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

(14.46)

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 the same.

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

(14.47)

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

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

(14.48)

If all devices have the identical energy efficiency, the optimal load distribution method is to keep the load of every operating device equal.

$${P}_{1}={P}_{2}=\dots ={P}_{n}$$

(14.49)

Using the inductive method, we can still draw the above conclusion

• (1) n = 2

Same as above. (omitted).

• (2) If n = k holds, prove that n = k + 1 still holds.

For system C composed of n devices with the identical energy efficiency function, if the optimization conclusion holds,

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{k}{P}_{i}\eta ({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{k}{P}_{i}={P}_{t} >0\\ {P}_{imax}\ge {P}_{i}>0\end{array}\end{array}$$

(14.50)

The optimal point is

$${P}_{1}={P}_{2}=\dots ={P}_{k}=\frac{{P}_{t}}{k}$$

(14.51)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}\eta (\frac{{P}_{t}}{k})$$

(14.52)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

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

(14.53)

For n = k + 1, we have

$$\begin{aligned}{W}_{t}&=\sum_{i=1}^{k+1}{P}_{i}\eta ({P}_{i})=\sum_{i=1}^{k}{P}_{i}\eta ({P}_{i})+{P}_{k+1}\eta \left({P}_{k+1}\right)\\&=({P}_{t}-{P}_{k+1})\eta (\frac{{P}_{t}-{P}_{k+1}}{k})+{P}_{k+1}\eta \left({P}_{k+1}\right)\end{aligned}$$

(14.54)

The optimization condition is

$${W'_t}\left({P}_{k+1}\right)=0$$

(14.55)

We have

$$\begin{aligned}&-\eta \left(\frac{{P}_{t}-{P}_{k+1}}{k}\right)-\frac{{P}_{t}-{P}_{k+1}}{k}{\eta }{\prime}\left(\frac{{P}_{t}-{P}_{k+1}}{k}\right)\\&+\eta \left({P}_{k+1}\right)+{P}_{k+1}{\eta }{\prime}\left({P}_{k+1}\right)=0\end{aligned}$$

(14.56)

It is easy to see that, the optimal point is

$$\begin{array}{c}\begin{array}{c}{P}_{k+1}=\frac{{P}_{t}-{P}_{k+1}}{k}\\ {P}_{k+1}=\frac{{P}_{t}}{k+1}\end{array}\\ {P}_{1}={P}_{2}=\dots {P}_{k}=\frac{{P}_{t}-{P}_{k+1}}{k}=\frac{{P}_{t}}{k+1}\end{array}$$

(14.57)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}\eta (\frac{{P}_{t}}{k+1})$$

(14.58)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

$$\text{max}{\eta }_{t(k+1)}\left({P}_{t}\right)=\eta (\frac{{P}_{t}}{k+1})$$

(14.59)

The above conclusion still works.

For system C composed of n devices with the similar energy efficiency function, if the optimization conclusion holds,

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{k}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{k}{P}_{i}={P}_{t} >0\\ {P}_{imax}\ge {P}_{i}>0\end{array}\end{array}$$

(14.60)

The optimal point is

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

(14.61)

The maximum total output energy W_t of system C is

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

(14.62)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

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

(14.63)

For n = k + 1, we have

$${W}_{t}=\sum_{i=1}^{k+1}{P}_{i}{\eta }_{i}\left({P}_{i}\right)=\left({P}_{t}-{P}_{k+1}\right){\eta }_{1}\left(\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right)+{P}_{k+1}{\eta }_{k+1}\left({P}_{k+1}\right)$$

$$=({P}_{t}-{P}_{k+1}){\eta }_{1}(\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}})+{P}_{k+1}{\eta }_{1}\left(\frac{{P}_{k+1}}{{\beta }_{k+1}}\right)$$

(14.64)

The optimization condition is

$${W'_t}\left({P}_{k+1}\right)=0$$

(14.65)

We have

$$\begin{aligned}&-{\eta }_{1}\left(\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right)-\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}}{\eta }_{1}{\prime}\left(\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right)\\&+{\eta }_{1}\left(\frac{{P}_{k+1}}{{\beta }_{k+1}}\right)+\frac{{P}_{k+1}}{{\beta }_{k+1}}{\eta }{\prime}\left(\frac{{P}_{k+1}}{{\beta }_{k+1}}\right)=0\end{aligned}$$

(14.66)

It is easy to see that, the optimal point is

$$\begin{array}{c}\begin{array}{c}\frac{{P}_{k+1}}{{\beta }_{k+1}}=\frac{{P}_{t}-{P}_{k+1}}{{\sum }_{i=1}^{k}{\beta }_{i}}\\ {P}_{k+1}=\frac{{\beta }_{k+1}{P}_{t}}{{\sum }_{i=1}^{k+1}{\beta }_{i}}\end{array}\\ {P}_{1}=\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{\beta }_{i}}\end{array}$$

(14.67)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{1}(\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{\beta }_{i}})$$

(14.68)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

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

(14.69)

The above conclusion works very well.

