Energy Efficiency Optimization of Heating and Cooling Systems

Abstract

In a heat source plant, there are many boilers. In the cooling system, there are many chillers. Whether it is a cooling system or a heating system, they are all thermal energy supply systems. There is a minimum energy consumption to provide constant thermal energy.

In a heat source plant, there are many boilers. In the cooling system, there are many chillers. Whether it is a cooling system or a heating system, they are all thermal energy supply systems. There is a minimum energy consumption to provide constant thermal energy.

21.1 Energy Efficiency of a Boiler or a Chiller

The energy efficiency of a boiler or chiller under constant output temperature difference Δt₀ (°C) is shown in Fig. 21.1. For the convenience of the following description, we refer to boilers or chillers collectively as equipment or devices.

Fig. 21.1

The energy efficiency function of a boiler or a chiller

In Fig. 21.1, Q is the flow rate (tons/hour), η is the efficiency, and Q_e is the flow rate at maximum efficiency. We have

$$ \begin{gathered} Q \ge 0 \hfill \\ \eta \left( Q \right) \ge 0 \hfill \\ \eta \left( 0 \right) = 0 \hfill \\ \eta {\prime \prime }(Q) < 0 \hfill \\ \end{gathered} $$

(21.1)

Approximately, we have

$$\eta (Q)={\sum }_{i=0}^{\infty }{a}_{i}{Q}^{i}=Q{\sum }_{i=1}^{\infty }({a}_{i}{Q}^{i-1})\approx Q({a}_{1}+{a}_{2}Q)$$

(21.2)

where

$$\begin{gathered} a_{1} + a_{2} Q > 0 \hfill \\ a_{1} > 0 \hfill \\ a_{2} < 0 \hfill \\ \end{gathered}$$

(21.3)

The equipment’s output energy is P(Q), which can be expressed as

$$P(Q)=\frac{k\Delta {t}_{0}Q}{\eta (Q)}$$

(21.4)

where k is a constant related to heat capacity.

21.2 Optimal Control of Heating Systems Composed of Equipment of the Identical Type

A thermal energy supply system has n devices of the identical model. The system is used to supply constant temperature Δt₀ water. The total water flow is Q_t. The flow of the i-th device is Q_i. Q_i is greater than zero. The water flow of all devices is variable. We have

$$\begin{gathered} Q_{i} > 0 \hfill \\ \mathop \sum \limits_{{i = 1}}^{n} Q_{i} = Q_{t} \hfill \\ \end{gathered}$$

(21.5)

η (Qi) is the energy efficiency of the i-th device at point (Q_i, Δt₀), η is the total energy efficiency of the thermal energy supply system, and P_t is the total energy consumption of the system.

$${P}_{t}=k\Delta t{\sum }_{i=1}^{n}{Q}_{i}\frac{1}{\eta ({Q}_{i})}$$

(21.6)

Using the contents of the previous chapters, it is not difficult to prove the following conclusions:

The optimal control method is to keep

$$ Q_{1} = Q_{2} = \ldots = Q_{n} = \frac{{Q_{t} }}{n} $$

(21.7)

The minimum value of total energy consumption is

$${\mathit{minP}}_{t}={P}_{0}\frac{1}{\eta (\frac{{Q}_{t}}{n})}$$

(21.8)

The ideal work P₀ is

$${P}_{0}=k{Q}_{t}\Delta {t}_{0}$$

(21.9)

The overall optimal efficiency is

$${\mathit{max\eta }}_{t}({Q}_{t})=\eta (\frac{{Q}_{t}}{n})$$

(21.10)

21.3 The Energy Efficiency Similarity of Different Equipment

We define the load rate γ as

$$\gamma =\frac{Q}{{Q}_{e}}$$

(21.11)

We call η_N(γ) as the normalized energy efficiency function of an equipment. The Normalized energy efficiency function η_N(γ) has a shape shown in Fig. 21.2.