14.7 Optimal Switching Theorem for Multi-Unit System, Yao Theorem 2

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

The total value of input energy A of system C is P_t, and there are n running devices with the identical energy efficiency, the highest overall energy efficiency of system C is

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

(14.70)

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

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

(14.71)

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

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

(14.72)

apparently

$$\frac{{P}_{t}}{n-1}>\frac{{P}_{t}}{n}>\frac{{P}_{t}}{n+1}$$

(14.73)

On the η₁(P₁) energy efficiency curve, the P_t/(n–1) point is on the right side of the P_t/n point, and the P_t/(n + 1) point is on the left side of the P_t/n point.

$$\begin{array}{c}{\eta }_{1}\left(\frac{{P}_{t}}{n}\right)\ge {\eta }_{1}\left(\frac{{P}_{t}}{n-1}\right)\\ {\eta }_{1}\left(\frac{{P}_{t}}{n}\right)\ge {\eta }_{1}\left(\frac{{P}_{t}}{n+1}\right)\end{array}$$

(14.74)

That is

$${\eta }_{1}\left(\frac{{P}_{t}}{n}\right)=max\left\{{\eta }_{1}\left(\frac{{P}_{t}}{n-1}\right),{\eta }_{1}\left(\frac{{P}_{t}}{n}\right),{\eta }_{1}\left(\frac{{P}_{t}}{n+1}\right)\right\}$$

(14.75)

Then the running number n is truly optimal.

As shown in Fig. 14.3, η₁(P_t/n) has the highest operating energy efficiency, and the P_t/n point is closer to the P_1e point than P_t/(n−1) and P_t/(n + 1),

Fig. 14.3

Energy efficiency comparison curve of the multi-unit system with the identical devices

In the above discussion, we have always regarded P_t as an invariable constant. In practical applications, P_t is a quantity that changes with the process requirements.

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

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

(14.76)

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

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

(14.77)

It should be switched from running on n devices to running on (n + 1) devices.

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

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

(14.78)

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

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

(14.79)

we should switch from running on n devices to running on n−1 devices.

Due to the limitation of P_1m, when P_t increases, P_t/n also increases until P_t/n = P_1m, which is still not satisfied

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

(14.80)

If P_t/n continues to increase, the device will be overloaded, and it is necessary to switch to (n + 1) devices forcibly at the point P_t/n = P_1m.

Due to the limitation of P_1m, when P_t decreases, P_t/n also increases until P_t/(n−1) = P_1m, which is still not satisfied

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

(14.81)

It is necessary to switch to (n–1) devices at the point P_t/(n−1) = P_1m.

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

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

(14.82)

For the same P_t, 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 C is

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

(14.83)

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

$${\eta }_{1}\left(\frac{{P}_{t}}{n{\sum }_{i=1}^{n}{\beta }_{i}}\right)=max\left\{{\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right),{\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k1}{\beta }_{i}}\right),{\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k2}{\beta }_{i}}\right),\dots \right\}$$

(14.84)

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 ${P}_{t}/{\sum }_{i=1}^{k1}{\beta }_{i}$ is the point closest to ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}$ to the left of point ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}$. Point ${P}_{t}/{\sum }_{i=1}^{k2}{\beta }_{i}$ is the point closest to ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}$ to the right of point ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}$, as shown in Fig. 14.4.

Fig. 14.4

Energy efficiency comparison curve of the multi-unit system with the similar efficiency devices

Point A is the point closest to B to the left of point B.

When P_t increases, the condition is satisfied

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

(14.85)

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

If there is no point ${P}_{t}/{\sum }_{i=1}^{k1}{\beta }_{i}$ that meets the conditions, the switching point is ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}={P}_{1m}$.

When P_t decreases, the condition is satisfied

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

(14.86)

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

If there is no point ${P}_{t}/{\sum }_{i=1}^{k2}{\beta }_{i}$ that meets the conditions, the switching point is ${P}_{t}/{\sum }_{i=1}^{k2}{\beta }_{i}={P}_{1m}$.

According to the system energy efficiency curve, the optimal switching method is given by observing the changing trend of the curve.

For the control process completed by such optimized switching, when the total load changes, the overall energy efficiency curve between two adjacent load switching points is the amplification of the energy efficiency curve of a single device, and the second derivative is less than zero, so such the solution is the only global optimal solution, and the overall energy efficiency value is also the global maximum efficiency value. The overall energy efficiency curve as shown in Fig. 14.5.

Fig. 14.5

The overall energy efficiency curve of multi-unit system

If there is no point of equal energy efficiency, it is at the point of maximum load of the devices.

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.

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

(14.87)

${P}_{t}/{\sum }_{i=1}^{k}{\beta }_{i}$ is the closest point to ${P}_{t}/{\sum }_{i=1}^{n}{\beta }_{i}$. If all devices have the identical energy efficiency, then k = n + 1 or k = n−1.

In practice, many devices need to be warmed up before they can be used, so preparations should be made in advance for starting up the device. Preheating also requires some energy.

14.8 Simulation Results

• 1. Assume system C has 3 devices in total, and all devices are devices with the same efficiency, there are

$$\begin{array}{c}{\eta }_{1}\left(P\right)={\eta }_{2}\left(P\right)={\eta }_{3}\left(P\right)=2.6P-2{P}^{2}\\ \begin{array}{c}{\eta }_{1e}={\eta }_{2e}={\eta }_{3e}=0.845\\ \begin{array}{c}{P}_{1e}={P}_{2e}={P}_{3e}=0.65\\ {P}_{1m}={P}_{2m}={P}_{3m}=1.1\end{array}\end{array}\end{array}$$

(14.88)

The total output W_t maximization expression of system C is

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{3}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{3}{P}_{i}={P}_{t} >0\\ {P}_{im}\ge {P}_{i}>0\end{array}\end{array}$$

(14.89)

Based on the above conclusions, the optimal control method when the two devices are running is to keep

$${P}_{1}={P}_{2}=\frac{{P}_{t}}{2}$$

(14.90)

The maximum value of W_t is

$${P}_{t}{\eta }_{1}\left(\frac{{P}_{t}}{2}\right)$$

(14.91)

The maximum value of the overall operating energy efficiency η_t2(P_t) is

$${\eta }_{1}\left(\frac{{P}_{t}}{2}\right)$$

(14.92)

Based on the above conclusions, the optimal control method for the operation of the three devices is to maintain

$${P}_{1}={P}_{2}={P}_{3}=\frac{{P}_{t}}{3}$$

(14.93)

The maximum value of W_t is

$${P}_{t}{\eta }_{1}(\frac{{P}_{t}}{3})$$

(14.94)

The maximum value of the overall operating energy efficiency η_t2(P_t) is

$${\eta }_{1}(\frac{{P}_{t}}{3})$$

(14.95)

Based on the above conclusions, the switching point P₁₂ of one operating device and two operating devices satisfies

$${\eta }_{1}\left({P}_{12}\right)={\eta }_{1}\left(\frac{{P}_{12}}{2}\right)$$

(14.96)

inferred

$${\text{P}}_{12}=0.8667$$

(14.97)

Based on the above conclusions, the switching point P₂₃ of 2 operating devices and 3 operating devices satisfies

$${\eta }_{1}\left(\frac{{P}_{23}}{2}\right)={\eta }_{1}\left(\frac{{P}_{23}}{3}\right)$$

(14.98)

inferred

$${\text{P}}_{23}=1.56$$

(14.99)

That is, when Pt changes:

0 < P_t <  = 0.8667, run with 1 device;

0.8667 < P_t <  = 1.56, run with 2 devices;

1.56 < P_t <  = 3.3, run with 3 devices.

When 0 < P_t <  = 0.8667, 0.8667 < P_t <  = 1.56 and 1.56 < P_t <  = 3.3，W_t ^‘’ < 0，so the obtained W_t is the maximum value.

In this example, if P_1m = 0.8, there is no equivalent switching point between the operation of one device and the operation of two devices, and only the maximum output point P_im of one device can be used as the switching point:

When 0 < P_t <  = 0.8, run with 1 device;

When 0.8 < P_t <  = 1.56, run with 2 devices;

When 1.56 < P_t <  = 2.4, run with 3 devices.

The overall energy efficiency curve of the system C is as shown in Fig. 14.6.