Fig. 21.2

The normalized efficiency function η_N (γ)

In Fig. 21.2, γ is the variable and η_N is the efficiency. η_N and η have the following relationship.

$$\eta (Q)=\eta (\gamma {Q}_{e})={\eta }_{N}(\gamma )$$

(21.12)

If the normalized efficiency functions of two different devices are identical, we have

$${\eta }_{N1}(\gamma )={\eta }_{N2}(\gamma )$$

(21.13)

We call them efficiency-similar devices. A thermal energy supply system containing efficiency-similar devices is called an efficiency-similar heating system.

Let γ_i be the load rate of the i-th device, and its form is:

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

(21.14)

For an efficiency-similar heat energy supply system with n-unit running equipment, the total energy consumption has the form

$${P}_{t}=k\Delta {t}_{0}{\sum }_{i=1}^{n}\frac{{Q}_{i}}{{\eta }_{i}({Q}_{i})}=k\Delta {t}_{0}{\sum }_{i=1}^{n}\frac{{\gamma }_{i}{Q}_{ie}}{{\eta }_{N}({\gamma }_{i})}$$

(21.15)

21.4 Optimal Control of an Efficiency-Similar Heating System

When the load rate of each running equipment is greater than zero, consider the minimization of the total energy consumption

$$ \begin{gathered} min\,P_{t} \hfill \\ s.t.\,\,\,\gamma_{i} > 0,i = 1,2, \ldots n \hfill \\ \mathop \sum \limits_{i = 1}^{n} \gamma_{i} Q_{ie} = Q_{t} \hfill \\ \end{gathered} $$

(21.16)

We consider three kinds of situation.

• (1) n = 2

There are two variables.

We have

$$\begin{gathered} \gamma_{1} Q_{1e} + \gamma_{2} Q_{2e} = Q_{t} \hfill \\ \gamma_{1} > 0 \hfill \\ \gamma_{2} > 0 \hfill \\ \end{gathered}$$

(21.17)

The objective function P_t is expressed as

$${P}_{t}=k\Delta {t}_{0}(\frac{{\gamma }_{1}{Q}_{1e}}{{\eta }_{N}({\gamma }_{1})}+\frac{{\gamma }_{2}{Q}_{2e}}{{\eta }_{N}({\gamma }_{2})})$$

(21.18)

The optimal condition is

$${{P}_{t}}{\prime}({\gamma }_{1})=0$$

(21.19)

We have

$$\frac{{Q}_{1e}{\eta }_{N}({\gamma }_{1})-{\gamma }_{1}{Q}_{1e}{\eta }_{N}{\prime}({\gamma }_{1})}{{\eta }_{N}^{2}({\gamma }_{1})}+\frac{-{Q}_{1e}{\eta }_{N}(\frac{{Q}_{t}-{\gamma }_{1}{Q}_{1e}}{{Q}_{2e}})+({Q}_{t}-{\gamma }_{1}{Q}_{1e})\frac{{Q}_{1e}}{{Q}_{2e}}{\eta }_{N}{\prime}(\frac{{Q}_{t}-{\gamma }_{1}{Q}_{1e}}{{Q}_{2e}})}{{\eta }_{N}^{2}(\frac{{Q}_{t}-{\gamma }_{1}{Q}_{1e}}{{Q}_{2e}})}=0$$

(21.20)

It is easy to see that

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

(21.21)

to be an optimal point.

The minimum value of total energy consumption is

$${\mathit{minP}}_{t}=k{Q}_{t}\Delta t\frac{1}{{\eta }_{N}(\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}})}=\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{Q}_{1e}+{Q}_{2e}})}$$

(21.22)

P₀ is the ideal work.

• (2) n = 3

There are three variables.