Fig. 14.6

The overall energy efficiency curve of system C with three devices

• 2. Assume that system C has 2 devices in total, and the two devices are devices with similar efficiency

Efficiency characteristics of device No. 1:

$$\begin{array}{c}{\eta }_{1}\left({P}_{1}\right)=2.6{P}_{1}-2{P}_{1}^{2}\\ \begin{array}{c}{\eta }_{1e}=0.845\\ \begin{array}{c}{P}_{1e}=0.65\\ {P}_{1m}=1.1\end{array}\end{array}\end{array}$$

(14.100)

Defined as β₁ = 1.

Efficiency characteristics of device No. 2:

$$\begin{array}{c}{\eta }_{2}\left({P}_{2}\right)=1.3{P}_{2}-0.5{P}_{2}^{2}\\ \begin{array}{c}{\eta }_{2e}=0.845\\ \begin{array}{c}{P}_{2e}=1.3\\ {P}_{2m}=2.2\end{array}\end{array}\end{array}$$

(14.101)

β₂ = 2, the output capacity of device No. 2 is larger than that of device No. 1.

The total output W_t maximization expression of system C is

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{2}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{2}{P}_{i}={P}_{t} >0\\ {P}_{im}\ge {P}_{i}>0\end{array}\end{array}$$

(14.102)

In this system, n = 1 has two combinations, and n = 2 has one combination.

When n = 1, there is a switching point between the first small device No. 1 and the second large device No. 2. According to the equivalent switching theorem, we have

$${\eta }_{1}\left({P}_{t}\right)={\eta }_{2}({P}_{t})$$

(14.103)

Obtain the switching point P₁₁ of a No. 1 small device and No. 2 large device

$${\text{P}}_{11}=0.8667$$

(14.104)

When n = 2, the optimal control method is to keep

$$\begin{array}{c}{P}_{1}=\frac{{P}_{t}}{{\sum }_{i=1}^{2}{\beta }_{i}}=\frac{{P}_{t}}{3}\\ {P}_{2}={\beta }_{2}{P}_{1}=\frac{{2P}_{t}}{3}\end{array}$$

(14.105)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{2}{\beta }_{i}}\right)={P}_{t}{\eta }_{1}(\frac{{P}_{t}}{3})$$

(14.106)

The maximum value of the overall operating energy efficiency η_t2(P_t) of system C is

$${\eta }_{t2}\left({P}_{t}\right)={\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{2}{\beta }_{i}}\right)={\eta }_{1}(\frac{{P}_{t}}{3})$$

(14.107)

Switching point P₁₂ of No. 1 equipment of 1 large device and 2 devices satisfies

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

(14.108)

from which we have

$${\text{P}}_{12}=1.56$$

(14.109)

When P₁₁ < P_t <  = P₁₂ and P₁₂ < P_t <  = 3.3，W_t ^‘’ < 0，so the obtained W_t is the maximum value.

The overall energy efficiency curve of the system C as shown in Fig. 14.7.

Fig. 14.7

The overall energy efficiency curve of system C with two devices

14.9 Quantum Optimization Method and Energy Efficiency Predictive Theory

There is no need to establish an accurate mathematical model of the system, based on the characteristics of the energy efficiency function, this chapter presents a constrained, nonlinear, integer-real-number hybrid energy efficiency optimization method for multi-unit systems.

This optimization method includes two theorems: optimal load distribution theorem and optimal switching theorem.

Optimal load distribution theorem: The optimal load distribution method of a multi-unit system is to keep the operating energy efficiency of each operating device equal, Yao Theorem 1.

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

(14.110)

Optimal switching theorem: 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, Yao Theorem 2.

$$\begin{aligned}{\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)&={\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right) or {\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{n}{\beta }_{i}}\right)\\&={\eta }_{1m} or {\eta }_{1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k}{\beta }_{i}}\right)={\eta }_{1m}\end{aligned}$$

(14.111)

We call this optimization method as quantum optimization method of multi-unit system.

We call this theory to solve energy efficiency optimization as energy efficiency predictive theory of multi-unit system.

These methods have the following advantages:

• (1) Easy to use, no needing to establish an accurate mathematical model of the system;

• (2) Strong versatility, including linear systems, nonlinear systems, multivariable systems, time invariant systems, time varying systems;

• (3) Integer and real optimizations are solved together.

14.10 The Second Definition of Similar Energy Efficiency Devices

We define the load rate γ_i of the i-th device as

$${\gamma }_{i}=\frac{{P}_{i}}{{P}_{ie}}$$

(14.112)

We call η_Ni(γ_i) as the normalization efficiency function of the i-th device. The normalization efficiency function η_Ni(γ) has a shape shown in Fig. 14.8.