Based on known conditions, we have

$$\begin{gathered} \gamma _{1} Q_{{1e}} + \gamma _{2} Q_{{2e}} + \gamma _{3} Q_{{3e}} = Q_{t} \hfill \\ \gamma _{1} > 0 \hfill \\ \gamma _{2} > 0 \hfill \\ \gamma _{3} > 0 \hfill \\ \end{gathered}$$

(21.23)

P_t expression becomes

$${P}_{t}=k\Delta {t}_{0}(\frac{{\gamma }_{1}{Q}_{1e}}{{\eta }_{N}({\gamma }_{1})}+\frac{{\gamma }_{2}{Q}_{2e}}{{\eta }_{N}({\gamma }_{2})}+\frac{{\gamma }_{3}{Q}_{3e}}{{\eta }_{N}({\gamma }_{3})})$$

(21.24)

Assume that γ₁ is a fixed optimal point and only γ₂ and γ₃ are variables. We have

$${\gamma }_{2}{Q}_{2e}+{\gamma }_{3}{Q}_{3e}={Q}_{t}-{\gamma }_{1}{Q}_{1e}=cons\mathit{tan}t$$

(21.25)

According to the conclusion drawn with n = 2, the optimal point is

$${\gamma }_{2}={\gamma }_{3}$$

(21.26)

Assume that γ₂ is a fixed optimal point and only γ₁ and γ₃ are variables. We have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{3}{Q}_{3e}={Q}_{t}-{\gamma }_{2}{Q}_{2e}=cons\mathit{tan}t$$

(21.27)

According to the conclusion at n = 2, the optimal point is at

$${\gamma }_{1}={\gamma }_{3}$$

(21.28)

Similarly, assume that γ₃ is fixed optimal point and only γ₁ and γ₂ are variables. We have

$${\gamma }_{1}{Q}_{1e}+{\gamma }_{2}{Q}_{2e}={Q}_{t}-{\gamma }_{3}{Q}_{3e}=cons\mathit{tan}t$$

(21.29)

According to the conclusion at n = 2, the optimal point is at

$${\gamma }_{1}={\gamma }_{2}$$

(21.30)

We have the optimal points

$${\gamma }_{1}={\gamma }_{2}={\gamma }_{3}=\frac{{Q}_{t}}{{\sum }_{i=1}^{3}{Q}_{ie}}$$

(21.31)

The minimum value of total energy consumption is

$${\mathit{minP}}_{t}=\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{3}{Q}_{ie}})}$$

(21.32)

• (3) n = k

There are n variables.

The above conclusion can be extended to the situation at n = k, the optimal point is

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

(21.33)

The minimum value of total energy consumption is

$${\mathit{minP}}_{t}=\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{k}{Q}_{ie}})}$$

(21.34)

21.5 Optimal Number of Running Units

If n is the optimal and all equipment are the identical model, there must be

$$\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n-1}{Q}_{ie}})}\ge \frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})}\le \frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n+1}{Q}_{ie}})}$$

(21.35)

Namely

$${\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n-1}{Q}_{ie}})\le {\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})\ge {\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n+1}{Q}_{ie}})$$

(21.36)

If n is the optimal and all devices have the identical energy efficiency, there must be

$$\frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n1}{Q}_{ie}})}\ge \frac{{P}_{0}}{{\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})}$$

(21.37)

That is

$${\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n1}{Q}_{ie}})\le {\eta }_{N}(\frac{{Q}_{t}}{{\sum }_{i=1}^{n}{Q}_{ie}})$$

(21.38)

where n1 is the operating number of devices in any combination.

21.6 Optimal Switch of Efficiency-Similar Heating System

Assume that there are M devices in a thermal energy supply system, n devices are running, and n is equal to or less than M. If n is optimal, the total energy consumption is the minimum.

$$ \begin{gathered} minP_{t} = \frac{{P_{0} }}{{\eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n} Q_{ie} }}} \right)}} \hfill \\ n \le M \hfill \\ \eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n1} Q_{ie} }}} \right) \le \eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n} Q_{ie} }}} \right) \ge \eta_{N} \left( {\frac{{Q_{t} }}{{\sum_{i = 1}^{n2} Q_{ie} }}} \right) \hfill \\ \gamma (Q_{t} ,n) = \frac{{Q_{t} }}{{\sum_{i = 1}^{n} Q_{ie} }} \hfill \\ \end{gathered} $$

(21.39)

n1 and n2 are the running number of equipment in any combinations.