Fig. 14.8

The normalization efficiency function η_Ni (γ)

In Fig. 14.8, γ is a variable, η_Ni is the efficiency. η_Ni and η_i have the relation as following.

$${\eta }_{i}({P}_{i})={\eta }_{i}({\gamma }_{i}{P}_{ie})={\eta }_{Ni}({\gamma }_{i})$$

(14.113)

If the normalization efficiency functions of the first device and the i-th device are the same, we have

$${\eta }_{N1}({\gamma }_{1})={\eta }_{N\text{i}}({\gamma }_{i})$$

(14.114)

We call them the similar energy efficiency devices.

14.11 Another Way to Prove Yao’s Theorem

Assume that P_i is greater than 0, that is, every operating device does work. The formula (14.1) is simplified as

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{n}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{n}{P}_{i}={P}_{t} >0\\ {P}_{imax}\ge {P}_{i}>0\end{array}\end{array}$$

(14.115)

Assuming that n is optimal, consider the following two situations:

• (1) n = 2

System C has two variables P₁ and P₂

$$\begin{array}{c}{P}_{1}+{P}_{2}={P}_{t}\\ \begin{array}{c}{P}_{1}>0\\ {P}_{2}>0\end{array}\end{array}$$

(14.116)

The objective function can be expressed as

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

(14.117)

The optimization condition is

$${W'_t}\left({P}_{1}\right)=0$$

(14.118)

According to known conditions

$${P}_{2}={\gamma }_{2}{P}_{1e}={P}_{t}-{{\gamma }_{1}P}_{1e}$$

(14.119)

We have

$${\eta }_{1}\left({P}_{1}\right)+{P}_{1}{\eta }_{1}{\prime}({P}_{1})-{\eta }_{2}\left({P}_{2}\right)-{P}_{2}{\eta }_{2}{\prime}\left({P}_{2}\right)=0$$

(14.120)

If two devices are similar energy efficiency devices, then we have

$$\begin{array}{c}\begin{array}{c}{\eta }_{1}\left({P}_{1}\right)={\eta }_{1}\left({\gamma }_{1}{P}_{1e}\right)={\eta }_{N1}\left({\gamma }_{1}\right)\\ {\eta }_{2}\left({P}_{2}\right)={\eta }_{2}\left({\gamma }_{2}{P}_{2e}\right)={\eta }_{N2}\left({\gamma }_{2}\right)\end{array}\\ {\eta }_{N1}\left(\gamma \right)={\eta }_{N2}\left(\gamma \right)\end{array}$$

(14.121)

The objective function can be expressed as

$${W}_{t}={P}_{1}{\eta }_{1}\left({P}_{1}\right)+{P}_{2}{\eta }_{2}\left({P}_{2}\right)={{\gamma }_{1}P}_{1e}{\eta }_{N1}\left({\gamma }_{1}\right)+{{\gamma }_{2}P}_{2e}{\eta }_{N2}\left({\gamma }_{2}\right)$$

(14.122)

The optimization condition is

$${W'_t}\left({\gamma }_{1}\right)=0$$

(14.123)

According to known conditions, we have

$${\gamma }_{2}=\frac{{P}_{t}-{\gamma }_{1}{P}_{1e}}{{P}_{2e}}$$

(14.124)

from which we know

$${\gamma }_{1}={\gamma }_{2}=\frac{{P}_{t}}{{P}_{1e}+{P}_{2e}}$$

(14.125)

is an optimization point.

The optimal control method is to keep

$$\begin{array}{c}{P}_{1}={\gamma }_{1}{P}_{1e}=\frac{{P}_{1e}}{{P}_{1e}+{P}_{2e}}{P}_{t}\\ {P}_{2}={\gamma }_{2}{P}_{2e}=\frac{{P}_{2e}}{{P}_{1e}+{P}_{2e}}{P}_{t}\end{array}$$

(14.126)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{N1}\left(\frac{{P}_{t}}{{P}_{1e}+{P}_{2e}}\right)$$

(14.127)

The maximum value of the overall operating energy efficiency η_t2(P_t) of system C is

$$\text{max}{\eta }_{t2}\left({P}_{t}\right)={\eta }_{N1}\left(\frac{{P}_{t}}{{P}_{1e}+{P}_{2e}}\right)$$

(14.128)

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. W_t has a maximum value, and the overall energy efficiency is a maximum value as well.