When Q_t changes, we consider three situations.

• (1) The γ (Q_t, n) is less than 1.

η_e is the maximum efficiency, and 1 is the load rate at η_e, as shown in Fig. 21.3.

Fig. 21.3

η_N (γ) curve when γ(Q_t,n) is less than 1

In Fig. 21.3, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n1\right)\right)={\eta }_{N}\left({Q}_{t},n\right)$$

(21.40)

When Q_t increases, the load rate γ(Q_t,n) increases, and η_N(Q_t,n) and η_N(Q_t,n1) also increase. However, when γ(Qt,n) > 1, Q_t increases, η_N(Q_t, n1)) still increases, but η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n1), n1 is optimal, we should change the number of running units from n to n1, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(21.41)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n1)$$

(21.42)

When Q_t decreases, the load rate γ(Q_t,n) decreases, and η_N(Q_t, n) decreases also. However, η_N(Q_t, n2) increases. When η_N(Q_t, n) < η_N(Q_t, n2), n2 is the optimal, we should change the number of running units from n to n2, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(21.43)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(21.44)

• (2) The γ(Q_t,n) is greater than 1, as shown in Fig. 21.4.

  Fig. 21.4

  

  η_N (γ) curve when γ(Q_t,n) is greater than 1

In Fig. 21.4, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n\right)$$

(21.45)

When Q_t increases, the load rate γ (Q_t, n) increases, and η_N(Q_t, n1) increases also, however η_N(Q_t, n) decreases. When η_N(Q_t, n)< η_N(Q_t, n1), n1 is the optimal, we should change the number of running units from n to n1, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(21.46)

The optimal switch point is

$${\eta }_{N}\left({Q}_{t},n\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(21.47)

When Q_t decreases, the load rate γ(Q_t, n) decreases, η_N(Q_t, n) and η_N(Q_t, n2) both increase. When $\gamma (Q_{t} ,n)$ < 1, Q_t decreases, η_N (Q_t, n2) still increases, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t,n2), n2 is the optimal, we should change the number of running units from n to n2, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(21.48)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(21.49)

• (3) The $\gamma (Q_{t} ,n)$ is equal to 1.

When Q_t increases, the load rate γ(Q_t,n) increases, η_N(Q_t, n1) increase also, however η_N(Q_t, n) decreases. When η_N(Q_t, n) < η_N(Q_t, n1), n1 is the optimal, we should change the number of running units from n to n1, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n1\right)$$

(21.50)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n1)$$

(21.51)

When Q_t decreases, the load rate γ (Q_t, n) decreases, η_N (Q_t, n) decreases also, however η_N (Q_t, n2) increases. When η_N(Q_t, n) < η_N(Q_t, n2), n2 is the optimal, we should change the number of running units from n to n2, we have

$$\mathit{max}\left({\eta }_{N}\left({Q}_{t},n1\right),{\eta }_{N}\left({Q}_{t},n\right),{\eta }_{N}\left({Q}_{t},n2\right)\right)={\eta }_{N}\left({Q}_{t},n2\right)$$

(21.52)

The optimal switch point is

$${\eta }_{N}({Q}_{t},n)={\eta }_{N}({Q}_{t},n2)$$

(21.53)

21.7 Conclusion

The proof of the optimal control and switching method given in this chapter is mainly based on the characteristics of the energy efficiency function, which can be approximated as a concave non-negative function through the origin. The optimal method has the following characteristics:

• (1) Includes linear and nonlinear systems,

• (2) No systematic mathematical model is required,

• (3) High versatility.