• (2) If n = k holds, prove that n = k + 1 still holds.

For system C composed of n devices with the similar energy efficiency function, if the optimization conclusion holds,

$$\begin{array}{c}{W}_{t}=\text{max}\sum_{i=1}^{k}{P}_{i}{\eta }_{i}({P}_{i}) \\ \begin{array}{c}s.t. \sum_{i=1}^{k}{P}_{i}={P}_{t} >0\\ {P}_{imax}\ge {P}_{i}>0\end{array}\end{array}$$

(14.129)

The optimal point is

$${\gamma }_{1}={\gamma }_{2}=\dots ={\gamma }_{k}=\frac{{P}_{t}}{{\sum }_{i=1}^{k}{P}_{ie}}$$

(14.130)

The optimal control method is to keep

$$\begin{array}{c}{P}_{1}=\frac{{P}_{1e}}{{\sum }_{i=1}^{k}{P}_{ie}}{P}_{t}\\ \begin{array}{c}\begin{array}{c}{P}_{2}=\frac{{P}_{2e}}{{\sum }_{i=1}^{k}{P}_{ie}}{P}_{t}\\ \dots \end{array}\\ {P}_{k}=\frac{{P}_{ke}}{{\sum }_{i=1}^{k}{P}_{ie}}{P}_{t}\end{array}\end{array}$$

(14.131)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{N1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k}{P}_{ie}}\right)={P}_{t}{\eta }_{1}\left(\frac{{P}_{1e}}{{\sum }_{i=1}^{k}{P}_{ie}}{P}_{t}\right)$$

(14.132)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

$$\text{max}{\eta }_{tk}\left({P}_{t}\right)={\eta }_{N1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k}{P}_{ie}}\right)={\eta }_{1}\left(\frac{{P}_{1e}}{{\sum }_{i=1}^{k}{P}_{ie}}{P}_{t}\right)$$

(14.133)

For n = k + 1, we have

$$\begin{aligned}{W}_{t}&=\sum_{i=1}^{k+1}{P}_{i}{\eta }_{i}\left({P}_{i}\right)=\left({P}_{t}-{{\gamma }_{k+1}P}_{\left(k+1\right)e}\right){\eta }_{N1}\left(\frac{{P}_{t}-{\gamma }_{k+1}{P}_{\left(k+1\right)e}}{{\sum }_{i=1}^{k}{P}_{ie}}\right)\\&+{{\gamma }_{k+1}P}_{\left(k+1\right)e}{\eta }_{N(k+1)}\left({\gamma }_{k+1}\right)\end{aligned}$$

(14.134)

The optimization condition is

$${W'_t}\left({\gamma }_{k+1}\right)=0$$

(14.135)

It is easy to see that, the optimal point is

$$\begin{array}{c}{\gamma }_{k+1}=\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{P}_{ie}}\\ {\gamma }_{1}={\gamma }_{2}=\dots ={\gamma }_{k}=\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{P}_{ie}}\end{array}$$

(14.136)

The optimal control method is to keep

$$\begin{array}{c}{P}_{1}=\frac{{P}_{1e}}{{\sum }_{i=1}^{k+1}{P}_{ie}}{P}_{t}\\ \begin{array}{c}\begin{array}{c}{P}_{2}=\frac{{P}_{2e}}{{\sum }_{i=1}^{k+1}{P}_{ie}}{P}_{t}\\ \dots \end{array}\\ {P}_{k+1}=\frac{{P}_{(k+1)e}}{{\sum }_{i=1}^{k+1}{P}_{ie}}{P}_{t}\end{array}\end{array}$$

(14.137)

The maximum total output energy W_t of system C is

$$\text{max}{W}_{t}={P}_{t}{\eta }_{N1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{P}_{ie}}\right)={P}_{t}{\eta }_{1}\left(\frac{{P}_{1e}}{{\sum }_{i=1}^{k+1}{P}_{ie}}{P}_{t}\right)$$

(14.138)

The maximum value of the overall operating energy efficiency η_tk(P_t) of system C is

$$\text{max}{\eta }_{t(k+1)}\left({P}_{t}\right)={\eta }_{N1}\left(\frac{{P}_{t}}{{\sum }_{i=1}^{k+1}{P}_{ie}}\right){\eta }_{1}\left(\frac{{P}_{1e}}{{\sum }_{i=1}^{k+1}{P}_{ie}}{P}_{t}\right)$$

(14.139)